Add files using upload-large-folder tool

.gitattributes

@@ -242,3 +242,4 @@ VtFOT4oBgHgl3EQf6zSP/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex
 0tFPT4oBgHgl3EQfTjTq/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf filter=lfs diff=lfs merge=lfs -text
 x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf filter=lfs diff=lfs merge=lfs -text
+M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf filter=lfs diff=lfs merge=lfs -text

2tFQT4oBgHgl3EQfGDUx/content/tmp_files/2301.13243v1.pdf.txt

@@ -0,0 +1,1623 @@
+arXiv:2301.13243v1  [cs.IT]  30 Jan 2023
+Reconfigurable Intelligent Surface-Aided NOMA
+with Limited Feedback
+Mojtaba Ahmadi Almasi and Hamid Jafarkhani
+Abstract—The design of feedback channels in frequency di-
+vision duplex (FDD) systems is a major challenge because of
+the limited available feedback bits. We consider non-orthogonal
+multiple access (NOMA) systems that incorporate reconfigurable
+intelligent surfaces (RISs). In limited feedback RIS-aided NOMA
+systems, the RIS-aided channel and the direct channel gains
+should be quantized and fed back to the transmitter. This
+paper investigates the rate loss of the overall RIS-aided NOMA
+systems suffering from quantization errors. We first consider
+random vector quantization for the overall RIS-aided channel
+and identical uniform quantizers for the direct channel gains.
+We then obtain an upper bound for the rate loss, due to the
+quantization error, as a function of the number of feedback bits
+and the size of RIS. Our numerical results indicate the sum rate
+performance of the limited feedback system approaches that of
+the system with full CSI as the number of feedback bits increases.
+I. INTRODUCTION
+Reconfigurable intelligent surfaces (RISs) are presumed as
+an attractive solution to enhance the spectral, power effi-
+ciency, and coverage of wireless communication systems [1].
+These surfaces consist of many passive and cost-effective
+elements capable of controlling the propagation environment
+by properly adjusting the direction of coming signals. These
+distinctive properties make RIS a promising solution for
+broad connectivity in the next generation of wireless networks.
+Previously, intelligent surfaces that are not reconfigurable [2],
+[3] and reconfigurable multiple-input multiple-output (MIMO)
+systems [4], [5] have been proposed for orthogonal multiple
+access (OMA) systems. Also, it is shown that RISs can have
+notable use cases and boost performance when merged with
+other emerging technologies such as non-orthogonal multiple
+access (NOMA) [6], [7].
+NOMA has been a topic of research as a promising new
+technology for the next generation of wireless communica-
+tions. Specifically, in the downlink, power-domain NOMA
+aims to serve two or more users by sharing the same
+time/frequency/code resource block [8]. At the transmitter side,
+NOMA squeezes the information signals using superposition
+coding. Before decoding the intended signal, the stronger user
+applies successive interference cancellation, i.e., first decodes
+the weaker user’s signal and then subtracts it from the received
+signal.
+In this paper, we incorporate the RIS in downlink NOMA
+to improve the quality of the weak user’s channel. Unlike
+many other time division duplex (TDD)-based RIS-assisted
+NOMA (RIS-NOMA) systems like [6], [7], [9], [10], we
+The authors are with Center for Pervasive Communications and Computing,
+University of California, Irvine. This work was supported in part by the NSF
+Award CNS-2008786.
+consider a frequency division duplex (FDD) system. The FDD-
+based RIS-NOMA is more challenging in the sense that the
+channel must be estimated at the receiver and fed back to the
+transmitter via a limited feedback channel. The availability
+of the quantized channel gains instead of perfect channel
+state information (CSI) creates the following two major issues.
+First, the quantized channel gains can result in a severe rate
+loss [11], [12]. This phenomenon is more harmful when the
+quantized channel gains inaccurately change the order of
+NOMA users. Second, the phase information obtained from
+the overall quantized channel restricts the performance of
+the RIS [13]. Motivated by this, we investigate the impact
+of quantizing the overall RIS-aided channel and the direct
+channel gains on the system’s rate loss.
+The limited feedback problem in RIS-aided systems is
+studied in [14]–[17]. Ref. [14] proposes a cascaded codebook
+design and bit partitioning strategy in the presence of line-
+of-sight (LoS) and non-LoS (NLoS) channels. In [15], the
+feedback is divided into two parts, channel feedback and angle
+information feedback. In particular, [15] uses a random vector
+quantizer (RVQ) codebook to quantize the channel followed by
+feeding back the indices related to the angle of arrival (AoA)
+and angle of departure (AoD) information of the cascade chan-
+nel matrix. Similarly, [16] designs a codebook-based limited
+feedback protocol for RIS using learning methods. In [17],
+authors aim to reconstruct the channel using the signal strength
+feedback and exploiting the sparsity and low-rank properties.
+None of the above limited feedback methods can be applied to
+our RIS-NOMA. Particularly, the feedback methods in [14]–
+[16] mainly try to send the normalized channel vectors to per-
+form beamforming at the transmitter. Further, [17] determines
+the channel direction while the channel gains are estimated
+based on the distance and not precisely. However, our RIS-
+NOMA system requires precise channel gains to accurately
+order the users and perform the superposition coding. Also, it
+needs the overall RIS-aided channel vector to adjust the RIS.
+The main contributions of this paper are as follows:
+• We provide a limited feedback framework for RIS-
+NOMA systems and analyze its performance.
+• We find an upper bound on the rate loss caused by
+quantization.
+We conduct numerical simulations to evaluate the sum rate
+and the rate loss of the limited feedback RIS-NOMA system.
+The results verify our theoretical derivations.
+Notations: In this paper, j = √−1. Small letters, bold
+letters, and bold capital letters designate scalars, vectors,
+and matrices, respectively. Superscripts (·)T and (·)† denote,
+respectively, the transpose and the transpose-conjugate oper-
+ations. Further, |x|, E[x], and V[x] denote the absolute, the
+
+es
+- Ajusting element phases using the phase vector
+Fig. 1: The limited feedback RIS-NOMA system model. B1 bits and B2+B′
+bits are allocated to User 1 and User 2, respectively.
+expected, and the variance of x, respectively. The operation
+∠x calculates the element-wise angle of the vector x and ⌊·⌋
+is the floor function. Finally, γ(n, x) =
+� x
+0 tn−1e−tdt denotes
+the lower incomplete gamma function.
+II. SYSTEM MODEL
+Our system model is shown in Fig. 1 in which a single-input
+single-output system model similar to [18], [19] is considered.
+NOMA is a suitable multiple access technique for a single-
+antenna setup because multi-user solutions are not applicable.
+The base station (BS) uses NOMA to simultaneously serve two
+users named User 1 and User 2.1 The distance from User 2
+to the BS is more than that of User 1. To determine which
+user is near, the BS captures the distance information through
+a channel quality indicator (CQI). The purpose of the RIS is
+to serve the far user to improve the channel quality [6], [7].
+The RIS is equipped with N antenna elements that ideally can
+direct the incident signal to any arbitrary directions in [− π
+2 , π
+2 ].
+Like [22], [23], it is assumed that the perfect CSI is
+estimated at the users. This assumption allows us to focus
+on studying the impact of the channel gain and phase vector
+quantization errors. Recently, [24] has investigated the impact
+of CSI impairments such as erroneous channel estimation
+or delay in feedback on beamforming in a RIS-NOMA sys-
+tem without considering quantization error. As a promising
+solution, our work on beamforming in relay networks with
+channel statistics [25] and quantized feedback [26], [27] can
+be extended to the underlying RIS-NOMA system to study the
+CSI impairments.
+A. Transmit Channel Model
+We recall that both users are capable of estimating their
+channels. That is, User 1 obtains h1 ∈ C and User 2 obtains
+h2 ∈ CN×1 and g ∈ CN×1 [28]. The channels capture
+the small-scale fading and path loss effects. For User 1,
+h1 = √L1h′
+1, where L1 = d−α1
+1
+is due to the path loss.
+The parameters d1 and α1 denote the distance between the
+BS and User 1 and the path loss factor, respectively. Further,
+h′
+1 ∼ CN(0, 1) denotes the small-scale Rayleigh fading. We
+define the channel gain H1 = |h1|2 with the probability den-
+sity function (pdf) of fH1(H1) =
+1
+L1 e− H1
+L1 . Also, the channel
+between the BS and the RIS is defined as h2 = √L2h′
+2, where
+1User pairing is out of the scope of this paper. We assume that the users
+are paired using one of the existing methods in the literature such as [10],
+[20], [21]. The complexity of rate loss calculation increases as the number of
+NOMA users grows.
+L2 = d−α2
+2
+and h′
+2 ∼ CN(0, I). The channel between the
+RIS and User 2 is given by g =
+�
+Lgg′, where Lg = d−αg
+g
+and g′ ∼ CN(0, I). We note that I represents the identity
+matrix of size N × N. In fact, the small-scale fading from the
+BS to User 2 is subject to the double-Rayleigh fading [18],
+[29], [30]. Another possible channel model from the BS to
+the RIS is Rician fading. Since the RIS is helping the far user,
+it is reasonable to assume that its distance from the BS is
+large [6], [7]. Thus, it is likely that the LoS channel is blocked
+by moving objects or buildings justifying a Rayleigh fading
+model. The parameters d2 and dg denote the distance between
+the transmitter and the RIS and the distance from the RIS to
+User 2, respectively. Further, the parameters α2 and αg denote
+the path loss factors. The effective overall channel between the
+BS and User 2 is defined as ˜h2 = gT Θh2. Correspondingly,
+the channel gain is ˜H2 =
+��gT Θh2
+��2 in which Θ = diag(θ),
+where θ = [ejφ1, · · · , ejφN ] and φi ∈ [− π
+2 , π
+2 ] reflect the
+impact of the RIS. The optimal values of Θ result in the
+maximum gain of H2 =
+��N
+i=1|h2,i||gi|
+�2
+. Deriving the
+exact pdf of H2 is complicated. For the sake of simplicity, we
+first use the following upper bound on the pdf of the random
+variable z = �N
+i=1 |h2,i||gi| [7]:
+fz(z) ≤ C1
+C2
+�
+z
+C2
+� 3N−2
+2
+e−2 z
+C2 ,
+(1)
+where C1 =
+2N π
+N
+2 ΓN( 3
+2)
+( 3N−2
+2
+)!
+and C2 =
+�
+L2Lg. Through exten-
+sive simulations, it is shown that this bound is tight [7]. With-
+out loss of generality, we assume N is an even number. Then,
+noting that H2 = z2, we have fH2(H2) =
+1
+2√H2 fz
+�√H2
+�
+.
+Finally, an upper bound on the pdf of H2 follows as:
+fH2(H2) ≤ C1
+C2
+1
+2√H2
+� √H2
+C2
+� 3N−2
+2
+e−
+2√
+H2
+C2 .
+(2)
+B. Feedback Channel
+In FDD systems, the downlink channel is estimated at the
+user side and then fed back to the BS and the other user using
+the limited feedback channel. In our system model shown in
+Fig. 1, User 1 quantizes the channel gain H1 and maps it
+to q(H1). Then, the index of q(H1) is fed back to the BS
+using B1 bits. Since User 2 communicates with the BS through
+the RIS, the phase information should be sent to the RIS as
+well. In this regard, first, User 2 maps the overall channel
+vector Gh2 to Q(Gh2) using B′ bits, where Q(·) is a RVQ
+and G = diag(g). User 2 then determines the phase vector
+θ of the quantized channel Gh2 denoted by θQ. The exact
+structure of quantizers q(·) and Q(·) is discussed in the next
+section, but does not change the overall characteristics of the
+feedback channel model. Next, User 2 quantizes the channel
+gain H2,Q = |θT
+QGh2|2, i.e., q(H2,Q), with B2 bits. User 2
+feeds the corresponding B2 + B′ bits back to the BS. The
+feedback link from User 2 to the BS is assumed to support
+B2 + B′ bits, although, the direct links might be blocked [14].
+The same explanation holds for the feedback link from User 1
+to the BS.
+In our system model, the BS, the RIS, and the users are
+fixed and the phase vector θQ and the channel gains H1 and
+H2 are only required once for every channel coherence time.
+
+B'
+RIS
+ Sending the phlase vector to RIS
+- Determining RIS phas
+- Allocating powers to users based on
+h2
+g
+- Determining the
+channel gains
+BS-RIS-User 2 gain
+- Determining the BS-User 1
+User 2
+Base Station
+h1
+channel gain
+B1
+B1
+User 1
+B2 +B'
+B2 +B'0
+δ
+sδ
+(2B  -1)δ
+(s+1)δ
+x
+q(x)
+2δ
+i
+Fig. 2: Applied uniform quantizer for quantizing channel gains (i = 1, 2).
+C. Sum Rate
+To maximize the sum rate by efficiently allocating the
+transmit power and subject to some minimum rate for each
+user, the following optimization problem can be defined
+maximize
+β
+R1 + R2
+(3a)
+subject to
+R1, R2≥ Rth,
+(3b)
+P1 + P2= P,
+(3c)
+where R1 = log2(1 + βPH1) and R2 = log2
+�
+1 + (1−β)P H2
+βP H2+1
+�
+. The
+power allocation can be parameterized by a factor β such
+that P1 = βP and P2 = (1 − β)P. In [31], given full
+CSI, Problem (3) is extended to an arbitrary number of users
+and individual minimum rate constrains. Using the solution
+given in [31, Eq. 15], we obtain the optimum factor β for
+H2 ≤ H1 as β∗ =
+P H2−ǫ
+(1+ǫ)P H2 where ǫ = 2Rth − 1. Hence,
+R1 = log2
+�
+1 + P H2H1−ǫH1
+H2(1+ǫ)
+�
+and R2 = Rth. The threshold Rth
+is the same for all users to ease the formulation but the
+approach works for arbitrary thresholds. Further, there are
+other useful objective functions to be considered. For example,
+similar to Ref. [11], we can study the rate fairness in our RIS-
+NOMA system with limited feedback and quantization error.
+III. UNIFORM AND RANDOM VECTOR QUANTIZERS
+In this section, we describe uniform quantizers and RVQs,
+used in our system. We use uniform quantizers to compress the
+scalar channel gains and RVQs to quantize the overall channel
+vector.
+To quantize H1, we define the following uniform quantizer
+q : R → R, shown in Fig. 2:
+q(x) =
+�
+⌊ x
+δ ⌋ × δ,
+x ≤
+�
+2B1 − 1
+�
+δ,
+�
+2B1 − 1
+�
+δ,
+x >
+�
+2B1 − 1
+�
+δ,
+(4)
+where x is any non-negative real number and δ denotes the
+size of quantization partitions. The index of q(H1) is fed back
+by B1 bits. The method in (4) quantizes the channel gain to the
+left boundary of the partition instead of the center point. When
+the gain is quantized to the center point, the quantized value
+might be higher than the true channel gain. This can frequently
+cause outage at the weak user, i.e., solving the optimization
+problem in (3) may result in allocating insufficient power to
+the weak user. Thus, Constraint (3b) may not hold for the
+quantized channel gain. To avoid the outage, we quantize the
+channel gain to the left boundary which guarantees the power
+allocation to the weak user is more than the needed optimal
+value.2 The uniform quantization is selected for simplicity but
+our approach works for non-uniform quantization as well.
+In general, there are two options for quantizing the vector
+Gh2: vector quantization and scalar quantization, applied to
+2The optimal power allocation, i.e., P ∗
+1 and P ∗
+2 , is obtained by determining
+β∗ in (3).
+vector’s elements. Since the number of elements in RISs can
+be large, scalar quantization will require a huge number of
+feedback bits and may not be practical. Inspired by this, we
+use random vector quantization in which the feedback bits can
+be far less than the number of elements3. We define the RVQ
+codebook W = {w1, w2, . . . , wM}, in which the codeword
+wi ∈ CN×1, is the quantized overall RIS-aided channel vector
+Gh2. The codebook W is generated by selecting each of M =
+2B′ vectors independently from a uniform distribution on the
+complex unit sphere [35].
+We aim to maximize the channel gain using the codebook
+such that
+Q(Gh2) = argmax
+w∈W
+|w†Gh2|2.
+(5)
+Further,
+we
+let
+φQ = ∠Q(Gh2)
+and
+θQ = ejφQ
+where
+φQ = [φ1,Q, · · · , φN,Q]T. The channel gain H2,Q =
+���θT
+QGh2
+���
+2
+takes any non-negative value and is mapped to q(H2,Q) using
+the quantizer in (4). As mentioned before, a uniform quantizer
+is applied to H2,Q instead of H2. Since H2 and H2,Q do
+not necessarily belong to the same partition, their quantized
+values might be different, i.e., q(H2,Q) ≤ q(H2). The index
+of q(H2,Q) is sent using B2 feedback bits. The uniform
+quantizers applied to H1 and H2,Q include the same number
+of partitions, i.e., B1 = B2 = B. Defining η = H2,Q
+H2 , we have
+η ∈ [0, 1]. Deriving the pdf of η is not straightforward, but
+needed in some analysis later. The exact pdf of η for a 2 × 1
+Rayleigh channel vector and a large number of feedback bits
+is derived in [36]. However, each element in Gh2 is a double-
+Rayleigh variable and its pdf is different from that of Rayleigh
+channels. Recently, the study of the pdf of
+��� �N
+i=1 |h2,i||gi|ejκi
+���
+2
+has been the topic of research in RIS-aided systems [18], [37].
+Note that κi does not necessarily equal to φi,Q. In [18, Lemma
+1], the pdf of the random variable
+��� �N
+i=1 |h2,i||gi|ejκi
+���
+2
+where
+κi is treated as a phase-noise is accurately approximated by
+a Gamma random variable. Following the same approach, our
+extensive empirical study shows that the pdf of the random
+variable η can be approximated by the pdf of a beta random
+variable with the shape parameters r1 and r2, i.e.,
+fη(η) ≈
+1
+B(r1,r2)ηr1−1(1 − η)r2−1,
+(6)
+where r1 =
+�
+E[η](1−E[η])
+V[η]
+�
+E[η], r2 =
+�
+E[η](1−E[η])
+V[η]
+�
+(1 − E[η]), and
+B(r1,r2) =
+Γ(r1+r2)
+Γ(r1)Γ(r2) represents a normalization constant that
+ensures the total probability is 1. The values of E[η] and V[η]
+depend on the RIS’s size N and the feedback bits B′. The
+empirical results show a close resemblance between the real
+pdf and the approximation, but we do not have space to show
+them in this paper.
+When the full CSI is available at the BS, the user ordering
+is always accurate and the rates are calculated using (3). In
+a limited feedback system, an inaccurate user ordering can
+impose severe rate loss. Even if the user ordering is accurate,
+3RVQ is a simple method but not the most efficient to quantize a vector
+with a limited number of feedback bits [32]. Other techniques such as Lloyd
+Algorithm [33] and variable-length limited feedback beamforming [34] which
+outperform the RVQ can be used to enhance the performance of the underlying
+limited feedback system although they increase the complexity.
+
+H1
+H2
+δ 
+η
+ 
+(2B-1)δ 
+η
+H2 =H1
+∞
+∞
+Fig. 3: Presentation of all the possible regions for the rate loss.
+the quantization can reduce the achievable sum rate.
+IV. RATE LOSS ANALYSIS
+Let us name the user with a higher channel gain the strong
+user. We calculate the rate loss only for the strong user because
+the weak user will not experience quantization rate loss. When
+instead of full CSI, the quantized channel gains are used in
+Problem (3), we call the resulting power allocation factor βq.
+In general, the strong user’s rate loss is obtained as
+∆R= Ri − Ri,q = log2(1 + PXi) − log2(1 + PXi,q)
+= log2
+�
+1 +
+P ∆X
+1+P Xi,q
+�
+≤ log2(1 + P∆X),
+(7)
+where ∆X = Xi − Xi,q denotes the normalized signal-to-
+noise ratio (SNR) loss. When user ordering is accurate, we
+have X1 = βH1 (X2 = βH2) and X1,q = βqH1 (X2,q = βqH2,Q).
+For an inaccurate user ordering, Xi is similar to the accurate
+one while X1,q = (1−βq)P q(H1)
+βqP q(H1)+1 and X2,q = (1−βq)P q(H2,Q)
+βqP q(H2,Q)+1 . Based
+on (7), an upper bound on the average rate loss is found as
+E[∆R] ≤ E[log2(1 + P∆X)] ≤ log2(1 + PE[∆X]).
+(8)
+The second inequality is due to Jensen’s inequality. Thus, to
+find the upper bound, we first derive ∆X and then calculate
+the expectation of ∆X, i.e., E[∆X]. However, the calculation
+of ∆X heavily depends on the values of H1 and H2. The main
+three Super Regions for this calculation, as shown in Fig. 3,
+are:
+• Super Region I: This consists of the conditions in which
+q(H1) = 0 and/or q(H2,Q) = 0 which results in βq = ∞.
+• Super Region II: This includes the main partitions of
+User 2’s uniform quantizer, i.e., δ
+≤
+q(H2,Q)
+<
+�
+2B − 1
+�
+δ.
+• Super Region III: This includes the upper marginal par-
+tition of User 2’s uniform quantizer, i.e., q(H2,Q) =
+�
+2B − 1
+�
+δ.
+We denote ∆X in Super Region I as ∆XI. In what follows,
+we calculate E[∆XI], i.e., the expected normalized SNR loss
+in each region.
+A. Super Region I
+Since βq = ∞, in this super region, NOMA is not feasible
+and we let E[∆XI] = 0.
+B. Super Region II
+Lemma 1. The total average normalized SNR loss of Super
+Region II is
+E[∆XII] ≤
+�
+(2B − 1)δ
+�
+C3 + δ
+�
+C4 + C5E1
+�
+δ
+L1
+���
++ C5
+√
+2δ,
+(9)
+where
+C3 = C6 + C7 + C8
+and
+C4 = C9 + C10.
+In
+detail,
+C5 ≥ C
+′
+5E
+�
+1
+√η
+�
+,
+C6 ≥ C
+′
+6E
+�
+1−η
+√η
+�
+,
+C7 ≥ C
+′
+7E
+�
+1−η
+√η
+�
+,
+C8 ≥ C
+′
+7E
+�
+(1 − η)√η
+�,
+C9 ≥ C
+′
+9E
+�
+1
+√η
+�
+,
+and
+C10 ≥ C
+′
+10E
+�
+1
+√η
+�
+in
+which
+C
+′
+5 =
+C1C2
+2
+3N+2
+2
+� 3N+2
+2
+�
+!,
+C
+′
+6 =
+C1C2
+2
+3N+2
+2
+� 3N+2
+2
+�
+! +
+C1L1
+2
+3N−2
+2
+C2
+� 3N−2
+2
+�
+!, C
+′
+7 =
+C1C3
+2
+2
+3N+6
+2
+L1
+� 3N+6
+2
+�
+!,
+C
+′
+9 =
+C1
+2
+3N−2
+2
+C2
+� 3N−2
+2
+�
+! +
+C1L1
+2
+3N−6
+2
+C3
+2
+� 3N−6
+2
+�
+!,
+and
+C
+′
+10 =
+C1C2
+2
+3N+2
+2
+L1
+� 3N+2
+2
+�
+!.
+Also,
+E1(x) =
+� ∞
+x
+e−t
+t dt
+denotes
+the exponential-integral function.
+Proof. Please see Appendix A.
+C. Super Region III
+Lemma 2. The total average normalized SNR loss for Super
+Region III is bounded by
+E[∆XIII] ≤ e−2
+√
+(2B−1)δ
+C2
+�
+2C11
+�
+1 + (2B−1)δ
+L1
+�
+×
+�
+1 +
+�
+2
+�
+(2B−1)δ
+C2
+2
+� 3N+2
+2
+�
++ C12
+�
+1 +
+�
+2
+�
+(2B−1)δ
+C2
+2
+� 3N−2
+2
+��
+. (10)
+where C11 =
+C1C2
+2
+2
+3N+4
+2
+� 3N+4
+2
+�
+! and C12 = C1C2L1
+2
+3N+2
+2
+� 3N
+2
+�
+!.
+Proof. Please see Appendix C.
+Finally, we have the following theorem on the expectation
+of the total rate loss for the quantized RIS-NOMA.
+Theorem 1. The total average rate loss for the quantized RIS-
+NOMA system with limited feedback is upper bounded by
+E[∆R] ≤ log2(1 + PE[∆X]),
+(11)
+where
+E[∆X]≤
+�
+(2B − 1)δ
+�
+C3 + δ
+�
+C4 + C5E1
+�
+δ
+L1
+���
++ C5
+√
+2δ
++e−2
+√
+(2B−1)δ
+C2
+�
+2C11
+�
+1 + (2B−1)δ
+L1
+��
+1 +
+�
+2
+�
+(2B−1)δ
+C2
+2
+� 3N+2
+2
+�
++C12
+�
+1 +
+�
+2
+�
+(2B−1)δ
+C2
+2
+� 3N−2
+2
+��
+.
+(12)
+Proof. We know that E[∆X] = E[∆XI]+E[∆XII]+E[∆XIII].
+Noting E[∆XI] = 0 and replacing E[∆XII] and E[∆XIII]
+with (9) and (10), respectively, results in (12).
+To guarantee that the rate loss approaches zero as B
+increases, one feasible solution is to define δ = ζ12−ζ2B for
+ζ1, ζ2 ∈ (0, 1). The parameters ζ1 and ζ2 are design parameters
+and should be optimized. Such a parameter optimization is out
+of the scope of this paper although the choice of ζ1 and ζ2 will
+not affect the system model. In simulations, we will intuitively
+select ζ1 and ζ2 to achieve good performance.
+
+8
+-H2=H1
+Region Ill.B
+Region Ill.C
+nH2-=H1
+V8
+nH2=H1
+Region Ill.A
+(2B-1)6
+n
+Region II.D
+Region II.C
+Region I
+Region Il.B
+Région Il.A
+H
+6
+(2B-1)620
+30
+40
+50
+60
+P (dBm)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Sum Rate (bits/s/Hz)
+NOMA, Full CSI
+OMA, Full CSI
+NOMA, B=2, B =2
+NOMA, B=4, B =2
+NOMA, B=6, B =2
+NOMA, B=2, B =4
+NOMA, B=4, B =4
+NOMA, B=6, B =4
+NOMA, B=2, B =8
+OMA, B =6
+OMA, B =12
+(a)
+20
+30
+40
+50
+60
+P (dBm)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Sum Rate (bits/s/Hz)
+NOMA, Full CSI
+OMA, Full CSI
+NOMA, B=2, B =2
+NOMA, B=4, B =2
+NOMA, B=6, B =2
+NOMA, B=2, B =4
+NOMA, B=4, B =4
+NOMA, B=6, B =4
+NOMA, B=2, B =8
+OMA, B =6
+OMA, B =12
+(b)
+Fig. 4: Performance of the sum rate versus the transmit power P for (a)
+ζ1 = 10−5 and ζ2 = 0.95 and (b) ζ1 = 0.5 × 10−5 and ζ2 = 0.95.
+V. NUMERICAL RESULTS
+We compare the sum rate performance of the RIS-NOMA
+system with limited feedback to that of the RIS-assisted or-
+thogonal multiple access (RIS-OMA) system. The RIS-NOMA
+system is described in Section II. For the RIS-OMA, we
+consider the same system model in Section II and replace
+NOMA with OMA. Furthermore, since power allocation is not
+required in the RIS-OMA, the channel gains are not fed back
+although the RIS-aided channel vector information should be
+fed back for beamforming at the RIS.
+The parameters are set according to [9] as follows. The
+distances are selected as d1=10 m, d2=40 m, and dg=10 m.
+Further, the path loss exponents are set as α1 = 3.5, α2 = 2.5,
+and αg = 2.5. The number of RIS elements is set to N = 10.
+We present the simulation results for the sum rate perfor-
+mance versus the total transmit power P for various feedback
+bits in Fig. 4. The transmit power depends on the users’
+path loss such that the power should compensate for the
+propagation loss. Simulation is conducted for the full CSI RIS-
+NOMA, full CSI RIS-OMA, limited feedback RIS-NOMA,
+and limited feedback RIS-OMA. To study the impact of δ
+where δ = ζ12−ζ2B, we set ζ2 = 0.95 and consider two
+different values for ζ1. We set ζ1 = 10−5 and 0.5 × 10−5
+in Figs. 4(a) and 4(b), respectively. The full CSI RIS-NOMA
+achieves the highest sum rate. When B and B′ increase,
+the limited feedback RIS-NOMA’s sum rate and the limited
+feedback RIS-OMA’s sum rate approach those of the full CSI
+RIS-NOMA and the full CSI RIS-OMA, respectively. This is
+consistent with our findings in Theorem 1. For instance, B = 6
+and B′ = 4, the sum rate is almost the same as that of the full
+CSI. Further, as we decrease the resolution of the quantizer,
+the length of the region in which we quantize the channel
+gain to 0 enlarges. Adopting a zero channel gain results in a
+zero sum rate as indicated in the low power portion of Fig. 4.
+However, in RIS-OMA, we do not impose any minimum rate
+constraint. This causes the RIS-OMA’s sum rate to be equal or
+slightly higher than the RIS-NOMA’s sum rate at low power
+levels although there is no guarantee for the minimum sum
+rate.
+We also observe that for B = 2 where B = B1 = B2 and
+B′ = 2, i.e., a total of 6 feedback bits, the limited feedback
+RIS-NOMA’s sum rate shows different behavior compared to
+the limited feedback RIS-OMA with B′ = 6. At low transmit
+powers, the limited feedback RIS-OMA’s sum rate is better
+than that of the limited feedback RIS-NOMA for both δ values.
+2
+4
+6
+8
+10
+12
+B (bit)
+10-1
+100
+Rate Loss (bits/s/Hz)
+B =2, P=40
+B =4, P=40
+B =8, P=40
+B =2, P=60
+B =4, P=60
+B =8, P=60
+Fig. 5: The average rate loss versus the number of the feedback bits.
+As we increase the power, the limited feedback RIS-NOMA
+improves the sum rate in comparison to the limited feedback
+RIS-OMA. However, at very high transmit power levels, the
+limited feedback RIS-NOMA’s slope is smaller than that of
+the limited feedback RIS-OMA, as shown in Fig. 4(a). When
+we select a smaller δ, as in Fig. 4(b), for any given B and B′,
+the limited feedback RIS-NOMA’s sum rate becomes higher
+than that of the limited feedback RIS-OMA. Another important
+observation is the impact of allocating B and B′ on the limited
+feedback RIS-NOMA’s sum rate. For instance, let us assume
+the total number of feedback bits is 12. When B = 4 and
+B′ = 4, in Fig. 4(a), the sum rate is higher than that of B = 2
+and B′ = 8. Whereas, given the same B and B′, in Fig. 4(b),
+these two schemes achieve almost the same sum rate.
+Fig. 5 compares the limited feedback RIS-NOMA’s average
+rate loss for ζ1 = 0.5 × 10−5 and ζ2 = 0.95. As the feedback
+bits and the power increase, the rate loss reduces. When B′ is
+fixed, by increasing B, the rate loss at power 60 dBm reduces
+faster than at power 40 dBm. In fact, increasing the power can
+compensate for the quantization error.
+VI. CONCLUSION
+In this paper, we studied an FDD-based RIS-NOMA system
+with a limited feedback channel. We used a RVQ to quantize
+the RIS-aided channel vector. Also, we considered a uniform
+quantizer for the channel gains. We then analyzed the rate
+loss of the strong user under the accurate and inaccurate
+user ordering conditions. Since the BS receives the quantized
+channel gains, inaccurate user ordering can happen often. We
+derived the rate loss resulted from quantization and showed
+that the rate loss essentially depends on the number of
+feedback bits, B and B′. From the simulations, we observed
+that the parameters B and B′ affect the sum-rate, the rate
+loss, and the probability that NOMA is not useful. As the
+number of feedback bits increases, the quantized RIS-NOMA’s
+performance approaches that of the full CSI RIS-NOMA.
+APPENDIX A
+PROOF OF LEMMA 1
+We divide this super region into Regions II.A-II.D, as
+shown in Fig. 3. Due to space limitations, we only provide
+the detailed calculation of the upper bound and constants for
+Region II.A. The upper bound and constants for other regions
+can be found similarly.
+Region II.A: In this region, H2 ≤ H1 which means User 1 is
+the strong user. It is clear that the output of the RVQ results
+in H2,Q ≤ H2 ≤ H1 and the uniform quantizer results in
+
+q(H2,Q) ≤ q(H1). Thus, the user ordering is accurate and
+the BS recognizes User 1 as the strong user. The average
+normalized SNR loss is upper bounded as
+E[∆XII.A] ≤ C6
+�
+(2B − 1)δ + C9δ
+�
+(2B − 1)δ.
+(13)
+The proof of (13) and the values of C6 and C9 are provided
+in Appendix B.
+Region II.B: In this region, H1 ≤ H2 indicates User 2 is
+stronger than User 1. It is possible that the RVQ leads to
+H2,Q ≤ H1 ≤ H2. Obviously, the uniform quantizer results
+in q(H1) ≥ q(H2,Q). Thus, the BS recognizes User 1 as the
+strong user which is not an accurate user ordering. The average
+normalized SNR loss is expressed as
+E[∆XII.B] ≤ C7
+�
+(2B − 1)δ,
+(14)
+It should be mentioned that C7 is a constant and bounded
+since E
+�
+1−η
+√η
+�
+is bounded. Also, E
+�
+1−η
+√η
+�
+can be approximated
+as
+B(r1− 1
+2 ,r2+1)
+B(r1,r2)
+using (6).
+Region II.C: In this region, H1 ≤ H2 and using the RVQ
+results in H1 < H2,Q ≤ H2. For H2,Q − H1 < δ, the uniform
+quantizer leads to q(H1) = q(H2,Q). In such a region, the user
+ordering is inaccurate and the BS picks User 1 as the strong
+user. Hence, the average normalized SNR loss is obtained as
+E[∆XII.C] ≤ C10δ
+�
+(2B − 1)δ + C5
+√
+2δ,
+(15)
+Region II.D: In this region, H1 ≤ H2 and the RVQ leads
+to H1 < H2,Q ≤ H2. If the uniform quantizer results in
+q(H1) < q(H2,Q), the user ordering is accurate and the BS
+selects User 2 as the strong user. Then, the average normalized
+SNR loss is bounded by
+E[∆XII.D] ≤ C8
+�
+(2B − 1)δ + C5δ
+�
+(2B − 1)δE1
+�
+δ
+L1
+�
+,
+(16)
+The constant C8 is bounded, and using (6) we obtain the
+approximated value of E
+�
+(1 − η)√η
+�
+as
+B(r1+ 1
+2 ,r2+1)
+B(r1,r2)
+which
+is finite.
+APPENDIX B
+PROOF OF (13)
+In Region II.A, the normalized SNR loss is obtained as
+∆XII.A= P H1H2−ǫH1
+P H2(1+ǫ)
+− P H1q(H2,Q)−ǫH1
+P q(H2,Q)(1+ǫ)
+(a)
+≤ P H1H2−ǫH1−P H1(H2,Q−δ)+ǫH1
+P H2(1+ǫ)
+(b)
+≤ H1H2−H1H2,Q+δH1
+H2
+(c)
+= H1H2−ηH1H2+δH1
+H2
+= (1 − η)H1 + δ H1
+H2 .
+(17)
+The inequality (a) follows from the fact that for any δ ≤
+q(H2,Q) <
+�
+2B − 1
+�
+δ, the inequalities q(H2,Q) ≥ H2,Q − δ
+and H2,Q ≥ q(H2,Q) hold. The inequality (b) is due to ǫ ≥ 0.
+Further, (c) is true because H2,Q = ηH2. For a given constant
+η, the expectation of ∆XII.A over the super region defined by
+δ
+η ≤ H2 < (2B−1)δ
+η
+and H2 ≤ H1 is given as
+E[∆XII.A|η] ≤ (1 − η)
+� (2B−1)δ
+η
+δ
+η
+� ∞
+H2
+H1fH1(H1)fH2(H2)dH1dH2
+�
+��
+�
+≡I1
++δ
+� (2B−1)δ
+η
+δ
+η
+� ∞
+H2
+H1
+H2
+fH1(H1)fH2(H2)dH1dH2
+�
+��
+�
+≡I2
+.
+(18)
+Next, we compute the integrals I1 and I2. For I1, we have
+I1=
+� (2B −1)δ
+η
+δ
+η
+e− H2
+L1 (H2 + L1)fH2(H2)dH2
+≤ C1
+2
+� (2B−1)δ
+η
+δ
+η
+�√H2
+C2
+� 3N
+2
+e
+−
+�
+H2
+L1 +
+2√
+H2
+C2
+�
+dH2
+�
+��
+�
+≡I1,1
++ C1L1
+2C2
+2
+� (2B−1)δ
+η
+δ
+η
+�√H2
+C2
+� 3N−4
+2
+e
+−
+�
+H2
+L1 +
+2√
+H2
+C2
+�
+dH2
+�
+��
+�
+≡I1,2
+.
+(19)
+The integral I1,1 is obtained as
+I1,1≤ � (2B−1)δ
+η
+0
+� √H2
+C2
+� 3N
+2 e
+−
+�
+H2
+L1 +
+2√
+H2
+C2
+�
+dH2
+(a)
+≤
+� (2B−1)δ
+η
+0
+� √H2
+C2
+� 3N
+2 e−
+2√
+H2
+C2 dH2
+(b)
+≤
+C2
+2
+3N
+2
+� 3N+2
+2
+�
+!
+�
+(2B−1)δ
+η
+.
+(20)
+The inequality (a) follows from the fact that e− H2
+L1 ≤ 1. The
+inequality (b) is due to the definition of the lower incomplete
+gamma function and using the upper bound γ(n, x) ≤ (n − 1)!x.
+Likewise, we obtain an upper bound on I1,2 as
+I1,2 ≤
+C2
+2
+3N−4
+2
+� 3N−2
+2
+�
+!
+�
+(2B−1)δ
+η
+.
+(21)
+Substituting (20) and (21) into (19) gives
+I1≤
+C1C2
+2
+3N+2
+2
+� 3N+2
+2
+�
+!
+�
+(2B−1)δ
+η
++
+C1L1
+2
+3N−2
+2
+C2
+� 3N−2
+2
+�
+!
+�
+(2B−1)δ
+η
+= C
+′
+6
+�
+(2B−1)δ
+η
+,
+(22)
+Also, we calculate I2 as
+I2 =
+� (2B−1)δ
+η
+δ
+η
+e− H2
+L1
+�
+1 + L1
+H2
+�
+fH2(H2)dH2 ≤ C
+′
+9
+�
+(2B−1)δ
+η
+,
+(23)
+Thus, the upper bound on E[∆XII.A|η] is found as
+E[∆XII.A|η] ≤ (1 − η)I1 + δI2,
+(24)
+where the upper bounds on I1 and I2 are derived in (22) and
+(23), respectively. In order to calculate E[∆XII.A], we have
+E[∆XII.A] ≤
+� 1
+0 (1 − η)I1fη(η)dη + δ
+� 1
+0 I2fη(η)dη
+≤ C
+′
+6
+�
+(2B − 1)δ
+� 1
+0
+1−η
+√η fη(η)dη
++C
+′
+9δ
+�
+(2B − 1)δ
+� 1
+0
+1
+√ηfη(η)dη
+= C
+′
+6E
+�
+1−η
+√η
+� �
+(2B − 1)δ + C
+′
+9E
+�
+1
+√η
+�
+δ
+�
+(2B − 1)δ
+≤ C6
+�
+(2B − 1)δ + C9δ
+�
+(2B − 1)δ,
+(25)
+Note that E
+�
+1−η
+√η
+�
+and E
+�
+1
+√η
+�
+are finite and for the approx-
+imation in (6) are, respectively, equal to
+B(r1− 1
+2 ,r2+1)
+B(r1,r2)
+and
+
+B(r1− 1
+2 ,r2)
+B(r1,r2) .
+APPENDIX C
+PROOF OF LEMMA 2
+We divide Super Region III into Regions III.A-III.C, as
+shown in Fig. 3.
+Region III.A: In this region, User 1 has a better channel
+compared to User 2, i.e., H2 ≤ H1. If the RVQ results in
+�
+2B − 1
+�
+δ ≤ H2,Q ≤ H2 ≤ H1, then uniform quantizer’s
+outcome will be q(H1) = q(H2,Q) =
+�
+2B − 1
+�
+δ and the
+ordering is accurate. In this region, the average normalized
+SNR loss is bounded by
+E[∆XIII.A] ≤e−2
+√
+(2B −1)δ
+C2
+�
+C11
+�
+1 +
+�
+2
+�
+(2B−1)δ
+C2
+2
+� 3N+2
+2
+�
++C12
+�
+1 +
+�
+2
+�
+(2B−1)δ
+C2
+2
+� 3N−2
+2
+��
+,
+(26)
+Region III.B: In this region, User 2 has a better channel
+and q(H1) =
+�
+2B − 1
+�
+δ, i.e.,
+�
+2B − 1
+�
+δ ≤ H1 ≤ H2. In
+such a region, the RVQ will result in either
+�
+2B − 1
+�
+δ ≤
+H1 ≤ H2,Q or
+�
+2B − 1
+�
+δ ≤ H2,Q ≤ H1. Also, the uniform
+quantizer will provide q(H1) = q(H2,Q) =
+�
+2B − 1
+�
+δ. Hence,
+the quantizers will inaccurately change the users’ order. The
+average normalized SNR loss for this region is bounded as
+E[∆XIII.B] ≤ C11e−2
+√
+(2B −1)δ
+C2
+�
+1 +
+�
+2
+�
+(2B−1)δ
+C2
+2
+� 3N+2
+2
+�
+.
+(27)
+Region III.C: In this region, User 1’s channel gain is lower
+than that of User 2 such that δ ≤ q(H1) < q(H2,Q) =
+�
+2B − 1
+�
+δ. Thus, User 2 is the strong user and the ordering
+is accurate. We obtain the expectation of the normalized SNR
+loss as
+E[∆XIII.C] ≤ 2 C11
+L1
+�
+2B − 1
+�
+δe−2
+√
+(2B−1)δ
+C2
+�
+1 +
+�
+2
+�
+(2B−1)δ
+C2
+2
+� 3N+2
+2
+�
+.
+(28)
+REFERENCES
+[1] C. Huang et al., “Reconfigurable intelligent surfaces for energy effi-
+ciency in wireless communication,” IEEE Trans. Wireless Commun.,
+vol. 18, no. 8, pp. 4157–4170, Aug. 2019.
+[2] L. Subrt and P. Pechac, “Intelligent walls as autonomous parts of smart
+indoor environments,” IET Commun., vol. 6, no. 8, pp. 1004–1010, May
+2012.
+[3] S. Hu, F. Rusek, and O. Edfors, “The potential of using large antenna
+arrays on intelligent surfaces,” in Proc. IEEE 85th Veh. Technol. Conf.
+(VTC Spring), pp. 1–6, 2017.
+[4] B. Cetiner et al., “Multifunctional reconfigurable MEMS integrated
+antennas for adaptive MIMO systems,” IEEE Commun. Mag., vol. 42,
+no. 12, pp. 62–70, Dec. 2004.
+[5] B. A. Cetiner and H. Jafarkhani, “Method and apparatus for an adap-
+tive multiple-input multiple-output (MIMO) wireless communications
+systems,” US Patent # 7,469,152, 2008.
+[6] Z. Ding and H. Vincent Poor, “A simple design of IRS-NOMA transmis-
+sion,” IEEE Commun. Lett., vol. 24, no. 5, pp. 1119–1123, May 2020.
+[7] Z. Ding, R. Schober, and H. V. Poor, “On the impact of phase shifting
+designs on IRS-NOMA,” IEEE Wireless Commun. Lett., vol. 9, no. 10,
+pp. 1596–1600, Oct. 2020.
+[8] 3rd Generation Partnership Project (3GPP), “Study on downlink mul-
+tiuser superposition transmission for LTE,” Mar. 2015.
+[9] Y. Cheng et al., “Downlink and uplink intelligent reflecting surface aided
+networks: NOMA and OMA,” IEEE Trans. Wireless Commun., vol. 20,
+no. 6, pp. 3988–4000, June 2021.
+[10] Z. Yang, Y. Liu, Y. Chen, and N. Al-Dhahir, “Machine learning for user
+partitioning and phase shifters design in RIS-aided NOMA networks,”
+IEEE Trans. Commun., vol. 69, no. 11, pp. 7414–7428, Nov. 2021.
+[11] X. Liu and H. Jafarkhani, “Downlink non-orthogonal multiple access
+with limited feedback,” IEEE Trans. Wireless Commun., vol. 16, no. 9,
+pp. 6151–6164, Sept. 2017.
+[12] X. Zou, M. Ganji, and H. Jafarkhani, “Downlink asynchronous non-
+orthogonal multiple access with quantizer optimization,” IEEE Wireless
+Communications Letters, vol. 9, no. 10, pp. 1606–1610, Oct. 2020.
+[13] P. Xu et al., “Reconfigurable intelligent surfaces-assisted communica-
+tions with discrete phase shifts: How many quantization levels are
+required to achieve full diversity?” IEEE Wireless Commun. Lett.,
+vol. 10, no. 2, pp. 358–362, Feb. 2021.
+[14] W. Chen et al., “Adaptive bit partitioning for reconfigurable intelligent
+surface assisted FDD systems with limited feedback,” IEEE Trans.
+Wireless Commun., vol. 21, no. 4, pp. 2488–2505, 2022.
+[15] D. Shen and L. Dai, “Channel feedback for reconfigurable intelligent sur-
+face assisted wireless communications,” in Proc. IEEE Global Commun.
+Conf., pp. 1–5, Dec. 2020.
+[16] J. Kim, S. Hosseinalipour, A. C. Marcum, T. Kim, D. J. Love, and C. G.
+Brinton, “Learning-based adaptive IRS control with limited feedback
+codebooks,” IEEE Trans. Wireless Commun., pp. 1–1, 2022.
+[17] N. Prasad, M. M. U. Chowdhury, and X. F. Qi, “Channel reconstruction
+with limited feedback in intelligent surface aided communications,” in
+Proc. IEEE 94th Veh. Technol. Conf. (VTC Fall), pp. 1–5, 2021.
+[18] F. A. P. de Figueiredo et al., “Large intelligent surfaces with discrete set
+of phase-shifts communicating through double-rayleigh fading channels,”
+IEEE Access, vol. 9, pp. 20 768–20 787, 2021.
+[19] Y. Zhang et al., “Reconfigurable intelligent surfaces with outdated
+channel state information: Centralized vs. distributed deployments,”
+IEEE Trans. Commun., vol. 70, no. 4, pp. 2742–2756, 2022.
+[20] W. Liang, Z. Ding, Y. Li, and L. Song, “User pairing for downlink non-
+orthogonal multiple access networks using matching algorithm,” IEEE
+Trans. Commun., vol. 65, no. 12, pp. 5319–5332, 2017.
+[21] D.-Y. Kim, H. Jafarkhani, and J.-W. Lee, “Low-complexity dynamic
+resource scheduling for downlink MC-NOMA over fading channels,”
+IEEE Trans. Wireless Commun., vol. 21, no. 5, pp. 3536–3550, 2022.
+[22] M. Fu et al., “Intelligent reflecting surface for downlink non-orthogonal
+multiple access networks,” in Proc. IEEE Globecom Workshops, pp. 1–6,
+Dec. 2019.
+[23] X. Mu et al., “Exploiting intelligent reflecting surfaces in NOMA
+networks: Joint beamforming optimization,” IEEE Trans. Wireless Com-
+mun., vol. 19, no. 10, pp. 6884–6898, Oct. 2020.
+[24] Y. Omid, S. Shahabi, C. Pan, Y. Deng, and A. Nallanathan, “Low-
+complexity beamforming design for IRS-aided NOMA communication
+system with imperfect CSI,” arXiv preprint arXiv:2203.03004, 2022.
+[25] Y. Jing and H. Jafarkhani, “Network beamforming with channel means
+and covariances at relays,” in 2008 IEEE International Conference on
+Communications.
+IEEE, 2008, pp. 3743–3747.
+[26] E. Koyuncu, Y. Jing, and H. Jafarkhani, “Distributed beamforming in
+wireless relay networks with quantized feedback,” IEEE J. Sel. Areas
+Commun., vol. 26, no. 8, pp. 1429–1439, Oct. 2008.
+[27] E. Koyuncu, C. Remling, X. Liu, and H. Jafarkhani, “Outage-optimized
+multicast beamforming with distributed limited feedback,” IEEE Trans.
+Wireless Commun., vol. 16, no. 4, pp. 2069–2082, April 2017.
+[28] Z. He and X. Yuan, “Cascaded channel estimation for large intelligent
+metasurface assisted massive MIMO,” IEEE Wireless Commun. Lett.,
+vol. 9, no. 2, pp. 210–214, Feb. 2020.
+[29] W. Khalid et al., “RIS-aided physical layer security with full-duplex
+jamming in underlay D2D networks,” IEEE Access, vol. 9, pp. 99 667–
+99 679, 2021.
+[30] I. Trigui et al., “Bit error rate analysis for reconfigurable intelligent
+surfaces with phase errors,” IEEE Commun. Lett., vol. 25, no. 7, pp.
+2176–2180, Apr. 2021.
+[31] Z. Yang, W. Xu, C. Pan, Y. Pan, and M. Chen, “On the optimality of
+power allocation for NOMA downlinks with individual QoS constraints,”
+IEEE Commun. Lett., vol. 21, no. 7, pp. 1649–1652, July 2017.
+[32] Y. T. Wu et al., “Comparison of codebooks for beamforming in limited
+feedback MIMO systems,” in Proc. IEEE Int. Conf. on Computer Sci.
+and Autom. Eng. (CSAE), vol. 2, pp. 32–36, May 2012.
+[33] A. Gersho and R. M. Gray, Vector quantization and signal compression.
+Springer, 1992.
+[34] E. Koyuncu and H. Jafarkhani, “Variable-length limited feedback beam-
+forming in multiple-antenna fading channels,” IEEE Trans. Inf. Theory,
+vol. 60, no. 11, pp. 7140–7165, Nov. 2014.
+[35] N. Jindal, “MIMO broadcast channels with finite-rate feedback,” IEEE
+Trans. Inf. Theory, vol. 52, no. 11, pp. 5045–5060, Nov. 2006.
+
+[36] C. R. Murthy and B. D. Rao, “Quantization methods for equal gain
+transmission with finite rate feedback,” IEEE Trans. Signal Process.,
+vol. 55, no. 1, pp. 233–245, Jan. 2007.
+[37] S. Atapattu et al., “Reconfigurable intelligent surface assisted two–way
+communications: Performance analysis and optimization,” IEEE Trans.
+Commun., vol. 68, no. 10, pp. 6552–6567, Oct. 2020.
+

6dAyT4oBgHgl3EQfpfjS/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:01135876539a8963ba025e860c1ef18eff4edb74dc96be2294ad91c0f382c3bb
+size 56267

8dAzT4oBgHgl3EQf-v5v/content/tmp_files/2301.01938v1.pdf.txt

@@ -0,0 +1,1158 @@
+Phase Transitions and Critical Phenomena for the FRW Universe in an Effective
+Scalar-Tensor Theory
+Haximjan Abdusattar,1, 2, ∗ Shi-Bei Kong,1, 2, † Hongsheng Zhang,3, 4, ‡ and Ya-Peng Hu1, 2, 5, §
+1College of Physics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China
+2Key Laboratory of Aerospace Information Materials and Physics (NUAA), MIIT, Nanjing 211106, China
+3School of Physics and Technology, University of Jinan,
+336 West Road of Nan Xinzhuang, Jinan, Shandong 250022, China
+4Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
+Chinese Academy of Sciences, Beijing 100190, China
+5Center for Gravitation and Cosmology, College of Physical Science
+and Technology, Yangzhou University, Yangzhou 225009, China
+We find phase transitions and critical phenomena of the FRW (Friedmann-Robertson-Walker)
+universe in the framework of an effective scalar-tensor theory that belongs to the Horndeski class.
+We identify the thermodynamic pressure (generalized force) P of the FRW universe in this theory
+with the work density W of the perfect fluid, which is a natural definition directly read out from
+the first law of thermodynamics. We derive the thermodynamic equation of state P = P(V, T) for
+the FRW universe in this theory and make a thorough discussion of its P-V phase transitions and
+critical phenomena. We calculate the critical exponents, and show that they are the same with the
+mean field theory, and thus obey the scaling laws.
+I.
+INTRODUCTION
+Since 1970’s, great efforts have been devoted to the investigations of black hole thermodynamics. Researchers
+come to several significant theoretical achievements in this area [1–3]. Nowadays, black hole thermodynamics takes a
+fundamental status in different modern advances in theoretical physics, for example in AdS/CFT correspondence [4, 5].
+However, it is still unsatisfactory for the developments of black hole thermodynamics in aspect of observations. The
+principal reason roots in the global property of traditional black hole thermodynamics, which requires the knowledge
+of global structure of the manifold [6]. To apply black hole thermodynamics in realistic astrophysics, one needs to
+obtain the properties in future time-like infinity, null infinity, and space-like infinity [7]. Besides gedanken experiments,
+it is impossible to obtain such information.
+In view of this situation, researchers develop quasi-local black hole thermodynamics, which only needs the structure of
+a patch of the manifold. This development makes black hole thermodynamics become tractable in realistic astrophysical
+environments. A natural concept related to a patch of a manifold is apparent horizon [7]. On the other hand, for black
+hole thermodynamics, the other crucial element is charge, and the desirable one is a conserved charge. Fortunately, for
+spherically symmetric spacetime, the Kodama vector leads to a conserved current, and further a conserved charge by
+integrating the current [8]. By using the properties of Kodama vector and apparent horizon, one arrives at the unified
+first law as a result of field equation [9]. In such an interpretation of field equation, a term related to a generalized
+force (thermodynamic pressure) appears in thermodynamic laws [10].
+FRW universe is a spherically symmetric spacetime, and has a thermal spectrum from its apparent horizon associated
+with a Hawking temperature [11, 12]. The existence of the apparent horizon is the main cause of having a self-
+consistent thermodynamics [13]. Ref.[14] first studied the connection between Friedmann equations and first law of
+thermodynamics for FRW universe [15], and subsequently Refs.[16, 17] extended similar studies to some alternative
+theories of gravity. However, some other important properties, including phase transition and critical behavior that
+have already been discovered in black holes, are rarely known for the FRW universe.
+Note that, the thermodynamic pressure plays an important role for phase transitions and critical behaviors. For
+example, in asymptotically AdS black hole, one usually treats the cosmological constant Λ as the thermodynamic
+variable analogous to the pressure P := −Λ/8π [18–20], and its conjugate quantity is the thermodynamic volume
+V . Moreover, one can further construct an equation of state P = P(V, T) for a black hole thermodynamic system,
+∗Electronic address: [email protected]
+†Electronic address: [email protected]
+‡Electronic address: sps [email protected]
+§Electronic address: [email protected]
+arXiv:2301.01938v1  [gr-qc]  5 Jan 2023
+
+2
+and investigate the P-V (van der Waals-like [21]) phase transition in the P-V phase diagram [22–25](see [26–40]
+for more related works, and also [41, 42] for reviews). For the FRW universe, the a significant issue is also related
+to the definition of its thermodynamic pressure P. In our recent papers [43–45], we have studied the first law of
+thermodynamics for the FRW universe and compared it with the usual standard form of first law dU = TdS − PdV ,
+and hence identified the proper thermodynamic pressure P with the work density W of the matter field which defined
+by Hayward [9]
+P ≡ W := −habT ab/2 ,
+(1.1)
+where hab and T ab are the 0, 1-components of the metric and the stress-tensor [16] with a, b = 0, 1, x0 = t, x1 = r. 1
+Using this definition of thermodynamic pressure, we further derived the thermodynamic equation of state P = P(V, T)
+for the FRW universe, and found an interesting P-V phase transition in a gravity with a generalized conformal scalar
+field [44]. These results indicate that the FRW universe might has a similarity with usual van der Waals thermodynamic
+system, and it is interesting and worthy of further investigations of its phase transitions in other modified theories of
+gravity.
+One of the most well-studied theories of modified gravity is the Lovelock gravity [47], which is a natural generalization
+of Einstein’s gravity. Because it gives covariant, conserved, second-order field equations, it is of particular interest.
+However, Lovelock terms usually do not have dynamical contribution to the field equations in four dimensional
+spacetime. Recently, a trick has been proposed to circumvent this limitation [48–50]. The trick is inspired by the
+novel 4D EGB gravity [50], where the coupling constant has been rescaled as α = α′/(D − 4), and taken the D → 4
+limit in the field equation. This procedure leaves nonvanishing contributions of the Lovelock terms on the equation of
+motion and thus one ends up with a seemingly novel theory of gravity in four dimension. However, this trick has been
+criticized to be ill-defined at the action level, and it can also result in divergence in the equations of motion and break
+the diffeomorphism of a general 4D spacetime [51–53]. To tackle this problem, a well-defined effective scalar tensor
+reformulation of Lovelock gravity was proposed [54].
+In the present paper, we would like to investigate the thermodynamics of the FRW universe in the effective
+scalar-tensor theory [54]. A scalar-tensor theory with high order derivatives is Horndeski gravity [55], whose equations
+of motion only have second-order derivatives, which eliminates any Ostrogradsky instability [56], which is similar to
+Lovelock gravity [47]. In Horndeski gravity, the P-V phase transition has been found to occur in black holes [22],
+which has raised interest in whether a similar phase transition can occur in the FRW universe. In this paper, we
+investigate the thermodynamic law of FRW universe in the most generic modified theory of gravity, i.e. effective
+scalar-tensor theory, and obtain its thermodynamic pressure P. Furthermore, we derive the thermodynamic equation
+of state P = P(V, T) of FRW universe in this theory, and find that it also represents a phase transition and critical
+behaviour around the critical point.
+This paper is organized as follows. In Sec.II, we briefly review the effective scalar-tensor theory and its Friedmann
+equations in a FRW universe. In Sec.III, we investigate the thermodynamics of the FRW universe in this modified
+gravity, and derive its thermodynamic equation of state P = P(V, T). In Sec.IV, we demonstrate the P-V phase
+transition and critical behaviors of the FRW universe in the effective scalar-tensor theory. In Sec.V, we make conclusions
+and discussion.
+II.
+A BRIEF INTRODUCTION OF THE EFFECTIVE SCALAR-TENSOR THEORY AND FRW
+UNIVERSE
+In this part we make a brief introduction on effective scalar-tensor reformulation of the regularized Lovelock gravity
+of Refs.[48–50] and it’s Friedmann’s equation’s in the FRW universe.
+The action from which this theory2 is given by [54]
+S =
+�
+dDx √−g L + Sm ,
+(2.1)
+1 For asymptotically AdS black holes, the thermodynamic pressure can still be defined by the work density once the cosmological constant
+term is treated as an effective stress-tensor, i.e. T eff
+µν
+≡ −Λgµν/8π, but the signs are different, i.e. P ≡ −W, which can be easily checked
+to be consistent with P := −Λ/8π. It should be emphasized that this definition is more general, since it is independent from the existence
+of the cosmological constant [43, 46].
+2 This theory can be viewed as a particular subclass of the Horndeski theory [57–60] (See also [61, 62] for more related works and for an
+extensive review [55, 63] of the literature).
+
+3
+where g is a determinant of the metric tensor gµν, Sm is the action associated with matter fields, and the Lagrangian 3
+L = α0 +
+�
+α1 − 2α′
+2Ke−2χ�
+R + α′
+2
+�
+−6K2e−4χ + 24Ke−2χX + 8X2 + 8X2χ + 4Gµνχµχν + χG
+�
++α′
+3
+�
+LH
+2 {192X3} + LH
+3 {−144X2} + LH
+4 {24X2} + LH
+5 {48X}
+�
+,
+(2.2)
+where R and Gµν are the four-dimensional Ricci scalar and Einstein tensor, respectively, and G is the Gauss-Bonnet
+combination of the four-dimensional curvature tensors, G = RµνγδRµνγδ − 4RµνRµν + R2. Here, χ is related to the
+metric in n-dimensional maximally symmetric space, χµ ≡ ∇µχ, X ≡ −χµχµ/2, K is the constant curvature, and LH
+i
+with i = 2, 3, 4, 5 are the corresponding terms in Horndeski theory. Note that the α0 and α1 correspond to Einstein’s
+theory with the cosmological constant, α′2, α′3 are the coupling constants with dimension of [length]2, [length]4, and
+we denote them, as α, β for simplicity in the following calculation.
+In the co-moving coordinate system {t, r, θ, ϕ}, the line-element of FRW universe is written as
+ds2 = −dt2 + a2(t)
+�
+dr2
+1 − kr2 + r2(dθ2 + sin2 θdϕ2)
+�
+,
+(2.3)
+where a(t) is the time-dependent scale factor, k is the spacial curvature. The matter field in the FRW universe, is
+usually treated as a perfect fluid with stress-tensor
+Tµν = (ρ + p)uµuν + pgµν ,
+(2.4)
+where ρ and p are the energy density and pressure, and uµ is the four-velocity of the fluid satisfying uµuµ = −1.
+Applying the modified theory of gravity (2.1) with (2.2) to the spatially flat (k = 0) (2.3) FRW universe with perfect
+fluid energy momentum stress-tensor (2.4), the Friedmann’s equations [13, 14, 54] are obtained as
+(1 + αH2 + βH4)H2 = 8π
+3 ρ ,
+(2.5)
+and
+(1 + 2αH2 + 3βH4) ˙H = −4π(ρ + p) ,
+(2.6)
+which are also satisfy the energy conservation equation
+˙ρ + 3H(ρ + p) = 0 ,
+(2.7)
+where H ≡ ˙a(t)/a(t) is the Hubble parameter, and “ · ” stands for the derivative with respect to the cosmic time.
+It should be emphasized that, if β = 0, the Eqs.(2.5) and (2.6) reduces to the Friedmann’s equations obtained in
+holographic cosmology [64–66], quantum corrected entropy-area relation [67], four dimensional Einstein-Gauss-Bonnet
+gravity [68] and gravity with a generalized conformal scalar field [44, 56].
+III.
+THERMODYNAMICS AND EQUATION OF STATE FOR THE FRW UNIVERSE IN THE
+EFFECTIVE SCALAR-TENSOR THEORY
+In this section, in the framework of the effective scalar-tensor theory, we give the first law of thermodynamics and
+construct an equation of state for the FRW universe from its Friedmann’s equations.
+For later convenience, we rewrite the line element of FRW universe (2.3) with areal radius R ≡ a(t)r to the spatially
+flat (k = 0) in the following form
+ds2 = habdxadxb + R2(dθ2 + sin2 θdϕ2) ,
+(3.1)
+where a, b = 0, 1 with x0 = t, x1 = r and hab = [−1, a2(t)]. For simplicity, we denote a ≡ a(t) in the following. For the
+dynamical spacetime there is an apparent horizon which the marginally trapped surface with vanishing expansion and
+3 If α′3 is set to zero, it is consistent with the result of the well-defined version of the four dimensional Einstein-Gauss-Bonnet theory
+[53, 56].
+
+4
+satisfies hab∂aR∂bR = 0 [9]. Apply this condition to the metric (3.1), one can easily obtain the radius of apparent
+horizon of the FRW universe [14]
+RA = 1
+H ,
+(3.2)
+whose time derivative is
+˙RA = −H ˙HR3
+A ,
+(3.3)
+which characterizes the time-dependent nature of the apparent horizon.
+With the expressions of the apparent horizon (3.2) and (3.3), one can rewrite the Friedmann’s equations (2.5) and
+(2.6) as
+ρ =
+3
+8πR2
+A
+�
+1 + α
+R2
+A
++ β
+R4
+A
+�
+,
+(3.4)
+p = −
+3
+8πR2
+A
+�
+1 + α
+R2
+A
++ β
+R4
+A
+�
++
+˙RA
+4πHR3
+A
+�
+1 + 2α
+R2
+A
++ 3β
+R4
+A
+�
+.
+(3.5)
+Substituting the above expressions into Eq.(1.1), we obtain the work density of the matter field for a FRW universe in
+this effective scalar-tensor theory
+W = 1
+2(ρ − p) =
+3
+8πR2
+A
+�
+1 + α
+R2
+A
++ β
+R4
+A
+�
+−
+˙RA
+8πHR3
+A
+�
+1 + 2α
+R2
+A
++ 3β
+R4
+A
+�
+.
+(3.6)
+The surface gravity on the apparent horizon of the spatially flat FRW universe [14]
+κ = − 1
+RA
+�
+1 −
+˙RA
+2
+�
+.
+(3.7)
+Thus the Hawking temperature associated with the apparent horizon of the spatially flat FRW universe is [14]
+T ≡ |κ|
+2π =
+1
+2πRA
+�
+1 −
+˙RA
+2
+�
+.
+(3.8)
+The total energy of matter inside the apparent horizon is usually defined by E = ρV [13, 16]. Accordingly, we use
+the Eq.(3.4) and thermodynamic volume V = 4πR3
+A/3 to obtain the energy for the FRW universe
+E = ρV = RA
+2 +
+α
+2RA
++
+β
+2R3
+A
+,
+(3.9)
+which will actually lead to a good thermodynamic first law to the FRW universe in this scalar-tensor theory as 4
+dE = −TdS + WdV ,
+(3.10)
+where T is the Hawking temperature (3.8), W is the work density (3.6). One immediate result of the above relation is
+the explicit form of the entropy
+S = πR2
+A + 4πα ln RA
+R0
+− 3πβ
+R2
+A
+= A
+4 + 2πα ln A
+A0
+− 12π2β
+A
+,
+(3.11)
+where A0, R0 are constants. The entropy S includes three terms, the first term is the Bekenstein-Hawking entropy, the
+second term is a logarithmic correction which often appears as the leading-order quantum correction [72–79], the third
+term represents further fluctuation of the entropy [80–82] could be regarded as an effective theory of quantum gravity.
+Note that the Eq.(3.11) recovers to the result of four dimensional regularized Gauss-Bonnet AdS black hole [56, 82], if
+4 This relation holds in nearly all of previous studies [43–45], where the energy E could be regarded as an effective Misner-Sharp energy
+[69–71].
+
+5
+β = 0. It should also be pointed that the minus sign5 before TdS in (3.10) arises from the treatment that the surface
+gravity (3.7) on the apparent horizon is negative [43, 46, 85].
+For a thermodynamic system, besides the laws of thermodynamics, equation of state like the van der Waals system
+usually also plays an important role. In order to clearly obtain the equation of state of the FRW universe in the
+effective scalar-tensor theory, we first compare Eq.(3.10) with the standard form of the thermodynamic first law
+dU = TdS − PdV ,
+(3.12)
+one can read out the internal energy U and thermodynamic pressure P, i.e.
+U ≡ −E ,
+(3.13)
+P ≡
+W .
+(3.14)
+Using Eqs.(3.8), (3.6) and (3.13), we further obtain the equation of state for the FRW universe in the effective
+scalar-tensor theory, i.e.6
+P =
+T
+2RA
+�
+1 + 2α
+R2
+A
++ 3β
+R4
+A
+�
++
+1
+8πR2
+A
+�
+1 − α
+R2
+A
+− 3β
+R4
+A
+�
+,
+(3.15)
+where RA = (3V/4π)1/3. The α and β terms may contain some new features of the FRW universe, which will be
+demonstrated in the following discussions. If α is absent, the equation of state is simplified to
+P =
+T
+2RA
++
+1
+8πR2
+A
++ 3βT
+2R5
+A
+−
+3β
+8πR6
+A
+.
+(3.16)
+IV.
+P-V CRITICALITY FOR THE FRW UNIVERSE IN THE EFFECTIVE SCALAR-TENSOR THEORY
+In this section, we would like to study the P-V phase transition and critical behaviors for the FRW universe in the
+effective scalar-tensor theory based on the equation of state (3.15). We first obtain the critical point and illustrate the
+behaviors in the P-V diagram. Then, we further calculate the critical exponents and discuss whether they satisfy the
+scaling laws or not.
+A.
+P-V phase transition and critical behavior
+The necessary conditions for P-V phase transition are [20, 22]
+�∂P
+∂V
+�
+T
+=
+�∂2P
+∂V 2
+�
+T
+= 0 ,
+(4.1)
+or equivalently
+� ∂P
+∂RA
+�
+T
+=
+� ∂2P
+∂R2
+A
+�
+T
+= 0 ,
+(4.2)
+have a critical-point solution T = Tc, P = Pc, RA = Rc.
+For the equation of state (3.15), the critical conditions (4.2) are
+9β
+4πR7c
+− 15βTc
+2R6c
++
+α
+2πR5c
+− 3αTc
+R4c
+−
+1
+4πR3c
+− Tc
+2R2c
+= 0 ,
+(4.3)
+− 63β
+4πR8c
++ 45βTc
+R7c
+−
+5α
+2πR6c
++ 12αTc
+R5c
++
+3
+4πR4c
++ Tc
+R3c
+= 0 .
+(4.4)
+5 The minus sign before T is a common feature of cosmological horizons such as the apparent horizon of the FRW universe and the event
+horizon of the de Sitter spacetime, but its nature and interpretation is a longstanding and puzzling problem. Recently, some interesting
+papers [83, 84] were dedicated to clarify this problem.
+6 If β is equal to zero, this equation reduces to that of a gravity with a generalized conformal scalar field [44], and if both α and β are set
+to zero, it recovers the one in Einstein gravity [43].
+
+6
+From the above two equations (4.3) and (4.4), one can obtain the equation for the critical radius of the apparent
+horizon
+− R8
+c + 12αR6
+c + 12α2R4
+c + 90βR4
+c + 132αβR2
+c + 135β2 = 0 .
+(4.5)
+It can be seen that the Eq.(4.5) contains higher-order terms of Rc, so it is very difficult to solve analytically. We first
+discuss a simple case, i.e. α = 0, β ̸= 0, and then the general case with α ̸= 0, β ̸= 0.
+1. α = 0,
+β ̸= 0
+In this situation, there is also a critical point if β is negative:
+Rc =
+4�
+3(15 − 4
+√
+15)β,
+Tc =
+1
+2 × 53/4π
+4�
+(1 − 4/
+√
+15)β
+,
+Pc = −
+�
+−(39 + 152/
+√
+15)β
+60πβ
+,
+(4.6)
+where Eqs.(3.16) and (4.2) have been used. From the above results, one can still get a dimensionless constant:
+2RcPc
+Tc
+= 2
+3 +
+1
+√
+15 ≈ 0.924866 ,
+(4.7)
+which is much larger than 3/8 (or 0.375) in the van der Waals system [20, 24].
+For convenience, one can define dimensionless pressure, temperature and radius as
+�P := P
+Pc
+,
+�R := RA
+Rc
+,
+�T := T
+Tc
+.
+(4.8)
+and rewrite (3.16) as
+�P = 2[3(4
+√
+5 − 5
+√
+3) �R4 −
+√
+3] �R �T + 5(4
+√
+3 − 3
+√
+5) �R4 +
+√
+5
+2
+�
+5
+√
+5 − 6
+√
+3
+� �R6
+.
+(4.9)
+In order to show the characteristic behaviors of the phase transition, we illustrate the corresponding �P- �R diagram
+based on Eq.(4.9) as shown in Fig.1.
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5 R
+˜
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+P
+˜
+T=1.1Tc
+T=Tc
+T=0.8Tc
+FIG. 1: Isothermal lines in the �
+P - �
+R diagram. The dashed blue line is the isothermal line with a higher temperature, i.e. T > Tc, where phase
+transition occurs; The red solid line is the isothermal line at the critical temperature T = Tc; The solid green line is the isothermal line with a
+lower temperature, i.e. T < Tc, where the system has only one phase thus no phase transition could occur. Note that the isothermal lines intersect
+at a thermodynamic singularity �
+Rs = (1 + 4/15)1/4.
+We can see from Fig.1 that, the phase transition occurs at the temperature larger7 than the critical temperature
+T > Tc, while the behavior is similar to an ideal gas for T < Tc. This behavior is different from that of a van der Waals
+7 It means that the phase transition of the FRW universe is a high energy phenomenon, which probably happened at the early stages of
+the Universe.
+
+7
+system and most of black holes system where coexistence phases appear below the critical temperature [20–22, 24].
+Moreover, there is a ‘thermodynamic singularity’, characterized by the pressure independence of temperature, i.e.
+(∂ �P/∂ �T)Rs = 0. It leads to a common point for different isothermal lines in the �P- �R diagram, which has also been
+found in many previous investigations [36, 37, 40].
+2. α ̸= 0,
+β ̸= 0
+In this case, the analytical solution of the critical radius can hardly be acquired, but it is relatively easy to discuss
+the parameter space that allows a physical solution of the critical radius.
+From Eqs.(3.15), (4.2) and (4.5), one can get the formal solutions of the critical temperature and critical pressure:
+Tc =
+−R4
+c + 2αR2
+c + 9β
+2πRc(R4c + 6αR2c + 15β),
+Pc = −7αR6
+c + 2R4
+c(5α2 + 33β) + 117αβR2
+c + 126β2
+8πR6c(R4c + 6αR2c + 15β)
+.
+(4.10)
+Note that the Rc, Tc and Pc are all should taken positive because of the critical point to be physical. This further
+constraints the allowed parameter space, see more detailed discussion in Appendix A. For α > 0, β > 0, we can
+distinguish the critical pressure Pc for given critical horizon radius Rc is negative, which means the system has no
+physical critical point corresponding to a van der Waals like phase transition in this situation. For α < 0, β < 0, we
+find that when β/α2 ≤ − 8
+√
+5+15
+75
+, there is a critical point once the values of coupling constants α and β are given.
+Therefore, we demonstrate some numerical results of these critical thermodynamic quantities as shown in Table I.
+TABLE I: The critical thermodynamic quantities for some coupling constants α and β
+β
+Rc
+Tc
+Pc
+RcPc/Tc
+α = −1
+−2
+1.551930
+0.075933
+0.019312
+0.394696
+−1
+1.388240
+0.083132
+0.022719
+0.373890
+− 8
+√
+5+15
+75
+1.242580
+0.089641
+0.026123
+0.362109
+α = − 1
+2
+−2
+1.430490
+0.084384
+0.024875
+0.421682
+−1
+1.245250
+0.096002
+0.031542
+0.409131
+− 8
+√
+5+15
+300
+0.878640
+0.126772
+0.052246
+0.362109
+We can see from Table I that for a specific α value, with the increase of β value, the critical radius decreases, while
+the critical temperature and critical pressure increase, and the rates of them RcPc/Tc decrease. In other words, when
+β is chosen at a fixed value, we find that as α increases, the critical radius decreases, and the critical temperature and
+pressure, and the rates of them RcPc/Tc are increase.
+B.
+critical exponents of the P-V phase transition
+In this part, we calculate the critical exponents near the critical point of P-V phase transition for a FRW universe
+in the framework of effective scalar-tensor theory.
+For a thermodynamic system, near the critical point of phase transition, there are four critical exponents (�α, �β, γ, δ)
+defined in the following [20, 22],
+CV = T
+�∂S
+∂T
+�
+V
+∝ |τ|−�α ,
+η = Vl − Vs
+Vc
+∼ ωl − ωs ∝ |τ|
+�β ,
+κT = − 1
+V
+�∂V
+∂P
+�
+T
+∝ |τ|−γ ,
+�P − 1 ∝ ωδ , (4.11)
+with
+τ = T
+Tc
+− 1 ,
+ω = RA
+Rc
+− 1 ,
+(4.12)
+where the labels ‘s’ and ‘l’ stand for ‘small’ and ‘large’ respectively.
+In the following, we will calculate the four critical exponents one by one for the FRW universe in the effective
+scalar-tensor theory.
+
+8
+As we have seen in previous section, the entropy of the FRW universe in the present work given in (3.11) is only a
+function of the thermodynamic volume V (or RA). Therefore, one can know that the heat capacity at constant volume
+CV is zero, which suggests that the first critical exponent ˜α = 0. To obtain the other three critical exponents, we use
+the expand the equation of state (3.15) near the critical point given by
+�P = 1 + a10τ + a11τω + a03ω3 + O(ω4, τω2) ,
+(4.13)
+with coefficients
+a10 = Tc(R4
+c + 2R2
+cα + 3β)
+2PcR5c
+,
+a11 = −Tc(R4
+c + 6R2
+cα + 15β)
+2PcR5c
+,
+a03 = (R4
+c + 2αR2
+c + 3β)(15β + αR2
+c)
+πPcR6c(R4c + 6αR2c + 15β)
+. (4.14)
+To clarity the critical behaviors of system, we demonstrate the coefficients of above equation in Table II by choosing
+some example values of the coupling constants (α, β).
+TABLE II: The coefficients in Eq.(4.13) for some example values of α and β
+β
+a10
+a11
+a03
+a11/a03
+α = 0
+−2
+−1.11672
+9.90859
+−6.39612
+−1.54915
+−1
+−1.11671
+9.90853
+−6.39580
+−1.54922
+−0.5
+−1.11670
+9.90844
+−6.39565
+−1.54925
+α = −1
+−2
+−1.09543
+8.44041
+−4.96216
+−1.70084
+−1
+−1.11428
+8.10766
+−4.55391
+−1.78038
+− 8
+√
+5+15
+75
+−1.16977
+7.79486
+−4.03490
+−1.93186
+α = − 1
+2
+−2
+−1.09273
+9.04763
+−5.59590
+−1.61683
+−1
+−1.09078
+8.76603
+−5.31302
+−1.64991
+− 8
+√
+5+15
+300
+−1.16974
+7.79473
+−4.03461
+−1.93196
+According to the Maxwell’s equal area law [31, 32], the end point of vapor and the starting point of liquid have the
+same pressure, i.e. �P ∗ = �Ps = �Pl, which indicates
+�P ∗ = a10τ + a11ωsτ + a03ω3
+s = a10τ + a11τωl + a03ω3
+l ,
+(4.15)
+or
+a11τ(ωl − ωs) + a03(ωl − ωs)3 = 0 .
+(4.16)
+Another relation from the Maxwell’s equal area law is that
+� s
+l �P dV = 0 in the P-V phase diagram [22, 31], which
+gives the following equation
+2a11τ(ω2
+l − ω2
+s) + 3a03(ω4
+l − ω4
+s) = 0 ,
+(4.17)
+where Eq.(4.13) has been used. From the above two Eqs. (4.16) and (4.17), one can get a nontrivial solution
+ωl =
+�
+−a11
+a03
+τ ,
+ωs = −
+�
+−a11
+a03
+τ ,
+(4.18)
+so
+ωl − ωs = 2
+�
+−a11
+a03
+τ ∝ |τ|1/2 ,
+(4.19)
+which shows that the second critical exponent �β = 1/2. Since a11/a03 is negative, we have τ > 0, which means that
+the coexistence phases in P-V diagram appear above the critical temperature T > Tc.8
+8 This is different from the behavior of the usual van der Waals systems and AdS black holes, where the coexistence phases are below the
+critical temperature [20–22].
+
+9
+The isothermal compressibility near the critical point can be calculated as follows
+κT = − 1
+Vc
+�∂V
+∂ �P
+�
+T
+����
+c
+∝ −
+�
+∂ �P
+∂ω
+�−1
+ω=0
+= −
+1
+a11τ ∝ τ −1 ,
+(4.20)
+which provides the third critical exponent γ = 1.
+When the temperature equals to the critical temperature T = Tc or τ = 0, from (4.13) we get
+�P − 1 ∝ ω3 ,
+(4.21)
+which gives the fourth critical exponent δ = 3.
+In summary, the four critical exponents associated with the P-V phase transition of the FRW universe in the
+effective scalar-tensor theory are obtained by
+�α = 0 ,
+�β = 1
+2 ,
+γ = 1 ,
+δ = 3 ,
+(4.22)
+which are consistent with the predications from the mean field theory [20–22], so the following scaling laws 9 are
+satisfied
+�α + 2�β + γ = 2 ,
+�α + �β(1 + δ) = 2 ,
+γ(1 + δ) = (2 − �α)(δ − 1) ,
+γ = �β(δ − 1) .
+(4.23)
+V.
+CONCLUSIONS AND DISCUSSION
+In this paper, we have studied the thermodynamic properties especially the P-V phase transitions of the FRW universe
+with a perfect fluid in an interesting effective scalar-tensor theory. We have studied the first law of thermodynamic for
+the FRW universe in this theory, and identified the thermodynamic pressure P with the work density of the perfect
+fluid, i.e. P := W. Using this identification, we have further derived the thermodynamic equation of state for the FRW
+universe P = P(V, T) in the effective scalar-tensor theory and found that there is a P-V phase transition. However,
+unlike the van der Waals system and most of the black holes system, the phase transitions in this framework occur
+above the critical temperature. Last but not the least, we have also calculated the corresponding critical exponents
+and found that they are the same as those in the van der Waals system and mean field theory, so they satisfy the
+scaling laws.
+Our results indicate that the van der Waals-type phase transition between a smaller phase and a larger phase for the
+FRW universe is a combined consequence of the matter fields, gravitational interactions and also the dynamical nature
+of the spacetime, etc. It should be pointed out that the thermodynamic pressure of the FRW universe with perfect
+fluid is clearly compatible with the first law of thermodynamics and the Friedmann equations, and the construction of
+the equation of state also relies on the definitions of the thermodynamic variables P, V, T. We are also curious about
+when these phase transitions occur in the evolution of the real Universe, and whether we can detect them through
+cosmological observations. Our results provide a theoretical platform for future astronomical observations. It would
+be also very interesting to study the thermodynamics and phase transitions of dynamical black holes in the same
+gravitational theory and compare the results with those of the FRW universe, and some universal properties may be
+found. These are open questions and will be studied in the future.
+Acknowledgment
+We are grateful for the stimulating discussions with Profs. Li-Ming Cao, Theodore A. Jacobson, Xiao-Mei Kuang,
+Yen Chin Ong, Shao-Wen Wei.
+This work is supported by the National Natural Science Foundation of China
+(NSFC) under grants No.12175105, No.11575083, No.11565017. H.Z. is Supported by the National Natural Science
+Foundation of China Grants No.12235019, and the National Key Research and Development Program of China (No.
+2020YFC2201400).
+9 It should be noted that there are only two independent ones.
+
+10
+Appendix A: Constraints on α and β from Critical Point
+In this appendix, based on Eqs.(4.5) and (4.10), we give conditions of the coupling constants α and β for which the
+critical apparent horizon radius Rc, critical temperature Tc, and critical pressure Pc are positive.
+By introducing
+X = R2
+c > 0 ,
+y = X
+α ,
+ξ = β
+α2 ,
+(A1)
+we can write Eq.(4.5) in the following form
+− y4 + 12y3 + 12y2 + 90ξy2 + 132ξy + 135ξ2 = 0 ,
+(A2)
+which has four roots y1, y2, y3, y4 that have complicated expressions, so we do not show them here. Since X and α2
+are positive, from Eq.(A1) one can see that the signs of y and ξ are only determined by α and β respectively. Based
+on this, we give the ranges of α and β in the following way.
+1. If α ∈ (−∞, 0) and β/α2 ∈ (−∞, −(8
+√
+5 + 15)/75] ∪ β/α2 = (8
+√
+5 − 15)/75, there is one positive critical radius
+Rc1 = √αy1.
+2. If α ∈ (−∞, 0) and β/α2 = (8
+√
+5 − 15)/75, or if α ∈ (0, +∞) and β/α2 ∈ (−∞, −(8
+√
+5 + 15)/75], there is one
+positive critical radius Rc2 = √αy2.
+3. If α ∈ (−∞, 0) and β/α2 ∈ [(8
+√
+5 − 15)/75, 1/15), or if α ∈ (0, +∞) and β/α2 = −(8
+√
+5 + 15)/75, there is one
+positive critical radius Rc3 = √αy3.
+4. If α ∈ (0, +∞) and β/α2 = −(8
+√
+5 + 15)/75 ∪ β/α2 ∈ [(8
+√
+5 − 15)/75, 1/15), there is one positive critical radius
+Rc4 = √αy4.
+The above results are shown in the following Table A:
+conditions ξ < − 8
+√
+5+15
+75
+ξ = − 8
+√
+5+15
+75
+ξ = 8
+√
+5−15
+75
+8
+√
+5−15
+75
+< ξ <
+1
+15
+α < 0
+Rc1
+Rc1
+Rc1, Rc2, Rc3
+Rc3
+α > 0
+Rc2
+Rc2, Rc3, Rc4
+Rc4
+Rc4
+In our previous work [44], there is no β term. In this case, the critical radius is very simple Rc =
+�
+(6 − 4
+√
+3)α.
+A natural question is that when β vanishes, whether previous results can be recovered. Through careful analysis,
+we found that only Rc1 could revert to the previous result, which is the interesting result to us. The root of (A2)
+corresponding to Rc1 is y1, which expression is
+y1 = 3 − 1
+α2
+�
+Aα2 + B − 1
+α2
+�
+2α2A − B − 24(3α2 + 7β)α4
+√
+Aα2 + B
+,
+(A3)
+where
+A ≡ 11α2 + 15β,
+B ≡ α4 − 18α2β + 45β2
+η1/3
++ η1/3α4 ,
+η ≡ 9β(3α4 + 17α2β − 75β2)
+α6
+− 6
+α6
+�
+β2(α2 − 3β)2(1075β2 + 450α2β − 19α4) − 1 .
+The critical temperature and critical pressure should be positive. When α > 0, β > 0, we find from Eq.(4.10) that,
+the critical pressure is negative, i.e. Pc < 0, so this case can be ruled out. Moreover, since we have chosen Rc1 as the
+critical horizon radius, we rule out the situation with α > 0. For α < 0, β > 0, from Tc > 0 (in Eq.(4.10)) we get
+�
+α2 > 3β ∩
+�
+Rc <
+�
+−
+�
+9α2 − 15β − 3α ∥
+��
+α2 + 9β + α < Rc <
+��
+9α2 − 15β − 3α
+��
+∥
+�
+α2 = 3β ∩
+�
+Rc <
+�
+−
+�
+9α2 − 15β − 3α ∥
+�
+−
+�
+9α2 − 15β − 3α < Rc <
+��
+9α2 − 15β − 3α
+��
+∥
+�
+α2 < 3β ∩ 3α2 > 5β ∩
+�
+Rc <
+��
+α2 + 9β + α ∥
+�
+−
+�
+9α2 − 15β − 3α < Rc <
+��
+9α2 − 15β − 3α
+��
+∥
+�
+3α2 ≤ 5β ∩
+��
+α2 + 9β + α > Rc
+�
+.
+(A4)
+
+11
+Unfortunately, it is difficult to obtain the constraints on α and β from Pc > 0. We think they are also determined by
+the critical temperature and pressure, so we don’t consider the situation with β > 0.
+For α < 0, β < 0, from Tc > 0 we get
+Rc <
+��
+9α2 − 15β − 3α .
+(A5)
+According to the Table A and Eq.(A5) with the numerical analysis, we find that the critical temperature and pressure
+are positive, if ξ = β/α2 ≤ − 8
+√
+5+15
+75
+in this situation.
+[1] S. W. Hawking, Commun. Math. Phys. 43, 199-220 (1975); S. W. Hawking, Nature 248, 30-31 (1974).
+[2] J. D. Bekenstein, Phys. Rev. D 9, 3292-3300 (1974); J. D. Bekenstein, Phys. Rev. D 7, 2333-2346 (1973).
+[3] J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31, 161-170 (1973).
+[4] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231-252 (1998) [arXiv:hep-th/9711200].
+[5] E. Witten, Adv. Theor. Math. Phys. 2, 505-532 (1998), [arXiv:hep-th/9803131].
+[6] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15 (1977), 2738-2751.
+[7] S. A. Hayward, Phys. Rev. D 53, 1938-1949 (1996), [arXiv:gr-qc/9408002].
+[8] T. Jacobson, Phys. Rev. Lett. 75, 1260-1263 (1995), [arXiv:gr-qc/9504004].
+[9] S. A. Hayward, Phys. Rev. D 49, 6467-6474 (1994);
+S. A. Hayward, Class. Quant. Grav. 15, 3147-3162 (1998), [arXiv:gr-qc/9710089].
+[10] T. Padmanabhan, Phys. Rept. 406, 49-125 (2005), [arXiv:gr-qc/0311036];
+T. Padmanabhan, Class. Quant. Grav. 19, 5387-5408 (2002), [arXiv:gr-qc/0204019].
+[11] R. G. Cai, L. M. Cao and Y. P. Hu, Class. Quant. Grav. 26 (2009), 155018, [arXiv:hep-th/0809.1554].
+[12] Y. P. Hu, Phys. Lett. B 701 (2011), 269-274, [arXiv:gr-qc/1007.4044].
+[13] Y. Gong and A. Wang, Phys. Rev. Lett. 99 (2007), 211301, [arXiv:hep-th/0704.0793].
+[14] R. G. Cai and S. P. Kim, JHEP 02, 050 (2005), [arXiv:hep-th/0501055].
+[15] R. G. Cai and L. M. Cao, Phys. Rev. D 75, 064008 (2007), [arXiv:gr-qc/0611071].
+[16] M. Akbar and R. G. Cai, Phys. Rev. D 75, 084003 (2007), [arXiv:hep-th/0609128].
+[17] M. Akbar and R. G. Cai, Phys. Lett. B 648, 243-248 (2007), [arXiv:gr-qc/0612089].
+[18] D. Kastor, S. Ray, and J. Traschen, Class. Quant. Grav. 26, 195011 (2009), [arXiv:hep-th/0904.2765].
+[19] B. P. Dolan, Class. Quant. Grav. 28, 125020 (2011), [arXiv:gr-qc/1008.5023];
+B. P. Dolan, Class. Quant. Grav. 28, 235017 (2011), [arXiv:gr-qc/1106.6260].
+[20] D. Kubiznak and R. B. Mann, JHEP 07, 033 (2012), [arXiv:hep-th/1205.0559].
+[21] David C Johnston, physics.chem-ph, [arXiv:cond-mat.soft/1402.1205].
+[22] Y. P. Hu, H. A. Zeng, Z. M. Jiang and H. Zhang, Phys. Rev. D 100, no.8, 084004 (2019), [arXiv:gr-qc/1812.09938].
+[23] Y. P. Hu, L. Cai, X. Liang, S. B. Kong and H. Zhang, Phys. Lett. B 822, 136661 (2021), [arXiv:gr-qc/2010.09363].
+[24] K. Bhattacharya and B. R. Majhi, Phys. Rev. D 95, no. 10, 104024 (2017), [arXiv:gr-qc/1702.07174];
+K. Bhattacharya, B. R. Majhi and S. Samanta, Phys. Rev. D 96, no.8, 084037 (2017), [arXiv:gr-qc/1709.02650].
+[25] S. H. Hendi and M. H. Vahidinia, Phys. Rev. D 88, no.8, 084045 (2013), [arXiv:hep-th/1212.6128];
+S. H. Hendi, R. B. Mann, S. Panahiyan and B. Eslam Panah, Phys. Rev. D 95, no.2, 021501 (2017), [arXiv:gr-qc/1702.00432].
+[26] S. Gunasekaran, R. B. Mann and D. Kubiznak, JHEP 11, 110 (2012), [arXiv:hep-th/1208.6251].
+[27] S. W. Wei and Y. X. Liu, Phys. Rev. D 87, no.4, 044014 (2013), [arXiv:gr-qc/1209.1707].
+[28] R. G. Cai, L. M. Cao, L. Li and R. Q. Yang, JHEP 09, 005 (2013), [arXiv:gr-qc/1306.6233].
+[29] M. H. Dehghani, S. Kamrani and A. Sheykhi, Phys. Rev. D 90 (2014) no.10, 104020, [arXiv:hep-th/1505.02386].
+[30] J. Xu, L. M. Cao and Y. P. Hu, Phys. Rev. D 91, no.12, 124033 (2015), [arXiv:gr-qc/1506.03578];
+R. G. Cai, Y. P. Hu, Q. Y. Pan and Y. L. Zhang, Phys. Rev. D 91, no.2, 024032 (2015), [arXiv:hep-th/1409.2369].
+[31] E. Spallucci and A. Smailagic, Phys. Lett. B 723, 436-441 (2013), [arXiv:hep-th/1305.3379].
+[32] B. R. Majhi and S. Samanta, Phys. Lett. B 773, 203 (2017), [arXiv:gr-qc/1609.06224].
+[33] P. Cheng, S. W. Wei and Y. X. Liu, Phys. Rev. D 94 (2016), 024025, [arXiv:gr-qc/1603.08694].
+[34] A. Dehyadegari and A. Sheykhi, Phys. Rev. D 98, no.2, 024011 (2018), [arXiv:gr-qc/1711.01151].
+[35] M. Estrada and R. Aros, Eur. Phys. J. C 80 (2020) no.5,395, [arXiv:gr-qc/1909.07280].
+[36] A. M. Frassino, D. Kubiznak, R. B. Mann and F. Simovic, JHEP 09 (2014), 080, [arXiv:hep-th/1406.7015].
+[37] R. A. Hennigar, E. Tjoa and R. B. Mann, JHEP 02, 070 (2017), [arXiv:hep-th/1612.06852].
+[38] A. Dehghani and M. R. Setare, [arXiv:hep-th/2204.01078].
+[39] K. Hegde, A. Naveena Kumara, C. L. A. Rizwan, A. K. M. and M. S. Ali, [arXiv:gr-qc/2003.08778].
+[40] R. Li and J. Wang, Phys. Lett. B 813, 136035 (2021), [arXiv:gr-qc/2009.09319].
+[41] N. Altamirano, D. Kubiznak, R. B. Mann and Z. Sherkatghanad, Galaxies 2 (2014), 89-159, [arXiv:hep-th/1401.2586].
+[42] D. Kubiznak, R. B. Mann and M. Teo, Class. Quant. Grav. 34, no.6, 063001 (2017), [arXiv:hep-th/1608.06147].
+[43] H. Abdusattar, S. B. Kong, W. L. You, H. Zhang and Y. P. Hu, JHEP 12, 168 (2022), [arXiv:gr-qc/2108.09407].
+[44] S. B. Kong, H. Abdusattar, Y. Yin and Y. P. Hu, Eur. Phys. J. C 82, no.11, 1047 (2022), [arXiv:gr-qc/2108.09411].
+[45] S. B. Kong, H. Abdusattar, H. Zhang and Y. P. Hu, [arXiv:gr-qc/2208.12603].
+
+12
+[46] H. Abdusattar, S. B. Kong, Y. Yin and Y. P. Hu, JCAP 08, no.08, 060 (2022), [arXiv:gr-qc/2203.10868].
+[47] D. Lovelock, J. Math. Phys. 12 (1971), 498-501; D. Lovelock, J. Math. Phys. 13 (1972), 874-876.
+[48] A. Casalino, A. Colleaux, M. Rinaldi and S. Vicentini, Phys. Dark Univ. 31, 100770 (2021), [arXiv:gr-qc/2003.07068].
+[49] A. Casalino and L. Sebastiani, Phys. Dark Univ. 31, 100771 (2021), [arXiv:gr-qc/2004.10229].
+[50] D. Glavan and C. Lin, Phys. Rev. Lett. 124, no.8, 081301 (2020), [arXiv:gr-qc/1905.03601].
+[51] J. Arrechea, A. Delhom and A. Jim´enez-Cano, Phys. Rev. Lett. 125, no.14, 149002 (2020), [arXiv:gr-qc/2009.10715];
+J. Arrechea, A. Delhom and A. Jim´enez-Cano, Chin. Phys. C 45, no.1, 013107 (2021), [arXiv:gr-qc/2004.12998].
+[52] M. G¨urses, T. C¸. S¸i¸sman and B. Tekin, Eur. Phys. J. C 80, no.7, 647 (2020), [arXiv:gr-qc/2004.03390].
+[53] R. A. Hennigar, D. Kubizˇn´ak, R. B. Mann and C. Pollack, JHEP 07, 027 (2020), [arXiv:gr-qc/2004.09472].
+[54] T. Kobayashi, JCAP 07 (2020), 013, [arXiv:gr-qc/2003.12771].
+[55] T. Kobayashi, Rept. Prog. Phys. 82 (2019) no.8, 086901, [arXiv:gr-qc/1901.07183].
+[56] P. G. S. Fernandes, Phys. Rev. D 103, no.10, 104065 (2021), [arXiv:gr-qc/2105.04687].
+[57] C. Gao, S. Yu and J. Qiu, [arXiv:gr-qc/2006.15586].
+[58] G. W. Horndeski, Int. J. Theor. Phys. 10 (1974), 363-384.
+[59] H. Lu and Y. Pang, Phys. Lett. B 809 (2020), 135717, [arXiv:gr-qc/2003.11552].
+[60] P. G. S. Fernandes, P. Carrilho, T. Clifton and D. J. Mulryne, Phys. Rev. D 102 (2020) no.2, 024025, [arXiv:gr-qc/2004.08362].
+[61] G. Alkac, G. D. Ozen and G. Suer, [arXiv:gr-qc/2203.01811].
+[62] K. Aoki, M. A. Gorji and S. Mukohyama, Phys. Lett. B 810 (2020), 135843, [arXiv:gr-qc/2005.03859];
+K. Aoki, M. A. Gorji and S. Mukohyama, JCAP 09 (2020), 014 [erratum: JCAP 05 (2021), E01], [arXiv:gr-qc/2005.08428].
+[63] P. G. S. Fernandes, P. Carrilho, T. Clifton and D. J. Mulryne, Class. Quant. Grav. 39 (2022) no.6, 063001, [arXiv:gr-
+qc/2202.13908].
+[64] P. S. Apostolopoulos, G. Siopsis and N. Tetradis, Phys. Rev. Lett. 102 (2009), 151301, [arXiv:hep-th/0809.3505].
+[65] N. Bilic, Phys. Rev. D 93 (2016) no.6, 066010, [arXiv:gr-qc/1511.07323].
+[66] J. E. Lidsey, Phys. Rev. D 88 (2013), 103519, [arXiv:hep-th/0911.3286].
+[67] R. G. Cai, L. M. Cao and Y. P. Hu, JHEP 08, 090 (2008), [arXiv:hep-th/0807.1232].
+[68] J. X. Feng, B. M. Gu and F. W. Shu, Phys. Rev. D 103 (2021), 064002, [arXiv:gr-qc/2006.16751].
+[69] H. Maeda and M. Nozawa, Phys. Rev. D 77 (2008), 064031, [arXiv:hep-th/0709.1199].
+[70] R. G. Cai, L. M. Cao, Y. P. Hu, and N. Ohta, Phys. Rev. D 80 (2009), 104016, [arXiv:hep-th/0910.2387].
+[71] R. G. Cai, L. M. Cao, Y. P. Hu and S. P. Kim, Phys. Rev. D 78 (2008), 124012, [arXiv:hep-th/0810.2610].
+[72] R. G. Cai, L. M. Cao and N. Ohta, JHEP 04, 082 (2010), [arXiv:hep-th/0911.4379].
+[73] S. Mukherji and S. S. Pal, JHEP 05, 026 (2002), [arXiv:hep-th/0205164].
+[74] A. Chatterjee and P. Majumdar, Phys. Rev. Lett. 92, 141301 (2004), [arXiv:gr-qc/0309026].
+[75] M. Domagala and J. Lewandowski, Class. Quant. Grav. 21, 5233-5244 (2004), [arXiv:gr-qc/0407051].
+[76] R. K. Kaul and P. Majumdar, Phys. Rev. Lett. 84, 5255-5257 (2000), [arXiv:gr-qc/0002040].
+[77] A. Sen, JHEP 04 (2013), 156, [arXiv:hep-th/1205.0971].
+[78] A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. 80, 904-907 (1998), [arXiv:gr-qc/9710007].
+[79] C. Rovelli, Phys. Rev. Lett. 77, 3288-3291 (1996), [arXiv:gr-qc/9603063].
+[80] A. Sheykhi, Phys. Rev. D 81 (2010), 104011, [arXiv:gr-qc/1004.0627];
+A. Sheykhi, Eur. Phys. J. C 69 (2010), 265-269, [arXiv:hep-th/1012.0383].
+[81] T. Zhu, J. R. Ren and M. F. Li, JCAP 08 (2009), 010, [arXiv:hep-th/0905.1838].
+[82] P. G. S. Fernandes, Phys. Lett. B 805 (2020), 135468, [arXiv:gr-qc/2003.05491].
+[83] B. Banihashemi and T. Jacobson, JHEP 07 (2022), 042, [arXiv:hep-th/2204.05324].
+[84] B. Banihashemi, T. Jacobson, A. Svesko and M. Visser, [arXiv:hep-th/2208.11706].
+[85] B. P. Dolan, D. Kastor, D. Kubiznak, R. B. Mann, and J. Traschen, Phys. Rev. D 87, no.10, 104017 (2013), [arXiv:hep-
+th/1301.5926].
+

CdE2T4oBgHgl3EQf9AmF/content/tmp_files/2301.04224v1.pdf.txt

@@ -0,0 +1,2042 @@
+Pix2Map: Cross-modal Retrieval for Inferring Street Maps from Images
+Xindi Wu 1*
+KwunFung Lau1
+Francesco Ferroni2
+Aljoˇsa Oˇsep1
+Deva Ramanan1,2
+1Carnegie Mellon University
+2Argo AI
+Abstract
+Self-driving vehicles rely on urban street maps for au-
+tonomous navigation. In this paper, we introduce Pix2Map,
+a method for inferring urban street map topology directly
+from ego-view images, as needed to continually update and
+expand existing maps. This is a challenging task, as we
+need to infer a complex urban road topology directly from
+raw image data. The main insight of this paper is that this
+problem can be posed as cross-modal retrieval by learning
+a joint, cross-modal embedding space for images and ex-
+isting maps, represented as discrete graphs that encode the
+topological layout of the visual surroundings. We conduct
+our experimental evaluation using the Argoverse dataset
+and show that it is indeed possible to accurately retrieve
+street maps corresponding to both seen and unseen roads
+solely from image data. Moreover, we show that our re-
+trieved maps can be used to update or expand existing maps
+and even show proof-of-concept results for visual localiza-
+tion and image retrieval from spatial graphs.
+1. Introduction
+We propose Pix2Map, a method for inferring road maps
+directly from images. More precisely, given camera im-
+ages, Pix2Map generates a topological map of the visible
+surroundings, represented as a spatial graph. Such maps en-
+code both geometric and semantic scene information such
+as lane-level boundaries and locations of signs [53] and
+serve as powerful priors in virtually all autonomous vehi-
+cle stacks. In conjunction with on-the-fly sensory measure-
+ments from lidar or camera, such maps can be used for lo-
+calization [3] and path planning [40]. As map maintenance
+and expansion to novel areas is challenging and expensive,
+often requiring manual effort [39,51], automated map main-
+tenance and expansion has been gaining interest in the com-
+munity [10,11,28,33,34,36,39,41].
+Why is it hard? To estimate urban street maps, we need to
+learn to map continuous images from ring cameras to dis-
+crete graphs with varying number of nodes and topology
+*Work done while at Carnegie Mellon University.
+Global Street Map
+Camera 
+Data
+Street Map
+Adjacency Matrix
+Figure 1.
+Illustration of our proposed Pix2Map for cross-
+modal retrieval. Given unseen 360◦ ego-view images collected
+from seven ring cameras (left), our Pix2Map predicates the lo-
+cal street map by retrieving from the existing street map library
+(right), represented as an adjacency matrix. Local street map can
+be further used for global high-definition map maintenance (top).
+in bird’s eye view (BEV). Prior works that estimate road
+topology from monocular images first process images using
+Convolutional Neural Networks or Transformers to extract
+road lanes and markings [10] or road centerlines [11] from
+images. These are used in conjunction with recurrent neural
+networks for the generation of polygonal structures [13] or
+heuristic post-processing [34] to estimate a spatial graph in
+BEV. This is a very difficult learning problem: such meth-
+ods need to jointly learn to estimate a non-linear mapping
+from image pixels to BEV, as well as to estimate the road
+layout and learn to generate a discrete spatial graph.
+Pix2Map.
+Instead, our core insight is to simply sidestep
+the problem of graph generation and 3D localization from
+monocular images by recasting Pix2Map as a cross-modal
+retrieval task: given a set of test-time ego-view images, we
+(i) compute their visual embedding and then (ii) retrieve
+a graph with the closest graph embedding in terms of co-
+sine similarity. Given recent multi-city autonomous vehi-
+cle datasets [14], it is straightforward to construct pairs of
+ego-view images and street maps, both for training and test-
+ing. We train image and graph encoders to operate in the
+same embedding space, making use of recent techniques
+for cross-modal contrastive learning [45]. Our key technical
+contribution is a novel but simple graph encoder, based on
+1
+arXiv:2301.04224v1  [cs.CV]  10 Jan 2023
+
+sequential transformer models from the language commu-
+nity (i.e., BERT [19]), that extracts fixed-dimensional em-
+beddings from street maps of arbitrary size and topology.
+In fact, even naive retrieval methods for finding the near-
+est neighbor in the training set perform comparably to the
+current leading techniques for map generation, specifically,
+finding the graph associated with the best-matching image.
+Further, we demonstrate the cross-modal retrieval efficacy
+of Pix2Map on real-world data from Argoverse [14].
+It
+outperforms baselines that learn to generate graphs directly
+from image cues [10, 11] due to its ability to learn graph
+embeddings that regularize the output space. In addition,
+cross-modal retrieval has the added benefit of allowing us
+to expand our graph library with unpaired graphs that lack
+camera data, which means it does not need the same paired
+(image, graph) training data used for learning the encoders.
+Interestingly, the training data (of image-graph pairs) used
+to learn the encoders may be different from the graph library
+used for retrieval. Moreover, we show that when expand-
+ing our graph library with unpaired graphs improves across
+all metrics. This suggests there is further potential in im-
+proving mapping accuracy as we expand our graph database
+with additional data, such as augmented road graph topolo-
+gies. Beyond mapping, we show pilot experiments for vi-
+sual localization, and the inverse method, Map2Pix, which
+retrieves a close-matching image from an image library
+given a graph. While not the primary focus of our work,
+approaches that extend Map2Pix may ultimately be useful
+for generating photorealistic simulated worlds [41].
+We summarize our main contributions as follows: (i)
+we show that dynamic street map construction from cam-
+eras can be posed as a cross-modal retrieval task and pro-
+pose an image-graph contrastive model based on this fram-
+ing. Building on recent advances in multimodal represen-
+tation learning, we train a graph encoder and an image en-
+coder with a shared latent space. We (ii) demonstrate empir-
+ically that this approach is effective in this new domain and
+perform ablation studies to highlight the impacts of archi-
+tectural decisions. Our approach outperforms the existing
+graph generation methods from image cues by a large mar-
+gin. We (iii) further show that it is possible to retrieve sim-
+ilar graphs to those in previously unseen areas without ac-
+cess to the ground truth graphs for those areas, and demon-
+strate the generalization ability to novel observations.
+2. Related Work
+Maps are ubiquitous in robotics: given a pre-built map
+of the environment, autonomous agents can localize them-
+selves via live sensory data and plan their future trajecto-
+ries [18]. Since the dawn of robotics, mapping and localiza-
+tion have been vibrant fields of research [54], tackled using
+different types of sensors, ranging from line-laser RGB-D
+sensors for indoor mapping [12, 24, 48], to lidar [4] and/or
+cameras [20,21,42], commonly used outdoors [9,14,22,52].
+In the following, we focus on map construction and main-
+tenance. For localization, we refer to prior work [40,54].
+Map Representation.
+Several map representations have
+been proposed in the community, ranging from full 3D
+maps, represented as meshes [48, 55], voxel grids [12, 24,
+32, 56], and (semantic) point clouds [4, 16]. In visual lo-
+calization [50], point clouds are often constructed using
+structure-from-motion methods [50] and additionally store
+visual descriptors that aid matching-based visual localiza-
+tion. The aforementioned representations can be used for
+highly-accurate 6-DoF camera localization. However, they
+are storage (and, consequentially, transmission) heavy [62],
+which limits their applicability in outdoor environments.
+High-Definition (HD) Maps.
+Alternatively, High Def-
+inition (HD) maps store key semantic information, such
+as road layout and traffic light sign positions [53], to-
+gether with their attributes and connectivity information. As
+shown in [40], such sparse and storage efficient maps can
+be used as priors for centimeter-precise vehicle localization
+in conjunction with vehicle sensors, such as cameras and li-
+dars. While immensely useful, HD maps are difficult to cre-
+ate and maintain [27,39,51] and often require human man-
+ual human annotations and post-processing, rendering map
+construction and maintenance costly. Therefore, a problem
+of great importance is the automation of map construction
+and maintenance directly from sensory data.
+HD Map Construction and Maintenance. Several meth-
+ods for HD map estimation rely on various sensor modali-
+ties. Li et al. [35] propose a method that generates a topo-
+logical map (represented as a spatial graph) of a city from
+satellite images. Wang et al. [57] propose a collaborative
+approach that fuses several sources of information (data
+from airplanes, drones, and cars), such that consequent
+manual human post-processing can be minimized. Several
+methods tackle map construction directly from on-board ve-
+hicle sensory data. Often, methods tackle this challenging
+problem by first detecting road features in images (e.g., seg-
+ment lanes) [5, 15, 38, 58, 59], and then utilize the camera
+and lidar sensory data to estimate precise road layout in 3D
+space. To generate spatial graphs, the aforementioned meth-
+ods employ generative recurrent neural networks [28, 36],
+or optimization-based approach [37]. Unlike the aforemen-
+tioned works, we estimate spatial graphs directly from im-
+ages, by-passing explicit lane estimation.
+Pixel Segmentation.
+State-of-the-art methods for graph
+generation from monocular images [10,11] tend to first seg-
+ment road lanes or centerlines in images, followed by graph
+generation using Polygon-RNN [13]. Instead of road lanes,
+HDMapNet [34] utilize methods for semantic/instance BEV
+maps (from cameras and/or lidar e.g., [23, 29, 47, 49, 60]),
+followed by heuristic post-processing to obtain vectorized
+2
+
+...
+...
+...
+...
+G1 ? I1 G1 ? I2 G1 ? I3
+...
+...
+G1 ? IN
+G2? I1 G2 ? I2 G2 ? I3
+...
+G2 ? IN
+G3? I1 G3? I2 G3? I3
+...
+G3? IN
+GN? I1 GN? I2 GN? I3
+...
+GN? IN
+...
+ (                           )
+...
+...
+...
+...
+...
+...
+...
+...
+...
+I1
+I2
+I3
+...
+IN
+G2
+G3
+GN
+G1
+...
+...
+...
+...
+...
+...
+...
+Figure 2. Pix2Map: The graph encoder (bottom) computes a graph embedding vector φgraph for each street map in a batch. The image
+encoder, (top) outputs an image embedding φimage for each corresponding image stack. We then build a similarity matrix for a batch, that
+contrasts the image and graph embeddings. We highlight that the adjacency matrix of a given graph is used as the attention mask for our
+transformer-based graph encoder.
+HD maps.
+HDMapGen [41] builds on recent develop-
+ments in generative graph modeling [61] to construct con-
+trol points of central lane lines and their connectivity in a hi-
+erarchical manner. However, generating graphs conditioned
+on a particular (image) input remains an open problem. Un-
+like HDMapGen, Pix2Map sidesteps generative modeling,
+and instead directly retrieves a graph from a large database
+whose embedding vector is most similar to image embed-
+dings in terms of cosine distance. We show that Pix2Map
+can also be used to keep HD maps up-to-date [6,7,33,44].
+3. Method
+In this section, we formalize our Pix2Map approach as
+a cross-modal retrieval task.
+As shown in Fig. 2, given
+training pairs of images and graphs, we learn image and
+graph encoders that map both inputs to a common fixed-
+dimensional space via contrastive learning. We then use the
+learned encoders to retrieve a graph (from a training library)
+with the most similar embedding to the test image.
+3.1. Problem Formulation
+We construct a library of image-graph pairs (I, G). Here,
+I is a list of 7 ego-view images from a camera ring and G is
+a street map represented as a graph G = (V, E). Each ver-
+(0, 0)
+(0, 0)
+(0, 0)
+(0, 0)
+Figure 3. Two street map examples from Pittsburgh (left) and Mi-
+ami (right) as segment graphs and our resampled node graph, with
+ego-vehicle origin at (0, 0).
+tex v ∈ V represents a lane node and E ∈ {0, 1}|V |×|V | that
+encodes the connectivity between nodes, stored in an adja-
+cency matrix. Lane nodes have a position attribute (x, y)
+in a local egocentric “birds-eye-view” coordinate frame,
+such that (0, 0) is the ego-vehicle location (Fig. 3). Im-
+portantly, different graphs G may have different numbers
+of lane nodes and connectivity information.
+Graph Representation. The Argoverse dataset represents
+a street map as a segment graph, where each vertex rep-
+resents a lane segment. Lane segments are represented as
+polylines with 10 (x, y) points. We convert this segment
+graph to a node graph by defining each (x, y) point as a
+graph node and adding a directed edge between successive
+points in a polyline (Fig. 3). We further resample the seg-
+3
+
+VectorMapWithSampling(PIT)
+20
+15
+10
+/coordinate
+5
+0
+-5
+-10
+15
+-20
+20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+x coordinateVectorMapWithSampling(PiT
+20
+15
+10
+/coordinate
+5
+0
+-10
+15
+-20
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+x coordinate7x3
+64
+64
+128
+256
+512
+512
+7
+1
+14
+28
+56
+112
+224VectorMapWithSampling(PIT)
+20
+15
+10
+coordinate
+5 -
+-0
+-5 
+-10
+-15
+-20
+-20
+-15
+-10
+-5
+0
+7
+5
+10
+15
+20
+x coordinateAdjacency Matrix
+20
+Node index
+40
+60
+80
+0
+20
+40
+60
+80
+Node indexVectorMapWithSampling(PIT)
+20
+15
+10
+coordinate
+5
+0
+y
+-5
+-10
+15
+-20
+-20
+-15
+-10
+-5
+0
+5
+1
+10
+15
+20
+x coordinateVectorMapWithoutSampling(PIT)
+20
+15
+10
+5
+/coordinate
+0
+-10
+-15
+-20
+20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+x coordinateVectorMapWithSampling(PIT)
+20
+15
+10
+/coordinate
+5
+0
+-5
+-10
+15
+-20
+20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+x coordinateVectorMapWithoutSampling(MiA
+20
+15
+10
+/coordinate
+5
+0
+-5 
+-10
+-15
+-20
+-20
+-15
+-10
+0
+1
+5
+-5
+10
+15
+20
+x coordinateVectorMapWithSampling(MIA
+20
+15
+10
+ycoordinate
+5
+0
+-5
+-10
+-15
+-20
+20
+15
+-10
+-5
+0
+5
+10
+15
+20
+x coordinatement graphs by fitting degree-3 spline curves to lane seg-
+ments, ensuring that connected nodes throughout the graph
+are approximately equidistant (2m). We use this library for
+training the image and graph encoders, as detailed below.
+3.2. Image Encoder
+Given an image I, we use ResNet18 [26] as a fea-
+ture extractor (without fully connected layers) as an im-
+age encoder that learns fixed-dimensional embedding vec-
+tors φimage(I) ∈ R512.
+To process n input images (we
+use n = 7 images throughout our work), we stack them
+channel-wise. We experiment with ImageNet-pre-trained
+weights, as well as training “from scratch”.
+In the pre-
+trained case, we replace the first convolution layer with one
+that stacks the original pre-trained filters n times, with each
+weight divided by n. We reuse the original weights when
+making the new convolutional filters so the benefits of the
+pre-trained weights will be preserved. We ablate different
+training strategies and alternative encoder architectures in
+Sec. 4.
+3.3. Graph Encoder
+Given a graph G = (V, E), we would like to produce a
+fixed dimensional embedding φgraph(G) ∈ R512, that is in-
+variant to orderings of graph nodes. Unlike pixels in an im-
+age or words in a sentence, nodes in graphs do not have an
+inherent order. We construct such a graph encoder using a
+Transformer architecture inspired by sequence-to-sequence
+architectures from language [19] defined on sequential to-
+kens. Our encoder treats lane nodes as a collection of tokens
+and edges as masks for attention processing:
+vl+1 =
+�
+{w:E(v,w)=1}
+Value(vl)Softmaxw[Query(vl)Key(w)], (1)
+where vl is the embedding for vertex v at layer l and v0 is
+initialized to its (x, y) position. We omit multiple attention
+heads and layer norm operations for brevity.
+Embeddings are fed into a Transformer that computes
+new embeddings by taking an attention-weighted average of
+embeddings from nodes w adjacent to v (as encoded in the
+adjacency matrix E). We apply M = 7 transformer layers
+(similar to BERT [19]). Finally, we average (or mean pool)
+all output embeddings to produce a final fixed-dimensional
+embedding for graph G, regardless of the number of nodes
+or their connectivity: φgraph(G) =
+1
+|V |
+�
+v∈V vM ∈ R512.
+In Sec. 4 we ablate various design choices for our graph
+encoder, including the usage of periodic positional embed-
+dings and encoding the edge connectivity information.
+3.4. Image-Graph Contrastive Learning
+To learn a joint embedding space we follow cross-modal
+contrastive formalism of [45], and briefly describe it here
+for completeness. Given N image-graph pairs (I, G) within
+a batch, our model jointly learns the encoders φimage(·) and
+φgraph(·) such that the cosine similarity of the N correct
+image-graph pairs will be high and the N 2 − N incorrect
+pairs will be low. We define cosine similarity between im-
+age i and graph j as:
+αij =
+⟨φimage(Ii), φgraph(Gj)⟩
+||φimage(Ii)||||φgraph(Gj)||.
+(2)
+We then compute bidirectional contrastive losses composed
+of an image-to-graph loss ℓ(I→G) and a graph-to-image loss
+ℓ(G→I), following the form of the InfoNCE loss [43]:
+ℓ(I→G)
+i
+= − log
+exp αii
+�
+j exp αij
+,
+(3)
+ℓ(G→I)
+i
+= − log
+exp αii
+�
+j exp αji
+.
+(4)
+The contrastive loss is then computed as a weighted com-
+bination of the two, averaged over all positive image-graph
+pairs in each minibatch:
+ℓcontrastive =
+1
+2N
+N
+�
+i=1
+�
+ℓ(I→G)
+i
++ ℓ(G→I)
+i
+�
+.
+(5)
+The above penalizes all incorrect image-graph pairs equally.
+We found it beneficial to penalize false matches between
+pairs with similar graphs (as measured using graph metrics,
+Sec. 4) less severely, as similar graphs should intuitively
+have similar embeddings. We measure this similarity of
+graphs after aligning vertices. Formally, given a ground
+truth graph, G0 and candidate match Gi, we first establish
+a correspondence between each vertex v ∈ V0 and its clos-
+est match πi(v) = vi ∈ Vi (in terms of Euclidean distance
+between vertices). Given such corresponding vertices, we
+compute both a Chamfer Distance [2] (CD) and a binary
+cross-entropy (BCE) loss between the ground-truth binary
+adjacency matrix E0 and the permuted matrix Ei:
+ℓchamfer =
+�
+v∈V0
+�
+i
+αiDistance(v, πi(v)),
+(6)
+ℓedge =
+�
+v,w∈V0
+BCE(
+�
+i
+αiEi(πi(v), πi(w)) + ϵ, E0(v, w)), (7)
+where αi = softmaxi αi0. The final loss is then:
+ℓ = ω1ℓcontrastive + ω2ℓchamfer + ω3ℓedge,
+(8)
+where ω1 = 1, ω2 = 1, ω3 = 1/10. To ensure the BCE
+loss remains finite, we add a small nonzero ϵ to ensure edge
+probabilities are strictly positive. To speed up the loss com-
+putation, we ignore edges v, w ∈ V0 that are missing for all
+4
+
+graphs in the batch Ei(πi(v), πi(w)) = 0, ∀i. A key diffi-
+culty in evaluating graph edge losses such as our own or the
+Rand Loss (described in Sec. 4.2) is that they assume that a
+vertex-wise correspondence is already known between the
+predicted and target graph. A more theoretically optimal
+framework may search over one-to-one vertex correspon-
+dences that jointly minimize the Chamfer and edge loss,
+e.g., by solving a bipartite matching problem [30].
+3.5. Pix2Map via Cross-Modal Retrieval
+Given the learned encodings above, we now use them
+for regressing maps from pixel image input via retrieval.
+Denoting a graph library as G, we retrieve image I ⇒ G∗,
+where G∗ = argmaxG∈Gretrieval⟨φimage(I), φgraph(G)⟩. Note
+that the graph library used for retrieval Gretrieval need not be
+the same as the one used to train the image-graph encoders.
+Formally, let encoders be trained on a collection of image-
+graph pairs, written as Dtrain := {(I, G)|I ∈ Itrain, G ∈
+Gtrain}. Gretrieval need not be equivalent to Gtrain, and moreo-
+ever, the set of corresponding images Iretrieval is not needed.
+This has several important properties. Firstly, we can
+populate the graph library Gretrieval with additional graphs,
+not available during training, or cull the library to a partic-
+ular subset of training graphs corresponding to a given city
+neighborhood. Secondly, we can enlarge the graph library
+with graphs that have no corresponding images, including
+augmented variants of real street maps that capture poten-
+tial map updates (such as the potential addition of a lane at
+a particular intersection, for which no real-world imagery
+would be available). Thirdly, the above algorithm returns a
+ranked list of graphs, including near-ties. This can be used
+to generate multiple graphs that could correspond to an im-
+age input. Finally, similar observations hold for retrieving
+street map images using a graph (i.e., Map2Pix). Map2Pix
+is more likely to be a one-to-many task since the same street
+geometry can be associated with different visual pixels de-
+pending on the time of day or weather conditions.
+4. Experiments
+In this section, we first discuss our evaluation test-bed
+(Sec. 4.1) that we use to conduct the experimental eval-
+uation.
+Then, we perform ablation studies to highlight
+each component’s contribution (Sec. 4.4). We compare our
+method to a recent state-of-the-art in Sec. 4.3 and, finally,
+highlight several use-cases of our Pix2Map to automated
+map maintenance and expansion, and vehicle localization.
+4.1. Evaluation Test-Bed
+Dataset.
+For evaluation we use Argoverse dataset [14],
+which provides seven ring camera images (1920 × 1200)
+recorded at 30 Hz with overlapping fields of view, providing
+360◦ coverage. Crucially, Argoverse contains street maps
+that capture the geometry and connectivity of road lanes.
+Such map annotations are not available in other autonomous
+vehicle datasets such as nuScenes [9]. We perform the ex-
+periments across two cities in the United States, including
+Pittsburgh (86km) and Miami (204km).
+Splits. Argoverse provides train, validation, and test splits.
+Note that validation and test splits may include regions that
+spatially overlap with the regions included in the training
+set. However, recordings of these regions were collected at
+different data collection runs at different times. To evaluate
+realistic applications of map-updating (where one trains on,
+e.g., Pittsburgh up to 2021 and tests on Pittsburgh 2022+)
+and map-expansion (where one trains on the neighborhood
+of Squirrel Hill and tests on Shadyside), we split up the
+union of (test+val) into those regions that spatially over-
+lap the trainset and those that do not. We refer to these as
+MapUpdate and MapExpand test sets (Fig. 5). We present
+results for both settings, but default to MapUpdate for diag-
+nostics unless otherwise specified.
+Map Preprocessing.
+The key component of HD maps is
+the central line of drivable lanes. We extract subgraphs cor-
+responding to 40m × 40m spatial windows. We use the ad-
+jacency matrix to represent the node connectivity. An edge
+connects two nodes if they are immediately reachable by
+following the flow of traffic, i.e., a subgraph of nodes in
+a given lane corresponds to a directed path. Moreover, an
+edge exists between two lanes if the first node of the second
+lane follows directly from the last node of the first, either
+because one lane continues to another or because one can
+turn from one lane to the other. We make sure to rotate the
+node positions and lanes to align with the driving direction.
+Implementation.
+We train on a single NVIDIA A100
+GPU, and the training dataset contains up to 512 samples
+in one batch. The model is trained for a total of 40 epochs,
+where a single epoch takes 40 minutes of wall-clock time.
+We make use of the Adam optimizer with a learning rate
+of 2e-4. We use a pretrained ResNet18 for the image en-
+coder. In order to support an input containing several im-
+ages, we duplicate and stack the filters of the input conv
+layer corresponding to the number of images. We then di-
+vide the parameters by the number of images per input ex-
+ample, giving us a model initially returning identical output
+to the original model. We extract the feature representation
+from immediately before the fully connected layer. For the
+graph encoder, We use bert model with mean pooling and
+no positional embeddings to model the pariwise intersec-
+tions between each of the nodes. For each node, we pass in
+its adjacencies and its coordinates. We apply the attention
+mask which indicates to the model which tokens should be
+attended to and which should not.
+5
+
+4.2. Metrics
+To quantitatively evaluate the quality of the retrieved
+graphs, we design three types of metrics to capture the dif-
+ference between the retrieved graph G1 = (V1, E1) and the
+ground truth G2 = (V2, E2).
+Spatial Point Discrepancy. We first introduce metrics that
+represent lane graph nodes v ∈ V as (x, y) points cor-
+responding the lane centroid, ignoring edge connectivity.
+We can then use metrics for measuring differences between
+point sets. Chamfer Distance computes the closest point
+in v2 ∈ V2 for every v1 ∈ V1 (and vice versa, to ensure
+symmetry). Maximum Mean Discrepancy (MMD) [25, 41]
+measures the squared distance between point centroids in
+a Hilbert space using Gaussian kernels ⟨ϕ(v1), ϕ(v2)⟩H =
+k(x1 − x2, y1 − y2):
+MMD(G1, G2) =
+�����
+1
+|V1|
+�
+v1∈V1
+ϕ(v1) −
+1
+|V2|
+�
+v2∈V2
+ϕ(v2)
+�����
+2
+H
+.
+Edge Connectivity. The above metrics evaluate the quality
+of only the retrieved graph nodes, but not their edge con-
+nectivity. We define a RandLoss similar to (6), as:
+RandLoss =
+�
+v,w∈V1
+1[E2(π(v),π(w))̸=E1(v,w)],
+where 1 is an indicator function for mismatching edge la-
+bels between a pair of nodes in the graph G1 and their cor-
+responding pair in graph G2. This metric is also known as
+the Rand index, commonly used in evaluating clustering al-
+gorithms [46].
+Urban Planning. We also report a set of metrics motivated
+by the urban planning literature [1, 17, 41], evaluating the
+degree to which we are able to reconstruct the following key
+properties of urban HD maps. Connectivity is the number
+of edges relative to the number of lane nodes. Density is
+the number of edges relative to max the possible number of
+edges. Reach is designed to capture urban development and
+is defined to be the total distance covered by lanes:
+Connectivity = ∥E∥0
+|V | ,
+Density =
+∥E∥0
+|V |(|V | − 1),
+Reach =
+�
+(v,w):E(v,w)=1
+len(v, w).
+We report the absolute relative error [41] for these metrics.
+4.3. Baselines
+We show a visual comparison with top performers in
+Fig. 4 and quantitative results in Tab. 1. We first report the
+performance of a naive nearest-neighbor baseline, which
+returns the graph associated with the closest training im-
+age example.
+This unimodal approach already performs
+Input image                  Ground Truth           Topo/PRNN              Topo/TR                   PINET                        Our
+Figure 4. Qualitative results. From left to right: input image,
+ground-truth maps, maps generated by state-of-the-art methods,
+and, in the last column, our method. As can be seen, the retrieved
+maps with our method have the highest visual fidelity.
+Methods
+Chamfer
+RandLoss
+MMD
+U. density
+U. reach
+U. conn.
+101
+10−2
+10−1
+10−1
+10−1
+10−1
+Unimodal
+4.3967
+9.0764
+4.1873
+1.8391
+3.2746
+1.7734
+PINET [31]
+4.9244
+10.8935
+4.2983
+2.8194
+7.4194
+2.9231
+TOPO-PRNN [11]
+7.4811
+9.2813
+5.7726
+3.9371
+6.8297
+1.3934
+TOPO-TR [11]
+3.0140
+7.1603
+4.6431
+2.2467
+3.3091
+1.1530
+Pix2Map (ours)
+2.0882
+7.7562
+3.9621
+1.4354
+3.2893
+1.5532
+Table 1. Baseline comparisons. To enable fair comparisons with
+prior art [11], we retrain Pix2Map on their train/test split and ad-
+dress their task of predicting the frontal 50m × 50m road-graph
+(as opposed to our default setting of predicting the surrounding
+40m × 40m). Importantly, in this comparison, our method still
+outperforms baselines by a large margin: 2.0882 in terms of
+Chamfer distance, as compared to 3.0140 obtained by the closest
+competitor, TOPO-TR [11].
+on-par with the state-of-the-art Transformer and Polygon-
+RNN [27] based methods TOPO [10], and PINET [31]
+based method to extract lane boundaries.
+Our diagnos-
+tics further explore the improvement from unimodal to
+cross-modal retrieval. Specifically, we experiment across
+our evaluation metrics and find Pix2Map improves greatly
+over several state-of-the-art baselines in Chamfer distance,
+MMD, urban density error, and urban connectivity error,
+while performing comparably in terms of RandLoss and ur-
+ban reach error. Moreover, our method is especially strong
+in terms of preserving the spatial point discrepancy, outper-
+forming baselines by a large margin. We note that Pix2Map
+was designed to fully utilize image data available in the
+camera ring, whereas baselines use only frontal view.
+4.4. Model Analysis
+In this section, we experiment with two graph represen-
+tations, evaluate different design decisions on the graph and
+6
+
+ANOMADTRIBETRow
+Eimg
+Attention
+Adjacency
+Positional
+Resampling
+Chamfer
+RandLoss
+MMD
+U. density
+U. reach
+U. conn.
+Mask
+Matrix
+Encoding
+101
+10−2
+10−1
+10−1
+10−1
+10−1
+1
+1 × RN18
+�
+�
+1.9241
+9.0446
+4.1804
+1.2058
+3.6333
+1.8398
+2
+1 × RN18
+�
+�
+1.9309
+8.3834
+3.8383
+1.1934
+3.7428
+1.8629
+3
+1 × RN18
+�
+�
+�
+1.5908
+7.3283
+3.0888
+0.7593
+3.2997
+0.8397
+4
+1 × RN18
+�
+�
+3.2663
+6.9943
+6.3704
+3.6883
+5.3219
+4.1658
+5
+1 × RN18
+�
+�
+�
+�
+2.1564
+9.1200
+8.8328
+0.8813
+3.4290
+1.5481
+6
+7 × RN18
+�
+�
+�
+4.8129
+11.2118
+9.7538
+3.9169
+6.7794
+2.5285
+Table 2. Image and graph encoder ablations. From left to right, we ablate a) encoding each of the seven ego-images separately or using
+an early-fusion multiview image encoder, b) restricting the transformer attention mask to the graph-adjacency matrix or using the default
+fully-connected attention, c) the inclusion of the corresponding row of the attention matrix as a node input feature for the model, d) adding
+a positional encoding to each graph vertex, and e) resampling graph vertices to be equidistant. We find results are dramatically improved
+by early fusion for image encoding (row3-vs-row6) and graph vertex resampling (row3-vs-row4). Results are marginally improved by
+restricting the attention mask (row2-vs-row3) and adding the adjacency matrix as an input feature (row1-vs-row3 and row2-vs-row3).
+Perhaps surprisingly, adding in positional vertex encodings slightly decreases performance (row3-vs-row5).
+Train(5701)
+MapUpdate(3114)
+MapExpand(1600)
+Train(7421)
+MapUpdate(3185)
+MapExpand(1284)
+Figure 5. Pittsburgh (left) and Miami (right) datasets. Including training data (blue), MapUpdate test data (red) that overlap the blue
+but are collected at different times, and MapExpand test data (yellow) that do not non-overlap. We denote the size of each dataset (in terms
+of the number of image-graph pairs) in parentheses.
+Lcon.
+Ledge
+Lchamf.
+Chamfer
+RandLoss
+MMD
+U. density
+U. reach
+U. conn.
+101
+10−2
+10−1
+10−1
+10−1
+10−1
+✓
+3.2249
+9.1727
+7.1070
+2.7387
+14.1033
+3.6998
+✓
+✓
+2.7967
+8.9951
+8.8717
+0.9808
+10.0251
+1.0512
+✓
+✓
+1.7440
+8.5186
+2.2480
+0.9841
+7.4534
+1.5664
+✓
+✓
+✓
+1.5908
+7.3283
+3.0888
+0.7593
+3.2997
+0.8397
+Table 3. Ablation on the training loss. Adding a partial credit
+for matching to graphs with low Chamfer distance (Lcham.) to the
+ground truth improves results considerably compared to vanilla
+contrastive loss (Lcon.).
+By adding in an additional edge loss
+(Ledge), we further improve the performance.
+image encoders, and further evaluate the contrastive com-
+ponent.
+Ablations. Tab. 2 ablates several design decisions on image
+(Sec. 3.2) and graph (Sec. 3.3) encoders. For the analysis,
+we focus on Chamfer distance as an illustrative metric as it
+appears most consistent with graph qualitative estimation.
+Image Encoder.
+As can be seen in Tab. 2, row3-vs-row6,
+early fusion (1×RN18) is significantly better compared to a
+late fusion (7×RN18) variant, where we separately encode
+images with a distinct ResNet18 model for each camera,
+Methods
+Chamfer
+RandLoss
+MMD
+U. density
+U. reach
+U. conn.
+101
+10−2
+10−1
+10−1
+10−1
+10−1
+Unimodal
+3.2168
+9.7596
+7.7671
+0.7365
+3.9452
+1.3661
+Ours
+1.5908
+7.3283
+3.0888
+0.7593
+3.2997
+0.8397
+Ours++
+1.5208
+6.1504
+3.0944
+0.7407
+3.2610
+0.8089
+Table 4. Cross-model retrieval (ours) significantly outperforms
+classical unimodal retrieval (i.e., the nearest neighbor on image
+encoder features).
+Cross-modal retrieval can exploit the graph
+embedding space, which appears to regularize retrieval results,
+while the unimodal approach does not utilize any graph embed-
+ding.
+Moreover, our cross-modal retrieval can take advantage
+of larger unpaired graph libraries, which further improve perfor-
+mance (ours++). Unimodal retrieval requires paired data.
+followed by fusion via average-pooling.
+Graph Encoder. We consider three primary variations on the
+graph encoder architecture. Results suggest (Tab. 2, row2-
+vs-row3) that restricting the attention mask marginally im-
+proves the results. Furthermore, encoding the edge con-
+nectivity information (adding the adjacency matrix as an
+input feature, row1-vs-row3) slightly improves the perfor-
+mance. Perhaps surprisingly, positional vertex encodings
+7
+
+4000
+3500
+3000
+coordinate
+2500
+2000
+1500
+1000
+0
+500
+x coordinate4500
+4000
+3500
+coordinate
+3000
+2500
+2000
+1500
+1000
+0
+5001000150020002500
+x coordinate4500
+4000
+3500
+coordinate
+3000
+2500
+2000
+1500-
+1000
+0
+5001000150020002500
+x coordinate4000
+3500
+3000
+coordinate
+2500
+2000
+1500
+1000
+0
+500
+x coordinate4500
+4000
+3500
+coordinate
+3000
+2500
+2000
+1500-
+1000
+0
+5001000150020002500
+x coordinate4000
+3500
+3000
+coordinate
+2500
+2000
+1500
+1000
+0
+500
+x coordinateGround Truth
+Retrieved
+Image
+A
+B
+C
+D
+E
+F
+G
+H
+Figure 6. Given a set of ego-view images (front camera, top), we plot the ground-truth graph (middle), followed by the Pix2Map predictions
+(bottom) for both the MapUpdate (left) and MapExpand (right) tasks for two cities, Pittsburgh (PIT) and Miami (MIA). In general,
+retrieval results are quite accurate, particularly for MapUpdate, where train and test samples are drawn from the same geographic regions.
+Inclement weather such as heavy rain is challenging (row PIT MapExpand, col D) due to the degraded visual signal.
+City
+Library Size
+Chamfer
+RandLoss
+MMD
+U. density
+U. reach
+U. conn.
+101
+10−2
+10−1
+10−1
+10−1
+10−1
+PIT
+5.7k
+1.5908
+7.3283
+3.0888
+0.7593
+3.2997
+0.8397
+10k
+1.6457
+7.6247
+3.2848
+0.7264
+4.5891
+1.6364
+20k
+1.5369
+6.5373
+3.1883
+0.7581
+3.2902
+1.0602
+30k
+1.5239
+6.6553
+3.0253
+0.8586
+4.0642
+0.9615
+40k
+1.5208
+6.1504
+3.0944
+0.7407
+3.2610
+0.8089
+MIA
+7.4k
+1.4747
+6.8693
+3.4033
+1.0948
+4.6253
+1.1910
+10k
+1.4991
+6.2315
+3.3118
+1.2784
+5.5209
+1.3679
+20k
+1.3878
+8.0234
+3.3910
+1.1290
+4.1237
+1.3249
+30k
+1.4012
+7.1898
+3.2773
+1.2444
+4.2471
+1.2298
+40k
+1.3878
+7.6305
+3.3351
+1.2523
+5.3894
+1.3385
+60k
+1.3080
+6.3369
+3.18879
+1.1972
+4.7578
+1.1977
+80k
+1.2711
+6.2852
+3.19506
+1.0123
+4.6827
+1.1651
+100k
+1.2462
+6.2740
+3.1277
+0.9884
+3.8521
+1.1397
+Table 5. Ablation with larger map-graph libraries. As we grow
+the graph retrieval library (including maps without corresponding
+image views), we observe performance grows consistently with
+the size of the retrieval library.
+slightly decrease performance (row3-vs-row5).
+Graph Representation.
+Tab. 2 (row3-vs-row4) shows that
+the proposed graph vertex resampling dramatically im-
+proves results. As we mentioned in Sec. 3.1, we resample
+the segment graphs and the connected nodes throughout the
+graph are approximately equidistant, with a distance of 2
+meters.
+Loss.
+Tab. 3 ablates our loss function (Sec. 3.4). Com-
+pared to the na¨ıve contrastive loss Lcontrastive, that weighs
+all incorrect image-graph matches equally, we find adding
+in partial credit for matching to graphs with low Cham-
+fer distance Lchamfer, to the ground-truth improves results
+considerably, while adding in an additional edge loss Ledge
+further improves performance. We use the most performant
+combination (row3) for further experiments.
+Unimodal v.s. Cross-modal.
+To quantify the benefits of
+the cross-modal training scheme, we compare Pix2Map to
+its image encoder alone, evaluating it as a unimodal image-
+encoding-based retriever. Specifically, in Pix2Map, we di-
+rectly retrieve graphs by finding the graph embedding which
+is closest to the input image embedding in the multimodal
+embedding space. However, in this ablation, we instead find
+the image embedding in the training set which is closest to
+the input image embedding, and return its corresponding
+graph. We find that using both modalities improves per-
+formance on almost all metrics, as shown in Tab. 4. The
+improvements range from a 37.0% decrease in RandLoss to
+a 60.2% decrease in MMD, with one increase of less than a
+percentage point in urban density error. Note, however, that
+while the unimodal ablation broadly performs worse than
+Pix2Map, it still performs better than any baseline shown in
+Tab. 1 in terms of Chamfer, MMD, and Urban Density.
+Augmenting the Graph Library.
+One of the benefits of
+our cross-modal retrieval approach is that we can match to
+(or retrieve from) arbitrary collections of graphs that are
+different from (or larger than) the training graphs used to
+learn cross-modal encoders. This allows us to make use
+of graphs which have no corresponding images. Interest-
+ingly, the Argoverse dataset provides such data, as maps
+include many locations for which no imagery is provided.
+By sampling random ego-vehicle positions in Miami and
+Pittsburgh, we grow both the Pittsburgh library and the Mi-
+ami library to 40k, thus significantly expanding our retrieval
+graph library. We summarize these results in Tab. 5. We ob-
+serve that performance improves across the suite of metrics
+as the graph retrieval library grows larger. This suggests
+there is a significant potential to further improve our results
+by simply growing our graph retrieval dataset using existing
+maps, without access to corresponding image pairs.
+4.5. Applications
+In this section, we discuss how our method can be used
+for practical purposes, and we show that graph library re-
+trieval can greatly improve various downstream applica-
+tions such as expansion (MapExpand) and update (MapUp-
+date) given existing maps, visual image-to-HD map local-
+ization and Map2Pix.
+8
+
+nd Truth
+-15
+1
+Retrieved
+-10 
+10 
+10
+0.6 
+06
+.620
+10
+0
+-10
+-20
+20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+20
+10
+0
+-10
+-20
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+2020
+20
+10
+10
+0
+0
+10
+-10
+-20
+-20
+-20
+15
+10
+-5
+0
+5
+10
+15
+20
+20
+15
+-10
+-5
+0
+5
+10
+15
+20
+20
+20
+10
+10
+0
+0
+-10
+-10
+-20
+-20
+-20
+15
+-10
+-5
+0
+10
+15
+20
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+2020
+10
+0
+-10
+-20
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+20
+10
+0
+-10
+-20
+20
+-15
+-10
+-5
+0
+5
+10
+15
+2020
+10
+0
+-10
+-20
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+20
+10
+0
+-10
+-20
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+2020
+10
+0
+-10
+-20
+-20
+15
+-10
+-5
+0
+5
+10
+15
+20
+20
+10
+-10
+-20
+-20
+15
+-10
+-5
+0
+5
+10
+15
+2020
+10
+0
+-10
+-20
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+20
+10
+0
+-10
+-20
+-20
+-15
+-10
+5
+0
+5
+10
+15
+2020
+10
+0
+-10
+-20
+-15
+-20
+-10
+-5
+0
+5
+10
+15
+20
+-
+20
+10
+0
+-10
++
+-20
+-20
+-15
+-10
+5
+0
+5
+10
+15
+20
+一Figure 7. Visual localization via Pix2Map. We overlay retrieval
+scores on the corresponding local graphs from the original city
+map, generating a graph “heatmap” of possible locations given in-
+stantaneous ego-view images. We plot the ground-truth location
+as a red dot. In general, ground-truth locations tend to lie in high-
+scoring (yellow) regions. For example, the top ground truth cor-
+responds to an intersection, while other high-scoring regions also
+tend to be graph intersections as well. Given a sequence of images,
+one may be able to reduce the ambiguity over time [8]
+City
+Task type
+Chamfer
+RandLoss
+MMD
+U. density
+U. reach
+U. conn.
+101
+10−2
+10−1
+10−1
+10−1
+10−1
+PIT
+MapUpdate
+1.5908
+7.3283
+3.0888
+0.7593
+3.2997
+0.8397
+MapExpand
+2.6654
+16.9768
+8.0468
+3.9482
+4.2949
+3.9699
+MIA
+MapUpdate
+1.4747
+6.8693
+3.4033
+1.0948
+5.5333
+1.1910
+MapExpand
+2.0637
+11.1354
+4.2605
+1.4922
+4.7318
+1.5940
+Table 6. Map update and expansion evaluation. As can be seen,
+map expansion to novel areas can be much harder than updating
+previously-seen areas.
+Map Expansion and Update. We use our graph retrieval
+method to mimic map expansion (MapExpand) and map up-
+date (MapUpdate) using data splits, as explained in Fig. 5.
+For map expansion, we retrieve local graphs corresponding
+to recordings obtained in a “new traversal” to expand the
+existing map. For map updates, we similarly retrieve local
+maps to update the global map.
+We qualitatively evaluate graph retrieval results in Fig. 6.
+Please see the caption for a detailed description, but gener-
+ally speaking, we find Pix2Map returns reasonable graphs
+similar to the ground truth. In Tab. 6, we evaluate the per-
+formance of map updating and map expansion across two
+cities (Pittsburgh and Miami). We do so by comparing ex-
+panded/updated maps with ground-truth maps using met-
+Figure 8. Qualitative results for Map2Pix. The goal is to re-
+trieve ego-camera images given a street map. Such image retrieval
+may be useful for simulator-based training and validation of au-
+tonomous stacks. A single street geometry might retrieve multiple
+consistent, realistic imagery.
+rics, described in Sec. 4.2. As shown above, map expansion
+to novel areas is harder than updating previously-seen areas.
+Localization. Furthermore, our method demonstrates great
+visual localization ability based on visual and geometric
+understanding. We use the cosine similarities of retrieved
+graphs to generate a heatmap of possible ego-vehicle loca-
+tions over a city-level map, showing the locations where
+their corresponding graphs are assigned a high likelihood,
+relative to the ground truth location shown as a red dot.
+While the ground truth is usually assigned a high likelihood,
+which indicates the promising performance of localization
+ability, the distribution becomes less sharp with respect to
+position when farther away from intersections. See Fig. 7
+for more details.
+Map2Pix.
+We further show that it is also possible to re-
+trieve egocentric camera data using street maps. Such tech-
+niques could be used in the future to synthesize virtual
+worlds consistent with the query road geometry. We pro-
+vide a few example image retrievals in Figure 8, visualizing
+the front views of top K = 2 images for each street map.
+As can be seen, the retrieved images correspond to rough
+geometric layouts encoded in the query graphs.
+9
+
+2500
+2000
+1500
+1000
+500
+0
+1000
+2000
+3000
+4000
+5000825
+800
+775
+750
+725
+700
+675
+650
+2100
+2125
+2150
+2175
+2200
+2225
+2250
+22751325
+2500
+1300
+2000
+1275
+1250
+1500
+1225
+1000
+1200 -
+500 
+1175
+1150
+1000
+2000
+3000
+4000
+5000
+2525 2550 2575 2600 2625 2650 2675 27001325
+1300
+1275
+1250
+1225
+1200
+1175
+1150
+2525
+2550
+2575
+2600
+2625
+2650
+2675
+27002500
+2000
+1500
+1000
+500
+0
+1000
+2000
+3000
+4000
+50002500
+2000
+1500
+1000
+500
+0
+1000
+2000
+3000
+4000
+50001550
+1525
+1500
+1475
+1450
+1425
+1400
+1375
+2800
+2825
+2850
+2875
+2900
+2925
+2950
+29751300
+1275
+1250
+1225
+1200
+1175
+1150
+1125
+2475
+2500
+2525
+2550
+2575
+2600
+2625
+26502500
+2000
+1500
+1000
+500
+0
+1000
+2000
+3000
+4000
+5000PU075. Conclusion
+Overall, our experiments suggest that learning a multi-
+modal embedding space for camera data and map data is a
+promising direction, and we hope our work can become an
+essential building block for map expansion and update in
+the autonomous driving field. Beyond map maintenance,
+we also show our approach can also be used as a novel
+form of visual localization. While these results are encour-
+aging, there are also many potentially impactful future di-
+rections possible. For example, instead of performing graph
+retrieval, one could use the latent space to generate new un-
+seen graphs using a graph-based decoder architecture. We
+believe our ablations regarding (resampled) graph repre-
+sentations, (partial-credit) training losses, and early-vs-late
+multiview vision will be useful for learning such decoders
+for future study.
+Acknowledgments. This work was supported by the CMU Argo AI Cen-
+ter for Autonomous Vehicle Research.
+References
+[1] Sawsan AlHalawani, Yong-Liang Yang, Peter Wonka, and
+Niloy J Mitra. What makes london work like london? Com-
+puter Graphics Forum, 2014. 6
+[2] Harry G Barrow, Jay M Tenenbaum, Robert C Bolles, and
+Helen C Wolf.
+Parametric correspondence and chamfer
+matching: Two new techniques for image matching. Tech-
+nical report, SRI International Menlo Park, CA AI Center,
+1977. 4
+[3] Ioan Andrei Barsan, Shenlong Wang, Andrei Pokrovsky, and
+Raquel Urtasun. Learning to localize using a lidar intensity
+map. arXiv preprint arXiv:2012.10902, 2020. 1
+[4] Jens Behley and Cyrill Stachniss.
+Efficient Surfel-Based
+SLAM using 3D Laser Range Data in Urban Environments.
+In RSS, 2018. 2
+[5] Karsten Behrendt and Ryan Soussan. Unsupervised labeled
+lane markers using maps. In ICCV Workshops, 2019. 2
+[6] Julie Stephany Berrio, Stewart Worrall, Mao Shan, and Ed-
+uardo Nebot. Long-term map maintenance pipeline for au-
+tonomous vehicles. IEEE TPAMI, 2021. 3
+[7] Sagar Ravi Bhavsar, Andrei Vatavu, Timo Rehfeld, and Gun-
+ther Krehl. Sensor fusion-based online map validation for
+autonomous driving. In IVS, 2020. 3
+[8] Marcus A Brubaker, Andreas Geiger, and Raquel Urta-
+sun. Map-based probabilistic visual self-localization. IEEE
+TPAMI, 2015. 9
+[9] Holger Caesar, Varun Bankiti, Alex H Lang, Sourabh Vora,
+Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Gi-
+ancarlo Baldan, and Oscar Beijbom.
+nuscenes: A multi-
+modal dataset for autonomous driving. In CVPR, 2020. 2,
+5
+[10] Yigit Baran Can, Alexander Liniger, Danda Pani Paudel, and
+Luc Van Gool. Structured bird’s-eye-view traffic scene un-
+derstanding from onboard images. In ICCV, 2021. 1, 2, 6
+[11] Yigit Baran Can, Alexander Liniger, Danda Pani Paudel, and
+Luc Van Gool. Topology preserving local road network esti-
+mation from single onboard camera image. In CVPR, 2022.
+1, 2, 6
+[12] Vincent Cartillier, Zhile Ren, Neha Jain, Stefan Lee, Irfan
+Essa, and Dhruv Batra. Semantic mapnet: Building allo-
+centric semanticmaps and representations from egocentric
+views. arXiv preprint arXiv:2010.01191, 2020. 2
+[13] Lluis Castrejon, Kaustav Kundu, Raquel Urtasun, and Sanja
+Fidler. Annotating object instances with a polygon-rnn. In
+CVPR, 2017. 1, 2
+[14] Ming-Fang Chang, John Lambert, Patsorn Sangkloy, Jag-
+jeet Singh, Slawomir Bak, Andrew Hartnett, De Wang, Peter
+Carr, Simon Lucey, Deva Ramanan, and James Hays. Argo-
+verse: 3d tracking and forecasting with rich maps. In CVPR,
+2019. 1, 2, 5
+[15] Wensheng* Cheng, Hao* Luo, Wen Yang, Lei Yu, Shoushun
+Chen, and Wei Li. Det: A high-resolution dvs dataset for
+lane extraction. In CVPR Workshops, 2019. 2
+[16] Sungjin Cho, Chansoo Kim, Jaehyun Park, Myoungho Sun-
+woo, and Kichun Jo. Semantic point cloud mapping of lidar
+based on probabilistic uncertainty modeling for autonomous
+driving. Sensors, 2020. 2
+[17] Hang Chu, Daiqing Li, David Acuna, Amlan Kar, Maria
+Shugrina, Xinkai Wei, Ming-Yu Liu, Antonio Torralba, and
+Sanja Fidler. Neural turtle graphics for modeling city road
+layouts. In ICCV, 2019. 6
+[18] Nachiket Deo, Eric Wolff, and Oscar Beijbom. Multimodal
+trajectory prediction conditioned on lane-graph traversals. In
+CoRL, 2022. 2
+[19] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina
+Toutanova.
+Bert:
+Pre-training of deep bidirectional
+transformers for language understanding.
+arXiv preprint
+arXiv:1810.04805, 2018. 2, 4
+[20] J. Engel, T. Sch¨ops, and D. Cremers. LSD-SLAM: Large-
+scale direct monocular SLAM. In ECCV, 2014. 2
+[21] J. Engel, J. St¨uckler, and D. Cremers.
+Large-scale direct
+SLAM with stereo cameras. In IROS, 2015. 2
+[22] Andreas Geiger, Philip Lenz, and Raquel Urtasun. Are we
+ready for autonomous driving? the KITTI vision benchmark
+suite. In CVPR, 2012. 2
+[23] Thomas Gilles, Stefano Sabatini, Dzmitry Tsishkou, Bog-
+dan Stanciulescu, and Fabien Moutarde. Gohome: Graph-
+oriented heatmap output for future motion estimation.
+In
+ICRA, 2022. 2
+[24] Margarita Grinvald, Fadri Furrer, Tonci Novkovic, Jen Jen
+Chung, Cesar Cadena, Roland Siegwart, and Juan Nieto.
+Volumetric instance-aware semantic mapping and 3d object
+discovery. IEEE RAL, 2019. 2
+[25] Ehsan
+Hajiramezanali,
+Arman
+Hasanzadeh,
+Krishna
+Narayanan, Nick Duffield, Mingyuan Zhou, and Xiaoning
+Qian.
+Variational graph recurrent neural networks.
+In
+NeurIPS, 2019. 6
+[26] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.
+Deep residual learning for image recognition.
+In CVPR,
+2016. 4
+[27] Namdar Homayounfar, Wei-Chiu Ma, Shrinidhi Kowshika
+Lakshmikanth, and Raquel Urtasun. Hierarchical recurrent
+attention networks for structured online maps.
+In CVPR,
+2018. 2, 6
+10
+
+[28] Namdar Homayounfar, Wei-Chiu Ma, Justin Liang, Xinyu
+Wu, Jack Fan, and Raquel Urtasun. Dagmapper: Learning to
+map by discovering lane topology. In ICCV, 2019. 1, 2
+[29] Anthony Hu, Zak Murez, Nikhil Mohan, Sof´ıa Dudas, Jef-
+frey Hawke, Vijay Badrinarayanan, Roberto Cipolla, and
+Alex Kendall. Fiery: Future instance prediction in bird’s-
+eye view from surround monocular cameras. In ICCV, 2021.
+2
+[30] Richard M Karp, Umesh V Vazirani, and Vijay V Vazirani.
+An optimal algorithm for on-line bipartite matching. In ACM
+STOC, 1990. 5
+[31] Yeongmin Ko, Younkwan Lee, Shoaib Azam, Farzeen Mu-
+nir, Moongu Jeon, and Witold Pedrycz. Key points estima-
+tion and point instance segmentation approach for lane de-
+tection. IEEE Trans. ITS, 2021. 6
+[32] Deyvid Kochanov, Aljoˇsa Oˇsep, J¨org St¨uckler, and Bastian
+Leibe. Scene flow propagation for semantic mapping and
+object discovery in dynamic street scenes. In IROS, 2016. 2
+[33] John Lambert and James Hays.
+Trust, but verify: Cross-
+modality fusion for hd map change detection. In NeurIPS,
+2021. 1, 3
+[34] Qi Li, Yue Wang, Yilun Wang, and Hang Zhao. Hdmapnet:
+An online hd map construction and evaluation framework.
+arXiv preprint arXiv:2107.06307, 2021. 1, 2
+[35] Zuoyue Li, Jan Dirk Wegner, and Aur´elien Lucchi. Topolog-
+ical map extraction from overhead images. In ICCV, 2019.
+2
+[36] Justin Liang, Namdar Homayounfar, Wei-Chiu Ma, Shen-
+long Wang, and Raquel Urtasun.
+Convolutional recurrent
+network for road boundary extraction. In CVPR, 2019. 1, 2
+[37] Martin Liebner, Dominik Jain, Julian Schauseil, David Pan-
+nen, and Andreas Hackel¨oer. Crowdsourced hd map patches
+based on road model inference and graph-based slam. In IVS,
+2019. 2
+[38] Lizhe Liu, Xiaohao Chen, Siyu Zhu, and Ping Tan. Cond-
+lanenet: a top-to-down lane detection framework based on
+conditional convolution. In ICCV, 2021. 2
+[39] Rong Liu, Jinling Wang, and Bingqi Zhang. High defini-
+tion map for automated driving: Overview and analysis. The
+Journal of Navigation, 2020. 1, 2
+[40] Wei-Chiu Ma, Ignacio Tartavull, Ioan Andrei Bˆarsan, Shen-
+long Wang, Min Bai, Gellert Mattyus, Namdar Homayoun-
+far, Shrinidhi Kowshika Lakshmikanth, Andrei Pokrovsky,
+and Raquel Urtasun. Exploiting sparse semantic hd maps for
+self-driving vehicle localization. In IROS, 2019. 1, 2
+[41] Lu Mi, Hang Zhao, Charlie Nash, Xiaohan Jin, Jiyang Gao,
+Chen Sun, Cordelia Schmid, Nir Shavit, Yuning Chai, and
+Dragomir Anguelov. Hdmapgen: A hierarchical graph gen-
+erative model of high definition maps. In CVPR, 2021. 1, 2,
+3, 6
+[42] R. Mur-Artal, J.M.M. Montiel, and J.D. Tardos.
+ORB-
+SLAM: A versatile and accurate monocular SLAM system.
+TRO, 2015. 2
+[43] Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Repre-
+sentation learning with contrastive predictive coding. arXiv
+preprint arXiv:1807.03748, 2018. 4
+[44] David Pannen, Martin Liebner, and Wolfram Burgard. Hd
+map change detection with a boosted particle filter. In ICRA,
+2019. 3
+[45] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya
+Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry,
+Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learn-
+ing transferable visual models from natural language super-
+vision. arXiv preprint arXiv:2103.00020, 2021. 1, 4
+[46] William M Rand.
+Objective criteria for the evaluation of
+clustering methods. Journal of the American Statistical as-
+sociation, 1971. 6
+[47] Thomas Roddick and Roberto Cipolla. Predicting semantic
+map representations from images using pyramid occupancy
+networks. In CVPR, 2020. 2
+[48] Radu Alexandru Rosu, Jan Quenzel, and Sven Behnke.
+Semi-supervised semantic mapping through label propaga-
+tion with semantic texture meshes. IJCV, 2020. 2
+[49] Avishkar Saha, Oscar Mendez, Chris Russell, and Richard
+Bowden. Translating images into maps. In ICRA, 2022. 2
+[50] Torsten Sattler, Bastian Leibe, and Leif Kobbelt. Fast image-
+based localization using direct 2d-to-3d matching. In ICCV,
+2011. 2
+[51] Heiko G Seif and Xiaolong Hu. Autonomous driving in the
+icity—hd maps as a key challenge of the automotive industry.
+Engineering, 2016. 1, 2
+[52] Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien
+Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou,
+Yuning Chai, Benjamin Caine, et al. Scalability in perception
+for autonomous driving: Waymo open dataset. In CVPR,
+2020. 2
+[53] Sebastian Thrun. Robotic mapping: A survey. Exploring
+artificial intelligence in the new millennium, 2002. 1, 2
+[54] Sebastian Thrun, Wolfram Burgard, and Dieter Fox. Prob-
+abilistic Robotics (Intelligent Robotics and Autonomous
+Agents). The MIT Press, 2005. 2
+[55] J.P.C. Valentin, S. Sengupta, J. Warrell, A. Shahrokni, and
+P.H.S. Torr. Mesh based semantic modelling for indoor and
+outdoor scenes. In CVPR, 2013. 2
+[56] V. Vineet, O. Miksik, M. Lidegaard, M. Niessner, S.
+Golodetz, V.A. Prisacariu, O. Kahler, D.W. Murray, S. Izadi,
+P. Peerez, and P.H.S. Torr.
+Incremental dense semantic
+stereo fusion for large-scale semantic scene reconstruction.
+In ICRA, 2015. 2
+[57] Shenlong Wang, Min Bai, Gellert Mattyus, Hang Chu, Wen-
+jie Luo, Bin Yang, Justin Liang, Joel Cheverie, Sanja Fidler,
+and Raquel Urtasun. Torontocity: Seeing the world with a
+million eyes. arXiv preprint arXiv:1612.00423, 2016. 2
+[58] Dong Wu, Manwen Liao, Weitian Zhang, and Xinggang
+Wang. Yolop: You only look once for panoptic driving per-
+ception. arXiv preprint arXiv:2108.11250, 2021. 2
+[59] Pan Xingang, Shi Jianping, Luo Ping, Wang Xiaogang, and
+Tang Xiaoou. Spatial as deep: Spatial cnn for traffic scene
+understanding. In AAAI, 2018. 2
+[60] Weixiang Yang, Qi Li, Wenxi Liu, Yuanlong Yu, Yuexin
+Ma, Shengfeng He, and Jia Pan. Projecting your view at-
+tentively: Monocular road scene layout estimation via cross-
+view transformation. In CVPR, 2021. 2
+11
+
+[61] Jiaxuan You, Rex Ying, Xiang Ren, William Hamilton, and
+Jure Leskovec. Graphrnn: Generating realistic graphs with
+deep auto-regressive models. In ICML, 2018. 3
+[62] Qunjie Zhou, Sergio Agostinho, Aljosa Osep, and Laura
+Leal-Taixe. Is geometry enough for matching in visual lo-
+calization? In ECCV, 2022. 2
+12
+

FtAyT4oBgHgl3EQf5Poe/content/tmp_files/2301.00799v1.pdf.txt

@@ -0,0 +1,856 @@
+Moir´e alchemy: artificial atoms, Wigner molecules, and emergent Kagome lattice
+Aidan P. Reddy, Trithep Devakul, and Liang Fu
+Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
+(Dated: January 3, 2023)
+Semiconductor moir´e superlattices comprise an array of artificial atoms and provide a highly
+tunable platform for exploring novel electronic phases. We introduce a theoretical framework for
+studying moir´e quantum matter, which treats intra-moir´e-atom interactions exactly and is controlled
+in the limit of large moir´e period. We reveal an abundance of new physics arising from strong electron
+interactions when multiple occupancy of moir´e atoms is involved. In particular, Coulomb interaction
+in three-electron moir´e atoms leads to a three-lobed “Wigner molecule”, which forms an emergent
+Kagome lattice at filling factor n = 3. Our work identifies two universal length scales characterizing
+the kinetic and interaction energies in moir´e materials and demonstrates a rich phase diagram due
+to their interplay.
+Introduction — The field of quantum science and en-
+gineering has long been fascinated with the creation of
+artificial atoms and artificial solids with desired prop-
+erties.
+Artifical atoms, such as quantum dots and su-
+perconducting qubits, exhibit discrete energy levels and
+provide a physical carrier of quantum information. An
+array of coupled artificial atoms defines an artificial solid,
+which may be used for quantum simulation and quantum
+computing. Recently, the advent of moir´e materals has
+provided a remarkably simple and robust realization of
+artificial solids, offering unprecedented opportunities to
+explore quantum phases of matter in two dimensions [1–
+3]. In particular, moir´e superlattices of semiconductor
+transition metal dichalcogenides (TMDs) host strongly
+interacting electrons in a periodic potential. When the
+moir´e period is large, doped electrons are spatially con-
+fined to the potential minima, leading to an array of arti-
+ficial atoms. Electron tunneling between adjacent moir´e
+atoms generates moir´e bands. The charge density can
+be easily varied across a range of moir´e band fillings by
+electrostatic gating, a level of tunability unprecedented
+in natural solids whose electron density is determined by
+the chemistry of their constituent atoms.
+The contin-
+uously tunable atomic number in semiconductor moir´e
+materials enables a modern form of alchemy.
+To date, much theoretical analysis has relied on an
+effective Hubbard model description of interacting elec-
+trons in the lowest few moir´e bands [4, 5].
+This ap-
+proach successfully explains and predicts many observed
+phenomena such as the emergence of Mott insulators at
+n = 1 [6, 7], incompressible Wigner crystals at fractional
+fillings n < 1 [6, 8–15] and charge transfer between dis-
+tinct species of moir´e atoms at n > 1 [5, 16–19] (n is
+the number of doped electrons or holes per moir´e unit
+cell). However, it is important to note that the charac-
+teristic Coulomb interaction energy within a moir´e atom
+is often several times larger than the single-particle su-
+perlattice gap. As a consequence, the low-energy Hilbert
+space can be substantially modified by interactions when
+multiple occupancy of moir´e atoms is involved. An accu-
+rate many-body theory for semiconductor moir´e systems
+therefore requires a proper treatment of the intra-moir´e-
+atom interaction.
+In this work, we predict new physics in semiconduc-
+tor moir´e systems arising from interaction effects at
+higher filling factors. We first develop an approach to
+modeling semiconductor moir´e systems that treats the
+short-range electronic correlations within a single multi-
+electron moir´e atom exactly. We model each moir´e atom
+as a potential well and solve the interacting few-electron
+atoms by exact diagonalization.
+Our approach is con-
+trolled in the “atomic limit” realized at large moir´e pe-
+riod where electrons are tightly bound to moir´e atoms
+and inter-atomic interactions can be neglected. For the
+three-electron atom (moir´e lithium), we find a distinctive
+equilateral triangle Wigner molecule charge configuration
+(see Fig 1d), which is stabilized by strong interaction and
+the threefold anisotropic moir´e crystal field. We further
+show that when the size of Wigner molecules becomes
+comparable to the moir´e period, an array of such Wigner
+molecules evolves into an emergent Kagome lattice of
+charges at filling factor n = 3, with electrons localized
+between moir´e potential minima. This emergent Kagome
+lattice arises due to the balance between Coulomb inter-
+action and moir´e potential.
+The change of charge configuration from triangular
+to Kagome lattice clearly demonstrates that the low-
+energy Hilbert space of moir´e systems is strongly filling-
+dependent due to interaction effects.
+Our work intro-
+duces parametrically controlled approximations for treat-
+ing strong electron-electron interactions in semiconduc-
+tor moir´e superlattices and reveals striking consequences
+of the interplay between quantum kinetic, moir´e poten-
+tial, and Coulomb interaction energies.
+Semiconductor moir´e continuum model— When two
+semiconductor TMD monolayers are stacked, a moir´e
+pattern appears due to lattice mismatch and/or twist
+angle. When the moir´e period aM is much larger than
+the monolayer lattice constant, the single-particle moir´e
+band structure is accurately described by a continuum
+model consisting of an effective mass approximation to
+the semiconductor band edge and a slowly-varying effec-
+tive periodic potential arising from band edge modulation
+throughout the moir´e unit cell. In this work, we always
+refer to the doped carriers as electrons, regardless of the
+true sign of their charge. The continuum model Hamilto-
+arXiv:2301.00799v1  [cond-mat.mes-hall]  2 Jan 2023
+
+2
+c
+a
+b
+d
+Image
+1a
+��
+�
+�
+����
+��
+�
+�
+����
+������
+��
+��
+�
+�
+max
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+����
+��
+�����
+��
+�
+�
+����
+��
+�
+�
+����
+������
+��
+��
+�
+�
+��
+��
+�
+�
+�
+����
+��
+��
+�
+�
+�
+����
+� � �
+����
+����
+����
+��
+�����
+�
+�
+�
+�
+�
+�
+���
+���
+���
+����
+����
+����
+������
+�� � ����
+��� �� �� � ��� ���� ��
+��� ���� ��
+��� �� ��
+��� �� ��
+��
+��
+�
+�
+�
+����
+��
+��
+�
+�
+�
+����
+� � �
+���
+���
+���
+��
+�����
+FIG. 1. Moir´e atoms and Wigner molecule (a) Schematic
+of moir´e superlattice and (b) corresponding moir´e potential at
+φ = 10◦. Its minima, moir´e atoms, form a triangular lattice.
+(c) Evolution of each of the high- and low-spin ground states
+of harmonic helium and lithium with the Coulomb coupling
+constant λ.
+The overall ground state of harmonic lithium
+transitions from low to high spin at λc = 4.34. (d) Charge
+density distribution of the high spin ground state of moir´e
+lithium including a crystal field corresponding to the contin-
+uum model parameters (V = 15meV, aM = 14nm, φ = 10◦,
+m∗ = 0.5me) without (left) and with (right) Coulomb inter-
+action.
+nian for TMD heterobilayers [4] (such as WSe2/WS2 and
+MoSe2/WSe2) and twisted Γ-valley homobilayers [20, 21]
+(such as twisted MoS2) assumes the form
+H = p2
+2m + ∆(r)
+(1)
+where ∆(r) = −2V �3
+i=1 cos(gi · r + φ) is an effective
+moir´e potential that has the translation symmetry of the
+superlattice in the first harmonic approximation, m is the
+effective mass, and gi =
+4π
+√
+3aM (sin 2πi
+3 , cos 2πi
+3 ) are the
+moir´e reciprocal lattice vectors (Fig 1(a)). The minima
+of ∆(r) define a periodic array of moir´e atoms to which
+doped charge is tightly bound in the atomic limit which
+we now examine.
+Moir´e atoms — We define an effective Hamiltonian for
+an electron confined to a moir´e atom by Taylor expanding
+∆(r) about the origin:
+∆(r) ≈ const. + 1
+2kr2 + c3 sin (3θ)r3 + . . .
+(2)
+where
+k
+=
+16π2V cos(φ)/a2
+M
+and
+c3
+=
+16π3 sin(φ)/(33/2a3
+M). The result is a circular oscillator
+with frequency ω =
+�
+k/m along with higher-order,
+rotation-symmetry-breaking corrections, which we call
+the moir´e crystal field. The effective Hamiltonian of an
+N-electron moir´e atom includes a Coulomb interaction
+e2/(ϵ|ri − rj|) between all of its electron pairs.
+Both kinetic energy and Coulomb energy favor charge
+delocalization, whereas the confinement potential fa-
+vors localization.
+The characteristic length ξ0
+≡
+�
+ℏ2/(mk)
+�1/4 at which the potential and kinetic energies
+of a harmonically-confined electron are equal defines the
+size of a single-electron moir´e atom. We further introduce
+the length scale at which the Coulomb and confinement
+energies of two classical point charges arranged symmet-
+rically about the origin of a harmonic potential are equal,
+ξc =
+�
+e2
+2ϵk
+�1/3
+. The ratio of these two length scales is
+directly related to the dimensionless coupling constant
+that is the ratio of the intra-atomic Coulomb energy to
+the atomic level spacing: λ ≡ e2/ϵξ0
+ℏω
+= 2(ξc/ξ0)3. Im-
+portantly, the size of the few-electron ground state of a
+moir´e atom, which we denote as ξ, is on the order of ξ0
+for λ ≪ 1 (weak interaction) and ξc for λ ≫ 1 (strong in-
+teraction), respectively. Therefore for general λ, we have
+ξ ∼ max{ξ0, ξc}.
+Importantly, we observe that the confinement potential
+weakens with increasing moir´e period: k ∝ a−2
+M . It thus
+follows that
+ξ0 ≡
+� ℏ2
+mk
+�1/4
+∝ a1/2
+M ; ξc ≡
+� e2
+2ϵk
+�1/3
+∝ a2/3
+M . (3)
+Consequently, at sufficiently large aM, the hierarchy of
+length scales aM > ξc > ξ0 is necessarily realized. Then,
+the size of the few-electron moir´e atom ξ is parametri-
+cally smaller than the distance between adjacent atoms
+aM, so that intra-atomic Coulomb interaction ∼ e2/ξ
+dominates over inter-atomic interaction ∼ e2/aM. This
+self-consistently justifies our treatment of isolated moir´e
+atoms as the first step to understanding moir´e solids.
+We begin by modeling each moir´e atom as purely har-
+monic, neglecting the influence of the crystal field. The
+single-particle eigenstates of the circular oscillator are la-
+beled by nodal and angular momentum quantum num-
+bers nr and lz with energy E = (2nr + |lz| + 1)ℏω. We
+identify n = 2nr + |lz| + 1 as the principal quantum
+number and refer to the circular oscillator eigenstates
+using electron configuration notation accordingly (i.e. s
+for lz = 0 and p for |lz| = 1 states).
+It is known rigorously that the ground state of two
+electrons with a time-reversal-symmetric Hamiltonian in-
+cluding a symmetric two-body interaction in arbitrary
+spatial dimensions is a spin singlet [22]. For all interac-
+tion strengths, the harmonic helium singlet ground state
+remains adiabatically connected to the 1s2, single-slater-
+determinant ground state at λ = 0, and the triplet first-
+excited state to the 1s12p1 state (see Fig1). Although
+the triplet-singlet energy gap remains positive for all λ, it
+asymptotically approaches 0 in the classical limit λ → ∞.
+As has been observed in the TMD moir´e platform and in
+GaAs quantum dots, Coulomb interactions allow for a
+modest magnetic field to induce a singlet-triplet transi-
+tion through the Zeeman effect [23–25].
+The above theorem does not apply to systems of more
+than two electrons. In the absence of a Coulomb interac-
+
+am3
+tion and moir´e crystal field, the moir´e lithium ground
+state configuration is a 1s22p spin-doublet with total
+spin and angular momentum quantum numbers (S, L) =
+(1, 1).
+At a critical coupling constant λc ≈ 4.34, we
+find that a ground state level crossing occurs between
+this low-spin doublet and a high-spin quartet state with
+(S, L) = (3/2, 0) originating from a 1s12p2 configuration
+(see Fig1)[26, 27]. After the level crossing, the energy
+difference between the low- and high-spin ground states
+remains small and asymptotes to 0 in the classical limit
+λ → ∞.
+The classical ground state of three interacting elec-
+trons in a harmonic potential is an equilateral triangle
+of side length ξT = (2/
+√
+3)1/3ξc centered about the ori-
+gin which spontaneously breaks the rotational symme-
+try, a configuration which we refer to as a triad.
+At
+finite λ, quantum fluctuation restores rotational symme-
+try, while preserving the pair correlations of the electron
+triad [27].
+On the other hand, the moir´e crystal field
+term sin(3θ)r3 breaks the rotational symmetry explic-
+itly. Since the threefold crystal anisotropy matches with
+the symmetry of the classical ground state, it stabilizes
+the triangular “Wigner molecule” in the presence of a
+Coulomb interaction. As we show in Fig. 1(c), the charge
+density of the (N, S) = (3, 3/2) state in the absence of
+the Coulomb interaction (λ = 0) is distorted only mildly
+by the crystal field, whereas, at λ = 6, it develops a
+local minimum at the origin and three distinct lobes at
+the corners of an equilateral triangle. The low-spin state
+S = 1/2 exhibits a similar density profile in the presence
+of the crystal field at moderate and large λ (see supple-
+ment), again in agreement with the classical limit where
+spin plays no role. The unique charge distribution of the
+Wigner molecule is a clear consequence of strong interac-
+tions and can be directly observed via a local probe such
+as scanning tunneling microscopy [28].
+Moir´e solids— Having established the physics of iso-
+lated moir´e atoms, we now turn to their crystalline en-
+sembles: moir´e solids. We reiterate the important obser-
+vation that the hierarchy of length scales aM > ξc > ξ0 is
+necessarily realized at sufficiently large aM. Equivalently,
+the energy scale of the moir´e potential depth V necessar-
+ily dominates over inter- and intra-atom Coulomb ener-
+gies e2/aM, e2/ξ as well as the quantum zero-point en-
+ergy ℏω ∝ a−1
+M associated with harmonic confinement.
+As a result, the ground state in this regime at integer
+filling n is an insulating array of n-electron moir´e atoms
+located at the moir´e potential minima, which is adiabat-
+ically connected to the decoupled limit aM → ∞.
+In the following, we investigate the intermediate regime
+aM ∼ ξc > ξ0 and reveal new physics that emerges from
+inter-atom coupling. We apply self-consistent Hartree-
+Fock theory to the continuum model (Eq.
+1), which
+treats intra- and inter-atomic interactions on equal foot-
+ing. In particular, at filling factor n = 3, we find that for
+realistic model parameters, electrons self-organize into
+an emergent Kagome lattice (Fig. 2). This is particu-
+larly striking, given that the Kagome lattice sites where
+����
+���
+���
+����
+����
+���
+���
+����
+�
+�
+�
+�����
+�
+�
+�
+�
+����
+����
+����
+����
+����
+����
+���
+(������
+�
+�
+�
+�
+����
+����
+����
+���
+�
+�������
+c
+a
+EF
+EF
+b
+d
+Final
+����
+���
+���
+����
+����
+���
+���
+����
+�
+�
+�
+�����
+����
+���
+���
+����
+����
+���
+���
+����
+�
+�
+�
+�
+�
+�����
+����
+���
+���
+����
+����
+���
+���
+����
+�
+�
+�
+�
+�
+�����
+FIG. 2. Emergent Kagome lattice at n = 3 (a) Quasi-
+particle band structure of self-consistent Hartree-Fock ground
+state with charge and spin density quantum numbers (n, sz) =
+(3, 3/2), resembling a Kagome lattice with broken inversion
+symmetry.
+Here we show only the spin-↑ bands.
+(b) Cor-
+responding real space electron density, exhibiting peaks that
+approximately form a Kagome lattice. A is the moir´e unit
+cell area. The parameters used are (V = 15meV, φ = 30◦,
+m = 0.5me, aM = 8nm, ϵ = 5). (c-d) Results for φ = 60◦
+where the continuum model is D6-symmetric and a perfect
+Kagome lattice emerges at n = 3.
+electrons localize are saddle points rather than minima
+of the moir´e potential.
+The origin of the emergent Kagome lattice can be
+understood in the regime ξ0 ≪ ξc and aM, which is
+smoothly connected to the classical limit ξ0 → 0 or equiv-
+alently m → ∞. In this regime, the ground state is de-
+termined by the competition between the moir´e potential
+and the Coulomb repulsion, which is controlled by the ra-
+tio aM/ξc. As previously discussed, the ground state at
+large aM/ξc is a lattice of electron triads of size ξc sep-
+arated by a distance of aM. This charge configuration,
+which consists of upper and lower triangles of size ∼ ξc
+and aM respectively, can be viewed as a precursor to the
+Kagome moir´e solid. As aM/ξc is reduced, the asymme-
+try in the size of the upper and lower triangles is naturally
+reduced, so that the charge configuration evolves towards
+the Kagome lattice.
+Our Hartree-Fock calculations fully support the above
+picture. For generic φ, due to the lack of D6 symmetry,
+the upper and lower triangles of the Kagome moir´e solid
+are distinct (Fig. 2b). Indeed, the corresponding quasi-
+particle band structure, Fig. 2a, resembles the Kagome
+lattice dispersion with broken inversion symmetry. As we
+show explicitly (see supplement), the three lowest quasi-
+particle bands are adiabatically connected to the s and
+px, py bands on a triangular lattice (as evidenced by the
+
+4
+�
+�
+�
+�
+�
+��
+��
+��
+��
+���
+(������
+a
+b
+����
+���
+���
+����
+����
+���
+���
+����
+�
+�
+�
+�
+�
+�����
+EF
+FIG. 3.
+Non-interacting bands.
+Non-interacting band
+structure and charge density at n = 3 for continuum model
+parameters (V = 15meV, φ = 60◦, m = 0.5me, aM = 8nm,
+ϵ = 5), which contrasts sharply with the interacting case
+shown in Fig. 2(c,d).
+twofold degeneracy at the γ point).
+This fully agrees
+with our picture of Wigner molecule array evolving into
+an asymmetric Kagome lattice.
+Even more interesting is the case φ = 60◦ as real-
+ized in twisted TMD homobilayers. Here, the underly-
+ing moir´e potential has two degenerate minima per unit
+cell forming a honeycomb lattice with D6 symmetry. At
+the filling factor n = 3, our Hartree-Fock calculation
+finds that charge assembles into a perfectly symmetric
+Kagome lattice (Fig. 2d). This is confirmed by the ap-
+pearance of a Dirac point in the Hartree-Fock band struc-
+ture Fig. 2c. Note that the Kagome band structure found
+here describes the dispersion of hole quasiparticles in the
+interaction-induced insulator at n = 3. In contrast, the
+non-interacting band structure at φ = 60◦, shown in
+Fig 3, is gapless at this filling. In addition, the Kagome
+moir´e solid features a symmetry-protected band degen-
+eracy at γ below the Fermi level, which is absent in the
+noninteracting case.
+It is interesting to note that the emergent Kagome lat-
+tice of charges minimizes neither the potential energy
+nor the Coulomb interaction energy. The potential en-
+ergy favors charges localized at honeycomb lattice sites,
+while the Coulomb interaction favors a triangular lat-
+tice Wigner crystal.
+Yet, remarkably, our calculations
+demonstrate that the Kagome lattice emerges as a com-
+promise due to their close competition for realistic mate-
+rial parameters.
+Although our Hartree-Fock results shown in Fig. 2 are
+for fully spin-polarized electrons, the emergent Kagome
+insulator at n = 3 persists regardless of its spin config-
+uration provided that ξc is comparable to aM and suffi-
+ciently large compared to ξ0. In this state, the low energy
+degrees of freedom are localized spins on the Kagome
+lattice.
+The effective spin interaction is an interesting
+problem that we leave to future study.
+In Fig. 4, we plot the length scales ξc, ξ0 and the cou-
+pling constant λ = 2(ξc/ξ0)3 as a function of the moir´e
+period aM for three representative TMD heterostruc-
+tures. These key quantities are calculated using contin-
+uum model parameters extracted from density functional
+�
+�
+��
+��
+�������
+���
+���
+���
+���
+���
+���
+����W������
+����W�������������������������
+��
+��
+�
+�
+��
+��
+�������
+�
+�
+�
+�
+�
+�
+�
+�6����6�
+�6���0�6��
+0�6��
+�� � ��
+FIG. 4. Key parameters for TMD moir´e heterostruc-
+tures.
+Length scales ξ0, ξc (left) and coupling constant λ
+(right) of several semiconductor moir´e systems calculated ac-
+cording to continuum model parameters extracted from den-
+sity functional theory [5, 20, 23].
+theory calculations. As the coupling constant λ increases
+with aM, the ground state at integer fillings evolves
+from a quantum solid where quantum effects dominate
+to a Wigner solid where classical effects dominate.
+In
+WSe2/WS2, the change between the two regimes corre-
+sponding to ξc = ξ0 occurs around aM ∼ 4.8nm.
+In
+twisted homobilayer MoSe2, we find ξc > ξ0 at all values
+of aM shown, owing in part to its larger effective mass at
+the Γ valley [20, 21]. Twisted homobilayers (WS2, WSe2,
+MoS2, MoSe2 and MoTe2) also have D6-symmetric moir´e
+potential, making them ideal candidates to realize the
+emergent Kagome solid at small twist angle.
+Our demonstration of the emergent Kagome lattice in
+TMD moir´e heterostructures paves the way for future
+investigation of magnetic and topological phases it may
+host. Geometric frustration makes the Kagome lattice
+a promising candidate for quantum spin liquid [29–31].
+The prospect that our emergent Kagome lattice may host
+such a phase deserves further investigation. Various in-
+teresting phases may also occur upon hole doping the
+emergent Kagome lattice, including exotic superconduc-
+tors [32], holon Wigner crystals [33], and fractional Chern
+insulators [34] due to nontrivial flat bands [35–38].
+Summary — Our work provides analytically controlled
+methods to treat strong interaction effect in moir´e su-
+perlattices and reveals novel phases of matter at higher
+filling factors n > 1. We have identified three key length
+scales that universally govern the physics of all moir´e ma-
+terials: the moir´e superlattice constant aM, the quantum
+confinement length ξ0, and crucially, the size of Wigner
+molecule ξc. ξ0 characterizes the strength of quantum ki-
+netic energy, and ξc the strength of Coulomb interaction.
+We have established two parameter regions in which the-
+oretical analysis is controlled even for strong interactions.
+First, when aM ≫ ξ0, ξc, moir´e atoms can be treated in
+isolation. By exactly solving the few-electron state of a
+single moir´e atom, we have predicted the existence of the
+Wigner molecule taking the form of an electron triad.
+
+5
+Second, when ξ0 ≪ ξc, aM, a self-organized electron lat-
+tice is formed to minimize the sum of potential and in-
+teraction energy. In particular, we predict an emergent
+Kagome lattice at the filling n = 3 for realistic material
+parameters that correspond to ξc ∼ aM. The interplay
+of these three length scales aM, ξ0 and ξc, in combina-
+tion with tunable electron filling n, presents a vast phase
+space and an organizing principle to explore moir´e quan-
+tum matter.
+Acknowledgements— It is our pleasure to thank Yang
+Zhang, Faisal Alsallom, and Allan MacDonald for valu-
+able discussions and related collaborations. This work
+was supported by the Air Force Office of Scientific Re-
+search (AFOSR) under award FA9550-22-1-0432.
+[1] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi,
+E. Kaxiras, and P. Jarillo-Herrero, Unconventional super-
+conductivity in magic-angle graphene superlattices, Na-
+ture 556, 43 (2018).
+[2] E. Y. Andrei and A. H. MacDonald, Graphene bilayers
+with a twist, Nature materials 19, 1265 (2020).
+[3] K. F. Mak and J. Shan, Semiconductor moir´e materials,
+Nature Nanotechnology 17, 686 (2022).
+[4] F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, Hub-
+bard model physics in transition metal dichalcogenide
+moir´e bands, Physical review letters 121, 026402 (2018).
+[5] Y. Zhang, N. F. Yuan, and L. Fu, Moir´e quantum chem-
+istry: charge transfer in transition metal dichalcogenide
+superlattices, Physical Review B 102, 201115 (2020).
+[6] E. C. Regan, D. Wang, C. Jin, M. I. Bakti Utama,
+B. Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yu-
+migeta, et al., Mott and generalized wigner crystal states
+in wse2/ws2 moir´e superlattices, Nature 579, 359 (2020).
+[7] Y. Tang, L. Li, T. Li, Y. Xu, S. Liu, K. Barmak,
+K. Watanabe, T. Taniguchi, A. H. MacDonald, J. Shan,
+et al., Simulation of hubbard model physics in wse2/ws2
+moir´e superlattices, Nature 579, 353 (2020).
+[8] Y. Xu, S. Liu, D. A. Rhodes, K. Watanabe, T. Taniguchi,
+J. Hone, V. Elser, K. F. Mak, and J. Shan, Correlated in-
+sulating states at fractional fillings of moir´e superlattices,
+Nature 587, 214 (2020).
+[9] H. Li, S. Li, E. C. Regan, D. Wang, W. Zhao, S. Kahn,
+K. Yumigeta, M. Blei, T. Taniguchi, K. Watanabe,
+et al., Imaging two-dimensional generalized wigner crys-
+tals, Nature 597, 650 (2021).
+[10] X. Huang, T. Wang, S. Miao, C. Wang, Z. Li, Z. Lian,
+T. Taniguchi, K. Watanabe, S. Okamoto, D. Xiao, et al.,
+Correlated insulating states at fractional fillings of the
+ws2/wse2 moir´e lattice, Nature Physics 17, 715 (2021).
+[11] C. Jin, Z. Tao, T. Li, Y. Xu, Y. Tang, J. Zhu, S. Liu,
+K. Watanabe, T. Taniguchi, J. C. Hone, et al., Stripe
+phases in wse2/ws2 moir´e superlattices, Nature Materials
+20, 940 (2021).
+[12] B. Padhi, R. Chitra, and P. W. Phillips, Generalized
+wigner crystallization in moir´e materials, Physical Re-
+view B 103, 125146 (2021).
+[13] N. Morales-Dur´an,
+P. Potasz, and A. H. MacDon-
+ald, Magnetism and quantum melting in moir´e-material
+wigner crystals, arXiv preprint arXiv:2210.15168 (2022).
+[14] Y. Zhou, D. Sheng, and E.-A. Kim, Quantum phases of
+transition metal dichalcogenide moir´e systems, Physical
+Review Letters 128, 157602 (2022).
+[15] B. A. Foutty, J. Yu, T. Devakul, C. R. Kometter,
+Y. Zhang, K. Watanabe, T. Taniguchi, L. Fu, and B. E.
+Feldman, Tunable spin and valley excitations of corre-
+lated insulators in γ-valley moir´e bands, arXiv preprint
+arXiv:2206.10631 (2022).
+[16] K. Slagle and L. Fu, Charge transfer excitations, pair
+density waves, and superconductivity in moir´e materials,
+Physical Review B 102, 235423 (2020).
+[17] W. Zhao, B. Shen, Z. Tao, Z. Han, K. Kang, K. Watan-
+abe, T. Taniguchi, K. F. Mak, and J. Shan, Gate-tunable
+heavy fermions in a moir\’e kondo lattice, arXiv preprint
+arXiv:2211.00263 (2022).
+[18] Y. Xu, K. Kang, K. Watanabe, T. Taniguchi, K. F. Mak,
+and J. Shan, A tunable bilayer hubbard model in twisted
+wse2, Nature Nanotechnology 17, 934 (2022).
+[19] H. Park, J. Zhu, X. Wang, Y. Wang, W. Holtzmann,
+T. Taniguchi, K. Watanabe, J. Y. Yan, L. Fu, T. Cao,
+D. Xiao, D. R. Gamelin, H. Yu, W. Yao, and X. Xu,
+Dipole ladders with giant hubbard u in a moir´e exciton
+lattice, unpublished (2022).
+[20] M. Angeli and A. H. MacDonald, γ valley transition
+metal dichalcogenide moir´e bands, Proceedings of the Na-
+tional Academy of Sciences 118, e2021826118 (2021).
+[21] Y. Zhang, T. Liu, and L. Fu, Electronic structures, charge
+transfer, and charge order in twisted transition metal
+dichalcogenide bilayers, Physical Review B 103, 155142
+(2021).
+[22] E. Lieb and D. Mattis, Theory of ferromagnetism and
+the ordering of electronic energy levels, in Inequalities
+(Springer, 2002) pp. 33–41.
+[23] C. R. Kometter, J. Yu, T. Devakul, A. P. Reddy,
+Y. Zhang, B. A. Foutty, K. Watanabe, T. Taniguchi,
+L. Fu, and B. E. Feldman, Hofstadter states and reen-
+trant charge order in a semiconductor moir\’e lattice,
+arXiv preprint arXiv:2212.05068 (2022).
+[24] R. Ashoori, H. Stormer, J. Weiner, L. Pfeiffer, K. Bald-
+win, and K. West, N-electron ground state energies of a
+quantum dot in magnetic field, Physical review letters
+71, 613 (1993).
+[25] M. Wagner, U. Merkt, and A. Chaplik, Spin-singlet–spin-
+triplet oscillations in quantum dots, Physical Review B
+45, 1951 (1992).
+[26] R. Egger, W. H¨ausler, C. Mak, and H. Grabert, Crossover
+from fermi liquid to wigner molecule behavior in quantum
+dots, Physical review letters 82, 3320 (1999).
+[27] S. A. Mikhailov, Quantum-dot lithium in zero magnetic
+field: Electronic properties, thermodynamics, and fermi
+liquid–wigner solid crossover in the ground state, Physi-
+cal Review B 65, 115312 (2002).
+[28] E. Wach, D. ˙Zebrowski, and B. Szafran, Charge den-
+sity mapping of strongly-correlated few-electron two-
+dimensional quantum dots by the scanning probe tech-
+nique, Journal of Physics: Condensed Matter 25, 335801
+(2013).
+[29] J.-W. Mei, J.-Y. Chen, H. He, and X.-G. Wen, Gapped
+spin liquid with z 2 topological order for the kagome
+heisenberg model, Physical Review B 95, 235107 (2017).
+
+6
+[30] P. Mendels and F. Bert, Quantum kagome frustrated
+antiferromagnets: One route to quantum spin liquids,
+Comptes Rendus Physique 17, 455 (2016).
+[31] C. Broholm, R. Cava, S. Kivelson, D. Nocera, M. Nor-
+man, and T. Senthil, Quantum spin liquids, Science 367,
+eaay0668 (2020).
+[32] W.-H. Ko, P. A. Lee, and X.-G. Wen, Doped kagome
+system as exotic superconductor, Physical Review B 79,
+214502 (2009).
+[33] H.-C. Jiang, T. Devereaux, and S. Kivelson, Holon wigner
+crystal in a lightly doped kagome quantum spin liquid,
+Physical Review Letters 119, 067002 (2017).
+[34] M.
+Kupczy´nski,
+B.
+Jaworowski,
+and
+A.
+W´ojs,
+Interaction-driven
+transition
+between
+the
+wigner
+crystal and the fractional chern insulator in topological
+flat bands, Physical Review B 104, 085107 (2021).
+[35] M. Kang, S. Fang, L. Ye, H. C. Po, J. Denlinger,
+C. Jozwiak, A. Bostwick, E. Rotenberg, E. Kaxiras,
+J. G. Checkelsky, et al., Topological flat bands in frus-
+trated kagome lattice cosn, Nature communications 11,
+1 (2020).
+[36] L. Ye, S. Fang, M. G. Kang, J. Kaufmann, Y. Lee, J. Den-
+linger, C. Jozwiak, A. Bostwick, E. Rotenberg, E. Kaxi-
+ras, et al., A flat band-induced correlated kagome metal,
+arXiv preprint arXiv:2106.10824 (2021).
+[37] M. Kang, S. Fang, J. Yoo, B. R. Ortiz, Y. M. Oey,
+J. Choi, S. H. Ryu, J. Kim, C. Jozwiak, A. Bostwick,
+et al., Charge order landscape and competition with su-
+perconductivity in kagome metals, Nature Materials , 1
+(2022).
+[38] M. Kang, S. Fang, J.-K. Kim, B. R. Ortiz, S. H. Ryu,
+J. Kim, J. Yoo, G. Sangiovanni, D. Di Sante, B.-G. Park,
+et al., Twofold van hove singularity and origin of charge
+order in topological kagome superconductor csv3sb5, Na-
+ture Physics 18, 301 (2022).
+

G9AyT4oBgHgl3EQfS_dT/content/tmp_files/2301.00096v1.pdf.txt

@@ -0,0 +1,774 @@
+International Journal of Engineering Trends and Technology 
+  
+                                      Volume 70 Issue 12, 281-288, December 2022 
+ISSN: 2231 – 5381 / https://doi.org/10.14445/22315381/IJETT-V70I12P226                                       © 2022 Seventh Sense Research Group®        
+          
+ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) 
+Original Article 
+ 
+Sentiment Analysis of COVID-19 Public Activity 
+Restriction (PPKM) Impact using BERT Method 
+ 
+Fransiscus1, Abba Suganda Girsang2 
+ 
+1,2 Computer Science Department, BINUS Graduate Program – Master of Computer Science,  
+Bina Nusantara University, Jakarta, Indonesia. 
+ 
+1Corresponding Author : [email protected]  
+ 
+Received: 06 August 2022             Revised: 13 December 2022         Accepted: 20 December 2022            Published: 24 December 2022 
+ 
+Abstract - Covid-19 has grown rapidly in all parts of the world and is considered an international disaster because of its wide-
+reaching impact. The impact of Covid-19 has spread to Indonesia, especially in the slowdown in economic growth. This was 
+influenced by the implementation of Community Activity Restrictions (PPKM) which limited community economic activities. 
+This study analyzes the mapping of public sentiment towards PPKM policies in Indonesia during the pandemic based on 
+Twitter data. Knowing the mapping of public sentiment regarding PPKM is expected to help stakeholders in the policy 
+evaluation process for each region. The method used is BERT with IndoBERT specific model. The results showed the 
+evaluation value of the IndoBERT f-1 score reached 84%, precision 86%, and recall 84%. Meanwhile, f-1 scores 70%, 72% 
+precision, and 70% recall for evaluating the use of SVM. Multinominal Naïve Bayes evaluation shows an f-1 score of 83%, 
+precision of 78%, and recall of 80%. In conclusion, the BERT method with the IndoBERT model is proven to be higher than 
+classical methods such as SVM and Multinominal Naïve Bayes. 
+Keywords - Sentiment Analysis, PPKM, BERT, SVM, Naïve Bayes. 
+1. Introduction  
+Based on research by Murad et al. 2020, the Covid-19 
+pandemic has grown rapidly in all parts of the world and is 
+considered an international disaster because of its wide-
+reaching impact. All lines of life are affected by the spread of 
+this virus in many countries. Various measures have been 
+taken to reduce the impact on people's health, socio-
+economic, education, and habits. Indonesia's economy 
+throughout the year slowed down to minus 5.3 percent in the 
+second quarter of 2020, and the average growth reached 
+minus 2.1 percent in 2020 [2]. 
+The implementation of PPKM will be considered 
+effective if viewed through the perspective of health analysis. 
+Still, their understanding will be contradictory if viewed 
+from an economic perspective experienced by the 
+community. The urgency of PPKM implementation must be 
+well conveyed to the public to support the percentage of 
+successful PPKM implementation, by setting sanctions for 
+people 
+who 
+violate 
+PPKM 
+can 
+help 
+smooth 
+the 
+implementation of PPKM policies. One of the sectors most 
+affected by the PPKM policy is micro-business in society. 
+Based on survey data on 206 small and medium 
+entrepreneurs (UMKM) in Greater Jakarta, as many as 82.9% 
+of UMKM entrepreneurs experienced a negative impact on 
+the pandemic conditions. The impact of a 30% decrease in 
+turnover was also experienced by 63.9% of the affected 
+UMKM [3]. The rapid growth of the positive number of the 
+Covid-19 virus and the policy of restrictions from the 
+government is a challenge for UMKM actors. Of course, this 
+can potentially produce a chain economic impact in the 
+future. 
+ Moreover, implementing government policies related to 
+pandemics must be in line with the conditions of the people 
+in various affected areas. The implementation of a unilateral 
+policy will certainly harm many parties. A Survey is needed 
+to see how the public responds to this policy, responses, 
+reactions, or opinions from the public. 
+In addition to survey data, people's opinions and 
+reactions share their views on various social media sites such 
+as Twitter [4]. Research related to Twitter sentiment on 
+government policies shows that by knowing public 
+sentiment, the government can take quick action to re-
+evaluate the policies taken, especially on topics with a high 
+negative rate based on opinions on Twitter [5]. 
+A huge number of Twitter users will certainly generate 
+many opinions related to many things every day, including 
+responses related to PPKM. Therefore, in this study, the 
+
+cC)
+000Fransiscus & Abba Suganda Girsang / IJETT, 70(12), 281-288, 2022 
+ 
+282 
+authors used Twitter data as a source. To get the data, the 
+author uses data mining techniques [6]. According to Gupta 
+and Chandra 2020, data mining is an efficient extraction 
+technique used to analyze qualitative data on a large scale. It 
+can be done repeatedly for the benefit of solving a problem. 
+This research uses data mining to extract public Twitter 
+information to analyze public opinion and sentiment on 
+PPKM policies in various regions. This opinion is then 
+processed into an expression using the sentiment analysis 
+method. 
+According to Diaz-Garcia, Ruiz, and Martin-Bautista 
+2020, sentiment analysis includes text mining techniques, 
+natural language processing, and automatic learning that 
+focuses on obtaining sentimental aspects from the text. The 
+sentiment result can be obtained using machine learning 
+methods [9]. Machine learning can be broadly defined as a 
+computational method that uses experience to improve 
+performance or make accurate predictions. The experience 
+referred to can be in the form of existing data (training set) or 
+the process of system interaction with the environment [10]. 
+In this study, the method used is Bidirectional Encoder 
+Representations from Transformers (BERT). BERT is a 
+machine learning model created to improve accuracy in 
+Natural Language Processing (NLP). BERT was developed 
+by Google AI Language researchers in 2018. This method 
+was developed by collaborating deep learning techniques 
+with methods such as UMLFiT, OpenAI Transformers, and 
+Transformers [11]. BERT is divided into two models such as 
+BERTBASE (12-layer encoder, 12 heads attention, hidden 
+size 768, and 110M parameters) and BERTLARGE (24 
+layers, 16 heads attention, hidden size 1024, and 340M 
+parameters) [12]. 
+Previous research related to sentiment analysis on social 
+media in China shows the accuracy of the BERT model 
+(75.65%) outperforms classical algorithms such as SVM 
+(70.66%), Naïve Bayes (66.97%), Logistic Regression 
+(70.02%), CNN (71.19%), and LSTM (57.73%). BERT is 
+effectively used in sentiment analysis because this algorithm 
+continues to be developed based on NLP, allowing it to be 
+applied in processing public opinion with big data [13]. 
+Previous study regarding identifying sentiments from 
+opinions related to Covid-19 vaccination based on Twitter 
+data. This study uses TF-IDF as a feature extraction method 
+and compares 6 classification algorithms, namely Naive 
+Bayes, Random Forest, SVM, Bi-LSTM, BERT, and CNN. 
+The performance measurement results show that the BERT 
+algorithm has the highest accuracy of 78.94%, and the lowest 
+performance is demonstrated by CNN, with an accuracy of 
+69.01%. The classification results show very high neutral 
+sentiment reaching 70% of the data, followed by positive 
+sentiment at 20%, then negative sentiment at 10% [14]. 
+Chandra and Saini 2021 analyzed sentiment on Twitter 
+data regarding presidential candidates Biden and Trump in 
+the 2020 election in the United States. This study uses the 
+LSTM and BERT methods to classify public sentiment 
+towards the two figures. The data used is quite large, namely 
+1.1 million tweets with geolocation. The study results show 
+that Biden's electability surpasses Trump, with performance 
+measurements showing that BERT is the best model, with an 
+accuracy of 87.45% and an F1 score of 75.7%. In 
+comparison, LSTM has an accuracy of 85.62% F1 score of 
+68.6%. 
+Previous research by Nugroho et al. 2021 [16] related to 
+sentiment analysis of user reviews of mobile-based 
+applications on Google Play using fine-tuned IndoBERT. In 
+this study, a comparison was made of the effectiveness of the 
+data labelling process on fine-tuning BERT such as BERT-M 
+and IndoBERT with traditional machine learning models 
+such as kNN, SVM, Naïve Bayes, Decision Tree, and 
+Random Forest. Two experiments were carried out with both 
+lexicon-based and scoring-based labeling on the training 
+data. The result is that the BERT method is superior, 
+especially IndoBERT with lexicon-based training data, 
+which has the highest accuracy of 84%. 
+Previous research involving the BERT method shows 
+that BERT is the best method with the best accuracy. 
+However, the accuracy is still not optimal because it still 
+depends on other supporting processes, such as data quality 
+on BERT training processes. However, when viewed from 
+the results and process, BERT tends to be more thorough, 
+especially in understanding the context and relevance of a 
+sentence. Therefore, the author chooses BERT as the suitable 
+method and is still very open to innovation and collaboration 
+to increase accuracy. 
+In this study, the author will use the IndoBERT 
+transformer model as a reference. IndoBERT is a 
+comprehensive Indonesian language benchmark consisting of 
+seven assignments for the Indonesian language. These 
+benchmarks are categorized into three pillars of NLP tasks, 
+namely morpho-syntax, semantics, and discourse. IndoBERT 
+works based on rich data where more than 200 million words 
+are trained [17]. Previous research by Nugroho et al. 2021 
+[16] shows high accuracy for IndoBERT with lexicon-based 
+training data preparation. However, lexicon-based training 
+data need more validation. In this study, the author adds a 
+manual validation process to verify that all labelled training 
+data is already accurate to build the final model. 
+ 
+2.1. Data Collection 
+Based on figure 1, it is known that the research stages 
+begin with data collection. As a classification-based 
+application, it requires a data source. The author uses the 
+Twitter dev API for the ad-hoc or stream data retrieval 
+process using Python. This data retrieval process can be done 
+
+Fransiscus & Abba Suganda Girsang / IJETT, 70(12), 281-288, 2022 
+ 
+283 
+with a lag time of 15 minutes and a maximum of up to 900 
+requests for data. Keywords used in this study ‘PPKM’ and 
+‘Jakarta’ with a total of up to 50,000 records starting from 
+January 1, 2021, until March 30, 2022. 
+ 
+2. Materials and Methods  
+ 
+Fig. 1 Research Process 
+ 
+2.2. Data Pre-processing  
+At this stage, the author makes a preprocessing process to 
+clean and normalize the input data to facilitate the 
+classification process. The preprocessing process consists of 
+several steps in it. 
+ 
+2.2.1 Tokenization 
+The purpose of this tokenization is to break a sentence 
+into pieces of words [18]. Tokenization is useful in order to 
+extract meaning from a text. For example, in a sentence, 
+authors want to detect nouns and verbs in the sentence, or 
+authors want to find the name of a person whose dimensions 
+are in a sentence. 
+ 
+2.2.2 Data Cleansing 
+Data cleansing is the process of cleaning data from noise 
+which aims to facilitate the labelling and classification 
+process on Twitter data. Data Cleansing consists of several 
+stages of data cleaning by applying Regular Expression 
+(RegEx), which aims to simplify the data classification 
+process. 
+ 
+2.2.3. Stopword Removal 
+Stopword removal is the process of removing noise in 
+the form of hashtags, symbols, URLs, content, and words 
+that have no clear meaning. The elimination of meaningless 
+words is done based on the dictionary in the program [19]. In 
+this study, stopwords were done by deleting meaningless 
+words 
+based 
+on 
+the 
+existing 
+dictionary 
+on 
+the 
+http://ranks.nl/stopwords/ site. 
+ 
+2.2.4 Manual Validation 
+After the data has been successfully cleaned at the data 
+cleansing and stopword stages, the author performs manual 
+validation to ensure that the initial data follows the PPKM 
+context using Microsoft Excel. At this stage, two validations 
+are necessary, such as data duplication and PPKM 
+correlation. Duplicate data will be eliminated to ensure the 
+tweet's unique value should be related to PPKM. In a special 
+case, some tweets mention the word PPKM, but the whole 
+tweet doesn't talk about PPKM. This case should be 
+eliminated to avoid invalid training data. 
+ 
+2.3. Classification Process 
+2.3.1 Data Training Initiation 
+After the cleansing process is completed, the next step is 
+training data initiation. The initial classification uses the 
+lexicon method with R programming. This lexicon is very 
+simple by calculating the weight of positive, negative, and 
+neutral words from tweets. Table 1 shows examples of 
+positive and negative words collected by the author from 
+http://ranks.nl/stopwords/indonesian.  
+ 
+After determining the dictionary of positive and negative 
+words, the author builds a simple classification model with R 
+programming. This model works by comparing positive and 
+negative word searches in tweets. Every time a negative 
+word is found, the sentiment score is -1; for every positive 
+word, the sentiment score is +1. Furthermore, this value will 
+be accumulated, and the calculation of the final value will be 
+carried out. A final score of more than 0 is classified as 
+positive, while a final score of less than 0 is classified as 
+negative, and a value equal to 0 is classified as neutral. 
+Sentiment classification results by lexicon are not entirely 
+accurate. Therefore, the author again conducted a manual 
+review to ensure the sentiment results were correct. This 
+process is essential to ensure the accuracy of the prediction 
+model.  
+Table 1. Positive and negative words corpus 
+Positive word 
+Negative word 
+bersuka cita 
+bersuka ria 
+bersyukur 
+dapat diandalkan 
+dapat dipercaya 
+dapat diraih 
+dapat disesuaikan 
+daya tarik 
+dengan mewah 
+dengan senang hati 
+dengan sopan 
+dermawan 
+barbar 
+basi 
+bau 
+bebal 
+beban 
+bejat 
+bekas 
+bekas luka 
+nepotisme 
+neraka 
+neurotik 
+ngambek 
+
+Twitter APIFransiscus & Abba Suganda Girsang / IJETT, 70(12), 281-288, 2022 
+ 
+284 
+ 
+Fig. 2 BERT Fine-Tuning process 
+ 
+ 
+2.3.2.. Model Initiation 
+The data resulting from the previous stage's lexicon-
+based classification became the IndoBERT model's input. 
+Figure 2 shows BERT fine-tuning process according to 
+Nugroho et al., 2021[2]. 
+ 
+The training process is not carried out in the early stages, 
+but the training process starts by using a model that has been 
+previously trained. In this study, the authors used the 
+IndoBERT-base p1 reference Hugging Face model as the 
+initial model. Then this model will be retrained for 
+optimization in a new task, namely sentiment classification 
+with PPKM context. This retraining process is referred to as 
+fine-tuning. In this process, the BERT model will receive a 
+sequence of words or sentences as input which will be 
+processed in the encoder stack. Each encoder applies self-
+attention and provides output through a feed-forward 
+network which the next encoder will follow. This process 
+will continue 12 times according to the IndoBERT-base 
+model. 
+ 
+In the next stage, after passing through all encoders, 
+each token in each position will provide an output in the 
+form of a vector with a size of 768. In the sentiment analysis 
+task, the output given to the first token position is [CLS]. At 
+the same time, a [SEP] token must be added at the end of the 
+sentence [16]. The last layer in the classifier layer produces 
+logits. Logits are output in the form of rough probability 
+predictions of the sentences to be classified. Next, softmax 
+will convert the logits into probabilities by taking the 
+exponents of each logit value so that the total probability is 
+exactly 1. This fine-tuning process can be done by adjusting 
+the hyperparameters. In the BERT training process, the 
+author uses 32 batch sizes, 10 epochs, and a learning rate 
+(Adam) 3e-6. 
+ 
+ 
+2.4. Testing & Visualization 
+ 
+The next stage is to try the model with input data. In 
+this study, the input data is in the form of tweets. Based on 
+this process, an entity identification process will be generated 
+where in this study, the author will focus on the location 
+entity. Of course, the process of introducing this entity will 
+depend on the quality of the training and previous training 
+data. Therefore, in this study, the authors will continue 
+improving the data quality and training process to produce 
+better accuracy. 
+Figure 3 shows the classification of sample flow data 
+using a pretrained IndoBERT model. Before fine-tuning, the 
+sample tweet will be tokenized using the IndoBERT input 
+format. Input formatting required a tokenizer by adding a 
+special token for each sentence. A [CLS] token must be 
+added at the beginning of each sentence for classification. At 
+the same time, a [SEP] token must be added at the end of the 
+sentence [16]. Tokenized sentences will be processed by a 
+transformer block that includes an encoder. The transformer 
+encoder learns and stores the tweet's semantic relationship 
+and grammatical structure information, and the Lexicon-
+based class is based on this input [20]. Finally, softmax 
+classifies the tweet as a sentiment result based on the 
+information. The sentence "Jualan saya rugi selama PPKM" 
+was classified as a Negative class in this case. 
+The author visualizes the data by making N-grams of 
+the uni-gram 
+type 
+and word 
+cloud. 
+N-grams 
+are 
+combinations of several words used together in the text. A 
+unigram has N=1, and the rationale behind n-grams is that 
+language structure [21]. A word cloud is essentially a visual 
+representation of text. Word clouds are used in many ways. It 
+can usually be done with pure text compression. In this case, 
+the word cloud represents words that occur more frequently 
+[22]. 
+ 
+Fig. 3 Sentiment Analysis IndoBERT Example 
+ 
+
+L-12(Output)
+Pos
+Predictions
+L=12
+Ntr
+Neg
+SoftMax
+H=768
+[CLS]  
+t1
+t.
+tn
+[SEP][PAD][PAD]
+BERTTokenizer
+Maxseguenceleneth=512Softmax
+Predictiveresult
+Classifier
+Negative
+Pretrained IndoBERT
+TransformerBlock
+.......
+Transformer Block
+TransformerBlock
+[CLS]
+Jualan
+saya
+rugi
+selama
+PPKM
+[SEP]Fransiscus & Abba Suganda Girsang / IJETT, 70(12), 281-288, 2022 
+ 
+285 
+ 
+Data visualization shows the most common issues in 
+tweets about PPKM in the Jakarta area. With this issue, the 
+author knows what opinions often arise according to their 
+respective sentiment classes. 
+2.5. Evaluation 
+After the model is formed and the classification results 
+are obtained, it is necessary to evaluate the model that has 
+been formed. This evaluation aims to assess the algorithm's 
+accuracy in the appointed case study to determine whether an 
+algorithm can work well. Result performance measurement 
+consists of a precision, recall, and F1-Score [23]. Precision 
+means the ratio of samples flagged as positive to samples 
+correctly flagged as positive. The recall is the ratio of the 
+number of positively flagged samples to the total number of 
+positively flagged samples. F-value is used as a scoring 
+metric to analyze the sentiment classification view [24]. It is 
+necessary to calculate the values to know the True Positive 
+(TP), False Positive (FP), True Negative (TN), and False 
+Negative (FN) values. They can be calculated as shown in 
+Eq.(1), Eq.(2) and Eq.(3). 
+ 
+𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛, 𝑝 =
+𝑇𝑃
+𝑇𝑃+𝐹𝑃          (1) 
+ 
+R𝑒𝑐𝑎𝑙𝑙, 𝑟 =
+𝑇𝑃
+𝑇𝑃+𝐹𝑁               (2) 
+ 
+𝐹 − 𝑆𝑐𝑜𝑟𝑒 =
+2𝑇𝑃
+2𝑇𝑃+𝐹𝑃+𝐹𝑁      (3) 
+ 
+The evaluation value should be compared with other 
+methods, and whether BERT has superior performance 
+compared to other classification methods that have been used 
+in the same cases and data can be seen. In this stage, the 
+authors compare the Naïve Bayes and SVM algorithms. In 
+the initial research process, the authors tried to execute the 
+Naïve Bayes and SVM algorithms, resulting in faster 
+execution times than current methods such as Convolution 
+Bi-directional Recurrent Neural Network (CBRNN) and Bi-
+directional Long Short-Term Memory (BiLSTM). As a 
+comparison, the average training time required in Naïve 
+Bayes and SVM training is under 15 minutes. Still, with the 
+same data, CBRNN and BiLSTM need a much longer time, 
+with the fastest time being 5 hours using the author's 
+hardware and software. This execution constraint several 
+times made the tools used by the author overloaded so that 
+they had to repeat the test and made the research process less 
+effective when applied to multilevel research such as 
+sentiment mapping. 
+Several studies have shown that SVM and Naïve Bayes 
+are still competitive in terms of performance compared to the 
+latest methods, such as LSTM. Research by Nikmah et al. 
+2022 shows that the accuracy of SVM (86.54%) and Naïve 
+Bayes (85.45%) can outperform LSTM (84.62%). In this 
+study, the quality of accurate training data is decisive. 
+Meanwhile, the authors used accurate training data in this 
+study by carrying out a manual validation process for all 
+data. 
+ 
+3. Results and Discussion  
+In this study, the author collects 50.000 raw data by 
+Twitter API with the keyword ‘PPKM’ and ‘Jakarta’. After 
+the Twitter data is collected, it needs to be preprocessed. This 
+stage consists of cleaning data, stopwords, and manual data 
+validation. As explained in chapter 2, this process aims to 
+clean and normalize the input data to simplify the 
+classification process. In addition, the data collected still 
+needs to be selected with several criteria that ensure the 
+authors only use data related to PPKM. All data that does not 
+fit into the criteria will be deleted manually, resulting in 
+5,315 data. 
+Before conducting the IndoBERT model training, the 
+author will carry out the process of classifying data has been 
+processed into 3 classes (positive, neutral, negative) with the 
+Lexicon method, where the results of this model 
+classification will be justified manually with the author to 
+ensure that the classification results are precise and accurate. 
+This is done to ensure that the data will produce maximum 
+output. Based on this process, the authors manually justified 
+sentiment classes in 5315 tweet data, resulting in 3,590 
+negative sentiments, 800 positive sentiments, and 925 neutral 
+sentiments. 
+The author continues to train the model with the 
+IndoBERT library as a classification algorithm. The author 
+divides the 5315 data into 4877 training data, 293 validation 
+data, and 145 testing data in the training process. The author 
+uses 10 epochs to build a model in the training process. 
+Figure 4 shows how the IndoBERT model of the accuracy of 
+the BERT fine-tuning learning process continues to increase 
+from epochs 1 to 10. 
+ 
+ 
+Fig. 4 BERT Fine-Tuning Accuracy History 
+
+Model Accuracy history
+1.0
+0.8
+Accuracy
+0.6
+0.4
+0.2
+trainaccuracy
+validationaccuracy
+testaccuracy
+0.0
+12
+0
+4
+6
+8
+EpochFransiscus & Abba Suganda Girsang / IJETT, 70(12), 281-288, 2022 
+ 
+286 
+ 
+Fig. 5 BERT Testing Sentiment Result 
+ 
+ 
+Fig. 6 Word Cloud 
+ 
+ 
+Fig. 7 Word N-gram 
+ 
+Testing and validation accuracy remains flat value due 
+to there are several data negative and positive training. 
+However, figure 5 shows the testing process is dominated by 
+negative sentiment. This shows a negative society response 
+to the PPKM policy in Jakarta based on tweet data. Figure 6 
+shows the word cloud that represents covid, ppkm, ‘enggak” 
+(means no) as a dominant value. The interesting word is 
+“pemerintah” (means government) also mentioned. Figure 7 
+shows N-gram visualization, same as the word cloud, N-
+gram shows covid and ppkm on the top list. Word 
+“pemerintah” also mentioned in this N-gram. By this 
+visualization, authors can identify the most common issues in 
+tweets about PPKM from the frequent word that appeared. 
+After defining the performance calculation, BERT needs 
+to be compared with other methods with the same data. 
+Compared with other methods, whether BERT performs 
+better than other classification methods used in the same case 
+and data can be seen. At this stage, the authors chose the 
+Multinominal Naïve Bayes algorithm and SVM as a 
+comparison because these two methods in several studies 
+showed good accuracy even though they were considered 
+classical methods [26]–[29]. Fig 8 shows the measurement 
+result between IndoBERT, SVM, and Multinominal Naïve 
+Bayes. 
+ 
+ 
+Fig. 8 Performance Comparison 
+ 
+Performance Comparison, BERT with IndoBERT model 
+is better performance than SVM and Multinominal Naïve 
+Bayes for the classification model. The higher gap is on the 
+Recall when BERT reaches 84%, Multinominal Naïve Bayes 
+at 80%, and SVM at 70%. As well as precision, BERT 
+reaches the highest recall, 84%, and F1, 84%. This value 
+shows how well BERT carries out the classification process. 
+ 
+4. Conclusion  
+This paper proposes the BERT algorithm with the 
+IndoBERT pre-trained model for sentiment analysis. An 
+experiment with Twitter data on the topic of PPKM Jakarta 
+shows that BERT, especially the IndoBERT model, is better 
+than other algorithms such as SVM and Multinominal Naïve 
+Bayes. In the modern era, understanding social media 
+sentiments is fundamental in seeing people's responses to a 
+topic. BERT is proven to help stakeholders understand the 
+community, especially in making the right policies based on 
+criticism and suggestions from the community.  
+However, related to huge model training structure and 
+corpus of BERT impact to long execution time. In future 
+research, the execution time issue should be solved by 
+simplifying 
+its 
+model 
+size 
+to 
+improve 
+efficiency. 
+Furthermore, sentiment analysis related to Covid-19 public 
+activity restriction should be explored by visualizing in 
+location mapping. The prediction could be reached using 
+Named Entity Recognition, which possibly extracts location 
+based on tweet text. 
+ 
+Funding Statement 
+Grants from Bina Nusantara University funded this 
+research.
+17%
+68%
+15%
+Positive
+Negative
+Neutral
+86%
+84%
+84%
+72%
+70%
+70%
+83%
+80%
+83%
+Precision
+Recall
+F1
+BERT IndoBERT
+SVM
+Multinominal Naïve Bayes
+
+masker
+banget
+kali
+penyebaran..covid
+sudah
+shat
+kota
+masuk
+sih
+kalo
+vaksin
+ppkm
+penyebaran viruspakai
+masyarakat
+enggak
+nan
+eg
+indonesiacepat
+karena
+pergi
+kesehatan
+daerab
+virus coronaJugi
+Covidyanya
+ppkmdarurat
+bagaimanaaminini
+ga
+orang
+an
+darura
+april
+sal
+kalau
+manusia
+lawan
+deh
+pemerintah
+dana
+Coronia covidWord FreguencyinTrain Data
+800
+600
+400-
+200
+wordFransiscus & Abba Suganda Girsang / IJETT, 70(12), 281-288, 2022 
+ 
+287 
+References 
+[1] Dina Fitria Murad et al., “The Impact of the COVID-19 Pandemic in Indonesia (Face to Face Versus Online Learning),” in 2020 Third 
+International Conference on Vocational Education and Electrical Engineering (ICVEE), pp. 1–4, 2020. Crossref, 
+https://doi.org/10.1109/ICVEE50212.2020.9243202 
+[2] Muhyiddin Muhyiddin and Hanan Nugroho, “A Year of Covid-19: A Long Road to Recovery and Acceleration of Indonesia’s 
+Development,” The Indonesian Journal of Development Planning, vol. 5, no. 1, pp. 1–19, 2021. Crossref,  
+https://doi.org/10.36574/jpp.v5i1.181 
+[3] Digitalization of MSMEs in the Midst of the Covid-19 Pandemic, Katadata, 2020. [Online]. Available: Https://Katadata.Co.Id/Umkm 
+[4] A. F. Thaha, “Impact of Covid-19 on MSMEs in Indonesia,” Jurnal Lentera Bisnis, vol. 2, no. 1, pp. 147–153, 2020, [Online]. 
+Available: https://ejournals.umma.ac.id/index.php/brand 
+[5] Eki Aidio Sukma et al., “Sentiment Analysis of the New Indonesian Government Policy (Omnibus Law) on Social Media Twitter,” 
+2020 International Conference on Informatics, Multimedia, Cyber and Information System (ICIMCIS), pp. 153–158, 2020. Crossref, 
+https://doi.org/10.1109/ICIMCIS51567.2020.9354287 
+[6] Digital 2021: The Latest Insights into the State of Digital, We Are Social, 2021. [Online]. Available: 
+https://Wearesocial.Com/Blog/2021/01/Digital-2021-the-Latest-Insights-Into-the-State-of-Digital 
+[7] Afreen Jaha, et al., "Text Sentiment Analysis Using Naïve Baye’s Classifier," International Journal of Computer Trends and 
+Technology, vol. 68, no. 4, pp. 261-265, 2020. Crossref, https://doi.org/10.14445/22312803/IJCTT-V68I4P141 
+[8] Himanshu Thakur and Aman Kumar Sharma, "Supervised Machine Learning Classifiers: Computation of Best Result of Classification 
+Accuracy," International Journal of Computer Trends and Technology, vol. 68, no. 10, pp. 1-8, 2020. Crossref, 
+https://doi.org/10.14445/22312803/IJCTT-V68I10P101 
+[9] Hamed Jelodar et al., “Deep Sentiment Classification and Topic Discovery on Novel Coronavirus or COVID-19 Online Discussions: 
+NLP Using LSTM Recurrent Neural Network Approach,” IEEE journal of biomedical and health informatics, vol. 24, no. 10, pp. 2733–
+2742, 2020. Crossref, https://doi.org/10.1109/JBHI.2020.3001216 
+[10] Mehryar Mohri, Afshin Rostamizadeh and Ameet Talwalkar, Foundations of Machine Learning, MIT Press, 2018. 
+[11] Jacob Devlin et al., “Bert: Pre-Training of Deep Bidirectional Transformers for Language Understanding,” Proceedings of NAACL-HLT 
+2019, pp. 4171-4186, 2018. Crossref, https://doi.org/10.48550/arXiv.1810.04805 
+[12] Quoc Thai Nguyen et al., “Fine-Tuning Bert for Sentiment Analysis of Vietnamese Reviews,” in 2020 7th NAFOSTED Conference on 
+Information and Computer Science (NICS), pp. 302–307, 2020. Crossref, https://doi.org/10.1109/NICS51282.2020.9335899 
+[13] Tianyi Wang et al., “COVID-19 Sensing: Negative Sentiment Analysis on Social Media in China via BERT Model,” IEEE Access, vol. 
+8, P. 1, 2020, Crossref, https://doi.org/10.1109/ACCESS.2020.3012595  
+[14] Liviu-Adrian Cotfas et al., “The Longest Month: Analyzing Covid-19 Vaccination Opinions Dynamics From Tweets in the Month 
+Following the First Vaccine Announcement,” IEEE Access, vol. 9, pp. 33203–33223, 2021. Crossref,  
+https://doi.org/10.1109/access.2021.3059821 
+[15] Afreen Jaha et al., “Text Sentiment Analysis Using Naïve Baye’s Classifier,” International Journal of Computer Trends and 
+Technology, vol. 68, no. 4, pp. 261-265, 2020. Crossref, 10.14445/22312803/IJCTT-V68I4P141 
+[16] Kuncahyo Setyo Nugroho et al., “BERT Fine-Tuning for Sentiment Analysis on Indonesian Mobile Apps Reviews,” in 6th International 
+Conference on Sustainable Information Engineering and Technology 2021, pp. 258–264, 2021. Crossref, 
+https://doi.org/10.48550/arXiv.2107.06802 
+[17] Fajri Koto et al., “Indolem and Indobert: A Benchmark Dataset and Pre-Trained Language Model for Indonesian NLP,” Proceedings of 
+the 28th International Conference on Computational Linguistics, Arxiv Preprint Arxiv:2011.00677, pp. 757-770, 2020. Crossref, 
+https://doi.org/10.48550/arXiv.2011.00677 
+[18] Saurav Pradha et al., “Effective Text Data Preprocessing Technique for Sentiment Analysis in Social Media Data,” 2019 11th 
+International Conference on Knowledge and Systems Engineering (KSE), pp. 1–8, 2019. Crossref, 
+https://doi.org/10.1109/KSE.2019.8919368 
+[19] Pooja Mehta and Dr.Sharnil Pandya, “A Review on Sentiment Analysis Methodologies, Practices and Applications,” International 
+Journal of Scientific and Technology Research, vol. 9, no. 2, pp. 601–609, 2020. 
+[20] Sarojadevi Palani, Prabhu Rajagopal and Sidharth Pancholi, “T-BERT--Model for Sentiment Analysis of Micro-Blogs Integrating Topic 
+Model and BERT,” Arxiv Preprint Arxiv:2106.01097, pp. 1-9, 2021. Crossref, https://doi.org/10.48550/arXiv.2106.01097 
+[21] Jay-Ar Lalata, Bobby Dioquino Gerardo, and Ruji Medina, “A Sentiment Analysis Model for Faculty Comment Evaluation Using 
+Ensemble Machine Learning Algorithms,” in Proceedings of the 2019 International Conference on Big Data Engineering, pp. 68–73, 
+2019. Crossref, https://doi.org/10.1145/3341620.3341638 
+[22] Sonia Saini et al., “Sentiment Analysis on Twitter Data Using R,” 2019 International Conference on Automation, Computational and 
+Technology Management (ICACTM), 2019, pp. 68–72. Crossref, https://doi.org/10.1109/ICACTM.2019.8776685 
+ 
+
+Fransiscus & Abba Suganda Girsang / IJETT, 70(12), 281-288, 2022 
+ 
+288 
+[23] RavinderAhuja et al., “The Impact of Features Extraction on the Sentiment Analysis,” Procedia Comput Science, vol. 152, pp. 341–348, 
+2019. Crossref, https://doi.org/10.1016/j.procs.2019.05.008 
+[24] P. Karthika, R. Murugeswari and R. Manoranjithem, “Sentiment Analysis of Social Media Network Using Random Forest Algorithm,” 
+2019 IEEE International Conference on Intelligent Techniques in Control, Optimization and Signal Processing (INCOS), pp. 1–5, 2019. 
+Crossref, https://doi.org/10.1109/INCOS45849.2019.8951367  
+[25] T. L. Nikmah, M. Z. Ammar, Y. R. Allatif, R. M. P. Husna, P. A. Kurniasari, and A. S. Bahri, “Comparison of LSTM, SVM, and Naive 
+Bayes for Classifying Sexual Harassment Tweets,” Journal of Soft Computing Exploration, vol. 3, no. 2, pp. 131–137, 2022. Crossref, 
+https://doi.org/10.52465/joscex.v3i2.85 
+[26] Usman Naseem et al., “Covidsenti: A Large-Scale Benchmark Twitter Data Set for COVID-19 Sentiment Analysis,” IEEE Trans 
+Comput Social System, vol. 8, no. 4, pp. 1003–1015, 2021. Crossref, https://doi.org/10.1109/TCSS.2021.3051189 
+[27] Heru Suroso et al., “Sentiment Analysis on ‘Homecoming Tradition Restriction’ Policy on Twitter,” 2020 3rd International Conference 
+on Computer and Informatics Engineering (IC2IE) pp. 75–79, 2020. https://doi.org/10.1109/IC2IE50715.2020.9274609 
+[28] Maryum Bibi et al., “A Cooperative Binary-Clustering Framework Based on Majority Voting for Twitter Sentiment Analysis,” IEEE 
+Access, vol. 8, pp. 68580–68592, 2020. Crossref, https://doi.org/10.1109/ACCESS.2020.2983859 
+[29] Meylan Wongkar and Apriandy Angdresey, “Sentiment Analysis Using Naive Bayes Algorithm of the Data Crawler: Twitter,” in 2019 
+Fourth International Conference on Informatics and Computing (ICIC), pp. 1–5, 2019. Crossref,  
+https://doi.org/10.1109/ICIC47613.2019.8985884  
+[30] M. K. Gupta and P. Chandra, “A Comprehensive Survey of Data Mining,” International Journal of Information Technology, vol. 12, no. 
+4, pp. 1243–1257, 2020. Crossref, https://doi.org/10.1007/s41870-020-00427-7 
+[31] Jose Angel Diaz-Garcia et al., “Non-Query-Based Pattern Mining and Sentiment Analysis for Massive Microblogging Online Texts,” 
+IEEE Access, vol. 8, pp. 78166–78182, 2020. Crossref, https://doi.org/10.1109/ACCESS.2020.2990461 
+[32] Rohitash Chandra and Ritij Saini, “Biden vs Trump: Modeling US General Elections Using BERT Language Model,” IEEE Access, vol. 
+9, pp. 128494–128505, 2021. Crossref, https://doi.org/10.1109/ACCESS.2021.3111035 
+ 
+ 
+

KtAyT4oBgHgl3EQf6Prq/content/tmp_files/2301.00820v1.pdf.txt

@@ -0,0 +1,1417 @@
+Draft version January 4, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+Classification of BATSE, Swift, and Fermi Gamma-Ray Bursts from Prompt Emission Alone
+Charles L. Steinhardt,1, 2 William J. Mann,1, 2 Vadim Rusakov,1, 2 Christian K. Jespersen,3
+1Cosmic Dawn Center (DAWN)
+2Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, DK-2100 Copenhagen Ø
+3Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
+ABSTRACT
+Although it is generally assumed that there are two dominant classes of gamma-ray bursts (GRB)
+with different typical durations, it has been difficult to unambiguously classify GRBs as short or long
+from summary properties such as duration, spectral hardness, and spectral lag. Recent work used
+t-distributed stochastic neighborhood embedding (t-SNE), a machine learning algorithm for dimen-
+sionality reduction, to classify all Swift gamma-ray bursts as short or long.
+Here, the method is
+expanded, using two algorithms, t-SNE and UMAP, to produce embeddings that are used to provide
+a classification for the 1911 BATSE bursts, 1321 Swift bursts, and 2294 Fermi bursts for which both
+spectra and metadata are available. Although the embeddings appear to produce a clear separation
+of each catalog into short and long bursts, a resampling-based approach is used to show that a small
+fraction of bursts cannot be robustly classified. Further, 3 of the 304 bursts observed by both Swift and
+Fermi have robust but conflicting classifications. A likely interpretation is that in addition to the two
+predominant classes of GRBs, there are additional, uncommon types of bursts which may require
+multi-wavelength observations in order to separate from more typical short and long GRBs.
+1. INTRODUCTION
+The most prominent feature of a γ-ray burst (GRB) is
+the short duration of its prompt emission, ranging from
+∼ 10−2s to ∼ 103s. The distribution of observed dura-
+tions is predominantly bimodal, leading to a standard
+division of GRBs into short and long bursts (cf. Kou-
+veliotou et al. (1993)). These two groups have been hy-
+pothesized to have distinct astrophysical origins, with
+short bursts associated with mergers of neutron stars
+(Tanvir et al. 2013; Berger et al. 2013; Ghirlanda et al.
+2018) and long bursts associated with core collapses of
+massive stars (Hjorth et al. 2003; Stanek et al. 2003).
+This would imply that it should be possible to cleanly
+separate the two types of bursts based on observed prop-
+erties.
+However, there is considerable overlap between the
+two distributions in duration. Thus, the standard di-
+viding line at T90 = 2s will miscategorize the shortest
+long bursts and the longest short bursts (Tavani et al.
+1998; Paciesas et al. 1999). The most prominent addi-
+tional observed features, spectral hardness (Kouveliotou
+et al. 1993) or spectral lag (Norris et al. 1986; Norris &
+Corresponding author: Charles Steinhardt
+[email protected]
+Bonnell 2006) do not provide a clean separation between
+short and long bursts. Environment has also been con-
+sidered as a possible distinguishing factor (Le´sniewska
+et al. 2022). More complex derived properties, or com-
+binations of the standard summary properties, have also
+been unable to provide a complete separation (Fruchter
+et al. 2006; Nakar 2007; Bromberg et al. 2011; Zhang
+et al. 2012; Bromberg et al. 2013).
+Recently, a new approach using the dimensionality
+reduction algorithm t-distributed Stochastic Neighbor
+Embedding (t-SNE) used the full Swift light curves to
+provide the first clear separation between short and
+long bursts (Jespersen et al. 2020). Although the same
+approach should be expected to successfully separate
+bursts from the BATSE and Fermi catalogs as well, the
+differences between these datasets require individualized
+tuning and the datasets cannot be combined into one
+t-SNE map without significant information loss. Thus,
+although this work uses very similar methodology to the
+Jespersen et al. (2020) pilot study, the methodology ap-
+plied to the other two catalogs is necessarily slightly
+distinct.
+This work expands upon that study in several signifi-
+cant ways:
+arXiv:2301.00820v1  [astro-ph.HE]  2 Jan 2023
+
+2
+• Bursts from BATSE and Fermi are also mapped,
+providing a complete t-SNE-based classification
+catalog combining all large datasets.
+• It is shown that the three catalogs yield similar
+separations, implying that the separation is truly
+astrophysical rather than due to selection effects,
+data reduction choices, or other systematics.
+• An additional algorithm, Uniform Manifold Ap-
+proximation and Projection (UMAP; McInnes
+et al. 2018) is considered as an alternative to t-
+SNE. UMAP and t-SNE are both dimensional-
+ity algorithms, and often produce similar results
+when well-tuned (Kobak & Berens 2019; Becht
+et al. 2019; Xiang et al. 2021). However, for most
+large, high-dimensionality datasets, UMAP pro-
+duces these results in significantly shorter compu-
+tation time (Hu et al. 2019)1.
+• A comparision of the classifications of bursts which
+appear in both the Swift and Fermi datasets is
+used both as a consistency check and to deter-
+mine whether additional subclassifications are sug-
+gested by the t-SNE and UMAP maps.
+• A new measure of uncertainty is introduced to de-
+scribe the stability of these classifications.
+In § 2, the methods used in Jespersen et al. (2020)
+are reviewed, modified as needed, and applied to the
+BATSE and Fermi catalogs. Corresponding choices for
+UMAP are also discussed. The resulting classifications
+are presented in § 3. In § 4, the robustness of this clas-
+sification is evaluated using a combination of multiple
+catalogs. The results and implications for future sur-
+veys are discussed in § 5. A full catalog including clas-
+sifications for objects in BATSE , Fermi , and Swift is
+included, as described in the Appendix.
+2. METHODOLOGY
+The methodology used in this work follows the gen-
+eral approach used in Jespersen et al. (2020) analysis
+of the Swift dataset, modified in order to accommodate
+the differences between various GRB observatories. Di-
+mensionality reduction is applied to the full set of light
+curves in every observed band. The resulting embedding
+is examined, and divided into cleanly-separated struc-
+tures. The objects in each structure are then considered
+to comprise a distinct class of GRBs.
+1 See
+also
+https://umap-learn.readthedocs.io/en/latest/
+performance.html
+In practice, there are several additional steps. First,
+the data must be standardized, removing incomplete or
+missing observations. Then, preprocessing is performed
+to remove or negate irrelevant information that these al-
+gorithms might otherwise interpret as meaningful. Af-
+terwards, dimensionality algorithms can be applied, and
+that application requires the choice of several hyperpa-
+rameters which must be selected individually for each
+dataset.
+Finally, it is necessary to determine which
+structures on the resulting embedding should be con-
+sidered distinct.
+Within the same approach, each of
+these steps requires different choices for Swift, BATSE,
+and Fermi. These are described in more detail in the
+subsections below.
+It is important to note that “unsupervised” algorithms
+such as the ones used here are in practice strongly de-
+pendent on the choice of hyperparameters. The proper
+interpretation of the embeddings presented here should
+not be that the hyperparameters we have chosen are cor-
+rect and others incorrect. Rather, every choice of hyper-
+parameters leads to a valid embedding, which contains
+potentially useful information about the distribution but
+is always incomplete. In that sense, it would be more
+like considering various projections. Unlike projections,
+however, there is no rigorous formalism such as principal
+component analysis for determining which will be most
+useful.
+Here, the choices rely on the physical assumption that
+GRBs can be divided into discrete groups due to distinct
+progenitors, but avoid asserting any specific number of
+progenitors.
+Rather, the hyperparameters which pro-
+duce the cleanest separations are selected. In that re-
+spect, the separation into two primary groups is a prop-
+erty of the dataset. However, because these embeddings
+focus on groups of a specific size, additional progenitors
+which are less common and thus have significantly fewer
+examples will not be revealed on these maps.
+Emebeddings focusing on small groups were consid-
+ered in Jespersen et al. (2020), but did not produce
+easily-interpretable, well-separated groups. Further, as
+described in § 4.1, the smaller substructures hinted at
+on the embeddings considered here do not appear to be
+meaningful.
+However, given the large space of possi-
+ble hyperparameters, of which only a small portion has
+been sampled in this work, it is likely that additional,
+astrophysically-interpretable groups could exist. Identi-
+fying how many and which of these groups are meaning-
+ful might require additional observaional information.
+2.1. Burst Selection
+One of the restrictions of t-SNE and UMAP is that
+they cannot be applied to data of different dimension or
+
+3
+labels.2 This is one of the reasons that Swift, BATSE,
+and Fermi are examined individually rather than as a
+group; the different bands and cadences of each set of
+observations do not allow a direct comparison.
+Here,
+bursts which at least one band is predominantly miss-
+ing or key metadata do not exist are rejected entirely.
+More minor flaws are generally adjusted or corrected in-
+stead of rejecting the burst. The specific choices for each
+individual dataset are included for reproducibility.
+For BATSE (Meegan 1997), the ASCII file is used
+as the canonical source of each light curve, due to
+the similar tte bfits files having various data prob-
+lems. Out of 2702 total bursts, 527 bursts with miss-
+ing ASCII files on the HEASARC server3 were there-
+fore rejected.
+Summary data including flux, fluence
+and T90 were taken from https://batse.msfc.nasa.gov/
+batse/grb/catalog/current/. An additional 264 bursts
+with missing summary data were rejected. In total, 791
+bursts were rejected and the remaining 1911 were in-
+cluded.
+For Swift (Lien et al. 2016), the same choices were
+made as in Jespersen et al. (2020). This work uses a
+more recent version of the catalog, which includes an
+additional 67 bursts.
+For Fermi (von Kienlin et al. 2020), the bcat file is
+used as the canonical source of each light curve.
+All
+but 34 bursts available at https://heasarc.gsfc.nasa.gov/
+FTP/fermi/data/gbm/bursts/ had a bcat file available,
+so 3108 were downloaded. The Fermi GBM Burst Cat-
+alog includes multiple models, and the best-fit average
+flux was used for each burst. These fluxes and t90 were
+taken from https://heasarc.gsfc.nasa.gov/W3Browse/
+fermi/fermigbrst.html. Unfortunately, nearly all of the
+bursts recorded after mid-2018 had not undergone spec-
+tral analysis, therefore lacking best-fit flux and fluence
+models. These 814 bursts were cut, leaving 2294 remain-
+ing for analysis.
+A potential concern is that the use of deconvolved flux
+lightcurves for Fermi but count rate lightcurves for the
+other observatories makes the three no longer directly
+comparable.
+However, this was already the case be-
+cause, e.g., Swift and Fermi provide different bands and
+thus different information about each bursts. Using dif-
+ferent data types can be thought of as simply another
+difference between datasets and pipelines. As shown in
+2 Or, at least, the distance metric selected must make a choice
+about how to handle the difference between missing and mea-
+sured dimensions in a consistent way, which is typically very dif-
+ficult unless the correct answer is already known.
+3 https://heasarc.gsfc.nasa.gov/FTP/compton/data/batse/
+ascii data/64ms/
+4, there is still strong agreement between the Swift and
+Fermi classifications, indicating that all of these differ-
+ences between observatories, reductions, and choice of
+lightcurves do not significantly alter the conclusions pre-
+sented here about the broad classification of GRB.
+2.2. Preprocessing
+Several preprocessing steps are required before dimen-
+sionality reduction algorithms can be used.
+Because
+these algorithms require identical formats and have dif-
+ficulty handling missing information, light curves are
+padded with additional zeros to produce data of iden-
+tical lengths as in Jespersen et al. (2020). The goal of
+additional preprocessing is to remove extraneous infor-
+mation which might otherwise contribute to the post-
+embedding positions of light curves while retaining as
+much information as possible.
+There are two key ex-
+traneous parameters: the overall brightness of the burst
+and the trigger time.
+Although the energy carried by a burst is physically
+meaningful, for most GRBs in the catalog, the redshift
+is unknown. Under the assumption that bursts with the
+same underlying astrophysical origins can exist at a wide
+range of redshift, the brightness will be a poor indicator
+of luminosity. Therefore, each burst is normalized by
+dividing by the total fluence.4 This retains hardness in-
+formation because the relative brightness between bands
+is preserved.
+The other issue is handling trigger time offsets, where
+the time that the burst was detected differs from the
+actual start of the burst.
+These offsets are typically
+due to instrumentation rather than due to the shape
+of the burst itself, and therefore should be discarded.
+To accomplish this, the same procedure is used as in
+Jespersen et al. (2020). A discrete-time Fourier trans-
+formation (DTFT) is performed on a concatenation of
+the lightcurves in all observed bands. An overall time
+shift will only change the phase information under this
+DTFT. However, relative time offsets between different
+bands, as well as other meaningful information such as
+duration, hardness, and spectral lag, will all contribute
+to the amplitudes as well. Therefore, the phase infor-
+mation is then discarded, and dimensionality reduction
+algorithms are run only on the amplitudes.
+2.3. Embedding
+Dimensionality reduction algorithms are then run on
+each of the datasets independently. It is necessary to
+perform different embeddings for Swift, BATSE, and
+4 For Fermi, we instead normalized by average flux, which we found
+to work better on the semi-processed bcat data.
+
+4
+Fermi, since they measure different bands with differ-
+ent cadences and report data in different formats. Two
+main algorithms are used here: t-SNE and UMAP. Both
+have hyperparameters which must be tuned for each in-
+dividual dataset. The primary hyperparameter of im-
+portance for t-SNE is perplexity. For UMAP, there are
+two hyperparameters which must be tuned, n neighbors
+and set op mix ratio.
+Dataset
+t-SNE
+UMAP
+UMAP
+perplexity
+n neighbors
+set op mix ratio
+BATSE
+0
+0
+0.25
+Fermi
+20
+25
+0.25
+Swift
+20
+20
+0.3
+Table 1. Hyperparameters chosen for the t-SNE and UMAP
+embeddings of each of the three GRB catalogs used in this
+work.
+In every case, a range of embeddings with different
+properties will result from various choices of hyperpa-
+rameters.
+Unfortunately, there is no rigorous mathe-
+matical formalism for choosing optimal hyperparame-
+ters. The multiple embeddings which can result from
+different choices are in some sense all correct and valid,
+yet simultaneously are all incomplete descriptions of the
+full structure. Naturally, some will be more useful for
+GRB classification. For each dataset, both t-SNE and
+UMAP were run with a variety of hyperparameters, and
+the ones which produced embeddings with the cleanest
+separations were chosen. The hyperparameters chosen
+are summarized in Table 1.
+Although these embeddings often show clear separa-
+tions between short and long GRB, that does not guar-
+antee that there are only those two classes of GRB. The
+perplexity and n neighbors hyperparameters for t-
+SNE and UMAP, respectively, essentially dictate the size
+of the groups which the embeddings focus on represent-
+ing properly. Thus, if there are two predominant groups
+and one or more tiny ones, the tiny groups might be
+attached to a larger one. An attempt to identify bursts
+which might not be standard short or longer GRBs is
+described in § 4.2.
+3. CLASSIFICATION
+For each catalog, two embeddings are produced, one
+with t-SNE and the other with UMAP, for a total of
+six maps.
+On each map, a division is made into two
+large groups, labeled short and long based on the typi-
+cal duration in each group. A small fraction of bursts are
+classified either as outliers, distinct from both groups, or
+as ambiguous, lying in the region between the short and
+long groups. The BATSE map is more complicated, pro-
+ducing an initial separation which produces a duration
+distribution qualitatively different than the other two
+catalogs. A more quantitative comparison is performed
+in § 4.
+3.1. Swift
+Since the technique used in this work is very similar to
+that in Jespersen et al. (2020), the embeddings and clas-
+sifications produced using the Swift catalog are nearly
+identical. There is a clear separation into two groups.
+one identified as short (orange, Fig. 1) and the other as
+long (purple).
+Figure 1. t-SNE (left) and UMAP (right) embeddings of
+1321 Swift lightcurves, colored by classification. The dura-
+tion distributions (bottom) are consistent with an interpre-
+tation as separation into short (orange) and long (purple)
+GRB rather than merely a classification purely by duration.
+The two embeddings agree on the classification of all but
+one burst (cyan). Three bursts are classified different in this
+work than in Jespersen et al. (2020) (black).
+Of the 1321 bursts, t-SNE classifies 114 as short and
+1207 as long. The UMAP embedding classifies just one
+burst differently: GRB090813 (cyan, Fig. 1) is long in
+the t-SNE embedding but short in UMAP. In addition,
+three bursts as classified differently here than in Jes-
+persen et al. (2020). GRB121226A and GRB180418A
+are long in the Jespersen catalog and short here, while
+GRB050724 switches classification from short to long.
+Although it may seem counterintuitive that a burst
+can switch classification when an identical technique is
+run on a superset of the data, this is indeed a property
+of both t-SNE and UMAP. Both algorithms assign a
+cost to placing every pair of objects at any specific dis-
+tance, such that similar objects are less costly at short
+distances and dissimilar objects less costly at large dis-
+tances, then seek to minimize the global sum of that
+
+Swift
+t-SNE
+UMAP
+200
+0
+0
+2
+0
+2
+logTgo
+logT9o5
+cost. The addition of a new point will typically result
+in a lowest-cost configuration that involves not merely
+placing that point on an existing map, but shifting their
+locations as well. For example, an analogous physical
+system might be one in which every object is attached
+to every other by a spring, with the stiffness of that
+spring depending upon their similarity. The addition of
+a new set of springs will likely result in all of the dis-
+tances changing in the equilbrium configuration.
+One consequence of the choice of perplexity (for t-
+SNE) and n neighbors (UMAP) is that both embed-
+dings attempt to place outliers into a cluster where
+plausible.
+Thus, bursts which are somewhat dissimi-
+lar to both short and long GRBs are often placed on
+the edges of whichever group is more similar. The addi-
+tion of a small number of similar objects can therefore
+change which group they are located close to.
+Thus,
+GRB050724, GRB121226A and GRB180418A are likely
+neither typical short bursts nor typical long bursts, but
+instead outliers, either for astrophysical reasons or due
+to a data processing artifact. An attempt to identify
+similar bursts in other datasets is described in § 4.2.
+3.2. Fermi
+The 2294 bursts in the Fermi catalog are arranged into
+the two embeddings shown in Fig. 2. These maps pro-
+duce the clearest separation of any of the three datasets
+using both t-SNE and UMAP. As a result, no bursts
+were unable to be clearly assigned to either group. A
+possible interpretation is that the higher-energy bands
+in the Fermi dataset are more useful for distinguish-
+ing between types of bursts than the bands available
+in BATSE and Swift. Had Fermi observed the full set
+of BATSE and Swift bursts, under this interpretation
+the Fermi embedding would be expected to look nearly
+identical on this larger dataset.
+The t-SNE map classifies 387 bursts as short and
+1907 as long. On the UMAP map, there are 385 short
+bursts and 1907 long bursts. Three bursts (cyan, Fig 2)
+are classified as short by t-SNE and long by UMAP:
+GRB090811696, GRB110719825, and GRB110728056.
+GRB080828189 (also shown in cyan) is the sole burst
+classified as long by UMAP but short by t-SNE. The re-
+maining 2290 classifiable bursts agree in both analyses.
+Still, given the completeness of the separation on both
+the t-SNE and UMAP maps, it is perhaps surprising
+that four bursts change classification between the two
+embeddings. Several possible causes of this reclassifica-
+tion are evaluated in § 4.2. As a result of that analysis,
+in the catalog presented here, a measure of the uncer-
+tainty in classification is developed. For the remainder
+of this section, bursts will be described as short or long
+Figure 2. t-SNE (left) and UMAP (right) embeddings of
+2294 Fermi lightcurves, colored by classification. The dura-
+tion distributions (bottom) are consistent with an interpreta-
+tion as separation into short (orange) and long (purple) GRB
+rather than merely a classification purely by duration. Al-
+though both embeddings show a clear separation, four bursts
+(cyan) are classified differently by t-SNE and UMAP.
+based on their most probable classification and the ap-
+parently clear separations in the embeddings in Figures
+2-3.
+The catalog associated with these work includes
+not only the central values but also estimated likeli-
+hoods for these classifications.
+In that catalog, 2.0%
+of Fermi bursts have between a 10% and 90% probabil-
+ity of being classified as short (or long). Such objects
+are therefore labeled as ambiguous rather than as short
+or long.
+3.3. BATSE
+The 1911 bursts in the reduced BATSE catalog are
+arranged into the embeddings shown in Fig. 3. Unlike
+Fermi and Swift, a significant number of BATSE bursts
+could not be included due to missing data, missing
+metadata, or high noise.
+In total, 791 of the 2702
+BATSE bursts were discarded, and many of the remain-
+ing ones have marginal quality and could not be well
+constrained.
+The t-SNE embedding classifies 491 bursts as short
+and 1420 as long. The UMAP embedding has similar
+size groups, with 484 long and 1427 long bursts. How-
+ever, unlike the Fermi and Swift embeddings, there is a
+more substantial disagreement in classification. 21 ob-
+jects are classified as short by UMAP and long by t-SNE,
+and 28 classified as long by UMAP and short by t-SNE
+(cyan, Fig.
+3).
+The individual objects which switch
+classification between embeddings are indicated in the
+catalog associated with this paper.
+The disagreement would have been far stronger us-
+ing the BATSE light curves obtained directly from the
+
+Fermi
+t-SNE
+UMAP
+500
+250
+0
+2
+0
+2
+0
+2
+logTgo
+logT9o6
+Figure 3. t-SNE (left) and UMAP (right) embeddings of
+1911 BATSE lightcurves, colored by classification. The dura-
+tion distributions (bottom) are consistent with an interpreta-
+tion as separation into short (orange) and long (purple) GRB
+rather than merely a classification purely by duration. The
+separation in BATSE is less robust than the other datasets,
+likely due to higher noise, missing data or metadata, and the
+necessity to perform additional background subtraction. As
+a result, 49 of the 1911 bursts (cyan) are classified differently
+by t-SNE and UMAP.
+HEASARC server. To produce a separation, it was nec-
+essary to re-process each light curve in order to fit and
+subtract a background. As a rudimentary subtraction, a
+linear fit was performed to the first (pre-burst) and last
+(well after T100) data in each of the four bands, then
+subtracted from the full light curve. The background-
+subtracted light curves were then used to produce the
+embeddings and catalog in this work.
+A more complete background subtraction would likely
+require rerunning or modifying the original processing
+pipeline. It is likely that at the end of such an effort, an
+improved and more robust separation would be possible.
+However, since higher-quality GRB data are available
+from newer observatories, here it is assumed that this
+would be of limited use. Thus, in this work the decision
+was made to include this approximate background sub-
+traction for completeness, but to focus on separating the
+Swift and Fermi datasets, as they will be most suitable
+for further analysis.
+4. CROSS-MATCHING AND VALIDATION
+A significant potential concern when using unsuper-
+vised machine learning methods is that because there
+is no training set, there is an inherent inability to val-
+idate the conclusions. Although the GRB light curves
+can cleanly be separated into two groups, the method-
+ology involved has no knowledge of astronomy or astro-
+physics. Thus, the statement that there are two distinct
+classes of GRB light curves does not necessarily mean
+that there are two astrophysical mechanisms for pro-
+ducing GRB. It could instead be that the two groups
+have been separated based on data artifacts or process-
+ing pipeline decisions. t-SNE and UMAP are remark-
+able tools for finding clusters and categories, but not for
+determining the causes of those categories.
+Jespersen
+et al. (2020) demonstrated that the Swift classifications
+line up well with previous progenitor hypothesis. For
+example, all Swift bursts with a known, associated su-
+pernova afterglow were classified as long, consistent with
+previous expectations (Hjorth & Bloom 2012; Cano et al.
+2017). However, only a few Swift GRB have observed
+afterglows, so it is difficult to rule out the possibility of
+this separation having been caused by data processing
+rather than astrophysical origin.
+A key goal of this work is to combine observations from
+all three available GRB observatories, with different
+bands, sensitivity, selection, and processing pipelines. If
+all produce a similar classification, this would validate
+the separation as being due to astrophysics. Here, two
+tests are performed using multiple catalogs in an effort
+to determine whether this classification is robust.
+4.1. Cross-matching Swift and Fermi GRB
+307 bursts were observed by both Swift and Fermi,
+allowing a comparison between the two catalogs. If the
+classification is robust and due to astrophysical origin,
+then bursts common to both catalogs must have the
+same label in both datasets. Of the 307, 298 have the
+same classification and only 9 disagree (Fig. 4), which
+strongly suggests that the separation is indeed based on
+the emission itself rather than artifacts induced by the
+data reduction.
+The same approach also allows an investigation of
+the several smaller clusters which appear on t-SNE and
+UMAP maps. The objects common to both telescopes
+which appear in a compact substructure of the long GRB
+groups (e.g., towards the top-left of the Swift map in Fig.
+1, shown as the green points on Fig. 4) do not com-
+prise a distinct substructure in the other dataset, but
+rather are merely part of the long GRB group. There-
+fore, these substructures are not interpreted as a distinct
+type of GRB or as having a distinct astrophysical origin,
+but rather as merely lying at one end of the parameter
+space of long GRBs.
+4.2. Classification Stability
+The 9 bursts classified differently in Swift and
+Fermi suggest that even the seemingly unambiguous sep-
+arations in these embeddings might not be entirely ro-
+bust. Here, three sources of potential instability in clas-
+sification are considered.
+
+BATSE
+t-SNE
+UMAP
+400
+200
+0
+2
+0
+2
+0
+2
+logT90
+logT9o7
+Figure 4. Locations of the 307 objects (various colors) com-
+mon to both Swift and Fermi within the t-SNE embedding
+(gray). 298 of the 306 are classified in the same way for both
+datasets (orange for short; green or purple for long), which
+implies that the separation is due to the GRB emission rather
+than artifacts introduced in data reduction. The remaining
+nine, which switch classification, are shown in cyan.
+Al-
+though there are possible substructures on the t-SNE maps,
+such as in the upper-left corner of the Swift embedding in Fig.
+1, these structures are not consistent between maps (green),
+and therefore are not interpreted as being of astrophysical
+origin.
+4.2.1. Data Ordering
+First, the exact embeddings produced by t-SNE and
+UMAP depend upon the order in which the bursts are
+fed into the algorithm. Although embeddings produced
+by different orders have almost identical structures, the
+actual locations of individual objects will vary (Fig.
+5). In order to investigate this, 1000 maps were gen-
+erated for each of the three datasets using burst lists
+sorted randomly into different orders.
+For each map,
+the bursts were divided into groups using spectral clus-
+tering (Fiedler 1973; Ng et al. 2001), directed to split
+into exactly two groups. In all 1000 trials, an identi-
+cal set of short and long bursts was produced by both
+t-SNE and UMAP. Thus, it can be concluded that the
+classification is resilient to changes in ordering.
+4.2.2. Outliers and Rare Bursts
+Perhaps a greater concern comes from the choice of hy-
+perparameters and resulting handling of outlier bursts.
+In this paper, the issue is described in terms of t-SNE hy-
+perparameters, but is common to both algorithms. Di-
+mensionality reduction algorithms must choose between
+preserving more local and more global structure, as the
+data are not truly two-dimensional and some informa-
+tion must be lost in the mapping. For t-SNE, this is
+controlled by the perplexity. The embeddings here have
+been tuned to focus on separating the two major groups
+of GRBs.
+During the gradient descent as t-SNE iteratively opti-
+mizes its embedding, each burst is attracted by similar
+Figure 5.
+t-SNE embeddings of the Swift GRB catalog
+using burst lists sorted randomly into 16 different orders.
+On each map, short bursts are indicated in orange and long
+bursts in purple. A comparison of 1000 maps generated for
+the Swift dataset showed an identical classification for every
+object, confirming that the separation is robust to a change
+in the list order.
+bursts and repelled by dissimilar ones. For a perplex-
+ity of N, t-SNE imposes a probability density function
+which can be thought of as optimizing for the typical ob-
+ject having N attractive neighbors. This will produce a
+clean separation between two groups each larger than N.
+However, tiny groups or individual objects with unique
+properties can be attached to the most similar group.
+If a burst is, e.g., far more similar to short bursts than
+to long bursts, it will be classified as short. However, if
+it has properties in common with both groups, then it
+will be attracted to both groups, and classified based on
+whichever set of attractors are stronger.
+In order to search for these groups, a resampling-based
+approach is adopted. Subsets of (600, 900, 1000) bursts
+are drawn from the full (Swift, BATSE, Fermi) cata-
+log, corresponding to ∼ 50% of the total bursts, and an
+embedding produced for each subset. As before, spec-
+tral clustering is applied to separate the embedding into
+two groups. With the smaller samples, there are not al-
+ways enough bursts to produce a clean separation with-
+out manual hyperparameter tuning.
+Thus, only sub-
+sets which produce a clean separation similar to the
+original grouping are included. Embeddings for which
+a Kolmogorov-Smirnov (KS) test indicates a p < 0.10
+
+Swift
+Fermi8
+probability of being drawn from the same distributions
+as the separation on the full catalog are rejected5. A
+KS test is chosen rather than a test such as Anderson-
+Darling in order to emphasize the bulk of the distri-
+bution rather than the tails, so that embeddings which
+move outliers will not be excluded.
+The hope is that if a burst is attracted by both groups,
+then it might change location. In some of the random
+trials, many of its closest neighbors from one group or
+the other will be excluded.
+However, a prototypical
+short burst will always be most similar to short bursts,
+even if some of its closest analogues are excluded.
+For the most part, this procedure indicates that the
+classification is stable (Fig. 6). In the Fermi catalog, the
+median resampling trial has 0.5% of bursts change loca-
+tion. 6.6% of bursts change location in at least one of
+the ∼ 750 trials, and 2.0% change location in more than
+10% of trials. This latter group are labeled as ambigu-
+ous in the catalog from this work. In the Swift catalog,
+0.5% change location in the median trial, 13.0% change
+location at least once and 2.6% are ambiguous. In the
+BATSE catalog, 2.8% of bursts change location in the
+median trial, 20.6% change location at least once and
+9.2% are ambiguous.
+4.2.3. Insufficient Information
+Of the 9 bursts classified differently by Swift and
+Fermi, 7 change location in at least one resampled map
+in at least one catalog and 6 change location more
+than 10% of the time (Table 2). The remaining bursts,
+GRB090531B and GRB130716A, are consistently clas-
+sified differently, in this case as long bursts by Swift and
+short bursts by Fermi.
+That is, in the Swift dataset
+alone, they are similar to typical long burst and in the
+Fermi dataset alone, they are similar to typical short
+bursts. A reasonable interpretation is that these are ex-
+tended emission bursts (Norris & Bonnell 2006; Kaneko
+et al. 2015), which are known to be shorter in the harder
+emission observed by Fermi and longer in Swift.
+The broader implication is that for some uncommon
+types of bursts, the information provided by one dataset
+alone is insufficient to classify them.
+Extended emis-
+sion bursts might only be detectable with a combina-
+tion of harder and softer emission, which currently no
+single telescope can provide.
+These bursts do appear
+distinct in an analysis which combines both Swift and
+Fermi.
+However, such a combination only exists for
+5 Note that by looking for similar distributions, this procedure es-
+timates the stability of the specific bimodal classification being
+evaluated. In principle, given computing resources well beyond
+those available to the authors, one could systematically search
+for the most stable bimodal classification.
+Burst
+Swift
+Fermi
+GRB090531B
+0.000
+1.000
+GRB090927
+0.070
+1.000
+GRB130716A
+0.000
+1.000
+GRB131004A
+0.194
+0.754
+GRB140209A
+0.682
+0.081
+GRB140320A
+1.000
+0.346
+GRB141205A
+0.257
+1.000
+GRB150120A
+0.146
+0.860
+GRB170318B
+0.926
+0.000
+Table 2. Probability that one of the resampled maps will
+classify a burst as short for the 9 bursts with different classifi-
+cations in the Swift and Fermi catalogs. 6 of the bursts have
+unstable classifications in at least one of the two catalogs,
+which implies that these are neither typical short nor long
+bursts. One additional burst, GRB090927, changes location
+in 7% of resampled Swift maps. However, two, GRB090531B
+and GRB130716A, have entirely stable but conflicting classi-
+fications. These are likely extended emission bursts, and in-
+dicate that there is not enough information in either dataset
+alone to determine that they are atypical.
+around 15% of these catalogs. Thus, if GRB090531B
+and GRB130716A are indeed extended emission bursts,
+there are likely an additional ∼ 15 undetected ex-
+tended emission bursts classified as long in Swift with
+no Fermi data, and a similar number of undetected ex-
+tended emission bursts classified as short in Fermi with
+no Swift data.
+4.3. Bulk Properties of Short and Long GRB
+Another way to evaluate whether the separations in
+these three catalogs are identical is to compare the bulk
+properties of the short and long populations in all three
+datasets.
+If GRB truly are correctly separated into
+classes by astrophysical origin, then the short and long
+GRB observed by each telescope should be drawn from
+identical distributions, and thus have identical distri-
+butions of properties. Further, with a clean separation
+between short and long GRB, it should now be possible
+to determine which other properties correlate with GRB
+type, something that would not have been possible for
+complete samples without this method.
+Such a comparison is significantly complicated by dif-
+ferent selection functions for each telescope. In partic-
+ular, the duration is dilated by a factor of (1 + z), and
+therefore telescopes sensitive to GRB at a wider range
+of redshift should also have a broader distribution of du-
+rations. Because the redshift is only known for a small
+fraction of GRBs, it is not possible to compare rest-
+frame durations for the full samples.
+
+9
+Figure 6. Stability of the classifications of bursts based on ∼ 750 random subsets of 600, 1000, and 900 random bursts drawn
+from the Swift (left), Fermi (center), and BATSE (right) catalogs, respectively. The vast majority of objects have an identical
+classification in every resampled embedding. 6.6% of the Fermi bursts change classification in at least one embedding, and 2.0%
+are classified differently in at least 10% of trials. Objects in this last group are labeled as ambiguous rather than short of long
+in the catalog. The fraction of ambiguous objects are larger for the Swift (2.6%) and BATSE (9.2%) catalogs.
+Still, a comparison of the observed durations indi-
+cates a similar separation in each dataset (Fig. 7). The
+most similar, given their similar frequency ranges, are
+BATSE and Fermi. The short GRB durations are sim-
+ilar in all three datasets, but the long GRBs in Swift
+include a longer-duration tail than in BATSE or Fermi.
+Thus, it is possible that all three telescopes are select-
+ing a similar set of short bursts down to redshift distri-
+butions and detection thresholds. However, the softer
+bands in Swift produce a dissimilar duration distribu-
+tion of long bursts when compared with BATSE and
+Fermi.
+The distribution of short and long GRBs on a
+hardness-duration plot further indicates that the t-SNE
+and UMAP classifications are separating objects with
+similar bulk properties (Fig.
+8).
+Only BATSE and
+Swift are shown here, as Fermi does not produce hard-
+ness as part of its data release. It should be noted that
+due to different available bands, the hardness measured
+by BATSE cannot be directly compared with that mea-
+sured by Swift. For both BATSE and Swift, short GRBs
+are generally harder and shorter than long GRBs, but
+with some overlap.
+This is again consistent with the
+hypothesis that short and long GRBs are robust cate-
+gories intrinsic to GRB emission rather than to details
+of the observatories or processing pipelines used to mea-
+sure their properties.
+5. DISCUSSION
+Following a pilot study using t-SNE to classify
+Swift gamma-ray bursts from the full observed light
+curves (Jespersen et al. 2020), here dimensionality re-
+duction is shown to be able to classify observations from
+Figure 7. Comparison of the observed duration distribu-
+tions of short and long bursts in the classifications from the
+Swift (top) Fermi (middle), andBATSE (bottom) datasets.
+The redshift distributions in these datasets are expected to
+be different, although there is insufficient redshift informa-
+tion to verify this. The distributions of short bursts are qual-
+itatively similar and could even be consistent with having
+been drawn from very similar distributions if it were possi-
+ble to correct for time dilation. However, the Swift selection
+of long GRB likely differs from BATSE and Fermi by more
+than redshift alone could account for, and is likely due to
+softer bands providing a different selection than the other
+two datasets.
+all three major GRB observatories, BATSE , Swift ,
+and Fermi. As with Swift, all three datasets produce
+a separation into two substantial groups. This fits with
+previous work proposing two distinct classes of progeni-
+tors: that short bursts are likely associated with mergers
+
+Stability
+Swift
+Fermi
+BATSE
+1.0
+0.9
+max[P(XEL), P(XES))
+0.8
+0.7
+0.6Swift
+short
+1.0
+long
+0.5
+0.0 -
+-2
+-1
+0
+1
+2
+3
+Count (Normalized)
+Fermi
+1.0
+short
+long
+0.5
+0.0
+-2
+-1
+0
+1
+2
+3
+BATSE
+0.75
+short
+long
+0.50
+0.25
+0.00
+-2
+-1
+0
+1
+2
+3
+logTgo10
+Figure 8. Distribution of objects in hardness and duration
+in the Swift (left) and BATSE (right) datasets. It should be
+noted that due to different bands used in the calculation, the
+Swift and BATSE hardness measurements are on different
+scales and cannot be compared directly. The Fermi catalog
+does not include a direct measure of hardness. The two cat-
+alogs exhibit a qualitatively similar, overlapping distribution
+of short and long GRBs. This is consistent with a selection
+that depends upon intrinsic GRB properties rather than de-
+tails of the observatory or processing pipeline.
+(Tanvir et al. 2013; Berger et al. 2013; Ghirlanda et al.
+2018) and long bursts with the core collapse of massive
+stars (Hjorth et al. 2003; Stanek et al. 2003).
+In previous studies, the definition of a short or long
+burst has generally been based on duration only. Since
+the two distributions overlap in duration, some bursts
+will therefore be misclassified. In this work, the separa-
+tion into short and long is based on the entire observed
+light curve, and produces a clean separation into two
+groups for each dataset without overlap. The hope is
+that this clean separation will correspond to physical
+properties, and that the resulting short and long bursts
+will indeed have distinct astrophysical origins. Jespersen
+et al. (2020) found that a comparison of their classifica-
+tion with both known and proposed supernova after-
+glows supports this interpretation.
+A significant issue with many machine learning meth-
+ods is that classifications can be based on extraneous
+information, confounding variables, or metadata. Thus,
+the clean separation in the Swift dataset reported by
+Jespersen et al. (2020) might occur due to different as-
+trophysical origins, as hoped, but could also be produced
+by differences in data processing. A comparison of all
+three catalogs shows a similar separation with similar
+properties. Further, nearly all of the objects observed
+by both Swift and Fermi are classified in the same way.
+Therefore, it can be concluded that the separations re-
+ported here are truly astrophysical in origin.
+5.1. Additional Classes
+Although 97% of the bursts common to Swift and
+Fermi are classified identically, 9 are not.
+Several of
+these bursts are part of the small fraction with un-
+stable classifications (§ 2).
+However, at least two
+bursts, GRB090531B and GRB130716A, have entirely
+stable but conflicting classifications. In the Swift data
+alone, they appear to be typical long bursts, but in
+the Fermi data alone, they appear to be typical short
+bursts. A strong possibility is that these are part of the
+previously-reported group of extended emission bursts
+(Norris & Bonnell 2006; Kaneko et al. 2015). If they
+have a distinct astrophysical origin, then GRBs should
+properly be divided into at least three groups, not two.
+However, their presence should also suggest that addi-
+tional classes of bursts might exist which are difficult to
+classify from any single one of these three observatories.
+It might have been hoped that the use of additional in-
+formation could separate them from more typical short
+or long bursts. However, dimensionality reduction algo-
+rithms which use all of the available information have
+been unable to do so.
+The authors of this work at-
+tempted to tune preprocessing and hyperparameters in
+order to separate these bursts from the others, but were
+unable to do so from any single catalog.
+Of course, given the combination of the Swift and
+Fermi observations, it is easy to identify these objects
+as the only ones which are classified differently.
+Per-
+
+Swift
+5
+3
+s(50 -
+1
+0
+-1
+0
+1
+2
+3
+logTgo
+BATSE
+5
+4
+3
+2
+(50 -
+s
+0 
+-1
+0
+1
+2
+3
+logTgo11
+haps this is not so surprising. Multi-wavelength astron-
+omy has proven to be far more powerful than any single
+observatory, and multi-messenger astronomy is poised
+to be similarly powerful. Thus, a key conclusion here
+is that the next generation of gamma-ray observatories
+should be constructed with multi-wavelength observa-
+tions in mind, and that the combination of Swift and
+Fermi has already proven to be more powerful than an
+improved version of either observatory would be alone.
+It is important to note that “unsupervised” algorithms
+such as the ones used here are in practice strongly de-
+pendent on the choice of hyperparameters. The proper
+interpretation of the embeddings presented here should
+not be that the hyperparameters we have chosen are cor-
+rect and others incorrect. Rather, every choice of hyper-
+parameters leads to a valid embedding, which contains
+potentially useful information about the distribution but
+is always incomplete. In that sense, it would be more
+like considering various projections. Unlike projections,
+however, there is no rigorous formalism such as princi-
+pal component analysis for determining which will be
+most useful. Thus, an inability in this work to find hy-
+perparameters which separate these possible extended
+emission bursts from a single catalog does not guaran-
+tee that insufficient information exists to do so.
+There is also considerable literature on the possibil-
+ity of a third class consisting of intermediate-duration
+bursts found in the BATSE catalog (Mukherjee et al.
+1998; Hakkila et al. 2003; Horv´ath et al. 2004; Chat-
+topadhyay et al. 2007; ˇR´ıpa et al. 2009; Zhang et al.
+2022). Thus, a search for hyperparameters that sepa-
+rate such an intermediate group is well motivated. Using
+the same techniques presented here, it was indeed pos-
+sible to produce an embedding that separates a group
+of intermediate duration using the BATSE 4B catalog.
+However, this group does not appear in a catalog re-
+stricted to post-4B BATSE bursts, and no similar group
+was found when tuning hyperparameters to search for
+one in the Fermi and Swift catalogs.
+As a result, a
+reasonable conclusion is that this group is not of astro-
+nomical origin (cf. Hakkila et al. 2000), but rather is
+related to instrumentation or data reduction techniques
+applied only to earlier part of the BATSE dataset.
+5.2. Physical Interpretations
+Norris et al. (2010) suggest that approximately 1/4
+of a sample of Swift short GRBs have signatures of ex-
+tended emission, and estimate that the true fraction may
+be as high as 50%. This is not observed in our classifi-
+cations, which could be due to detector effects, but pos-
+sibly also due to the ”choking” mechanism suggested by
+Bucciantini et al. (2012), which would reduce the rate
+of observed extended emission bursts.
+All the bursts
+classified here as short but with T90 > 2s are within or
+close to the approximate theoretical boundary of ≈ 100s
+suggested by Metzger et al. (2008); Bucciantini et al.
+(2012).
+Although the classification presented here does not
+immediately allow for distinctions between different pro-
+genitor scenarios for the extended emission, there are
+several signatures that will be exciting to follow with
+the next generation of GRB observatories. Currently,
+the only viable way to distinguish between the two pri-
+mary proposed progenitor scenarios, neutron star - neu-
+tron star (NS-NS) mergers and accretion induced col-
+lapse (AIC), is by the signature of the elements produced
+during the event (Metzger et al. 2008). This was done
+for a NS-NS merger by Watson et al. (2019), who identi-
+fied strontium in the spectra of the afterglow. However,
+these spectra have low signal-to-noise ratios, and need
+to be taken within a few days, making them hard both
+to obtain and analyze.
+Another possible way of distinguishing between dif-
+ferent short GRB extended emission progenitor mech-
+anisms would be to rely on the extra polarization that
+would be produced during the prompt emission in the
+AIC scenario. The proposed Daksha mission will carry
+X-ray polarimeters which will be able to measure the po-
+larization of the prompt emission shortly after a trigger
+(Bhalerao et al. 2022). Including the polarization would
+then allow t-SNE/UMAP to further subdivide the short
+group, corresponding to either NS-NS mergers (without
+extended emission) or AICs (with extended emission), if
+both progenitors classes do exist.
+This approach would also lend itself to distinguish-
+ing between different progenitor mechanisms for LGRBS
+(Toma et al. 2009), but will require having a large statis-
+tical sample due to the large uncertainties in observed
+prompt emission polarizations (Kole et al. 2020). For
+the planned Daksha mission, the lowest energy that will
+be detectable is currently 1 keV, but based on the mod-
+els of Metzger et al. (2008); Bucciantini et al. (2012),
+this should ideally be even lower, preferably past the
+Swift 0.3 keV limit, in order to best distinguish between
+different progenitor classes.
+5.3. Robustness of Machine Learning for Astronomical
+Problems
+It is surprising that even though the embeddings pro-
+duce clear separations, the classifications are not en-
+tirely stable. Even though the results of Jespersen et al.
+(2020) appeared unambiguous, and have been repro-
+duced by independent groups from the same codebase,
+a few objects may still have been misclassified.
+Re-
+
+12
+sampling shows that in a study measuring 1000 similar
+bursts, a small fraction of bursts could end up being clas-
+sified as either short or long. Even for the Fermi catalog,
+with the most robust classification, 2.0% of the bursts
+change classification in at least 10% of trials. However,
+in any individual embedding, there is a clear separation
+providing an unambiguous assignment of each GRB as
+either short or long.
+This is one example of a more generic issue when us-
+ing machine learning methods in astronomy. Statistical
+methods typically are associated with theorems proven
+from a set of axioms. Thus, it is known with certainty
+that if those axioms apply to a dataset, a particular
+method will produce, e.g., the minimum variance unbi-
+ased estimator. Such a method needs a proven theorem
+in order to be considered valid.
+However, machine learning algorithms are often in-
+stead validated via benchmark problems and datasets.
+An algorithm which outperforms previous attempts at
+those problems is considered successful. However, there
+is rarely a rigorous formalism proving optimality. One
+of the advantages of UMAP over t-SNE is that there is
+a stronger mathematical argument for its embedding.
+Further, often these benchmark problems are on ide-
+alized, noiseless datasets. One of the standard dimen-
+sionality reduction problems is to classify handwritten
+digits in the MNIST dataset (LeCun et al. 1998). These
+digits are noiseless and have no missing pixels. Indeed,
+a change as simple as inverting the colors, which leaves
+the digits equally legible for humans, defeats many state-
+of-the-art machine learning-based classification schemes
+(Sun et al. 2021).
+However, in scientific applications, both the central
+value and uncertainty are essential. Without the latter
+one cannot determine whether data are consistent with a
+model. Dimensionality reduction algorithms do not pro-
+duce uncertainties in the embedded locations, and there
+is no widely-adopted standard for doing so. The resam-
+pling method used here was developed by the authors in
+an attempt to estimate robustness in a non-parametric
+way. 6
+Thus, a significant challenge in using machine learning
+for astronomical purposes will be developing methods
+which can properly account for uncertainties.
+Rather
+than turning perfect data into central values,
+as-
+tronomers must turn noisy data with known/estimated
+uncertainties into central values with known/estimated
+uncertainties. As shown in this study, the results can be
+surprising.
+The authors would like to thank Kasper Heintz and
+Darach Watson for helpful discussions.
+The Cosmic
+Dawn Center (DAWN) is funded by the Danish National
+Research Foundation under grant No. 140.
+REFERENCES
+Becht, E., McInnes, L., Healy, J., et al. 2019, Nature
+biotechnology, 37, 38
+Berger, E., Fong, W., & Chornock, R. 2013, ApJL, 774,
+L23, doi: 10.1088/2041-8205/774/2/L23
+Bhalerao, V., Vadawale, S., & Tendulkar, S. 2022, in
+American Astronomical Society Meeting Abstracts,
+Vol. 54, American Astronomical Society Meeting
+Abstracts, 348.13
+Bromberg, O., Nakar, E., & Piran, T. 2011, ApJL, 739,
+L55, doi: 10.1088/2041-8205/739/2/L55
+Bromberg, O., Nakar, E., Piran, T., & Sari, R. 2013, The
+Astrophysical Journal, 764, 179,
+doi: 10.1088/0004-637x/764/2/179
+Bucciantini, N., Metzger, B. D., Thompson, T. A., &
+Quataert, E. 2012, MNRAS, 419, 1537,
+doi: 10.1111/j.1365-2966.2011.19810.x
+6 A better approach might have been to construct a large number of
+simulated datasets based on estimated uncertainties and repeat
+the procedure for each. However, even for datasets of this size,
+the runtime of t-SNE and UMAP is far too long to allow such a
+Monte Carlo.
+Cano, Z., Wang, S.-Q., Dai, Z.-G., & Wu, X.-F. 2017,
+Advances in Astronomy, 2017, 8929054,
+doi: 10.1155/2017/8929054
+Chattopadhyay, T., Misra, R., Chattopadhyay, A. K., &
+Naskar, M. 2007, ApJ, 667, 1017, doi: 10.1086/520317
+Fiedler, M. 1973, Czechoslovak mathematical journal, 23,
+298
+Fruchter, A. S., Levan, A. J., Strolger, L., et al. 2006,
+Nature, 441, 463, doi: 10.1038/nature04787
+Ghirlanda, G., Nappo, F., Ghisellini, G., et al. 2018, A&A,
+609, A112, doi: 10.1051/0004-6361/201731598
+Hakkila, J., Giblin, T. W., Roiger, R. J., et al. 2003, ApJ,
+582, 320, doi: 10.1086/344568
+Hakkila, J., Haglin, D. J., Pendleton, G. N., et al. 2000,
+ApJ, 538, 165, doi: 10.1086/309107
+Hjorth, J., & Bloom, J. S. 2012, The Gamma-Ray Burst -
+Supernova Connection (Cambridge University Press),
+169–190
+Hjorth, J., Sollerman, J., Møller, P., et al. 2003, Nature,
+423, 847, doi: 10.1038/nature01750
+
+13
+Horv´ath, I., M´esz´aros, A., Bal´azs, L. G., & Bagoly, Z. 2004,
+in Astronomical Society of the Pacific Conference Series,
+Vol. 312, Gamma-Ray Bursts in the Afterglow Era, ed.
+M. Feroci, F. Frontera, N. Masetti, & L. Piro, 82
+Hu, Y., Wang, X., Hu, B., et al. 2019, PLoS biology, 17,
+e3000365
+Jespersen, C. K., Severin, J. B., Steinhardt, C. L., et al.
+2020, ApJL, 896, L20, doi: 10.3847/2041-8213/ab964d
+Kaneko, Y., Bostancı, Z. F., G¨o˘g¨u¸s, E., & Lin, L. 2015,
+MNRAS, 452, 824, doi: 10.1093/mnras/stv1286
+Kobak, D., & Berens, P. 2019, Nature Communications, 10,
+5416, doi: 10.1038/s41467-019-13056-x
+Kole, M., De Angelis, N., Berlato, F., et al. 2020, A&A,
+644, A124, doi: 10.1051/0004-6361/202037915
+Kouveliotou, C., Meegan, C. A., Fishman, G. J., et al.
+1993, ApJL, 413, L101, doi: 10.1086/186969
+LeCun, Y., Bottou, L., Bengio, Y., Haffner, P., et al. 1998,
+Proceedings of the IEEE, 86, 2278
+Le´sniewska, A., Micha�lowski, M. J., Kamphuis, P., et al.
+2022, ApJS, 259, 67, doi: 10.3847/1538-4365/ac5022
+Lien, A., Sakamoto, T., Barthelmy, S. D., et al. 2016, ApJ,
+829, 7, doi: 10.3847/0004-637X/829/1/7
+McInnes, L., Healy, J., & Melville, J. 2018, arXiv e-prints,
+arXiv:1802.03426. https://arxiv.org/abs/1802.03426
+Meegan, C. A. 1997, The BATSE Catalog of Gamma-Ray
+Bursts, Tech. rep., NASA STI
+Metzger, B. D., Quataert, E., & Thompson, T. A. 2008,
+MNRAS, 385, 1455,
+doi: 10.1111/j.1365-2966.2008.12923.x
+Mukherjee, S., Feigelson, E. D., Jogesh Babu, G., et al.
+1998, ApJ, 508, 314, doi: 10.1086/306386
+Nakar, E. 2007, PhR, 442, 166,
+doi: 10.1016/j.physrep.2007.02.005
+Ng, A., Jordan, M., & Weiss, Y. 2001, in Advances in
+Neural Information Processing Systems, ed.
+T. Dietterich, S. Becker, & Z. Ghahramani, Vol. 14 (MIT
+Press). https://proceedings.neurips.cc/paper/2001/file/
+801272ee79cfde7fa5960571fee36b9b-Paper.pdf
+Norris, J. P., & Bonnell, J. T. 2006, ApJ, 643, 266,
+doi: 10.1086/502796
+Norris, J. P., Gehrels, N., & Scargle, J. D. 2010, ApJ, 717,
+411, doi: 10.1088/0004-637X/717/1/411
+Norris, J. P., Share, G. H., Messina, D. C., et al. 1986, ApJ,
+301, 213, doi: 10.1086/163889
+Paciesas, W. S., Meegan, C. A., Pendleton, G. N., et al.
+1999, ApJS, 122, 465, doi: 10.1086/313224
+Stanek, K. Z., Matheson, T., Garnavich, P. M., et al. 2003,
+ApJL, 591, L17, doi: 10.1086/376976
+Sun, Y., Zeng, Y., & Zhang, T. 2021, iScience, 24, 102880,
+doi: 10.1016/j.isci.2021.102880
+Tanvir, N. R., Levan, A. J., Fruchter, A. S., et al. 2013,
+Nature, 500, 547, doi: 10.1038/nature12505
+Tavani, M., Kniffen, D., Mattox, J., Paredes, J., & Foster,
+R. 1998, The Astrophysical Journal Letters, 497, L89
+Toma, K., Sakamoto, T., Zhang, B., et al. 2009, ApJ, 698,
+1042, doi: 10.1088/0004-637X/698/2/1042
+von Kienlin, A., Meegan, C. A., Paciesas, W. S., et al. 2020,
+ApJ, 893, 46, doi: 10.3847/1538-4357/ab7a18
+ˇR´ıpa, J., M´esz´aros, A., Wigger, C., et al. 2009, A&A, 498,
+399, doi: 10.1051/0004-6361/200810913
+Watson, D., Hansen, C. J., Selsing, J., et al. 2019, Nature,
+574, 497, doi: 10.1038/s41586-019-1676-3
+Xiang, R., Wang, W., Yang, L., et al. 2021, Frontiers in
+genetics, 12, 646936
+Zhang, F.-W., Shao, L., Yan, J.-Z., & Wei, D.-M. 2012, The
+Astrophysical Journal, 750, 88
+Zhang, S., Shao, L., Zhang, B.-B., et al. 2022, ApJ, 926,
+170, doi: 10.3847/1538-4357/ac4753
+
+14
+APPENDIX
+A data table containing the classification of each burst is given here and available online in a machine-readable
+format. For each burst, a classification is given for each telescope which observed. The possible classifications are the
+following types:
+• S (short): a burst identified as short by both t-SNE and UMAP and in at least 90% of resampled t-SNE maps.
+• L (long): a burst identified as long by both t-SNE and UMAP in at least 90% of resampled t-SNE maps.
+• A (ambiguous): a burst which was classified as both short and long in at least 10% of resampled t-SNE maps.
+• D (disagreement): a burst which is classified differently by t-SNE and UMAP, but which is classified consistently
+in at least 90% of resampled t-SNE maps.
+In addition, there is an overall classification given for each burst. If the burst is observed by only one telescope,
+the overall classification is same as for that telescope. For bursts observed by both Swift and Fermi, if both give
+the same classification, this also the overall classification. A burst which is S or L in one catalog and A or D in the
+other is classified as S or L based on the unambiguous classification. The three bursts which are classified as L by
+Swift but S by Fermi, GRB090531B, GRB090927, and GRB130716A, are classified as D (disagreement) in the overall
+classification. These may be extended emission bursts, as discussed in the main text.
+The primary name used for each burst the name given each individual catalog, with the Fermi name used for objects
+observed by both Fermi and Swift. Probabilities given are the probability that any given burst is short in resampled
+data.
+Name
+Fermi Name
+Type
+BATSE Type
+BATSE Prob.
+Swift Type
+Swift Prob.
+Fermi Type
+Fermi Prob.
+GRB910421
+L
+L
+0.0
+GRB910423
+L
+L
+0.0
+GRB910424
+L
+L
+0.01
+GRB910425A
+L
+L
+0.0
+GRB910425B
+L
+L
+0.0
+GRB910426
+L
+L
+0.0
+GRB910427
+L
+L
+0.0
+GRB910429
+L
+L
+0.0
+GRB910430
+L
+L
+0.0
+GRB910501
+L
+L
+0.0
+· · ·
+GRB090927
+GRB090927422
+D
+L
+0.07
+S
+1.0
+GRB090928
+GRB090928646
+L
+L
+0.01
+GRB090929
+GRB090929190
+L
+L
+0.0
+GRB090929A
+L
+L
+0.01
+GRB090929B
+L
+L
+0.0
+GRB091002
+GRB091002685
+L
+L
+0.02
+GRB091003
+GRB091003191
+L
+L
+0.0
+GRB091005
+GRB091005679
+L
+L
+0.0
+GRB091006
+GRB091006360
+S
+S
+0.99
+· · ·
+Table 3. Classifications based on this work for GRBs in the BATSE, Swift, and Fermi catalogs. Bursts with missing data or
+metadata which were excluded from the analysis are not included in the table. A full, machine-readable version is available
+online.
+

LtE2T4oBgHgl3EQfBAZB/content/tmp_files/2301.03597v1.pdf.txt

@@ -0,0 +1,464 @@
+arXiv:2301.03597v1  [cs.LG]  9 Jan 2023
+On the Minimax Regret for Linear Bandits in a
+wide variety of Action Spaces
+Debangshu Banerjee1 and Aditya Gopalan2
+1,2 Department of Electrical and Communication Engineering,
+Indian Institute of Science, India
+November 2022
+Abstract
+As noted in the works of Lattimore and Szepesv´ari [2020], it has been
+mentioned that it is an open problem to characterize the minimax regret
+of linear bandits in a wide variety of action spaces.
+In this article we
+present an optimal regret lower bound for a wide class of convex action
+spaces.
+1
+Introduction
+Minimax regret bounds in bandit environments are a well studied problem and
+results typically are limited to a particular action set, namely the l1 and l∞
+balls in Rd. We include in this article that display that the methods introduced
+by Lattimore and Szepesv´ari [2020] in Chapter 24 can indeed be generalized to
+a wide variety of action spaces, namely to any lp ball where p is in the range
+(1, ∞).
+2
+Key Result
+Note that the result we include in 2.1, is optimal in the bandit setting, in sense
+that algorithms like LinUCB achieve this.
+Theorem 2.1. Let X be the Lp ball defined as X = {x ∈ Rd
+: ∥x∥p ⩽ c},
+where 1 < p < ∞. Assume d ⩽ (2cn2)
+p
+2 . Then there exists a parameter θ ∈ Rd
+with ∥θ∥p
+p =
+1
+(c4
+√
+3)p
+d2
+n
+p
+2 such that Rn(θ) ⩾ d√n
+16
+√
+3.
+1
+
+Proof. We chose θ ∈ {+∆, −∆}d and note that the regret, defined as, Rn(θ), is
+Rn(θ) =
+n
+�
+t=1
+x∗⊤θ − x⊤
+t θ
+=
+n
+�
+t=1
+d
+�
+i=1
+x∗
+i θi − xtiθi
+= ∆
+n
+�
+t=1
+d
+�
+i=1
+c
+d
+1
+p − xtisign(θi)
+⩾ ∆d
+1
+p
+2c
+n
+�
+t=1
+d
+�
+i=1
+� c
+d
+1
+p − xtisign(θi)
+�2
+,
+(1)
+where the third equality follows from Lemma A.1 and the last inequality follows
+from Lemma A.2. The remainder of the proof follows the same idea as that pre-
+sented in the proof of the Unit Ball in Section 24.2 of Lattimore and Szepesv´ari
+[2020]. We present it here for the sake of completeness. We define a stopping
+time τi = n ∧ min {t : �t
+s=1 x2
+si ⩾ nc2
+d
+2
+p }. Thus
+Rn(θ) ⩾ ∆d
+1
+p
+2c
+d
+�
+i=1
+τi
+�
+t=1
+� c
+d
+1
+p − xtisign(θi)
+�2
+.
+Define a Random Variable Ui(σ) = �τi
+t=1
+�
+c
+d
+1
+p − xtiσ
+�2
+where σ ∈ {+1, −1}
+and note that
+Ui(1) =
+τi
+�
+t=1
+� c
+d
+1
+p − xti
+�2
+⩽ 2
+τi
+�
+t=1
+c2
+d
+2
+p + 2
+τi
+�
+t=1
+x2
+ti ⩽ 4nc2
+d
+2
+p
++ 2 ,
+(2)
+where the last inequality follows from the definition of τi.
+Now we fix an i, and make a perturbed version of θ′, which is the same as θ
+except in the ith position where θ′
+i = −θi. Thus, applying Pinsker’s inequality,
+Eθ[Ui(1)] ⩾ Eθ′[Ui(1)] −
+�4nc2
+d
+2
+p
++ 2
+��
+1
+2KL(Pθ||Pθ′)
+(3)
+⩾ Eθ′[Ui(1)] − ∆
+2
+�4nc2
+d
+2
+p
++ 2
+��
+�
+�
+�
+τi
+�
+t=1
+x2
+ti
+⩾ Eθ′[Ui(1)] − ∆
+2
+�4nc2
+d
+2
+p
++ 2
+��
+nc2
+d
+2
+p + 1
+2
+
+⩾ Eθ′[Ui(1)] − 4
+√
+3n∆c2
+d
+2
+p
+�
+nc2
+d
+2
+p ,
+where the last inequality follows under the assumption d ⩽ (2nc2)
+p
+2 . Thus
+Eθ[Ui(1)] + Eθ′[Ui(−1)] ⩾ Eθ′[Ui(1)) + Ui(−1)] − 4
+√
+3n∆c2
+d
+2
+p
+�
+nc2
+d
+2
+p
+= 2Eθ′
+�τic2
+d
+2
+p +
+τi
+�
+t=1
+x2
+ti
+�
+− 4
+√
+3n∆c2
+d
+2
+p
+�
+nc2
+d
+2
+p ⩾ nc2
+d
+2
+p ,
+where the last inequality follows from the definition of τi and setting the value
+of ∆ as
+1
+4
+√
+3
+�
+d
+2
+p
+nc2 . Using an average hammering trick
+�
+θ∈{±∆}d
+Rn(θ) ⩾ ∆d
+1
+p
+2c
+d
+�
+i=1
+�
+θ∈{±∆}d
+Eθ[Ui(sign(θi)]
+= ∆d
+1
+p
+2c
+d
+�
+i=1
+�
+θ−i∈{±∆}d−1
+�
+θi∈{±∆}
+Eθ[Ui(sign(θi)]
+⩾ ∆d
+1
+p
+2c
+d
+�
+i=1
+�
+θ−i∈{±∆}d−1
+nc2
+d
+2
+p = 2d−2∆ncd1− 1
+p .
+Hence there exists a θ in {±∆}d, such that
+Rn(θ) ⩾ nc∆d1− 1
+p
+4
+= d√n
+16
+√
+3.
+Remark 2.2. The results in 2.1 are interesting because with regards to the
+dimensionality d and time horizon n dependence it is exact.
+Remark 2.3. This result also shows that the minimum eigen value of the design
+matrix and regret are fundamentally different quantities Banerjee et al. [2022].
+For example note that for lp balls for p > 2, the minimum eigen value can grow
+at a rate lower than Ω(√n, whereas, the minimax regert remains bounded as
+Ω(√n.
+3
+Conclusion
+We expect that similar results can hold for general convex bodies and not just
+for lp balls.
+3
+
+References
+D. Banerjee, A. Ghosh, S. R. Chowdhury, and A. Gopalan. Exploration in linear
+bandits with rich action sets and its implications for inference. arXiv e-prints,
+pages arXiv–2207, 2022.
+T. Lattimore and C. Szepesv´ari.
+Bandit algorithms.
+Cambridge University
+Press, 2020.
+A
+Appendix
+A.1
+Technical Lemmas
+Lemma A.1 (Optimal Reward in Lp Ball). Let X be the Lp ball defined as
+X = {x ∈ Rd : ∥x∥p ⩽ c}. We compute the optimal reward for the linear bandit
+model
+max x⊤θ
+s.t.
+x ∈ X
+(4)
+The solution to the optimization problem 4 is
+1
+�
+�d
+i=1 |θi|
+p
+p−1
+� 1
+p
+�d
+i=1 c|θi|
+p
+p−1
+Proof. Note that the solution x∗ satisfies the following relation for any i ∈ [d]
+x∗
+i =
+�|θi|
+λ
+�
+1
+p−1 sign(θi) ,
+(5)
+where λ ⩾ 0 is the Lagrangian variable.
+Solving for λ using the constraint
+equation of the problem with now equality instead of inequality. (Because the
+optimal solution lies at the boundary)
+λ =
+��d
+i=1 |θi|
+p
+p−1
+cp
+� p−1
+p
+.
+(6)
+Now solving for x∗⊤θ = �d
+i=1 x∗
+i θi gives the result.
+Lemma A.2.
+d
+�
+i=1
+c
+d
+1
+p − xtisign(θi) ⩾ d
+1
+p
+2c
+d
+�
+i=1
+� c
+d
+1
+p − xtisign(θi)
+�2
+(7)
+4
+
+Proof.
+d
+�
+i=1
+� c
+d
+1
+p − xtisign(θi)
+�2
+= c2d1− 2
+p − 2
+d
+�
+i=1
+c
+d
+1
+p xtisign(θi) +
+d
+�
+i=1
+x2
+ti
+⩽ 2c2d1− 2
+p − 2
+d
+�
+i=1
+c
+d
+1
+p xtisign(θi)
+= 2c
+d
+1
+p
+d
+�
+i=1
+c
+d
+1
+p − xtisign(θi) ,
+(8)
+where the inequality follows from Lemma A.3. Rearranging gives the lemma.
+Lemma A.3. If ∥x∥p ⩽ c, then ∥x∥2
+2 ⩽ c2d1− 2
+p
+Proof.
+d
+�
+i=1
+x2
+i ⩽ (
+d
+�
+i=1
+|xi|p)
+2
+p d1− 2
+p ⩽ c2d1− 2
+p ,
+(9)
+where the first inequality follows from Holder’s inequality and the second in-
+equality follows from the hypothesis.
+5
+

LtE2T4oBgHgl3EQfBAZB/content/tmp_files/load_file.txt

@@ -0,0 +1,74 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf,len=73
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='03597v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='LG]  9 Jan 2023  On the Minimax Regret for Linear Bandits in a  wide variety of Action Spaces  Debangshu Banerjee1 and Aditya Gopalan2  1,2 Department of Electrical and Communication Engineering,  Indian Institute of Science, India  November 2022  Abstract  As noted in the works of Lattimore and Szepesv´ari [2020], it has been  mentioned that it is an open problem to characterize the minimax regret  of linear bandits in a wide variety of action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  In this article we  present an optimal regret lower bound for a wide class of convex action  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  1  Introduction  Minimax regret bounds in bandit environments are a well studied problem and  results typically are limited to a particular action set, namely the l1 and l∞  balls in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' We include in this article that display that the methods introduced  by Lattimore and Szepesv´ari [2020] in Chapter 24 can indeed be generalized to  a wide variety of action spaces, namely to any lp ball where p is in the range  (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  2  Key Result  Note that the result we include in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='1, is optimal in the bandit setting, in sense  that algorithms like LinUCB achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Let X be the Lp ball defined as X = {x ∈ Rd  : ∥x∥p ⩽ c},  where 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Assume d ⩽ (2cn2)  p  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Then there exists a parameter θ ∈ Rd  with ∥θ∥p  p =  1  (c4  √  3)p  d2  n  p  2 such that Rn(θ) ⩾ d√n  16  √  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' We chose θ ∈ {+∆, −∆}d and note that the regret, defined as, Rn(θ), is  Rn(θ) =  n  �  t=1  x∗⊤θ − x⊤  t θ  =  n  �  t=1  d  �  i=1  x∗  i θi − xtiθi  = ∆  n  �  t=1  d  �  i=1  c  d  1  p − xtisign(θi)  ⩾ ∆d  1  p  2c  n  �  t=1  d  �  i=1  � c  d  1  p − xtisign(θi)  �2  ,  (1)  where the third equality follows from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='1 and the last inequality follows  from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' The remainder of the proof follows the same idea as that pre-  sented in the proof of the Unit Ball in Section 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='2 of Lattimore and Szepesv´ari  [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' We present it here for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' We define a stopping  time τi = n ∧ min {t : �t  s=1 x2  si ⩾ nc2  d  2  p }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Thus  Rn(θ) ⩾ ∆d  1  p  2c  d  �  i=1  τi  �  t=1  � c  d  1  p − xtisign(θi)  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  Define a Random Variable Ui(σ) = �τi  t=1  �  c  d  1  p − xtiσ  �2  where σ ∈ {+1, −1}  and note that  Ui(1) =  τi  �  t=1  � c  d  1  p − xti  �2  ⩽ 2  τi  �  t=1  c2  d  2  p + 2  τi  �  t=1  x2  ti ⩽ 4nc2  d  2  p  + 2 ,  (2)  where the last inequality follows from the definition of τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  Now we fix an i, and make a perturbed version of θ′, which is the same as θ  except in the ith position where θ′  i = −θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Thus, applying Pinsker’s inequality,  Eθ[Ui(1)] ⩾ Eθ′[Ui(1)] −  �4nc2  d  2  p  + 2  ��  1  2KL(Pθ||Pθ′)  (3)  ⩾ Eθ′[Ui(1)] − ∆  2  �4nc2  d  2  p  + 2  ��  �  �  �  τi  �  t=1  x2  ti  ⩾ Eθ′[Ui(1)] − ∆  2  �4nc2  d  2  p  + 2  ��  nc2  d  2  p + 1  2  ⩾ Eθ′[Ui(1)] − 4  √  3n∆c2  d  2  p  �  nc2  d  2  p ,  where the last inequality follows under the assumption d ⩽ (2nc2)  p  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Thus  Eθ[Ui(1)] + Eθ′[Ui(−1)] ⩾ Eθ′[Ui(1)) + Ui(−1)] − 4  √  3n∆c2  d  2  p  �  nc2  d  2  p  = 2Eθ′  �τic2  d  2  p +  τi  �  t=1  x2  ti  �  − 4  √  3n∆c2  d  2  p  �  nc2  d  2  p ⩾ nc2  d  2  p ,  where the last inequality follows from the definition of τi and setting the value  of ∆ as  1  4  √  3  �  d  2  p  nc2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Using an average hammering trick  �  θ∈{±∆}d  Rn(θ) ⩾ ∆d  1  p  2c  d  �  i=1  �  θ∈{±∆}d  Eθ[Ui(sign(θi)]  = ∆d  1  p  2c  d  �  i=1  �  θ−i∈{±∆}d−1  �  θi∈{±∆}  Eθ[Ui(sign(θi)]  ⩾ ∆d  1  p  2c  d  �  i=1  �  θ−i∈{±∆}d−1  nc2  d  2  p = 2d−2∆ncd1− 1  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  Hence there exists a θ in {±∆}d, such that  Rn(θ) ⩾ nc∆d1− 1  p  4  = d√n  16  √  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' The results in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='1 are interesting because with regards to the  dimensionality d and time horizon n dependence it is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' This result also shows that the minimum eigen value of the design  matrix and regret are fundamentally different quantities Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  For example note that for lp balls for p > 2, the minimum eigen value can grow  at a rate lower than Ω(√n, whereas, the minimax regert remains bounded as  Ω(√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  3  Conclusion  We expect that similar results can hold for general convex bodies and not just  for lp balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  3  References  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Banerjee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Ghosh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Chowdhury, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Gopalan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Exploration in linear  bandits with rich action sets and its implications for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' arXiv e-prints,  pages arXiv–2207, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Lattimore and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Szepesv´ari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  Bandit algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  Cambridge University  Press, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  A  Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='1  Technical Lemmas  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='1 (Optimal Reward in Lp Ball).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Let X be the Lp ball defined as  X = {x ∈ Rd : ∥x∥p ⩽ c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' We compute the optimal reward for the linear bandit  model  max x⊤θ  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  x ∈ X  (4)  The solution to the optimization problem 4 is  1  �  �d  i=1 |θi|  p  p−1  � 1  p  �d  i=1 c|θi|  p  p−1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Note that the solution x∗ satisfies the following relation for any i ∈ [d]  x∗  i =  �|θi|  λ  �  1  p−1 sign(θi) ,  (5)  where λ ⩾ 0 is the Lagrangian variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  Solving for λ using the constraint  equation of the problem with now equality instead of inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' (Because the  optimal solution lies at the boundary)  λ =  ��d  i=1 |θi|  p  p−1  cp  � p−1  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  (6)  Now solving for x∗⊤θ = �d  i=1 x∗  i θi gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  d  �  i=1  c  d  1  p − xtisign(θi) ⩾ d  1  p  2c  d  �  i=1  � c  d  1  p − xtisign(θi)  �2  (7)  4  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  d  �  i=1  � c  d  1  p − xtisign(θi)  �2  = c2d1− 2  p − 2  d  �  i=1  c  d  1  p xtisign(θi) +  d  �  i=1  x2  ti  ⩽ 2c2d1− 2  p − 2  d  �  i=1  c  d  1  p xtisign(θi)  = 2c  d  1  p  d  �  i=1  c  d  1  p − xtisign(θi) ,  (8)  where the inequality follows from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' Rearranging gives the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content=' If ∥x∥p ⩽ c, then ∥x∥2  2 ⩽ c2d1− 2  p  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  d  �  i=1  x2  i ⩽ (  d  �  i=1  |xi|p)  2  p d1− 2  p ⩽ c2d1− 2  p ,  (9)  where the first inequality follows from Holder’s inequality and the second in-  equality follows from the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}
+page_content='  5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE2T4oBgHgl3EQfBAZB/content/2301.03597v1.pdf'}

M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:543c678224c720f77ef96f9d565c7dfd0a667946b56fac41f5e008eb3b340c6d
+size 388242

MdE4T4oBgHgl3EQfjA06/content/tmp_files/2301.05138v1.pdf.txt

@@ -0,0 +1,702 @@
+arXiv:2301.05138v1  [quant-ph]  12 Jan 2023
+Canonical description of quantum dynamics
+Martin Bojowald∗
+Institute for Gravitation and the Cosmos,
+The Pennsylvania State University,
+104 Davey Lab, University Park, PA 16802, USA
+Abstract
+Some of the important non-classical aspects of quantum mechanics can be de-
+scribed in more intuitive terms if they are reformulated in a geometrical picture
+based on an extension of the classical phase space. This contribution presents vari-
+ous phase-space properties of moments describing a quantum state and its dynamics.
+An example of a geometrical reformulation of a non-classical quantum effect is given
+by an equivalence between conditions imposed by uncertainty relations and centrifu-
+gal barriers, respectively.
+1
+Introduction
+Although classical and quantum mechanics are usually presented based on different mathe-
+matical methods and with a marked contrast in the underlying physical concepts, there are
+formulations that bring the two theories closer to each other. For instance, the Koopman
+wave function [1] can be used to make classical mechanics look quantum. Effective theories
+reverse the direction and make quantum mechanics look classical.
+Given the richer nature of quantum mechanics with more degrees of freedom of a state
+than in classical mechanics, there are different ways in which this “classicalization” can be
+achieved. A method of importance in elementary particle physics is the low-energy effective
+action [2] which determines the dynamics of a quantum system around the ground state,
+and the related Coleman–Weinberg potential [3]. Such low-energy methods are useful in
+situations in which only a few particle degrees of freedom are excited, as in basic scattering
+processes of interest in particle physics. Methods that do not rely on low-energy regimes are
+geometric ones or phase-space descriptions of quantum mechanics, going back for instance
+to the symplectic formulations of [4, 5, 6]. These approaches make use of the fact that the
+imaginary part of an inner product is antisymmetric and non-degenerate, and therefore
+can be interpreted as a symplectic form on the space of states.
+Here, we use a geometrical picture of quantum dynamics derived from a suitable param-
+eterization of states. In some ways, it is related to symplectic phase-space descriptions, but
+it includes a suitable generalization that makes it possible to formulate effective theories in
+which certain degrees of freedom contained in a full quantum state are suppressed. (The
+parameterization of states to be introduced determines which degrees of freedom may be
+∗e-mail address: [email protected]
+1
+
+suppressed.) Most of these truncations cannot be described by symplectic methods because
+removing some degrees of freedom, in general, introduces degenerate directions in phase
+space from the point of view of symplectic geometry. Accordingly, the truncated phase-
+space geometry is Poisson, making use of a Poisson bracket with a non-invertible Poisson
+tensor, as opposed to an invertible symplectic form. This method was first described in
+[7] within the setting of Poisson geometry, but with the benefit of hindsight it had several
+independent precursors [8, 9, 10, 11] that can now be recognized as low-order versions of
+Poisson spaces.
+As an instructive example, consider the uncertainty relation
+(∆x)2(∆p)2 − C2
+xp ≥ U
+(1)
+where U = ℏ2/4 in quantum mechanics. The fluctuations and covariance on the left-hand
+side are part of a parameterization of states by moments. The constant U on the right-hand
+side can be seen as a parameter that controls quantum features: It is positive for quantum
+states but may be smaller if the moments belong to a classical state, as encountered for
+instance in classical dynamics if it is expressed in term of the Koopman wave function.
+There are three moments of second order for a single classical degree of freedom, all seen
+on the left-hand side of the uncertainty relation (1). If one aims to find a quasiclassical or
+phase-space description of these and only these degrees of freedom, one has to work with
+an odd-dimensional phase space that cannot be symplectic but may well (and indeed does)
+have a Poisson bracket.
+Since moments are defined as polynomial expressions of basic expectation values, we
+may introduce a Poisson bracket by first defining
+{⟨ ˆO1⟩ρ, ⟨ ˆO2⟩ρ} = ⟨[ ˆO1, ˆO2]⟩ρ
+iℏ
+(2)
+for expectation values of operators in some state pure or mixed ρ. This operation defines an
+antisymmetric bracket on expectation values ⟨ ˆO⟩ρ, interpreted as a function on state space
+for any fixed ˆO. The bracket also satisfies the Jacobi identity because the commutator on
+the right-hand side does so. We can extend it to products of expectation values, as needed
+for moments, by imposing the Leibniz rule.
+If we include only moments up to some finite order in our quasiclassical description,
+(2) defines a Poisson bracket that, in general, is not symplectic. Nevertheless, dynamics
+on this phase space is well defined if the system is characterized by a Hamilton operator
+ˆH. The standard equation
+d⟨ ˆO⟩ρ
+dt
+= ⟨[ ˆO, ˆH]⟩ρ
+iℏ
+(3)
+is then equivalent to Hamilton’s equation
+d⟨ ˆO⟩ρ
+dt
+= {⟨ ˆO⟩ρ, Heff}
+(4)
+2
+
+on phase space with the effective Hamiltonian
+Heff(ρ) = ⟨ ˆH⟩ρ ,
+(5)
+again interpreted as a function on the space of states. In general, this equation couples
+infinitely many independent expectation values to one another.
+A truncation is therefore required for manageable approximations which, depending
+on the truncation order of moments, defines the corresponding effective description. For
+instance, we may use a semiclassical approximation of order N in ℏ if we include only
+expectation values ⟨ˆqaˆpb⟩ for a + b ≤ 2N, a ≥ 0 and b ≥ 0. It is more convenient to
+reformulate this truncation as using basic expectation values q = ⟨ˆq⟩ and p = ⟨ˆp⟩ together
+with central moments in a completely symmetric (or Weyl) ordering,
+∆(qapb) = ⟨(ˆq − q)a(ˆp − p)b⟩symm .
+(6)
+This ordering is defined by summing all (a+b)! permutations of (ˆq−q)a(ˆp−p)b and dividing
+by (a + b)!. For a = b = 1, this implies the standard covariance
+Cqp = ∆(qp) = 1
+2⟨ˆqˆp + ˆpˆq⟩ − qp
+(7)
+while, for instance,
+∆(q2p)
+=
+1
+3⟨(ˆq − q)2(ˆp − p) + (ˆq − q)(ˆp − p)(ˆq − q) + (ˆp − p)(ˆq − q)2⟩
+=
+1
+3⟨ˆq2ˆp + ˆqˆpˆq + ˆpˆq2⟩ − ⟨ˆq2⟩p − ⟨ˆqˆp + ˆpˆq⟩q + 2q2p
+(8)
+In this paper, we further analyze the underlying Poisson geometry and present new re-
+sults that demonstrate how well-known quantum parameters, such as U in (1), are equipped
+with an elegant geometrical interpretation as centrifugal barriers for quantum degrees of
+freedom. Physical applications related to tunneling will also be described, and we will
+compare these methods with adiabatic or low-energy ones.
+2
+Geometry of the free particle
+The quantum dynamics of the free particle can be described (in this case, exactly) by the
+following geometrical picture. We have the effective Hamiltonian
+Heff(p, ∆(p2)) =
+� ˆp2
+2m
+�
+= p2
+2m + ∆(p2)
+2m
+.
+(9)
+An immediate question in a quasiclassical picture is whether the last term, ∆(p2)/2m in
+which ∆(p2) is often suggestively written as the square (∆p)2 of a momentum fluctuation
+∆p, is another kinetic energy in addition to p2/(2m), or maybe contributes to the potential.
+3
+
+2.1
+Moment systems
+To answer this question, we consider the relevant Poisson brackets
+{∆(q2), ∆(p2)}
+=
+4∆(qp)
+(10)
+{∆(q2), ∆(qp)}
+=
+2∆(q2)
+(11)
+{∆(qp), ∆(p2)}
+=
+2∆(p2)
+(12)
+of moments of the same order as ∆(p2) in the new term of the effective Hamiltonian. The
+Poisson bracket is defined in (2), also making use of the Leibniz rule. For instance, to
+derive the first bracket, (10), we use
+{⟨ˆq2⟩, ⟨ˆp2⟩} = ⟨[ˆq2, ˆp2]⟩
+iℏ
+= 2⟨ˆqˆp + ˆpˆq⟩
+(13)
+directly from (2). The remaining terms obtained from expanding
+{∆(q2), ∆(p2)}
+=
+{⟨ˆq2⟩ − q2, ⟨ˆp2⟩ − p2}
+=
+{⟨ˆq2⟩, ⟨ˆp2⟩} − {⟨ˆq2⟩, p2} − {q2, ⟨ˆp2⟩} + {q2, p2}
+(14)
+require the Leibniz rule:
+{⟨ˆq2⟩, p2}
+=
+2p{⟨ˆq2⟩, ⟨ˆp⟩} = 4qp
+(15)
+{q2, ⟨ˆp2⟩}
+=
+2q{⟨ˆq⟩, ⟨ˆp2⟩} = 4qp
+(16)
+{q2, p2}
+=
+4qp{⟨ˆq⟩, ⟨ˆp⟩} = 4qp .
+(17)
+Combining all terms, we obtain (10).
+At generic orders, central moments have several welcome but also some unwelcome prop-
+erties. They are always real, by definition, facilitate the definition of a general semiclassical
+hierarchy ∆(qapb) ∝ ℏ(a+b)/2, and are always Poisson orthogonal to the basic expectation
+values: {q, ∆(qapb)} = 0 = {p, ∆(qapb)}.
+However, complications can sometimes arise
+because they are coordinates on a phase space with boundaries, given by uncertainty rela-
+tions including those of higher orders, and because their Poisson brackets are not canonical.
+While there is a closed-form expression for the Poisson bracket of two moments of canonical
+operators ˆq and ˆp, given by [7, 12]
+{∆(qapb), ∆(qcpd)} = ad∆(qa−1pb)∆(qcpd−1) − bc∆(qapb−1)∆(qc−1pd)
++
+�
+n
+�iℏ
+2
+�n−1
+Kn
+abcd ∆(qa+c−npb+d−n)
+(18)
+where 1 ≤ n ≤ Min(a + c, b + d, a + b, c + d) and
+Kn
+abcd :=
+n
+�
+m=0
+(−1)mm!(n − m)!
+� a
+m
+� �
+b
+n − m
+� �
+c
+n − m
+� � d
+m
+�
+,
+(19)
+4
+
+it is rather tedious to evaluate. Moreover, the non-canonical nature can make it difficult
+to determine algebraic or physical properties, as already seen in the example of the free
+particle.
+The non-canonical nature of these brackets implies that ∆p is not an obvious canon-
+ical momentum. However, the Casimir–Darboux theorem [13] states that every Poisson
+manifold (that is, any space equipped with a Poisson bracket) has local coordinates qi,
+pi, CI such that {qi, pj} = δij and {qi, CI} = 0 = {pi, CI}. The qi and pj are then pairs
+of canonical configuration coordinates and momenta, while the Casimir variables CI are
+conserved by any Hamiltonian. (If the manifold is symplectic, all CI are zero.)
+It turns out that the brackets of second-order moments have Casimir–Darboux variables
+(s, ps, C) where
+∆(q2) = s2
+,
+∆(qp) = sps
+,
+∆(p2) = p2
+s + C
+s2 .
+(20)
+The canonical bracket {s, ps} = 1 and {s, C} = 0 = {ps, C}, required for Casimir–Darboux
+variables, can directly be confirmed after inverting the mapping to obtain
+s =
+�
+∆(q2)
+,
+ps =
+∆(qp)
+�
+∆(q2)
+,
+C = ∆(q2)∆(p2) − ∆(qp)2 .
+(21)
+The last equation shows that
+C = ∆(q2)∆(p2) − ∆(qp)2 ≥ ℏ2/4
+(22)
+is bounded from below by Heisenberg’s uncertainty relation (or the Schr¨odinger–Robertson
+version). These properties have been found independently in a variety of contexts, includ-
+ing quantum field theory [8], quantum chaos [10], quantum chemistry [11] and models of
+quantum gravity [14].
+The canonical formulation (20) or (21) can be found if one combines the dynamics
+implied by the Hamiltonian (9) with the fact that C as defined in terms of moments by
+(22) is conserved by any Hamiltonian that depends only on basic expectation values and
+second-order moments; see for instance [11]. Hamiltonian dynamics gives
+d∆(q2)
+dt
+= {∆(q2), Heff(p, ∆(p2))} = 2
+m∆(qp)
+(23)
+using the Poisson bracket (10). If we define s =
+�
+∆(q2), we obtain
+˙s = 1
+2s
+d∆(q2)
+dt
+= ∆(qp)
+ms
+(24)
+and therefore ∆(qp)/s appears as a momentum ps of s, as in (20). The equation for ∆(p2)
+can then be obtained by solving C = ∆(q2)∆(p2) − ∆(qp)2 for this moment.
+In general, however, the canonical structure is more difficult to discern and requires
+more systematic derivations, for instance when one considers higher moments or more than
+5
+
+one classical pair of degrees of freedom. It may still be possible to derive the momenta
+of some moments using Hamiltonian dynamics, but deriving a complete mapping of all
+independent degrees of freedom to canonical form requires more care.
+While there is
+no completely systematic procedure, Poisson geometry helps to organize the derivation
+[15, 16]. Again in the example of second-order moments (20), we are still free to choose
+one degree of freedom as the first canonical variable, s =
+�
+∆(q2). Any Poisson bracket
+with s then takes the form
+{s, f} = ∂f
+∂ps
+(25)
+with the momentum ps to be determined, where f is any function of the moments. By
+evaluating this equation for sufficiently many functions f, or just for the basic moments, it
+provides differential equations in the independent variable ps, with s as a parameter, that
+can be solved to reveal the ps-dependence of moments.
+For instance, choosing f = ∆(qp), we obtain
+∂∆(qp)
+∂ps
+= {s, ∆(qp)} = 1
+2s{∆(q2), ∆(qp)} = ∆(q2)
+s
+= s .
+(26)
+The solution, ∆(qp) = sps + c(s) with a free s-dependent function c(s) is consistent with
+(20). The free function can be removed by a canonical transformation, replacing ps with
+ps + c(s)/s. For ∆(p2), we obtain the differential equation ∂∆(p2)/∂ps = 2∆(qp)/s = 2ps.
+The solution is again consistent with (20), where the free s-dependent function is here fixed
+by using C as a conserved quantity.
+If there are more than three moments, the procedure has to be iterated. After finding
+the first canonical pair analogous to (s, ps), the next step consists in rewriting the remaining
+moments in terms of quantities that have vanishing Poisson brackets with both s and ps,
+and are therefore canonically independent. For second-order moments of a single classical
+degree of freedom, this step merely consists in identifying C as a conserved quantity, but it
+can be more challenging in higher-dimensional phase spaces. At this point, the procedure
+is no longer fully systematic and usually requires special considerations of the moments
+system and its Poisson brackets in order to proceed in a tractable manner.
+2.2
+Free dynamics
+In Casimir–Darboux variables, the effective Hamiltonian (9) of the free particle takes the
+form
+Heff(p, s, ps; C) = p2
+2m + p2
+s
+2m +
+C
+2ms2
+(27)
+with, as we see now, contributions to both the kinetic and potential energies.
+It may
+seem counter-intuitive that the free particle is subject to a potential, but if the fluctuation
+direction with coordinate s is included in the configuration space, uncertainty relations
+must imply some kind of repulsive potential that prevents s from reaching zero. The new
+term C/(2ms2) where C ≥ ℏ2/4 ̸= 0 is precisely of this form.
+6
+
+Nevertheless, we can rephrase the dynamics as manifestly free if we further extend our
+configuration space. We can interpret the 1/s2 potential as a centrifugal one in an auxiliary
+plane with coordinates (X, Y ), such that X = s cos φ, Y = s sin φ with a spurious angle φ.
+This transformation is canonical if we relate
+ps = XpX + Y pY
+√
+X2 + Y 2
+,
+pφ = XpY − Y pX .
+(28)
+Inverting the usual derivation of the centrifugal potential by transforming from Cartesian
+to polar coordinates, we obtain the effective Hamiltonian in the form
+Heff
+=
+p2
+2m + p2
+s
+2m +
+p2
+φ
+2ms2
+=
+p2
+2m + (XpX + Y pY )2 + (XpY − Y pX)2
+2m(X2 + Y 2)
+=
+p2
+2m + p2
+X
+2m + p2
+Y
+2m .
+(29)
+There is no potential, but trajectories still are not allowed to reach s =
+√
+X2 + Y 2 = 0
+because the interpretation of C/(2ms2) as a centrifugal potential for motion in the (X, Y )-
+plane implies that the angular momentum l = pφ =
+√
+C ≥ ℏ/2 in the plane inherits a
+lower bound from C. The uncertainty relation is therefore re-expressed as a centrifugal
+barrier for motion that is required to have non-zero angular momentum in an auxiliary
+plane. Going through the geometry of Fig. 1 shows that we obtain the correct solutions
+s(t) = s(0)
+�
+1 +
+Ct2
+m2s(0)4
+(30)
+depending on the constant C ≥ ℏ2/2, in agreement with quantum fluctuations of a free
+particle. For a Gaussian state, C = ℏ/2 while (30) with C > ℏ/2 is also valid for non-
+Gaussian states.
+2.3
+Two classical degrees of freedom
+For a pair of classical degrees of freedom, x1 and x2 with momenta p1 and p2, there are ten
+second-order moments. In terms of canonical variables, the three position moments can be
+written as [15, 16]
+∆(x2
+1) = s2
+1
+,
+∆(x2
+2) = s2
+2
+,
+∆(x1x2) = s1s2 cos β
+(31)
+with a new parameter β that describes position correlations in the form of an angle. The
+canonical momentum pβ of β appears in ∆(x1p2) and in
+∆(p2
+1) = p2
+s1 + U1
+s2
+1
+(32)
+7
+
+l/P
+Pt/m
+P
+X
+Y
+s
+Figure 1: The auxiliary plane of free-particle motion. Since the effective Hamiltonian (29)
+has no potential contribution, trajectories in the (X, Y )-plane are straight lines. Invoking
+rotational symmetry, a single trajectory may be chosen to point in the X-direction. Along
+this trajectory, X(t) = Pt/m where P is the conserved momentum in the X-direction,
+assuming the initial condition X(0) = 0. The impact parameter of the trajectory relative
+to the origin equals angular momentum divided by the linear momentum, l/P. (The impact
+parameter is the distance s(0) between the origin and the trajectory at a point where the
+trajectory is orthogonal to the radius vector. Therefore, l = s(0)P or s(0) = l/P.) The
+right-angled triangle directly implies the solution (30) for s(t), also using l =
+√
+C and the
+initial-value relationship P =
+√
+C/s(0) that follows from the impact parameter.
+where
+U1 = (pα − pβ)2 +
+1
+2 sin2 β
+�
+(C1 − 4p2
+α) −
+�
+C2 − C2
+1 + (C1 − 4p2α)2 sin (α + β)
+�
+.
+(33)
+Here, a fourth canonical parameter, α, shows up together with its momentum pα. The eight
+degrees of freedom (s1, ps1; s2, ps2; α, pα; β, pβ) are completed to ten independent degrees of
+freedom by two Casimir variables, C1 and C2 in ∆(p2
+1) and with a similar appearance in
+∆(p2
+2).
+The quasiclassical interpretation of uncertainty as rotation still holds for two degrees of
+freedom: If we assume that pα and
+4√C2 are much smaller than pβ and √C1, the dependence
+of U1 on α disappears, and we have
+∆(p2
+1) = p2
+s1 + p2
+β
+s2
+1
++
+C1
+2s2
+1 sin2 β .
+(34)
+This expression is equivalent to the kinetic energy in spherical coordinates with angles ϑ =
+β and spurious ϕ, with constant amgular momentum pϕ =
+�
+C1/2. Quantum uncertainty
+can therefore be modeled as a centrifugal barrier in a 3-dimensional auxiliary space with
+coordinates (X, Y, Z) related to (s1, β, ϕ) by a standard transformation between Cartesian
+and spherical coordinates.
+For this argument, we have to ignore the variables α, pα and C2. A few indications
+exist as to their possible physical meaning. First, they turn out to be undetermined by
+a minimization of the effective potential for two degrees of freedom subject to a generic
+classical potential [16]. However, minimum energy should be realized in the ground state,
+which should be given by unique wave function, a pure state. Since moments refer to
+8
+
+any state, pure or mixed, it is conceivable that α, pα and C2 describe moment degrees
+of freedom related to the impurity of a state. To test this conjecture, one would have to
+work out all relevant boundaries on the space of second-order moments, in addition to the
+standard uncertainty relation for each classical pair. In order to produce a unique and
+pure ground state, these boundaries would have to be such that the ground-state moments
+are situated in a corner where α, pα and C2 can no longer vary when the energy is held
+fixed.
+These manifold questions in a 10-dimensional phase space are quite tricky and
+remain to be worked out. If the relationship with impurity can be made more precise, the
+canonical variables would provide an interesting quasiclassical dynamics for mixed states.
+The related effective potential could suggest new ways to control impurity.
+3
+Effective potentials
+In general, we may expand the effective Hamiltonian as
+Heff
+=
+⟨ ˆH⟩ = ⟨H(q + (ˆq − q), p + (ˆp − p))⟩
+(35)
+=
+H(q, p) +
+�
+a+b≥2
+1
+a!b!
+∂a+bH(q, p)
+��qa∂pb
+∆(qapb)
+if ˆH is given in completely symmetric ordering. (Otherwise, there will be re-ordering terms
+that explicitly depend on ℏ.) The infinite series over a and b is reduced to a finite sum if
+ˆH is polynomial in ˆq and ˆp. In this case, it just rewrites ⟨ ˆH⟩ in terms of central moments.
+For non-polynomial Hamiltonians, the series is expected to be asymptotic rather than
+convergent.
+If we truncate the moment order for semiclassical states, the series is also reduced to a
+finite sum. This semiclassical interpretation is based on a broad definition of semiclassical
+states where moments are assumed to be analytic in ℏ, such that they obey the hierarchy
+∆(qapb) ∼ O(ℏ(a+b)/2). The interpretation does not presuppose a specific shape of states,
+such as Gaussians. If ˆH =
+1
+2m ˆp2 + V (ˆq), the effective Hamiltonian expanded to second
+order in moments reads
+Heff(q, p, s, ps)
+=
+p2
+2m + V (q) + 1
+2m∆(p2) + 1
+2V ′′(q)∆(q2) + · · ·
+=
+1
+2m(p2 + p2
+s) + V (q) + 1
+2V ′′(q)s2 +
+C
+2ms2 + · · ·
+(36)
+using as before Casimir–Darboux variables such that ∆(q2) = s2 and ∆(p2) = p2
+s + C/s2.
+An application to tunneling immediately follows because we always have V ′′(q) < 0
+around local maxima of the potential, provided it is twice differentiable. The term 1
+2V ′′(q)s2
+in the effective potential therefore lowers the potential barrier in the new s-direction; see
+Fig. 2 for an example. Tunneling can then be described by quasiclassical motion in an
+extended phase space, bypassing the barrier with conserved energy [17, 16]. (See also [18]
+for an analysis of the same effect directly in terms of moments.)
+9
+
+-5
+-4
+-3
+-2
+-1
+ 0
+ 1
+ 2
+ 3
+x
+ 0
+ 0.5
+ 1
+ 1.5 2
+ 2.5
+ 3
+s
+-4
+-2
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+Figure 2: Second-order effective potential for a cubic classical potential. As the contour
+lines indicate, the classical barrier is lowered in the s-direction, making it possible for
+trajectories to bypass it while conserving energy. The lines also show that there is still
+a trapped region in this second-order truncation of the quantum potential. Higher-order
+moments would have to be included in order to describe complete tunneling at all energies.
+10
+
+This application of moment methods shows the importance of keeping s as a degree
+of freedom that evolves independently of q (while being coupled to it in a specific way).
+Other effective methods often replace independent degrees of freedom such as s with new q-
+dependent quantum corrections in an effective Hamiltonian, which in general are of higher-
+derivative form. Such higher-derivative or adiabatic corrections may be derived as a further
+approximation within our quasiclassical systems. For instance, the general equation
+˙ps = −∂Heff
+∂s
+(37)
+provides a differential equation for ps depending on q and s on the right-hand side. If
+we consider the full effective potential as in Fig. 2, the equation for ˙ps is part of the
+equations of motion that describe a trajectory in the (q, s)-plane in which s is kept as a
+degree of freedom independent of q. Several other effective methods, such as those based
+on path integrals as in [2], do not exhibit new independent degrees of freedom but rather
+use additional approximations that (explicitly or implicitly) assume adiabatic evolution of
+quantum variables such as s, while there is no such restriction on classical variables such
+as q.
+If the evolution of s and its momentum ps is completely ignored, the left-hand side
+of (37) vanishes, and the equation is turned into an algebraic equation relating s = s0(q)
+at an extremum of the effective potential, where ∂Heff/∂s = 0. If this extremum is a
+local minimum, the solution is stable, but it does not evolve in s which merely follows
+the evolution of q in an adiabatic manner. If we insert s = s0(q) as well as ps = m˙s =
+m ˙qds0(q)/dq in the effective Hamiltonian, we obtain a position-dependent mass correction
+of the classical kinetic energy 1
+2m ˙q2.
+For small oscillations around the minimum, close to the ground state, one may assume
+that ˙ps is small but not exactly zero, and include corresponding deviations δs of s =
+s0(q) + δs from its value at the minimum. If one expands the right-hand side of (37) in
+δs, it implies a linear equation for δs as a function of ˙ps, which in turn is related to ¨s by
+Hamilton’s equation for s. Inserting s = s0(q) + δs with the solutions for s0(q) and δs in
+the effective Hamiltonian implies a higher-derivative correction in an effective Hamiltonian
+that now depends on second-order derivatives. At higher orders in a systematic adiabatic
+expansion, derivatives of arbitrary orders appear. In a second-order moment description,
+all these terms are replaced by the coupled dynamics of a single quantum degree of freedom,
+s.
+4
+Conclusions
+We have presented several examples in which a canonical description of quantum degrees
+of freedom can lead to new geometrical insights. We have presented examples in which
+uncertainty relations are replaced by centrifugal barriers in a quasiclassical phase space
+equipped with non-classical dimensions. Such descriptions may make it easier to study the
+interplay between uncertainty relations and the dynamics.
+11
+
+The novel formulation presented here, based on the mathematical subject of Poisson
+geometry, unifies and extends several previous approaches from various fields. Quantum
+degrees of freedom, up to a given order in ℏ, are represented by independent dynamical
+variables rather than higher-derivative corrections of classical terms obtained from an adi-
+abatic or derivative expansion. Physically, effective Hamiltonians that describe quantum
+evolution by maintaining the classical number of degrees of freedom require higher deriva-
+tive terms because quantum dynamics is non-local in time (and also in space in quantum
+field theory) if one considers only the classical degrees of freedom. Because a wave function
+is in general spread out, it can have a non-negligible effect on distant points even before one
+would expect the classical position to reach there. By maintaining fluctuations and higher
+moments as independent variables, however, the system can still be described by local
+evolution. The formalism described here therefore provides an efficient and often intuitive
+description of what would appear as non-adiabatic and non-local quantum effects in other
+effective descriptions. If extensions to higher orders and multiple degrees of freedom are
+feasible, there are promising advantages for numerical quantum evolution.
+The methods also make it possible to characterize states and to distinguish systemat-
+ically between classical and quantum states, which may be of advantage in hybrid treat-
+ments where some degrees of freedom can be considered classical. Their moments can then
+be ignored, while other degrees of freedom would couple to their own moments. As seen in
+our discussion of uncertainty relations, Casimir variables also play a role because they are
+subject to different bounds in classical and quantum systems. The standard uncertainty
+relation has a lower bound of C = 0 in classical physics because a distribution on phase
+space may be sharply peaked in both q and p, but this is no longer possible in quantum
+physics. In this way, boundaries of quasiclassical phase spaces provide a model independent
+way to distinguish between classical and quantum systems or their hybrid combinations.
+Acknowledgements
+The author thanks Cesare Tronci for an invitation to the 746. WE-Heraeus-Seminar “Koop-
+man Methods in Classical and Classical-Quantum Mechanics” where these results were
+presented. This work was supported in part by NSF grant PHY-2206591.
+References
+[1] B. O. Koopman, Hamiltonian Systems and Transformations in Hilbert Space, PNAS
+17 (1931) 315–318
+[2] F. Cametti, G. Jona-Lasinio, C. Presilla, and F. Toninelli, Comparison between quan-
+tum and classical dynamics in the effective action formalism, In Proceedings of the
+International School of Physics “Enrico Fermi”, Course CXLIII, pages 431–448, Am-
+sterdam, 2000. IOS Press, [quant-ph/9910065]
+12
+
+[3] S. Coleman and E. Weinberg,
+Radiative corrections as the origin of spontaneous
+symmetry breaking, Phys. Rev. D 7 (1973) 1888–1910
+[4] F. Strocchi, Complex coordinates and quantum mechanics, Rev. Mod. Phys. 38 (1966)
+36–40
+[5] T. W. B. Kibble, Geometrization of quantum mechanics, Commun. Math. Phys. 65
+(1979) 189–201
+[6] A. Heslot, Quantum mechanics as a classical theory, Phys. Rev. D 31 (1985) 1341–1348
+[7] M. Bojowald and A. Skirzewski, Effective Equations of Motion for Quantum Systems,
+Rev. Math. Phys. 18 (2006) 713–745, [math-ph/0511043]
+[8] R. Jackiw and A. Kerman, Time Dependent Variational Principle And The Effective
+Action, Phys. Lett. A 71 (1979) 158–162
+[9] F. Arickx, J. Broeckhove, W. Coene, and P. van Leuven,
+Gaussian Wave-packet
+Dynamics, Int. J. Quant. Chem.: Quant. Chem. Symp. 20 (1986) 471–481
+[10] R. A. Jalabert and H. M. Pastawski, Environment-independent decoherence rate in
+classically chaotic systems, Phys. Rev. Lett. 86 (2001) 2490–2493
+[11] O. Prezhdo, Quantized Hamiltonian Dynamics, Theor. Chem. Acc. 116 (2006) 206
+[12] M. Bojowald, D. Brizuela, H. H. Hernandez, M. J. Koop, and H. A. Morales-T´ecotl,
+High-order quantum back-reaction and quantum cosmology with a positive cosmolog-
+ical constant, Phys. Rev. D 84 (2011) 043514, [arXiv:1011.3022]
+[13] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1997
+[14] T. Vachaspati and G. Zahariade, A Classical-Quantum Correspondence and Backre-
+action, Phys. Rev. D 98 (2018) 065002, [arXiv:1806.05196]
+[15] B. Bayta¸s, M. Bojowald, and S. Crowe, Faithful realizations of semiclassical trunca-
+tions, Ann. Phys. 420 (2020) 168247, [arXiv:1810.12127]
+[16] B. Bayta¸s, M. Bojowald, and S. Crowe, Effective potentials from canonical realizations
+of semiclassical truncations, Phys. Rev. A 99 (2019) 042114, [arXiv:1811.00505]
+[17] O. Prezhdo and Yu.˜V. Pereverzev, Quantized Hamilton Dynamics, J. Chem. Phys.
+113 (2000) 6557
+[18] L. Aragon-Mu˜noz, G. Chacon-Acosta, and H. Hern´andez-Hern´andez, Effective quan-
+tum tunneling from a semiclassical momentous approach, Int. Mod. J. Phys. B 34
+(2020) 2050271, [arXiv:2004.00118]
+13
+

MdE4T4oBgHgl3EQfjA06/content/tmp_files/load_file.txt

@@ -0,0 +1,292 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf,len=291
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='05138v1  [quant-ph]  12 Jan 2023  Canonical description of quantum dynamics  Martin Bojowald∗  Institute for Gravitation and the Cosmos,  The Pennsylvania State University,  104 Davey Lab, University Park, PA 16802, USA  Abstract  Some of the important non-classical aspects of quantum mechanics can be de-  scribed in more intuitive terms if they are reformulated in a geometrical picture  based on an extension of the classical phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' This contribution presents vari-  ous phase-space properties of moments describing a quantum state and its dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  An example of a geometrical reformulation of a non-classical quantum effect is given  by an equivalence between conditions imposed by uncertainty relations and centrifu-  gal barriers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  1  Introduction  Although classical and quantum mechanics are usually presented based on different mathe-  matical methods and with a marked contrast in the underlying physical concepts, there are  formulations that bring the two theories closer to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' For instance, the Koopman  wave function [1] can be used to make classical mechanics look quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Effective theories  reverse the direction and make quantum mechanics look classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  Given the richer nature of quantum mechanics with more degrees of freedom of a state  than in classical mechanics, there are different ways in which this “classicalization” can be  achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' A method of importance in elementary particle physics is the low-energy effective  action [2] which determines the dynamics of a quantum system around the ground state,  and the related Coleman–Weinberg potential [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Such low-energy methods are useful in  situations in which only a few particle degrees of freedom are excited, as in basic scattering  processes of interest in particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Methods that do not rely on low-energy regimes are  geometric ones or phase-space descriptions of quantum mechanics, going back for instance  to the symplectic formulations of [4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' These approaches make use of the fact that the  imaginary part of an inner product is antisymmetric and non-degenerate, and therefore  can be interpreted as a symplectic form on the space of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  Here, we use a geometrical picture of quantum dynamics derived from a suitable param-  eterization of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' In some ways, it is related to symplectic phase-space descriptions, but  it includes a suitable generalization that makes it possible to formulate effective theories in  which certain degrees of freedom contained in a full quantum state are suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' (The  parameterization of states to be introduced determines which degrees of freedom may be  ∗e-mail address: bojowald@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='edu  1  suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=') Most of these truncations cannot be described by symplectic methods because  removing some degrees of freedom, in general, introduces degenerate directions in phase  space from the point of view of symplectic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Accordingly, the truncated phase-  space geometry is Poisson, making use of a Poisson bracket with a non-invertible Poisson  tensor, as opposed to an invertible symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' This method was first described in  [7] within the setting of Poisson geometry, but with the benefit of hindsight it had several  independent precursors [8, 9, 10, 11] that can now be recognized as low-order versions of  Poisson spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  As an instructive example, consider the uncertainty relation  (∆x)2(∆p)2 − C2  xp ≥ U  (1)  where U = ℏ2/4 in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The fluctuations and covariance on the left-hand  side are part of a parameterization of states by moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The constant U on the right-hand  side can be seen as a parameter that controls quantum features: It is positive for quantum  states but may be smaller if the moments belong to a classical state, as encountered for  instance in classical dynamics if it is expressed in term of the Koopman wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  There are three moments of second order for a single classical degree of freedom, all seen  on the left-hand side of the uncertainty relation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' If one aims to find a quasiclassical or  phase-space description of these and only these degrees of freedom, one has to work with  an odd-dimensional phase space that cannot be symplectic but may well (and indeed does)  have a Poisson bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  Since moments are defined as polynomial expressions of basic expectation values, we  may introduce a Poisson bracket by first defining  {⟨ ˆO1⟩ρ, ⟨ ˆO2⟩ρ} = ⟨[ ˆO1, ˆO2]⟩ρ  iℏ  (2)  for expectation values of operators in some state pure or mixed ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' This operation defines an  antisymmetric bracket on expectation values ⟨ ˆO⟩ρ, interpreted as a function on state space  for any fixed ˆO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The bracket also satisfies the Jacobi identity because the commutator on  the right-hand side does so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' We can extend it to products of expectation values, as needed  for moments, by imposing the Leibniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  If we include only moments up to some finite order in our quasiclassical description,  (2) defines a Poisson bracket that, in general, is not symplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Nevertheless, dynamics  on this phase space is well defined if the system is characterized by a Hamilton operator  ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The standard equation  d⟨ ˆO⟩ρ  dt  = ⟨[ ˆO, ˆH]⟩ρ  iℏ  (3)  is then equivalent to Hamilton’s equation  d⟨ ˆO⟩ρ  dt  = {⟨ ˆO⟩ρ, Heff}  (4)  2  on phase space with the effective Hamiltonian  Heff(ρ) = ⟨ ˆH⟩ρ ,  (5)  again interpreted as a function on the space of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' In general, this equation couples  infinitely many independent expectation values to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  A truncation is therefore required for manageable approximations which, depending  on the truncation order of moments, defines the corresponding effective description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' For  instance, we may use a semiclassical approximation of order N in ℏ if we include only  expectation values ⟨ˆqaˆpb⟩ for a + b ≤ 2N, a ≥ 0 and b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' It is more convenient to  reformulate this truncation as using basic expectation values q = ⟨ˆq⟩ and p = ⟨ˆp⟩ together  with central moments in a completely symmetric (or Weyl) ordering,  ∆(qapb) = ⟨(ˆq − q)a(ˆp − p)b⟩symm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  (6)  This ordering is defined by summing all (a+b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' permutations of (ˆq−q)a(ˆp−p)b and dividing  by (a + b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' For a = b = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' this implies the standard covariance  Cqp = ∆(qp) = 1  2⟨ˆqˆp + ˆpˆq⟩ − qp  (7)  while,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' for instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  ∆(q2p)  =  1  3⟨(ˆq − q)2(ˆp − p) + (ˆq − q)(ˆp − p)(ˆq − q) + (ˆp − p)(ˆq − q)2⟩  =  1  3⟨ˆq2ˆp + ˆqˆpˆq + ˆpˆq2⟩ − ⟨ˆq2⟩p − ⟨ˆqˆp + ˆpˆq⟩q + 2q2p  (8)  In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' we further analyze the underlying Poisson geometry and present new re-  sults that demonstrate how well-known quantum parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' such as U in (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' are equipped  with an elegant geometrical interpretation as centrifugal barriers for quantum degrees of  freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Physical applications related to tunneling will also be described, and we will  compare these methods with adiabatic or low-energy ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  2  Geometry of the free particle  The quantum dynamics of the free particle can be described (in this case, exactly) by the  following geometrical picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' We have the effective Hamiltonian  Heff(p, ∆(p2)) =  � ˆp2  2m  �  = p2  2m + ∆(p2)  2m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  (9)  An immediate question in a quasiclassical picture is whether the last term, ∆(p2)/2m in  which ∆(p2) is often suggestively written as the square (∆p)2 of a momentum fluctuation  ∆p, is another kinetic energy in addition to p2/(2m), or maybe contributes to the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='1  Moment systems  To answer this question, we consider the relevant Poisson brackets  {∆(q2), ∆(p2)}  =  4∆(qp)  (10)  {∆(q2), ∆(qp)}  =  2∆(q2)  (11)  {∆(qp), ∆(p2)}  =  2∆(p2)  (12)  of moments of the same order as ∆(p2) in the new term of the effective Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The  Poisson bracket is defined in (2), also making use of the Leibniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' For instance, to  derive the first bracket, (10), we use  {⟨ˆq2⟩, ⟨ˆp2⟩} = ⟨[ˆq2, ˆp2]⟩  iℏ  = 2⟨ˆqˆp + ˆpˆq⟩  (13)  directly from (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The remaining terms obtained from expanding  {∆(q2), ∆(p2)}  =  {⟨ˆq2⟩ − q2, ⟨ˆp2⟩ − p2}  =  {⟨ˆq2⟩, ⟨ˆp2⟩} − {⟨ˆq2⟩, p2} − {q2, ⟨ˆp2⟩} + {q2, p2}  (14)  require the Leibniz rule:  {⟨ˆq2⟩, p2}  =  2p{⟨ˆq2⟩, ⟨ˆp⟩} = 4qp  (15)  {q2, ⟨ˆp2⟩}  =  2q{⟨ˆq⟩, ⟨ˆp2⟩} = 4qp  (16)  {q2, p2}  =  4qp{⟨ˆq⟩, ⟨ˆp⟩} = 4qp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  (17)  Combining all terms, we obtain (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  At generic orders, central moments have several welcome but also some unwelcome prop-  erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' They are always real, by definition, facilitate the definition of a general semiclassical  hierarchy ∆(qapb) ∝ ℏ(a+b)/2, and are always Poisson orthogonal to the basic expectation  values: {q, ∆(qapb)} = 0 = {p, ∆(qapb)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  However, complications can sometimes arise  because they are coordinates on a phase space with boundaries, given by uncertainty rela-  tions including those of higher orders, and because their Poisson brackets are not canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  While there is a closed-form expression for the Poisson bracket of two moments of canonical  operators ˆq and ˆp, given by [7, 12]  {∆(qapb), ∆(qcpd)} = ad∆(qa−1pb)∆(qcpd−1) − bc∆(qapb−1)∆(qc−1pd)  +  �  n  �iℏ  2  �n−1  Kn  abcd ∆(qa+c−npb+d−n)  (18)  where 1 ≤ n ≤ Min(a + c, b + d, a + b, c + d) and  Kn  abcd :=  n  �  m=0  (−1)mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' (n − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  � a  m  � �  b  n − m  � �  c  n − m  � � d  m  �  ,  (19)  4  it is rather tedious to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Moreover, the non-canonical nature can make it difficult  to determine algebraic or physical properties, as already seen in the example of the free  particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  The non-canonical nature of these brackets implies that ∆p is not an obvious canon-  ical momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' However, the Casimir–Darboux theorem [13] states that every Poisson  manifold (that is, any space equipped with a Poisson bracket) has local coordinates qi,  pi, CI such that {qi, pj} = δij and {qi, CI} = 0 = {pi, CI}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The qi and pj are then pairs  of canonical configuration coordinates and momenta, while the Casimir variables CI are  conserved by any Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' (If the manifold is symplectic, all CI are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=')  It turns out that the brackets of second-order moments have Casimir–Darboux variables  (s, ps, C) where  ∆(q2) = s2  ,  ∆(qp) = sps  ,  ∆(p2) = p2  s + C  s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  (20)  The canonical bracket {s, ps} = 1 and {s, C} = 0 = {ps, C}, required for Casimir–Darboux  variables, can directly be confirmed after inverting the mapping to obtain  s =  �  ∆(q2)  ,  ps =  ∆(qp)  �  ∆(q2)  ,  C = ∆(q2)∆(p2) − ∆(qp)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  (21)  The last equation shows that  C = ∆(q2)∆(p2) − ∆(qp)2 ≥ ℏ2/4  (22)  is bounded from below by Heisenberg’s uncertainty relation (or the Schr¨odinger–Robertson  version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' These properties have been found independently in a variety of contexts, includ-  ing quantum field theory [8], quantum chaos [10], quantum chemistry [11] and models of  quantum gravity [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  The canonical formulation (20) or (21) can be found if one combines the dynamics  implied by the Hamiltonian (9) with the fact that C as defined in terms of moments by  (22) is conserved by any Hamiltonian that depends only on basic expectation values and  second-order moments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' see for instance [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Hamiltonian dynamics gives  d∆(q2)  dt  = {∆(q2), Heff(p, ∆(p2))} = 2  m∆(qp)  (23)  using the Poisson bracket (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' If we define s =  �  ∆(q2), we obtain  ˙s = 1  2s  d∆(q2)  dt  = ∆(qp)  ms  (24)  and therefore ∆(qp)/s appears as a momentum ps of s, as in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The equation for ∆(p2)  can then be obtained by solving C = ∆(q2)∆(p2) − ∆(qp)2 for this moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  In general, however, the canonical structure is more difficult to discern and requires  more systematic derivations, for instance when one considers higher moments or more than  5  one classical pair of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' It may still be possible to derive the momenta  of some moments using Hamiltonian dynamics, but deriving a complete mapping of all  independent degrees of freedom to canonical form requires more care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  While there is  no completely systematic procedure, Poisson geometry helps to organize the derivation  [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Again in the example of second-order moments (20), we are still free to choose  one degree of freedom as the first canonical variable, s =  �  ∆(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Any Poisson bracket  with s then takes the form  {s, f} = ∂f  ∂ps  (25)  with the momentum ps to be determined, where f is any function of the moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' By  evaluating this equation for sufficiently many functions f, or just for the basic moments, it  provides differential equations in the independent variable ps, with s as a parameter, that  can be solved to reveal the ps-dependence of moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  For instance, choosing f = ∆(qp), we obtain  ∂∆(qp)  ∂ps  = {s, ∆(qp)} = 1  2s{∆(q2), ∆(qp)} = ∆(q2)  s  = s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  (26)  The solution, ∆(qp) = sps + c(s) with a free s-dependent function c(s) is consistent with  (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The free function can be removed by a canonical transformation, replacing ps with  ps + c(s)/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' For ∆(p2), we obtain the differential equation ∂∆(p2)/∂ps = 2∆(qp)/s = 2ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  The solution is again consistent with (20), where the free s-dependent function is here fixed  by using C as a conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  If there are more than three moments, the procedure has to be iterated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' After finding  the first canonical pair analogous to (s, ps), the next step consists in rewriting the remaining  moments in terms of quantities that have vanishing Poisson brackets with both s and ps,  and are therefore canonically independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' For second-order moments of a single classical  degree of freedom, this step merely consists in identifying C as a conserved quantity, but it  can be more challenging in higher-dimensional phase spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' At this point, the procedure  is no longer fully systematic and usually requires special considerations of the moments  system and its Poisson brackets in order to proceed in a tractable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='2  Free dynamics  In Casimir–Darboux variables, the effective Hamiltonian (9) of the free particle takes the  form  Heff(p, s, ps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' C) = p2  2m + p2  s  2m +  C  2ms2  (27)  with, as we see now, contributions to both the kinetic and potential energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  It may  seem counter-intuitive that the free particle is subject to a potential, but if the fluctuation  direction with coordinate s is included in the configuration space, uncertainty relations  must imply some kind of repulsive potential that prevents s from reaching zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The new  term C/(2ms2) where C ≥ ℏ2/4 ̸= 0 is precisely of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  6  Nevertheless, we can rephrase the dynamics as manifestly free if we further extend our  configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' We can interpret the 1/s2 potential as a centrifugal one in an auxiliary  plane with coordinates (X, Y ), such that X = s cos φ, Y = s sin φ with a spurious angle φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  This transformation is canonical if we relate  ps = XpX + Y pY  √  X2 + Y 2  ,  pφ = XpY − Y pX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  (28)  Inverting the usual derivation of the centrifugal potential by transforming from Cartesian  to polar coordinates, we obtain the effective Hamiltonian in the form  Heff  =  p2  2m + p2  s  2m +  p2  φ  2ms2  =  p2  2m + (XpX + Y pY )2 + (XpY − Y pX)2  2m(X2 + Y 2)  =  p2  2m + p2  X  2m + p2  Y  2m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  (29)  There is no potential, but trajectories still are not allowed to reach s =  √  X2 + Y 2 = 0  because the interpretation of C/(2ms2) as a centrifugal potential for motion in the (X, Y )-  plane implies that the angular momentum l = pφ =  √  C ≥ ℏ/2 in the plane inherits a  lower bound from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The uncertainty relation is therefore re-expressed as a centrifugal  barrier for motion that is required to have non-zero angular momentum in an auxiliary  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Going through the geometry of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' 1 shows that we obtain the correct solutions  s(t) = s(0)  �  1 +  Ct2  m2s(0)4  (30)  depending on the constant C ≥ ℏ2/2, in agreement with quantum fluctuations of a free  particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' For a Gaussian state, C = ℏ/2 while (30) with C > ℏ/2 is also valid for non-  Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='3  Two classical degrees of freedom  For a pair of classical degrees of freedom, x1 and x2 with momenta p1 and p2, there are ten  second-order moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' In terms of canonical variables, the three position moments can be  written as [15, 16]  ∆(x2  1) = s2  1  ,  ∆(x2  2) = s2  2  ,  ∆(x1x2) = s1s2 cos β  (31)  with a new parameter β that describes position correlations in the form of an angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The  canonical momentum pβ of β appears in ∆(x1p2) and in  ∆(p2  1) = p2  s1 + U1  s2  1  (32)  7  l/P  Pt/m  P  X  Y  s  Figure 1: The auxiliary plane of free-particle motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Since the effective Hamiltonian (29)  has no potential contribution, trajectories in the (X, Y )-plane are straight lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Invoking  rotational symmetry, a single trajectory may be chosen to point in the X-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Along  this trajectory, X(t) = Pt/m where P is the conserved momentum in the X-direction,  assuming the initial condition X(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The impact parameter of the trajectory relative  to the origin equals angular momentum divided by the linear momentum, l/P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' (The impact  parameter is the distance s(0) between the origin and the trajectory at a point where the  trajectory is orthogonal to the radius vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Therefore, l = s(0)P or s(0) = l/P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=') The  right-angled triangle directly implies the solution (30) for s(t), also using l =  √  C and the  initial-value relationship P =  √  C/s(0) that follows from the impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  where  U1 = (pα − pβ)2 +  1  2 sin2 β  �  (C1 − 4p2  α) −  �  C2 − C2  1 + (C1 − 4p2α)2 sin (α + β)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  (33)  Here, a fourth canonical parameter, α, shows up together with its momentum pα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The eight  degrees of freedom (s1, ps1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' s2, ps2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' α, pα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' β, pβ) are completed to ten independent degrees of  freedom by two Casimir variables, C1 and C2 in ∆(p2  1) and with a similar appearance in  ∆(p2  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  The quasiclassical interpretation of uncertainty as rotation still holds for two degrees of  freedom: If we assume that pα and  4√C2 are much smaller than pβ and √C1, the dependence  of U1 on α disappears, and we have  ∆(p2  1) = p2  s1 + p2  β  s2  1  +  C1  2s2  1 sin2 β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  (34)  This expression is equivalent to the kinetic energy in spherical coordinates with angles ϑ =  β and spurious ϕ, with constant amgular momentum pϕ =  �  C1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Quantum uncertainty  can therefore be modeled as a centrifugal barrier in a 3-dimensional auxiliary space with  coordinates (X, Y, Z) related to (s1, β, ϕ) by a standard transformation between Cartesian  and spherical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  For this argument, we have to ignore the variables α, pα and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' A few indications  exist as to their possible physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' First, they turn out to be undetermined by  a minimization of the effective potential for two degrees of freedom subject to a generic  classical potential [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' However, minimum energy should be realized in the ground state,  which should be given by unique wave function, a pure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Since moments refer to  8  any state, pure or mixed, it is conceivable that α, pα and C2 describe moment degrees  of freedom related to the impurity of a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' To test this conjecture, one would have to  work out all relevant boundaries on the space of second-order moments, in addition to the  standard uncertainty relation for each classical pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' In order to produce a unique and  pure ground state, these boundaries would have to be such that the ground-state moments  are situated in a corner where α, pα and C2 can no longer vary when the energy is held  fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  These manifold questions in a 10-dimensional phase space are quite tricky and  remain to be worked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' If the relationship with impurity can be made more precise, the  canonical variables would provide an interesting quasiclassical dynamics for mixed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  The related effective potential could suggest new ways to control impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  3  Effective potentials  In general, we may expand the effective Hamiltonian as  Heff  =  ⟨ ˆH⟩ = ⟨H(q + (ˆq − q), p + (ˆp − p))⟩  (35)  =  H(q, p) +  �  a+b≥2  1  a!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  ∂a+bH(q, p)  ∂qa∂pb  ∆(qapb)  if ˆH is given in completely symmetric ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' (Otherwise, there will be re-ordering terms  that explicitly depend on ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=') The infinite series over a and b is reduced to a finite sum if  ˆH is polynomial in ˆq and ˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' In this case, it just rewrites ⟨ ˆH⟩ in terms of central moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  For non-polynomial Hamiltonians, the series is expected to be asymptotic rather than  convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  If we truncate the moment order for semiclassical states, the series is also reduced to a  finite sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' This semiclassical interpretation is based on a broad definition of semiclassical  states where moments are assumed to be analytic in ℏ, such that they obey the hierarchy  ∆(qapb) ∼ O(ℏ(a+b)/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The interpretation does not presuppose a specific shape of states,  such as Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' If ˆH =  1  2m ˆp2 + V (ˆq), the effective Hamiltonian expanded to second  order in moments reads  Heff(q, p, s, ps)  =  p2  2m + V (q) + 1  2m∆(p2) + 1  2V ′′(q)∆(q2) + · · ·  =  1  2m(p2 + p2  s) + V (q) + 1  2V ′′(q)s2 +  C  2ms2 + · · ·  (36)  using as before Casimir–Darboux variables such that ∆(q2) = s2 and ∆(p2) = p2  s + C/s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  An application to tunneling immediately follows because we always have V ′′(q) < 0  around local maxima of the potential, provided it is twice differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The term 1  2V ′′(q)s2  in the effective potential therefore lowers the potential barrier in the new s-direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' 2 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Tunneling can then be described by quasiclassical motion in an  extended phase space, bypassing the barrier with conserved energy [17, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' (See also [18]  for an analysis of the same effect directly in terms of moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=')  9  5  4  3  2  1  0  1  2  3  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='5 2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='5  3  s  4  2  0  2  4  6  8  10  Figure 2: Second-order effective potential for a cubic classical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' As the contour  lines indicate, the classical barrier is lowered in the s-direction, making it possible for  trajectories to bypass it while conserving energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The lines also show that there is still  a trapped region in this second-order truncation of the quantum potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Higher-order  moments would have to be included in order to describe complete tunneling at all energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  10  This application of moment methods shows the importance of keeping s as a degree  of freedom that evolves independently of q (while being coupled to it in a specific way).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  Other effective methods often replace independent degrees of freedom such as s with new q-  dependent quantum corrections in an effective Hamiltonian, which in general are of higher-  derivative form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Such higher-derivative or adiabatic corrections may be derived as a further  approximation within our quasiclassical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' For instance, the general equation  ˙ps = −∂Heff  ∂s  (37)  provides a differential equation for ps depending on q and s on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' If  we consider the full effective potential as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' 2, the equation for ˙ps is part of the  equations of motion that describe a trajectory in the (q, s)-plane in which s is kept as a  degree of freedom independent of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Several other effective methods, such as those based  on path integrals as in [2], do not exhibit new independent degrees of freedom but rather  use additional approximations that (explicitly or implicitly) assume adiabatic evolution of  quantum variables such as s, while there is no such restriction on classical variables such  as q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  If the evolution of s and its momentum ps is completely ignored, the left-hand side  of (37) vanishes, and the equation is turned into an algebraic equation relating s = s0(q)  at an extremum of the effective potential, where ∂Heff/∂s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' If this extremum is a  local minimum, the solution is stable, but it does not evolve in s which merely follows  the evolution of q in an adiabatic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' If we insert s = s0(q) as well as ps = m˙s =  m ˙qds0(q)/dq in the effective Hamiltonian, we obtain a position-dependent mass correction  of the classical kinetic energy 1  2m ˙q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  For small oscillations around the minimum, close to the ground state, one may assume  that ˙ps is small but not exactly zero, and include corresponding deviations δs of s =  s0(q) + δs from its value at the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' If one expands the right-hand side of (37) in  δs, it implies a linear equation for δs as a function of ˙ps, which in turn is related to ¨s by  Hamilton’s equation for s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Inserting s = s0(q) + δs with the solutions for s0(q) and δs in  the effective Hamiltonian implies a higher-derivative correction in an effective Hamiltonian  that now depends on second-order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' At higher orders in a systematic adiabatic  expansion, derivatives of arbitrary orders appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' In a second-order moment description,  all these terms are replaced by the coupled dynamics of a single quantum degree of freedom,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  4  Conclusions  We have presented several examples in which a canonical description of quantum degrees  of freedom can lead to new geometrical insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' We have presented examples in which  uncertainty relations are replaced by centrifugal barriers in a quasiclassical phase space  equipped with non-classical dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Such descriptions may make it easier to study the  interplay between uncertainty relations and the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  11  The novel formulation presented here, based on the mathematical subject of Poisson  geometry, unifies and extends several previous approaches from various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Quantum  degrees of freedom, up to a given order in ℏ, are represented by independent dynamical  variables rather than higher-derivative corrections of classical terms obtained from an adi-  abatic or derivative expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Physically, effective Hamiltonians that describe quantum  evolution by maintaining the classical number of degrees of freedom require higher deriva-  tive terms because quantum dynamics is non-local in time (and also in space in quantum  field theory) if one considers only the classical degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Because a wave function  is in general spread out, it can have a non-negligible effect on distant points even before one  would expect the classical position to reach there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' By maintaining fluctuations and higher  moments as independent variables, however, the system can still be described by local  evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The formalism described here therefore provides an efficient and often intuitive  description of what would appear as non-adiabatic and non-local quantum effects in other  effective descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' If extensions to higher orders and multiple degrees of freedom are  feasible, there are promising advantages for numerical quantum evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  The methods also make it possible to characterize states and to distinguish systemat-  ically between classical and quantum states, which may be of advantage in hybrid treat-  ments where some degrees of freedom can be considered classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Their moments can then  be ignored, while other degrees of freedom would couple to their own moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' As seen in  our discussion of uncertainty relations, Casimir variables also play a role because they are  subject to different bounds in classical and quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' The standard uncertainty  relation has a lower bound of C = 0 in classical physics because a distribution on phase  space may be sharply peaked in both q and p, but this is no longer possible in quantum  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' In this way, boundaries of quasiclassical phase spaces provide a model independent  way to distinguish between classical and quantum systems or their hybrid combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  Acknowledgements  The author thanks Cesare Tronci for an invitation to the 746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' WE-Heraeus-Seminar “Koop-  man Methods in Classical and Classical-Quantum Mechanics” where these results were  presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' This work was supported in part by NSF grant PHY-2206591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  References  [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Koopman, Hamiltonian Systems and Transformations in Hilbert Space, PNAS  17 (1931) 315–318  [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Cametti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Jona-Lasinio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Presilla, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Toninelli, Comparison between quan-  tum and classical dynamics in the effective action formalism, In Proceedings of the  International School of Physics “Enrico Fermi”, Course CXLIII, pages 431–448, Am-  sterdam, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' IOS Press, [quant-ph/9910065]  12  [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Coleman and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Weinberg,  Radiative corrections as the origin of spontaneous  symmetry breaking, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' D 7 (1973) 1888–1910  [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Strocchi, Complex coordinates and quantum mechanics, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' 38 (1966)  36–40  [5] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Kibble, Geometrization of quantum mechanics, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' 65  (1979) 189–201  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Heslot, Quantum mechanics as a classical theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' D 31 (1985) 1341–1348  [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Bojowald and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Skirzewski, Effective Equations of Motion for Quantum Systems,  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' 18 (2006) 713–745, [math-ph/0511043]  [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Jackiw and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Kerman, Time Dependent Variational Principle And The Effective  Action, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' A 71 (1979) 158–162  [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Arickx, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Broeckhove, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Coene, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' van Leuven,  Gaussian Wave-packet  Dynamics, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' : Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' 20 (1986) 471–481  [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Jalabert and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Pastawski, Environment-independent decoherence rate in  classically chaotic systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' 86 (2001) 2490–2493  [11] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Prezhdo, Quantized Hamiltonian Dynamics, Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' 116 (2006) 206  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Bojowald, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Brizuela, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Hernandez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Koop, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Morales-T´ecotl,  High-order quantum back-reaction and quantum cosmology with a positive cosmolog-  ical constant, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' D 84 (2011) 043514, [arXiv:1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='3022]  [13] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Arnold, Mathematical Methods of Classical Mechanics, Springer, 1997  [14] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Vachaspati and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Zahariade, A Classical-Quantum Correspondence and Backre-  action, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' D 98 (2018) 065002, [arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='05196]  [15] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Bayta¸s, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Bojowald, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Crowe, Faithful realizations of semiclassical trunca-  tions, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' 420 (2020) 168247, [arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='12127]  [16] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Bayta¸s, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Bojowald, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Crowe, Effective potentials from canonical realizations  of semiclassical truncations, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' A 99 (2019) 042114, [arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='00505]  [17] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Prezhdo and Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='˜V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Pereverzev, Quantized Hamilton Dynamics, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='  113 (2000) 6557  [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Aragon-Mu˜noz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Chacon-Acosta, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Hern´andez-Hern´andez, Effective quan-  tum tunneling from a semiclassical momentous approach, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content=' B 34  (2020) 2050271, [arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}
+page_content='00118]  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE4T4oBgHgl3EQfjA06/content/2301.05138v1.pdf'}

OdE2T4oBgHgl3EQfrAj4/content/tmp_files/2301.04046v1.pdf.txt

@@ -0,0 +1,2762 @@
+Numerical study of conforming space-time methods for
+Maxwell’s equations
+Julia I.M. Hauser1, Marco Zank2
+1Institut für Wissenschaftliches Rechnen,
+Technische Universität Dresden,
+Zellescher Weg 25, 01217 Dresden, Germany
+[email protected]
+2Fakultät für Mathematik, Universität Wien,
+Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
+[email protected]
+Abstract
+Time-dependent Maxwell’s equations govern electromagnetics. Under certain condi-
+tions, we can rewrite these equations into a partial differential equation of second order,
+which in this case is the vector wave equation. For the vectorial wave, we investigate
+the numerical application and the challenges in the implementation. For this purpose,
+we consider a space-time variational setting, i.e. time is just another spatial dimension.
+More specifically, we apply integration by parts in time as well as in space, leading to a
+space-time variational formulation with different trial and test spaces. Conforming dis-
+cretizations of tensor-product type result in a Galerkin–Petrov finite element method that
+requires a CFL condition for stability. For this Galerkin–Petrov variational formulation,
+we study the CFL condition and its sharpness. To overcome the CFL condition, we use
+a Hilbert-type transformation that leads to a variational formulation with equal trial and
+test spaces. Conforming space-time discretizations result in a new Galerkin–Bubnov finite
+element method that is unconditionally stable. In numerical examples, we demonstrate
+the effectiveness of this Galerkin–Bubnov finite element method. Furthermore, we inves-
+tigate different projections of the right-hand side and their influence on the convergence
+rates.
+This paper is the first step towards a more stable computation and a better under-
+standing of vectorial wave equations in a conforming space-time approach.
+1
+Introduction
+The time-dependent Maxwell’s equations and their discretization are required in many electro-
+magnetic applications. One application is the modeling of an electric motor which is governed
+by Ampère’s circuital law. Ampère’s circuital law relates the time-dependent electric field with
+the current density and the magnetic field. If the corresponding space-time domain is star-like
+with respect to a ball, then we can rewrite Maxwell’s system into the vectorial wave equation.
+Application of this technique in the static case is investigated in [38]. Extensive studies of
+the static case can be found not only for analytic methods, but also for the approximation by
+1
+arXiv:2301.04046v1  [math.NA]  10 Jan 2023
+
+finite elements, see e.g. [8], or in a more applied static nonlinear example [20]. Moreover, the
+quasi-static case is examined, see [7].
+However, the complete space-time setting of these equations has been studied in less detail.
+Most approaches for time-dependent Maxwell’s equation are either dealing with the frequency
+domain, see [30], or using time-stepping methods, see [22, Chapter 12.2].
+The latter, i.e.
+the time-stepping approach, observes instabilities. One solution is to stabilize the systems
+which stem from time-stepping methods. The development of such stabilized methods is an
+active research field, see e.g.
+[39] on energy-preserving meshing for FDTD schemes or [4]
+for the Cole–Cole model which involves polarization. Other time-stepping methods are the
+so-called leapfrog and the Newmark-beta methods, [31], where the first is a special case of the
+second. A comparison and improvement of these methods can be found in [9]. Another class
+of time-stepping approaches are locally implicit time integrators, see e.g. [19] and references
+there.
+An alternative to time-stepping methods is a space-time approach. Most approaches apply
+discontinuous Galerkin methods to Maxwell’s equations, see [1, 11, 12, 26, 40] and references
+there. Note that these methods are non-conforming in general.
+In this paper, we derive conforming space-time finite element discretizations for the vecto-
+rial wave equation without introducing additional unknowns, i.e. without a reformulation of
+the vectorial wave equation as a first-order system. The advantage of staying with the second-
+order formulation and of conforming methods is the need for a smaller number of degrees
+of freedom compared with first-order formulations or discontinuous Galerkin methods. First,
+we state the variational formulation for different trial and test spaces, i.e. a Galerkin–Petrov
+formulation. Second, using the so-called modified Hilbert transformation HT , introduced in
+[35, 43], we present a variational formulation with equal trial and test spaces, i.e. a Galerkin–
+Bubnov formulation. The main goal of this paper is to derive and describe in detail these
+two conforming space-time methods, the assembling of the resulting linear systems and their
+comparison concerning stability and convergence. In more detail, we investigate the Galerkin–
+Petrov formulation and the corresponding CFL condition. Additionally, we elaborate on a
+Galerkin–Bubnov formulation using the modified Hilbert transformation, where all necessary
+mathematical tools are developed.
+In the end, we compare the results of the conforming
+Galerkin–Petrov and Galerkin–Bubnov finite element methods with respect to their stability
+and convergence behavior by considering numerical examples. Moreover, we investigate the
+convergence behavior of both conforming space-time approaches when the right-hand side is
+approximated by two different projections.
+Before we introduce the space-time approach we state its derivation from Maxwell’s equa-
+tions to get a better understanding of the properties of the vectorial wave equation. For this
+purpose, we need to rewrite Maxwell’s equations using differential forms in a Lipschitz domain
+Q ⊂ R4. Hence, we define the Faraday 2-form
+F := e ∧ dt + b
+where e is the 1-form corresponding to the electric field E and b the 2-form corresponding to
+the magnetic flux density B. Additionally, we define the Maxwell’s 2-form G and the source
+3-form J by
+G := h ∧ dt − ˜d,
+J := ˜j ∧ dt − ˜ρ,
+2
+
+where h is the 1-form corresponding to the magnetic field H, ˜d is the 2-form corresponding to
+the electric flux density D, ˜j is the 2-form corresponding to the given electric current density
+j and ˜ρ is the 3-form corresponding to the given charge density ρ.
+Then, Maxwell’s equations can be written, using the four-dimensional exterior derivative,
+as
+d F
+=
+0,
+d G
+=
+J ,
+G
+=
+⋆ϵ,−µ−1F.
+�
+�
+�
+(1)
+For the derivation of these equations, consider [37]. Here, the operator ⋆ϵ,−µ−1 is a weighted
+Hodge star operator.
+In R4 with the Euclidean metric, ϵ is the weight in the direction
+⋆(dx01, dx02, dx03)⊤ and (−µ−1) in the direction ⋆(dx23, dx31, dx12)⊤.
+Since we assume that the domain is star-like with respect to a ball, the Poincaré lemma
+[24, Theorem 4.1] is applicable. Hence, the closed form F is exact and there is a potential A,
+which is a 1-form such that dA = F. From this we also derive the following relations in the
+Euclidean metric
+E = −∂tA + ∇xA0,
+B = curlx A,
+where A0 is the time component and A := (A1, A2, A3)⊤ the spatial component of A. If we
+insert this result into the second equation of (1), combined with the third of (1), we get
+d ⋆ϵ,−µ−1 dA = J .
+(2)
+Note that the potential A is not unique. Adding to A the exterior derivative of any 0-form φ is
+again a suitable potential ˜
+A = A + dφ which solves equation (2). This is equivalent to adding
+the space-time gradient ∇(t,x)φ of a function φ ∈ H1
+0(Q) to (A0, A)⊤, with the usual Sobolev
+space H1
+0(Q). In physics, this is called the gauge invariance, which ’is a manifestation of (the)
+nonobservability of A’, see [21, p. 676]. To make the potential A unique, we use gauging.
+A survey of different gauges can be found in [21]. The choice of the gauge determines the
+resulting partial differential equation.
+To derive the vectorial wave equation from (2), we
+consider the Weyl gauge, also called the temporal gauge, where the component of A in the
+time direction is zero, i.e. A0 = 0.
+Before we state the vectorial wave equation in terms of the Euclidean metric, we introduce
+some notation that is used in the rest of the paper. First, for d = 2, 3, the spatial bounded
+Lipschitz domain Ω ⊂ Rd with boundary ∂Ω admits the outward unit normal nx. Second, for
+a given terminal time T > 0, we define the space-time cylinder Q := (0, T) × Ω ⊂ Rd+1 and
+its lateral boundary Σ := [0, T] × ∂Ω ⊂ Rd+1. Next, we fix some notation, which differs for
+d = 2 and d = 3.
+To consider problems for d = 2, i.e. Ω ⊂ R2, we define a × b = a1b2 − a2b1 for vectors a =
+(a1, a2)⊤, b = (b1, b2)⊤ ∈ R2. With this, we set the curl of a sufficiently smooth vector-valued
+function v: Ω → R2 with v = (v1, v2)⊤ as the scalar function curlx v = ∇x×v = ∂x1v2−∂x2v1.
+This curl operator is sometimes called rot. In addition, for a sufficiently smooth scalar function
+w: Ω → R, we define the vector-valued function curlx w = (∂x2w, −∂x1w)⊤. Note that the
+vector-valued curl is the adjoint operator of the scalar-valued curl operator.
+In the case of d = 3, i.e. Ω ⊂ R3, the curl of a sufficiently smooth vector-valued function
+v: Ω → R3 with v = (v1, v2, v3)⊤ is given by the vector-valued function curlx v = ∇x × v =
+3
+
+(∂x2v3 − ∂x3v2, ∂x3v1 − ∂x1v3, ∂x1v2 − ∂x2v1)⊤ , where × denotes the usual cross product of
+vectors in R3.
+In the whole work, the curl acts only with respect to the spatial variable x ∈ Ω for functions
+in (t, x) ∈ Q ⊂ Rd+1, d = 2, 3.
+With this notation and the Weyl gauge, we rewrite equation (2) into the vectorial wave
+equation for d = 2, 3 simply by inserting the definition of the exterior derivative in the Eu-
+clidean metric following [2, Chapter 2]. Hence, we want to find a function A: Q → Rd such
+that
+∂t (ϵ∂tA) + curlx
+�
+µ−1 curlx A
+�
+=
+j
+in Q,
+A(0, ·)
+=
+0
+in Ω,
+∂tA(0, ·)
+=
+0
+in Ω,
+γtA
+=
+0
+on Σ,
+�
+�
+�
+�
+�
+�
+�
+(3)
+where γt is the tangential trace operator, j : Q → Rd is a given current density, ϵ: Ω → Rd×d
+is a given permittivity, and µ: Ω → R for d = 2 and µ: Ω → R3×3 for d = 3 is a given
+permeability. However, equation (2) is a four-dimensional equation for a four-dimensional
+vector potential (A0, A)⊤. The fourth equation in (2) is divx(ϵ(∂tA)) = −ρ in Q. On the
+other hand, this equation holds as long as ∂tA satisfies the initial condition divx(ϵ∂tA)(0, ·) =
+−ρ(0, ·) in Ω, see [17] for more details. So, in case of ∂tA(0, ·) = 0 in Ω we are limited to
+examples where ρ(0, ·) = 0 in Ω. However, using the same technique used for inhomogeneous
+Dirichlet boundary conditions in finite element methods for the Poisson’s equation, see e.g. [14],
+we can extend the results of this paper to inhomogeneous initial data.
+With this knowledge, we give an outline of this paper.
+In Section 2, we recall well-
+known function spaces, which are needed to state the trial and test spaces of the variational
+formulations. In Section 3, we introduce the modified Hilbert transformation which allows
+for a Galerkin–Bubnov formulation. Then, we state different variational formulations and
+their properties in Section 4. In Section 5, we define the finite element spaces that we use for
+the space-time finite element discretizations in Section 6, where we also examine a resulting
+CFL condition. Numerical examples for a two-dimensional spatial domain are presented in
+Section 7, which show the sharpness of the CFL condition and the different behavior of the
+numerical solutions when changing the projection of the right-hand side. Finally, we draw
+conclusions in Section 8.
+2
+Classical function spaces
+For completeness, we recall classical function spaces, see [3, 13, 30, 43] for further details and
+references. First, for a bounded Lipschitz domain D ⊂ Rm with m ∈ N, the space L2(D) is
+the classical Lebesgue spaces for real-valued functions v: D → R with the standard Hilber-
+tian norm ∥·∥L2(D).
+Second, the Lebesgue space L∞(D) of measurable functions bounded
+almost everywhere is equipped with the usual norm ∥·∥L∞(D). Third, for k ∈ N, the space
+Hk(D) ⊂ L2(D) is the classical Sobolev space with the Hilbertian norm ∥·∥Hk(D). The sub-
+space H1
+0(D) := {v ∈ H1(D) : v|∂D = 0} ⊂ H1(D) is endowed with the Hilbertian norm
+∥·∥H1
+0(D) := |·|H1(D) := ∥∇x(·)∥L2(D), which is actually a norm in H1
+0(D) due to the Poincaré
+inequality.
+For the interval (0, T), we write L2(0, T) := L2((0, T)), Hk(0, T) := Hk((0, T)), and the
+4
+
+subspaces
+H1
+0,(0, T) := {v ∈ H1(0, T) : v(0) = 0} ⊂ H1(0, T),
+H1
+,0(0, T) := {v ∈ H1(0, T) : v(T) = 0} ⊂ H1(0, T)
+are equipped with the Hilbertian norm |·|H1(0,T) := ∥∂t(·)∥L2(0,T), which again, is actually
+a norm in H1
+0,(0, T) and H1
+,0(0, T) due to Poincaré inequalities. By interpolation, we intro-
+duce Hs
+0,(0, T) := [H1
+0,(0, T), L2(0, T)]s and Hs
+,0(0, T) := [H1
+,0(0, T), L2(0, T)]s for s ∈ [0, 1].
+Further, the aforementioned spaces are generalized from real-valued functions v: D → R
+to vector-valued functions v: D → X for a Hilbert space X with inner product (·, ·)X. In
+particular, for X = Rn with n ∈ N, the space L2(D; Rn) is the usual Lebesgue space for vector-
+valued functions v: D → Rn endowed with the inner product (v, w)L2(D) := (v, w)L2(D;Rn) :=
+�
+D(v(x), w(x))Rndx for v, w ∈ L2(D; Rn) and the induced norm ∥·∥L2(D) := ∥·∥L2(D;Rn) :=
+�
+(·, ·)L2(D). Analogously, the vector-valued spaces H1
+0,(0, T; X) and H1
+,0(0, T; X) are intro-
+duced over a Hilbert space X.
+With this notation, for a bounded Lipschitz domain Ω ⊂ Rd with d ∈ {2, 3}, we define the
+vector-valued Sobolev spaces
+H(div; Ω) :=
+�
+v ∈ L2(Ω; Rd) : divx v ∈ L2(Ω; R)
+�
+endowed with the Hilbertian norm
+∥v∥H(div;Ω) :=
+�
+∥v∥2
+L2(Ω) + ∥divx v∥2
+L2(Ω)
+�1/2
+,
+where divx is the (distributional) divergence. Similarly, with the (distributional) curl, we set
+H(curl; Ω) :=
+��
+v ∈ L2(Ω; R2) : curlx v ∈ L2(Ω; R)
+�
+,
+d = 2,
+�
+v ∈ L2(Ω; R3) : curlx v ∈ L2(Ω; R3)
+�
+,
+d = 3,
+equipped with their natural Hilbertian norm
+∥v∥H(curl;Ω) :=
+�
+∥v∥2
+L2(Ω) + ∥curlx v∥2
+L2(Ω)
+�1/2
+.
+Next, we introduce the tangential trace operator γt, see [3], [13, Section 4.3], [15, The-
+orem 2.11] and [30, Subsection 3.5.3] for details.
+For d = 3, we define the continuous
+mapping γt : H(curl; Ω) → H−1/2(∂Ω; R3) as unique extension of the vector-valued function
+γtv = v|∂Ω ×nx for v ∈ H1(Ω; R3), which is defined by the Green’s identity for the curl opera-
+tor. Analogously, for d = 2, we set the continuous mapping γt : H(curl; Ω) → H−1/2(∂Ω; R) as
+unique extension of the scalar function γtv = v|∂Ω · τ x for v ∈ H1(Ω; R2), where τ x is the unit
+tangent vector, i.e. τ x · nx = 0. Here, H−1/2(∂Ω; Rm) is the dual space of the fractional-order
+Sobolev space H1/2(∂Ω; Rm), and (·)|∂Ω is the usual trace operator mapping from H1(Ω; Rm)
+onto H1/2(∂Ω; Rm) for m ∈ N. Note that the tangential trace is not surjective for d = 3. For
+a detailed discussion about the range of the different tangential traces in H(curl; Ω) for d = 3,
+consider [6]. Having the tangential trace operator γt, we introduce the closed subspace
+H0(curl; Ω) := {v ∈ H(curl; Ω) : γtv = 0} ⊂ H(curl; Ω)
+with the Hilbertian norm ∥·∥H0(curl;Ω) := ∥·∥H(curl;Ω). Last, we recall that the set C∞
+0 (Ω; Rd)
+of smooth functions with compact support is dense in H0(curl; Ω) with respect to ∥·∥H0(curl;Ω),
+see [15, Theorem 2.12].
+5
+
+3
+Modified Hilbert transformation
+In this section, we introduce the modified Hilbert transformation HT as developed in [35, 43],
+and state its main properties, see also [34, 36, 44, 45]. The modified Hilbert transformation
+acts in time only.
+Hence, in this section, we consider functions v: (0, T) → R, where a
+generalization to functions in (t, x) is straightforward because of the tensor product structure
+of the domain Q.
+For v ∈ L2(0, T), we consider the Fourier series expansion
+v(t) =
+∞
+�
+k=0
+vk sin
+��π
+2 + kπ
+� t
+T
+�
+,
+vk := 2
+T
+� T
+0
+v(t) sin
+��π
+2 + kπ
+� t
+T
+�
+dt,
+and we define the modified Hilbert transformation HT as
+(HT v)(t) =
+∞
+�
+k=0
+vk cos
+��π
+2 + kπ
+� t
+T
+�
+,
+t ∈ (0, T).
+(4)
+Note that the functions t �→ sin
+�� π
+2 + kπ
+� t
+T
+�
+, k ∈ N0, form an orthogonal basis of L2(0, T)
+and H1
+0,(0, T), whereas the functions t �→ cos
+�� π
+2 + kπ
+� t
+T
+�
+, k ∈ N0, form an orthogonal basis
+of L2(0, T) and H1
+,0(0, T). Hence, the mapping HT : Hs
+0,(0, T) → Hs
+,0(0, T) is an isomorphism
+for s ∈ [0, 1], where the inverse is the L2(0, T)-adjoint, i.e.
+(HT v, w)L2(0,T) = (v, H−1
+T w)L2(0,T)
+(5)
+for all v, w ∈ L2(0, T). In addition, the relations
+(v, HT v)L2(0,T) > 0
+for 0 ̸= v ∈ Hs
+0,(0, T), 0 < s ≤ 1,
+(6)
+∂tHT v = −H−1
+T ∂tv in L2(0, T)
+for v ∈ H1
+0,(0, T)
+(7)
+hold true. For the proofs of these aforementioned properties, we refer to [35, 43, 34, 36, 44,
+45]. Furthermore, the modified Hilbert transformation (4) allows a closed representation [35,
+Lemma 2.8] as Cauchy principal value integral, i.e. for v ∈ L2(0, T),
+(HT v)(t) = v.p.
+� T
+0
+1
+2T
+�
+1
+sin π(s+t)
+2T
++
+1
+sin π(s−t)
+2T
+�
+v(s) ds,
+t ∈ (0, T).
+Further integral representations of HT are contained in [36, 45].
+4
+Variational formulations
+In this section, we state space-time variational formulations of the vectorial wave equation (3).
+First, we recall a variational setting with different trial and test spaces. Second, we introduce a
+new space-time variational formulation of the vectorial wave equation (3) with equal trial and
+test spaces. The second space-time variational formulation is derived by using the modified
+Hilbert transformation HT for the temporal part as introduced in Section 3.
+Before we state these variational formulations, however, we make the following assump-
+tions, which are assumed for the rest of this work.
+6
+
+Assumption 4.1. Let the spatial domain Ω ⊂ Rd, d = 2, 3, be given such that
+• Ω is a bounded Lipschitz domain,
+• and Q := (0, T) × Ω is a star-like domain with respect to a ball B, i.e. the convex hull
+of B and each point in Q is contained in Q.
+Further, let j, ϵ and µ be given functions, which fulfill:
+• The function j : Q → Rd satisfies j ∈ L2(Q; Rd).
+• The permittivity ϵ: Ω → Rd×d is symmetric, bounded, i.e. ϵ ∈ L∞(Ω; Rd×d), and uni-
+formly positive definite, i.e.
+ess inf
+x∈Ω
+inf
+0̸=ξ∈Rd
+ξ⊤ϵ(x)ξ
+ξ⊤ξ
+> 0.
+• For d = 2, the permeability µ: Ω → R satisfies µ ∈ L∞(Ω; R) and
+ess inf
+x∈Ω µ(x) > 0.
+For d = 3, the permeability µ: Ω → R3×3 is symmetric, bounded, i.e. µ ∈ L∞(Ω; R3×3),
+and uniformly positive definite, i.e.
+ess inf
+x∈Ω
+inf
+0̸=ξ∈R3
+ξ⊤µ(x)ξ
+ξ⊤ξ
+> 0.
+Note that in Assumption 4.1, the given functions j, ϵ and µ are real-valued and that the
+functions ϵ, µ do not depend on the temporal variable t.
+4.1
+Different trial and test spaces
+In this subsection, we derive a space-time variational formulation of the vectorial wave equa-
+tion (3), where the trial and test spaces are different. With the aforementioned assumptions
+on the functions ϵ and µ the function spaces L2(Ω; Rd) and H0(curl; Ω) are endowed with the
+inner products
+(v, w)L2ϵ(Ω) :=
+�
+Ω
+(ϵ(x)v(x), w(x))Rd dx,
+v, w ∈ L2(Ω; Rd),
+and
+(v, w)H0,ϵ,µ(curl;Ω) := (v, w)L2ϵ(Ω) + (v, w)H0,µ(curl;Ω),
+v, w ∈ H0(curl; Ω),
+respectively, where
+(v, w)H0,µ(curl;Ω) :=
+�
+Ω
+�
+µ(x)−1 curlx v(x), curlx w(x)
+�
+Rd dx,
+v, w ∈ H0(curl; Ω).
+The norms ∥·∥L2ϵ(Ω) and ∥·∥H0,ϵ,µ(curl;Ω), induced by (·, ·)L2ϵ(Ω) and (·, ·)H0,ϵ,µ(curl;Ω), respec-
+tively, are equivalent to the standard norms ∥·∥L2(Ω) and ∥·∥H0(curl;Ω) due to Assumption 4.1.
+Moreover, we introduce the seminorm |·|H0,µ(curl;Ω) :=
+�
+(·, ·)H0,µ(curl;Ω).
+7
+
+Next, we define the space-time Sobolev spaces, which are used for the variational formu-
+lation. We consider
+Hcurl;1
+0;0,
+(Q) :=L2(0, T; H0(curl; Ω)) ∩ H1
+0,(0, T; L2(Ω; Rd)),
+(8)
+Hcurl;1
+0;,0
+(Q) :=L2(0, T; H0(curl; Ω)) ∩ H1
+,0(0, T; L2(Ω; Rd)),
+(9)
+which are endowed with the Hilbertian norm
+|v|Hcurl;1(Q) := ∥v∥Hcurl;1
+0;0,
+(Q) := ∥v∥Hcurl;1
+0;,0
+(Q)
+:=
+�� T
+0
+∥∂tv(t, ·)∥2
+L2ϵ(Ω)dt +
+� T
+0
+|v(t, ·)|2
+H0,µ(curl;Ω) dt
+�1/2
+.
+Note that |·|Hcurl;1(Q) is actually a norm in Hcurl;1
+0;0,
+(Q) and Hcurl;1
+0;,0
+(Q), since the Poincaré
+inequality
+∥v∥L2(Q) ≤ 2T
+π ∥∂tv∥L2(Q)
+holds true for all v ∈ H1
+0,(0, T; L2(Ω; Rd)) and for all v ∈ H1
+,0(0, T; L2(Ω; Rd)). This Poincaré
+inequality follows from
+∥v∥2
+L2(Q) =
+d
+�
+j=1
+�
+Ω
+∥vj(·, x)∥2
+L2(0,T)dx ≤ 4T 2
+π2
+d
+�
+j=1
+�
+Ω
+∥∂tvj(·, x)∥2
+L2(0,T)dx = 4T 2
+π2 ∥∂tv∥2
+L2(Q)
+for all v = (v1, . . . , vd)⊤ with v ∈ H1
+0,(0, T; L2(Ω; Rd)) or v ∈ H1
+,0(0, T; L2(Ω; Rd)), where [43,
+Lemma 3.4.5] is applied.
+With these space-time Sobolev spaces, we motivate a space-time variational formulation
+of the vectorial wave equation (3). To derive this variational formulation, we multiply the
+vectorial wave equation (3) by a test function v, integrate over the space-time domain Q and
+then use integration by parts with respect to space and time. This leads to the following
+variational formulation:
+Find A ∈ Hcurl;1
+0;0,
+(Q) such that
+− (ϵ∂tA, ∂tv)L2(Q) + (µ−1 curlx A, curlx v)L2(Q) = (j, v)L2(Q)
+(10)
+for all v ∈ Hcurl;1
+0;,0
+(Q).
+Note that the first initial condition A(0, ·) = 0 is fulfilled in the strong sense, whereas the
+second initial condition ∂tA(0, ·) = 0 is incorporated in a weak sense in the right-hand side of
+(10). On the other hand, the boundary condition γtA = 0 is satisfied in the strong sense with
+A(t, ·) ∈ H0(curl; Ω) for almost all t ∈ (0, T).
+The unique solvability of the variational formulation (10) is proven in [18], [16, Theo-
+rem 3.8], and in [17] as a special case of Theorem 2, which is summarized in the next theorem.
+Theorem 4.2. Let Assumption 4.1 be satisfied. Then, a unique solution A ∈ Hcurl;1
+0;0,
+(Q) of
+the variational formulation (10) exists and the stability estimate
+|A|Hcurl;1(Q) ≤ T
+��j
+��
+L2(Q)
+holds true.
+Note that the trial and test spaces of the variational formulation (10) are different. To
+get equal trial and test spaces, the modified Hilbert transformation HT of Section 3 is used,
+which is investigated in the next two subsections.
+8
+
+4.2
+Basis representations
+In this subsection, we recall basis representations of functions in L2(Ω; Rd) and H0(curl; Ω).
+These representations are needed to define the modified Hilbert transformation acting on
+the space-time function space Hcurl;1(Q). To derive the basis representations, we define the
+subspace
+H(div ϵ0; Ω) :=
+�
+f ∈ L2(Ω; Rd) : divx(ϵf) = 0
+�
+⊂ L2(Ω; Rd)
+endowed with the weighted inner product (·, ·)L2ϵ(Ω), whereas the subspace
+X0,ϵ(Ω) := H0(curl; Ω) ∩ H(div ϵ0; Ω) ⊂ H0(curl; Ω)
+is equipped with the inner product (·, ·)H0,ϵ,µ(curl;Ω). With this notation, we recall a crucial
+decomposition result, the Helmholtz-Weyl decomposition of L2(Ω; Rd) and H0(curl; Ω).
+Lemma 4.3. Let Assumption 4.1 be satisfied. Then, the orthogonal decomposition
+L2(Ω; Rd) = ∇xH1
+0(Ω) ⊕ H(div ϵ0; Ω)
+(11)
+is valid, where the orthogonality holds true with respect to (·, ·)L2ϵ(Ω). Moreover, the orthogonal
+decomposition
+H0(curl; Ω) = ∇xH1
+0(Ω) ⊕ X0,ϵ(Ω)
+(12)
+is true, where the orthogonality holds true with respect to (·, ·)H0,ϵ,µ(curl;Ω), (·, ·)H0,µ(curl;Ω) and
+(·, ·)L2ϵ(Ω).
+Proof. For d = 3, the result is stated in [3, Proposition 7.4.3]. Additionally, for d ∈ {2, 3}, the
+decompositions follow from properties of the corresponding de Rham complexes, see e.g. [2,
+Section 2.3], [32, Lemma 2.7, Lemma 3.6, Theorem 5.5].
+Remark 4.4. Note that Lemma 4.3 also yields that ∇xH1
+0(Ω), H(div ϵ0; Ω) are closed sub-
+spaces of L2(Ω; Rd) and ∇xH1
+0(Ω), X0,ϵ(Ω) are closed subspaces of H0(curl; Ω).
+To define the modified Hilbert transformation on space-time function spaces, we need a
+basis representation of the underlying spatial Hilbert spaces. For that purpose, we consider
+the orthonormal basis {φi ∈ H1
+0(Ω) : i ∈ N} of H1
+0(Ω) with respect to the inner product
+�
+∇x(·), ∇x(·)
+�
+L2ϵ(Ω). This orthonormal basis fulfills the condition
+∀v ∈ H1
+0(Ω) :
+(∇xφi, ∇xv)L2ϵ(Ω) = λi(φi, v)L2(Ω),
+∥∇xφi∥L2ϵ(Ω) = 1,
+i ∈ N,
+for a nondecreasing sequence of related eigenvalues λi > 0, satisfying λi → ∞ as i → ∞, see
+[23, Section 4 in Chapter 4]. Additionally, we investigate the orthonormal basis {ei ∈ X0,ϵ(Ω) :
+i ∈ N0} of H(div ϵ0; Ω) with respect to (·, ·)L2ϵ(Ω), fulfilling
+∀v ∈ X0,ϵ(Ω) :
+(ei, v)H0,ϵ,µ(curl;Ω) = σi(ei, v)L2ϵ(Ω),
+∥ei∥L2ϵ(Ω) = 1,
+i ∈ N0,
+(13)
+with a nondecreasing sequence of related eigenvalues σi ≥ 1, satisfying σi → ∞ as i → ∞, see
+[3, Theorem 8.2.4]. For i ∈ N0, the property σi ≥ 1 follows from
+0 ≤ (ei, ei)H0,µ(curl;Ω) = (ei, ei)H0,ϵ,µ(curl;Ω) − (ei, ei)L2ϵ(Ω) = (σi − 1)(ei, ei)L2ϵ(Ω) = σi − 1,
+9
+
+where we use the variational formulation (13) for v = ei.
+Moreover, the set {σ−1/2
+i
+ei ∈
+X0,ϵ(Ω) : i ∈ N0} is an orthonormal basis of X0,ϵ(Ω) with respect to (·, ·)H0,ϵ,µ(curl;Ω) by
+construction and since X0,ϵ(Ω) ⊂ H(div ϵ0; Ω). Note that the set {ei ∈ X0,ϵ(Ω) : i ∈ N0} is
+also orthogonal with respect to (·, ·)H0,µ(curl;Ω).
+Next, we use the orthogonal decomposition of L2(Ω; Rd), given in (11), to construct an
+orthonormal basis of L2(Ω; Rd). Then, the desired orthonormal basis of L2(Ω; Rd) with respect
+to (·, ·)L2ϵ(Ω) is the set {φi ∈ H0(curl; Ω) : i ∈ Z} with
+φi =
+�
+ei,
+i ∈ N0,
+∇xφ−i,
+−i ∈ N.
+(14)
+Analogously, we construct an orthogonal basis of H0(curl; Ω) by using the orthogonal decom-
+position of H0(curl; Ω) given in (12). Then, the orthogonal basis of H0(curl; Ω) with respect
+to (·, ·)H0,ϵ,µ(curl;Ω) is again the set {φi ∈ H0(curl; Ω) : i ∈ Z} in (14). Note that the set
+{φi ∈ H0(curl; Ω) : i ∈ Z} from (14) is also orthogonal with respect to (·, ·)H0,µ(curl;Ω).
+The basis {φi ∈ H0(curl; Ω) : i ∈ Z} in (14) leads to the norm representation through
+Parseval’s identity
+∥v∥2
+L2ϵ(Ω) =
+∞
+�
+i=−∞
+|vi|2
+(15)
+for v ∈ L2(Ω; Rd), with the coefficients vi = (v, φi)L2ϵ(Ω), i ∈ Z, and the basis representation
+v(x) =
+∞
+�
+i=−∞
+viφi(x),
+x ∈ Ω,
+which converges in L2(Ω; Rd). Analogously, for w ∈ H0(curl; Ω), the seminorm |·|H0,µ(curl;Ω)
+and the norm ∥·∥H0,ϵ,µ(curl;Ω) admit the representations
+|w|2
+H0,µ(curl;Ω) =
+∞
+�
+i=0
+(σi − 1)
+� �� �
+≥0
+|wi|2 ,
+∥w∥2
+H0,ϵ,µ(curl;Ω) =
+∞
+�
+i=0
+σi
+����
+≥1
+|wi|2 +
+∞
+�
+i=1
+|w−i|2 ,
+(16)
+where the representation
+w(x) =
+∞
+�
+i=−∞
+wiφi(x),
+x ∈ Ω,
+converges in H0(curl; Ω) with the coefficients wi = (w, φi)L2ϵ(Ω), i ∈ Z.
+4.3
+Equal trial and test spaces
+The goal of this subsection is to state a space-time variational formulation of the vectorial wave
+equation (3) with equal trial and test spaces. For this purpose, we extend the definition of the
+modified Hilbert transformation HT of Section 3 to the vector-valued functions in Hcurl,1
+0;0,
+(Q)
+and Hcurl,1
+0;,0
+(Q), where the extension is denoted again by HT . More precisely, we define the
+mapping HT : L2(Q; Rd) → L2(Q; Rd) by
+(HT v)(t, x) :=
+∞
+�
+i=−∞
+∞
+�
+k=0
+vi,k cos
+��π
+2 + kπ
+� t
+T
+�
+φi(x),
+(t, x) ∈ Q,
+(17)
+10
+
+where the given function v ∈ L2(Q; Rd) is represented by
+v(t, x) =
+∞
+�
+i=−∞
+∞
+�
+k=0
+vi,k sin
+��π
+2 + kπ
+� t
+T
+�
+φi(x),
+(t, x) ∈ Q,
+(18)
+with the spatial eigenfunctions φi defined in (14). The inverse H−1
+T : L2(Q; Rd) → L2(Q; Rd)
+is defined by
+(H−1
+T w)(t, x) =
+∞
+�
+i=−∞
+∞
+�
+k=0
+wi,k sin
+��π
+2 + kπ
+� t
+T
+�
+φi(x),
+(t, x) ∈ Q,
+where the given function w ∈ L2(Q; Rd) is represented by
+w(t, x) =
+∞
+�
+i=−∞
+∞
+�
+k=0
+wi,k cos
+��π
+2 + kπ
+� t
+T
+�
+φi(x),
+(t, x) ∈ Q.
+(19)
+Next, we prove properties of this definition of the modified Hilbert transformation HT . These
+properties are then used for the derivation of a Galerkin–Bubnov method for the space-time
+variational formulation of the vectorial wave equation (3). Recall the definition of the vector-
+valued space-time Sobolev spaces Hcurl;1
+0;0,
+(Q), Hcurl;1
+0;,0
+(Q) defined in (8), (9).
+Lemma 4.5. The mapping HT : Hcurl;1
+0;0,
+(Q) → Hcurl;1
+0;,0
+(Q) is bijective and norm preserving,
+i.e. |v|Hcurl;1(Q) = |HT v|Hcurl;1(Q) for all v ∈ Hcurl;1
+0;0,
+(Q).
+Proof. We follow the arguments in [43, Subsection 3.4.5]. Let v ∈ Hcurl;1
+0;0,
+(Q) be fixed with
+the Fourier representations (18). The relation HT v ∈ Hcurl;1
+0;,0
+(Q) and the bijectivity follow
+by definition. To prove the equality |v|Hcurl;1(Q) = |HT v|Hcurl;1(Q), we use the representations
+(15), (16) of the spatial norms and analogous representations of the temporal norms.
+As HT acts only with respect to time, the properties (5), (7) of Section 3 remain valid.
+For completeness, we prove the following lemma.
+Lemma 4.6. The relations
+(ϵHT v, w)L2(Q) = (ϵv, H−1
+T w)L2(Q)
+(20)
+for all v ∈ L2(Q; Rd) and w ∈ L2(Q; Rd), and
+H−1
+T ∂tv = −∂tHT v
+(21)
+in L2(Q; Rd) for all v ∈ H1
+0,(0, T; L2(Ω; Rd)) are true.
+Proof. First, we prove property (20). Let v ∈ L2(Q; Rd) and w ∈ L2(Q; Rd) be fixed with the
+Fourier representations (18) and (19), respectively. Using the orthogonality properties of the
+involved basis functions yield
+(ϵHT v, w)L2(Q) = T
+2
+∞
+�
+i=−∞
+∞
+�
+k=0
+vi,kwi,k = (ϵv, H−1
+T w)L2(Q),
+11
+
+and thus, property (20).
+Second, to prove property (21), fix again v ∈ H1
+0,(0, T; L2(Ω; Rd)) with the Fourier repre-
+sentations (18), which converges in H1
+0,(0, T; L2(Ω; Rd)). Thus, the Fourier series
+H−1
+T ∂tv(t, x) = 1
+T
+∞
+�
+i=−∞
+∞
+�
+k=0
+vi,k
+�π
+2 + kπ
+�
+sin
+��π
+2 + kπ
+� t
+T
+�
+φi(x),
+(t, x) ∈ Q,
+is convergent with respect to L2(Q; Rd). Analogously, as the Fourier series (17) converges in
+H1
+,0(0, T; L2(Ω; Rd)), the series
+∂tHT v(t, x) = − 1
+T
+∞
+�
+i=−∞
+∞
+�
+k=0
+vi,k
+�π
+2 + kπ
+�
+sin
+��π
+2 + kπ
+� t
+T
+�
+φi(x),
+(t, x) ∈ Q,
+is also convergent in L2(Q; Rd). Comparing these Fourier representations gives the equality
+H−1
+T ∂tv = −∂tHT v in L2(Q; Rd).
+Last, we need the following result.
+Lemma 4.7. For v ∈ L2(Q; Rd), the equality
+HT (ϵv) = ϵHT v
+in L2(Q; Rd)
+holds true.
+Proof. Let v ∈ L2(Q; Rd) be a fixed function. With the normalized eigenfunction ψk(t) =
+�
+2
+T sin
+��
+π
+2 + kπ
+�
+t
+T
+�
+, we have the representations
+(ϵv)(t, x) =
+∞
+�
+i=−∞
+∞
+�
+k=0
+(ϵϵv, ψkφi)L2(Q)ψk(t)φi(x),
+(t, x) ∈ Q,
+and
+ϵ(x)v(t, x) =
+∞
+�
+i=−∞
+∞
+�
+k=0
+(ϵv, ψkφi)L2(Q)ψk(t)ϵ(x)φi(x),
+(t, x) ∈ Q,
+which leads to the equality
+∀k ∈ N0 :
+∞
+�
+i=−∞
+(ϵϵv, ψkφi)L2(Q)φi(x) =
+∞
+�
+i=−∞
+(ϵv, ψkφi)L2(Q)ϵ(x)φi(x)
+for almost all x ∈ Ω. Then we derive
+ϵ(x)(HT v)(t, x) =
+�
+T
+2
+∞
+�
+k=0
+cos
+��π
+2 + kπ
+� t
+T
+�
+∞
+�
+i=−∞
+(ϵv, ψkφi)L2(Q)ϵ(x)φi(x)
+=
+�
+T
+2
+∞
+�
+k=0
+cos
+��π
+2 + kπ
+� t
+T
+�
+∞
+�
+i=−∞
+(ϵϵv, ψkφi)L2(Q)φi(x) = HT (ϵv)(t, x)
+for almost all (t, x) ∈ Q, and hence, the assertion.
+12
+
+With the mapping HT : Hcurl;1
+0;0,
+(Q) → Hcurl;1
+0;,0
+(Q), the variational formulation (10) is equiv-
+alent to:
+Find A ∈ Hcurl;1
+0;0,
+(Q) such that
+− (ϵ∂tA, ∂tHT v)L2(Q) + (µ−1 curlx A, curlx HT v)L2(Q) = (j, HT v)L2(Q)
+(22)
+for all v ∈ Hcurl;1
+0;0,
+(Q).
+Next, we rewrite the variational formulation (22) using the properties (20), (21) to get:
+Find A ∈ Hcurl;1
+0;0,
+(Q) such that
+(ϵHT ∂tA, ∂tv)L2(Q) + (µ−1 curlx A, curlx HT v)L2(Q) = (j, HT v)L2(Q)
+(23)
+for all v ∈ Hcurl;1
+0;0,
+(Q), where also Lemma 4.7 is applied. The variational formulation (23)
+admits a unique solution, as the equivalent variational formulation (10) is uniquely solvable,
+see Theorem 4.2, and due to the fact that HT : Hcurl;1
+0;0,
+(Q) → Hcurl;1
+0;,0
+(Q) is an isometry, see
+Lemma 4.5.
+5
+Finite element spaces
+In this section, we state the finite element spaces, which are used for discretizations of the
+variational formulations in Section 4. For this purpose, we consider a temporal mesh T t and a
+spatial mesh T x and define related finite element spaces, where we assume that the bounded
+Lipschitz domain Ω ⊂ Rd is polygonal for d = 2, or polyhedral for d = 3.
+First, for a parameter α ∈ N0, we define the temporal mesh as a set T t := T t
+α = {τℓ}Nt
+ℓ=1
+by the decomposition
+0 = t0 < t1 < · · · < tNt−1 < tNt = T
+of the time interval (0, T), where Nt is the number of temporal elements τℓ = (tℓ−1, tℓ) ⊂ R,
+ℓ = 1, . . . , Nt. The local mesh sizes are ht,ℓ = tℓ − tℓ−1, ℓ = 1, . . . , Nt, whereas the maximal
+mesh size is ht := maxℓ=1,...,Nt ht,ℓ. In this paper, we consider a mesh sequence (T t
+α)α∈N0,
+where α is the level of refinement. We define the space of piecewise constant functions in time
+S0(T t
+α) :=
+�
+vht ∈ L2(0, T) : ∀ℓ ∈ {1, . . . , Nt} : vht|τℓ ∈ P0
+1(τℓ)
+�
+= span{ϕ0
+ℓ}Nt
+ℓ=1,
+with the elementwise characteristic functions ϕ0
+ℓ as basis functions, and the space of piecewise
+linear, globally continuous functions
+S1(T t
+α) :=
+�
+vht ∈ C[0, T] : ∀ℓ ∈ {1, . . . , Nt} : vht|τℓ ∈ P1
+1(τℓ)
+�
+= span{ϕ1
+ℓ}Nt
+ℓ=0
+with the usual nodal basis functions ϕ1
+ℓ, satisfying ϕ1
+ℓ(tκ) = δℓκ for ℓ, κ = 0, . . . , Nt. Here, for
+n ∈ {1, 2, 3}, Pp
+n(B) is the space of polynomials on a subset B ⊂ Rn of global degree at most
+p ∈ N0, and δℓκ is the Kronecker delta. Moreover, we introduce the subspaces
+S1
+0,(T t
+α) := S1(T t
+α) ∩ H1
+0,(0, T) = span{ϕ1
+ℓ}Nt
+ℓ=1,
+S1
+,0(T t
+α) := S1(T t
+α) ∩ H1
+,0(0, T) = span{ϕ1
+ℓ}Nt−1
+ℓ=0 .
+13
+
+Second, the spatial domain Ω ⊂ Rd is decomposed into Nx elements ωi ⊂ Rd for i =
+1, . . . , Nx, satisfying
+Ω =
+Nx
+�
+i=1
+ωi.
+Thus, for a parameter ν ∈ N0, the spatial mesh is the set T x := T x
+ν = {ωi}Nx
+i=1 with local
+mesh sizes hx,i =
+��
+ωi 1dx
+�1/d
+, i = 1, . . . , Nx, the maximal mesh size hx := hx,max(T x
+ν ) :=
+maxi=1,...,Nx hx,i, and the minimal mesh size hx,min(T x
+ν ) := mini=1,...,Nx hx,i. Here, for sim-
+plicity, the spatial elements ωi ⊂ Rd, i = 1, . . . , Nx, are triangles for d = 2 and tetrahedra for
+d = 3.
+In the rest of the paper, we assume that we have a shape-regular, globally quasi-uniform
+sequence (T x
+ν )ν∈N0 of admissible spatial meshes T x
+ν . The shape-regularity of the mesh sequence
+(T x
+ν )ν∈N0 ensures a constant cF > 0 such that
+∀ν ∈ N0 : ∀ω ∈ T x
+ν :
+sup
+x,y∈ω
+∥x − y∥2 ≤ cF rω,
+where ∥·∥2 is the Euclidean norm in Rd, and rω > 0 is the radius of the largest ball that can be
+inscribed in the element ω. The constant cF influences the conditional stability, namely a CFL
+condition. To derive this CFL condition, we also need an inverse inequality for the spatial curl
+operator. The global quasi-uniformity is affecting the inverse inequality. The mesh sequence
+(T x
+ν )ν∈N0 of decompositions of Ω is globally quasi-uniform if a constant cG ≥ 1 exists such
+that
+∀ν ∈ N0 :
+hx,max(T x
+ν )
+hx,min(T x
+ν ) ≤ cG.
+Next, we introduce vector-valued finite element spaces.
+For this purpose, we set the
+polynomial spaces for a spatial element ω ∈ T x
+ν
+RT 0(ω) :=
+�
+v ∈ P1
+d(ω)d : ∀x ∈ ω : v(x) = a + bx with a ∈ Rd, b ∈ R
+�
+and
+N 0
+I (ω) :=
+�
+v ∈ P1
+2(ω)2 : ∀(x1, x2) ∈ ω : v(x1, x2) = a + b · (−x2, x1)⊤ with a ∈ R2, b ∈ R
+�
+for d = 2, and
+N 0
+I (ω) :=
+�
+v ∈ P1
+3(ω)3 : ∀x ∈ ω : v(x) = a + b × x with a ∈ R3, b ∈ R3�
+for d = 3, see [13, Section 14.1, Section 15.1]. With this notation, we define the space of
+vector-valued piecewise constant functions
+S0
+d(T x
+ν ) :=
+�
+vhx ∈ L2(Ω; Rd) : ∀ω ∈ T x
+ν : vhx|ω ∈ P0
+d(ω)d�
+= span{ψ0
+k}N0
+x
+k=1,
+the lowest-order Raviart–Thomas finite element space
+RT 0(T x
+ν ) :=
+�
+vhx ∈ H(div; Ω) : ∀ω ∈ T x
+ν : vhx|ω ∈ RT 0(ω)
+�
+= span{ψRT
+k
+}NRT
+x
+k=1 ,
+14
+
+and the lowest-order Nédélec finite element space of the first kind
+N 0
+I (T x
+ν ) :=
+�
+vhx ∈ H(curl; Ω) : ∀ω ∈ T x
+ν : vhx|ω ∈ N 0
+I (ω)
+�
+,
+where we refer to [13, Section 19.2], [27, Section 3.5.1] or [30, Section 5.5] for more details.
+Here, the N0
+x basis functions ψ0
+k are the componentwise characteristic functions with respect
+to the spatial elements, the NRT
+x
+basis functions ψRT
+k
+are attached to the edges for d = 2 and
+the faces for d = 3 of the spatial mesh T x
+ν . Further, we consider the subspace
+N 0
+I,0(T x
+ν ) := N 0
+I (T x
+ν ) ∩ H0(curl; Ω) = span{ψN
+k }NN
+x
+k=1
+with NN
+x basis functions ψN
+k attached to the edges of the spatial mesh T x
+ν .
+Last, the temporal and spatial meshes T t
+α = {τℓ}Nt
+ℓ=1, T x
+ν = {ωi}Nx
+i=1 lead to a decomposition
+Q = [0, T] × Ω =
+Nt
+�
+ℓ=1
+τℓ ×
+Nx
+�
+i=1
+ωi
+of the space-time cylinder Q ⊂ Rd+1 with Nt · Nx space-time elements, i.e. the Cartesian
+product T t
+α × T x
+ν is a space-time mesh. To this space-time mesh, we relate space-time finite
+element spaces of tensor-product type
+S1
+0,(T t
+α) ⊗ N 0
+I,0(T x
+ν )
+and
+S1
+,0(T t
+α) ⊗ N 0
+I,0(T x
+ν ),
+(24)
+where ⊗ denotes the Hilbert tensor-product. Any function Ah ∈ S1
+0,(T t
+α) ⊗ N 0
+I,0(T x
+ν ) admits
+the representation
+Ah(t, x) =
+Nt
+�
+ℓ=1
+NN
+x
+�
+l=1
+Aℓ
+lϕ1
+ℓ(t)ψN
+l (x),
+(t, x) ∈ Q,
+(25)
+with coefficients Aℓ
+l ∈ R. Further, for a given function f ∈ L2(Q; Rd), we introduce the L2(Q)
+projection Π0
+h : L2(Q; Rd) → S0(T t
+α) ⊗ S0
+d(T x
+ν ) to find Π0
+hf ∈ S0(T t
+α) ⊗ S0
+d(T x
+ν ) such that
+∀wh ∈ S0(T t
+α) ⊗ S0
+d(T x
+ν ) :
+(Π0
+hf, wh)L2(Q) = (f, wh)L2(Q),
+(26)
+and the L2(Q) projection ΠRT ,1
+h
+: L2(Q; Rd) → S1(T t
+α)⊗RT 0(T x
+ν ) to find ΠRT ,1
+h
+f ∈ S1(T t
+α)⊗
+RT 0(T x
+ν ) such that
+∀wh ∈ S1(T t
+α) ⊗ RT 0(T x
+ν ) :
+(ΠRT ,1
+h
+f, wh)L2(Q) = (f, wh)L2(Q).
+(27)
+6
+FEM for the vectorial wave equation
+In this section, we state conforming space-time discretizations for the variational formula-
+tions (10), (23), using the notation of Section 5. For this purpose, let the bounded Lipschitz
+domain Ω ⊂ Rd be polygonal for d = 2, or polyhedral for d = 3. Additionally, we assume
+Assumption 4.1 and the assumptions on the mesh stated in Section 5.
+15
+
+6.1
+Galerkin–Petrov FEM
+In this subsection, we state the conforming discretization of the variational formulation (10),
+using the tensor-product spaces (24), by a Galerkin–Petrov finite element method:
+Find Ah ∈ S1
+0,(T t
+α) ⊗ N 0
+I (T x
+ν ) such that
+− (ϵ∂tAh, ∂twh)L2(Q) + (µ−1 curlx Ah, curlx wh)L2(Q) = (Πhj, wh)L2(Q)
+(28)
+for all wh ∈ S1
+,0(T t
+α) ⊗ N 0
+I (T x
+ν ). Hence, we approximate the solution A ∈ Hcurl;1
+0;0,
+(Q) of the
+variational formulation (10) by
+A(t, x) ≈ Ah(t, x) =
+Nt
+�
+κ=1
+NN
+x
+�
+k=1
+Aκ
+kϕ1
+κ(t)ψN
+k (x),
+(t, x) ∈ Q,
+(29)
+using the representation (25). The operator Πh in (28) is either the L2(Q) projection Π0
+h
+onto the space of piecewise constant functions defined in (26) or the L2(Q) projection ΠRT ,1
+h
+onto piecewise linear functions in time and Raviart–Thomas functions in space given in (27).
+The reason to replace the right-hand side j with Πhj in (28) is the comparison with the
+Galerkin–Bubnov finite element method (46), using the modified Hilbert transformation HT .
+An alternative is the use of formulas for numerical integration applied to (j, wh)L2(Q). Note
+that such formulas for numerical integration have to be sufficiently accurate to preserve the
+convergence rates of the finite element method, see the numerical examples in Section 7.
+The discrete variational formulation (28) is equivalent to the linear system
+(−Aht ⊗ Mhx + Mht ⊗ Ahx)A = J
+(30)
+with the temporal matrices
+Aht[ℓ, κ] = (∂tϕ1
+κ, ∂tϕ1
+ℓ)L2(0,T),
+Mht[ℓ, κ] = (ϕ1
+κ, ϕ1
+ℓ)L2(0,T)
+(31)
+for ℓ = 0, . . . , Nt − 1, κ = 1, . . . , Nt and the spatial matrices
+Ahx[l, k] = (µ−1 curlx ψN
+k , curlx ψN
+l )L2(Ω),
+Mhx[l, k] = (ϵψN
+k , ψN
+l )L2(Ω)
+(32)
+for k, l = 1, . . . , NN
+x . Here, the degrees of freedom are ordered such that the vector of coeffi-
+cients in (29) reads as
+A = (A1, A2, . . . , ANt)⊤ ∈ RNtNN
+x
+(33)
+with
+Aκ = (Aκ
+1, Aκ
+2, . . . , Aκ
+NN
+x )⊤ ∈ RNN
+x
+for κ ∈ {1, . . . , Nt}.
+In the same way, the right-hand side in (30) is given by
+J = (f0, f1, . . . , fNt−1)⊤ ∈ RNtNN
+x ,
+where we write
+fℓ = (fℓ
+1, fℓ
+2, . . . , fℓ
+NN
+x )⊤ ∈ RNN
+x
+for ℓ = 0, . . . , Nt − 1
+with
+fℓ
+l = (Πhj, ϕ1
+ℓψN
+l )L2(Q)
+for ℓ = 0, . . . , Nt − 1, l = 1, . . . , NN
+x .
+(34)
+16
+
+Note that the system matrix of the linear system (30) is sparse and allows for a realiza-
+tion as a two-step method, due to the sparsity pattern of the temporal matrices Aht, Mht
+in (31). Further, this sparsity pattern forms the basis of classical stability analysis of the
+Galerkin–Petrov finite element method (28) in the framework of time-stepping or finite dif-
+ference methods. See [35, Section 5] for the scalar wave equation, where the ideas can be
+extended to the vectorial wave equation, see the discussion in [16, Section 3.5.2] or [18]. Here,
+we skip details and only state that the Galerkin–Petrov finite element method (28) is stable
+if the temporal mesh is uniform with mesh size ht and the CFL condition
+ht <
+�
+12
+cI
+hx
+(35)
+is satisfied with the constant cI > 0 of the spatial inverse inequality
+∀vhx ∈ N 0
+I (T x
+ν ) :
+��curlx vhx
+��2
+L2(Ω) ≤ cIh−2
+x
+��vhx
+��2
+L2(Ω).
+(36)
+Bounds for the constant cI > 0 are given in [16, Lemma A.2] or [17, 18].
+As an example domain, which is also used in Section 7, we consider the unit square
+Ω = (0, 1) × (0, 1) ⊂ R2. Using uniform triangulations with isosceles right triangles as in
+Figure 1 yields the constant cI = 18. This result is proven by adapting the arguments of
+the proof of (36), given in [16, Lemma A.2] or [18], of the general situation to the case of
+isosceles right triangles. More precisely, in the proof of [16, Lemma A.2], the matrix JlJ⊤
+l
+is
+diagonal for isosceles right triangles, where Jl denotes the Jacobian matrix of the geometric
+mapping from the reference element to the physical one. Thus, the eigenvalues of JlJ⊤
+l
+are
+known explicitly. So, in this situation, the CFL condition (35) reads as
+ht <
+�
+12
+18hx ≈ 0.81649658 hx.
+(37)
+Numerical examples indicate the sharpness of the CFL condition (37), see Subsection 7.1.
+x1
+x2
+0
+1
+1
+Figure 1: Uniform triangulations of the unit square with isosceles right triangles.
+6.1.1
+Using the L2(Q) projection Πh = Π0
+h for the right-hand side J
+We present the calculation of the right-hand side J of the linear system (30) when Πh is the
+L2(Q) projection onto S0(T t
+α) ⊗ S0
+d(T x
+ν ) satisfying
+(Π0
+hj, wh)L2(Q) = (j, wh)L2(Q)
+17
+
+for all wh ∈ S0(T t
+α) ⊗ S0
+d(T x
+ν ). Then the projection Π0
+hj is the solution of the linear system
+M0
+ht ⊗ M0
+hx(j1, . . . , jNt)⊤ = (ˆj
+1, . . . , ˆj
+Nt
+)⊤
+(38)
+with the matrices and vectors
+M0
+ht[ℓ, κ] = (ϕ0
+κ, ϕ0
+ℓ)L2(0,T),
+ℓ, κ = 1, . . . , Nt,
+M0
+hx[l, k] = (ψ0
+k, ψ0
+l )L2(Ω),
+l, k = 1, . . . , N0
+x,
+jκ[k] = jκ
+k,
+κ = 1, . . . , Nt, k = 1, . . . , N0
+x,
+ˆj
+κ[k] = (j, ϕ0
+κψ0
+k)L2(Q),
+κ = 1, . . . , Nt, k = 1, . . . , N0
+x,
+through the representation
+Π0
+hj(t, x) =
+Nt
+�
+κ=1
+N0
+x
+�
+k=1
+jκ
+kϕ0
+κ(t)ψ0
+k(x),
+(t, x) ∈ Q.
+Thus, using this representation for the right-hand side (34) of the space-time Galerkin–Petrov
+method (30) yields
+fℓ
+l = (Π0
+hj, ϕ1
+ℓψN
+l )L2(Q) =
+Nt
+�
+κ=1
+N0
+x
+�
+k=1
+jκ
+k (ϕ0
+κ, ϕ1
+ℓ)L2(0,T)
+�
+��
+�
+=M1,0
+ht [ℓ,κ]
+(ψ0
+k, ψN
+l )L2(Ω)
+�
+��
+�
+=MN ,0
+hx [l,k]
+= F[l, ℓ]
+for ℓ = 0, . . . , Nt − 1, l = 1, . . . , NN
+x with the matrix
+F = MN,0
+hx J(M1,0
+ht )⊤ ∈ RNN
+x ×Nt,
+where
+MN,0
+hx [l, k] = (ψ0
+k, ψN
+l )L2(Ω)
+l = 1, . . . , NN
+x , k = 1, . . . , N0
+x,
+(39)
+J[k, κ] = jκ
+k,
+k = 1, . . . , N0
+x, κ = 1, . . . , Nt,
+(40)
+M1,0
+ht [ℓ, κ] = (ϕ0
+κ, ϕ1
+ℓ)L2(0,T),
+ℓ = 0, . . . , Nt − 1, κ = 1, . . . , Nt.
+6.1.2
+Using the L2(Q) projection Πh = ΠRT ,1
+h
+for the right-hand side J
+Analogously to Subsection 6.1.1, we present the calculation of the right-hand side J of the
+linear system (30) when Πh is the L2(Q) projection onto S1(T t
+α) ⊗ RT 0(T x
+ν ) given in (27).
+The projection ΠRT ,1
+h
+j is the solution of the linear system
+M1
+ht ⊗ MRT
+hx (j0, j1, . . . , jNt)⊤ = (ˆj
+0, ˆj
+1, . . . , ˆj
+Nt
+)⊤
+(41)
+with the matrices and vectors
+M1
+ht[ℓ, κ] = (ϕ1
+κ, ϕ1
+ℓ)L2(0,T),
+ℓ, κ = 0, . . . , Nt,
+(42)
+MRT
+hx [l, k] = (ψRT
+k
+, ψRT
+l
+)L2(Ω),
+l, k = 1, . . . , NRT
+x
+,
+jκ[k] = jκ
+k,
+κ = 0, . . . , Nt, k = 1, . . . , NRT
+x
+,
+ˆj
+κ[k] = (j, ϕ1
+κψRT
+k
+)L2(Q),
+κ = 0, . . . , Nt, k = 1, . . . , NRT
+x
+,
+18
+
+by the representation
+ΠRT ,1
+h
+j(t, x) =
+Nt
+�
+κ=0
+NRT
+x�
+k=1
+jκ
+kϕ1
+κ(t)ψRT
+k
+(x),
+(t, x) ∈ Q.
+Thus, using this representation for the right-hand side (34) yields
+fℓ
+l = (ΠRT ,1
+h
+j, ϕ1
+ℓψN
+l )L2(Q) =
+Nt
+�
+κ=0
+NRT
+x�
+k=1
+jκ
+k (ϕ1
+κ, ϕ1
+ℓ)L2(0,T)
+�
+��
+�
+=�
+Mht[ℓ,κ]
+(ψRT
+k
+, ψN
+l )L2(Ω)
+�
+��
+�
+=MN ,RT
+hx
+[l,k]
+= F[l, ℓ]
+for ℓ = 0, . . . , Nt − 1, l = 1, . . . , NN
+x with the matrix
+F = MN,RT
+hx
+J(�
+Mht)⊤ ∈ RNN
+x ×Nt,
+where
+MN,RT
+hx
+[l, k] = (ψRT
+k
+, ψN
+l )L2(Ω)
+l = 1, . . . , NN
+x , k = 1, . . . , NRT
+x
+,
+(43)
+J[k, κ] = jκ
+k,
+k = 1, . . . , NRT
+x
+, κ = 0, . . . , Nt,
+(44)
+�
+Mht[ℓ, κ] = (ϕ1
+κ, ϕ1
+ℓ)L2(0,T),
+ℓ = 0, . . . , Nt − 1, κ = 0, . . . , Nt.
+(45)
+Note that the matrices Mht ∈ RNt×Nt in (31) and �
+Mht ∈ RNt×(Nt+1) in (45) are submatrices
+of the matrix M1
+ht ∈ R(Nt+1)×(Nt+1) in (42).
+6.2
+Galerkin–Bubnov FEM
+In this subsection, we discretize the variational formulation (23) by a tensor-product ansatz,
+using the conforming finite element spaces of Section 5. In greater detail, we consider the
+Galerkin–Bubnov finite element method to find Ah ∈ S1
+0,(T t
+α) ⊗ N 0
+I (T x
+ν ) such that
+(ϵHT ∂tAh, ∂tvh)L2(Q) + (µ−1 curlx Ah, curlx HT vh)L2(Q) = (Πhj, HT vh)L2(Q)
+(46)
+for all vh ∈ S1
+0,(T t
+α) ⊗ N 0
+I (T x
+ν ). Again, Πh is either the L2(Q) projection Π0
+h defined in (26) or
+the L2(Q) projection ΠRT ,1
+h
+given in (27).
+The discrete variational formulation (46) is equivalent to the linear system
+(AHT
+ht ⊗ Mhx + MHT
+ht
+⊗ Ahx)A = J HT
+(47)
+with the spatial matrices Ahx, Mhx given in (32) and the temporal matrices
+AHT
+ht [ℓ, κ] = (HT ∂tϕ1
+κ, ∂tϕ1
+ℓ)L2(0,T),
+MHT
+ht [ℓ, κ] = (ϕ1
+κ, HT ϕ1
+ℓ)L2(0,T)
+(48)
+for ℓ, κ = 1, . . . , Nt, where HT , defined in Section 3, acts solely on time-dependent functions.
+As in Subsection 6.1, we use again the representation (29) of Ah and the vector A ∈ RNtNN
+x
+of its coefficients (33). The right-hand side of the linear system (47) is given by
+J HT = (J 1, J 2, . . . , J Nt)⊤ ∈ RNtNN
+x
+19
+
+with
+J ℓ = (J ℓ
+1 , J ℓ
+2 , . . . , J ℓ
+NN
+x )⊤ ∈ RNN
+x
+for ℓ = 1, . . . , Nt,
+where
+J ℓ
+l = (Πhj, ψN
+l HT ϕ1
+ℓ)L2(Q)
+for ℓ = 1, . . . , Nt, l = 1, . . . , NN
+x .
+(49)
+The temporal matrices AHT
+ht , MHT
+ht
+in (48) are positive definite due to property (6). In
+addition, the spatial matrix Mhx in (32) is also positive definite, whereas the spatial matrix
+Ahx in (32) is only positive semi-definite. Hence, the Kronecker product AHT
+ht ⊗Mhx is positive
+definite. On the other hand, the product MHT
+ht
+⊗ Ahx is positive semi-definite. Adding both
+results in a positive definite system matrix of the linear system (47). In other words, the linear
+system (47) is uniquely solvable. Note that, compared to the static case, we do not need any
+stabilization to get unique solvability of the linear system (47) and hence of the corresponding
+discrete variational formulation (46). However, further details on the numerical analysis of the
+Galerkin–Bubnov finite element method (46) are far beyond the scope of this contribution, we
+refer to [28, 29] for the case of the scalar wave equation.
+In addition, note that the temporal matrices AHT
+ht , MHT
+ht
+in (48) are dense, whereas the
+spatial matrices Ahx, Mhx in (32) are sparse. Thus, the linear system (47) does not allow for a
+realization as a multistep method. However, the application of fast (direct) solvers, as known
+for heat and scalar wave equations [25, 41, 42], is possible, which is the topic of future work.
+6.2.1
+Using the L2(Q) projection Πh = Π0
+h for the right-hand side J HT
+In this subsection, we present the calculation of the right-hand side J HT of the linear sys-
+tem (47) when Πh is the L2(Q) projection onto S0(T t
+α) ⊗ S0
+d(T x
+ν ) given in (26). Since this
+subsection is similar to Subsection 6.1.1, we skip the details.
+The entries (49) admit the representation
+J ℓ
+l = (Π0
+hj, ψN
+l HT ϕ1
+ℓ)L2(Q) = F HT [l, ℓ]
+for ℓ = 1, . . . , Nt, l = 1, . . . , NN
+x with the matrix
+F HT = MN,0
+hx J(MHT 1,0
+ht
+)⊤ ∈ RNN
+x ×Nt,
+where MN,0
+hx
+is defined in (39), J is given in (40) and
+MHT 1,0
+ht
+[ℓ, κ] = (ϕ0
+κ, HT ϕ1
+ℓ)L2(0,T),
+ℓ = 1, . . . , Nt, κ = 1, . . . , Nt.
+6.2.2
+Using the L2(Q) projection Πh = ΠRT ,1
+h
+for the right-hand side J HT
+Analogously to Subsection 6.2.1, we present the calculation of the right-hand side J HT of the
+linear system (47) when Πh is the L2(Q) projection onto S1(T t
+α)⊗RT 0(T x
+ν ) given in (27). As
+for Subsection 6.2.1, we skip the details, which are analogous to Subsection 6.1.2.
+The entries (49) admit the representation
+J ℓ
+l = (ΠRT ,1
+h
+j, ψN
+l HT ϕ1
+ℓ)L2(Q) = F HT [l, ℓ]
+for ℓ = 1, . . . , Nt, l = 1, . . . , NN
+x with the matrix
+F HT = MN,RT
+hx
+J(�
+MHT
+ht )⊤ ∈ RNN
+x ×Nt,
+20
+
+where MN,RT
+hx
+is defined in (43), J is given in (44) and
+�
+MHT
+ht [ℓ, κ] = (ϕ1
+κ, HT ϕ1
+ℓ)L2(0,T),
+ℓ = 1, . . . , Nt, κ = 0, . . . , Nt.
+(50)
+Note that the matrix MHT
+ht
+∈ RNt×Nt in (48) is a submatrix of the matrix �
+MHT
+ht
+∈ RNt×(Nt+1)
+in (50).
+7
+Numerical examples for the vectorial wave equation
+In this section, we give numerical examples of the conforming space-time finite element meth-
+ods (28) and (46) for a spatially two-dimensional domain Ω ⊂ R2, i.e.
+d = 2.
+For this
+purpose, we consider the unit square Ω = (0, 1) × (0, 1), and set ϵ(x) =
+�1
+0
+0
+1
+�
+, µ(x) = 1
+for x ∈ Ω. The spatial meshes T x
+ν are given by uniform decompositions of the spatial domain
+Ω into isosceles right triangles, where a uniform refinement strategy is applied, see Figure 2.
+We investigate the terminal times T ∈
+�√
+2, 3
+2
+�
+to examine the CFL condition (37) of the
+0
+0.2
+0.4
+0.6
+0.8
+1 x1
+0
+0.2
+0.4
+0.6
+0.8
+1
+x2
+Level 0
+Figure 2: Spatial meshes T x
+ν : Starting mesh T x
+0 and the mesh T x
+2 after two uniform refinement
+steps.
+Galerkin–Petrov finite element method (28). The temporal meshes T t
+α are defined by tℓ = Tℓ
+Ntα
+for ℓ = 0, . . . , Nt
+α, where Nt
+α = 5 · 2α, α = 0, . . . , 4. Note that for this choice of spatial and
+temporal meshes, we are in the framework of the CFL condition (37).
+In the following numerical examples, we measure the error of the space-time finite element
+methods (28) and (46) in the space-time norms ∥·∥L2(Q) and |·|Hcurl;1(Q). In particular, we
+state the numerical results for ∥A − Ah∥L2(Q) and
+|A − Ah|Hcurl;1(Q) =
+�
+∥∂tA − ∂tAh∥2
+L2(Q) + ∥curlx A − curlx Ah∥2
+L2(Q)
+�1/2
+,
+where A ∈ Hcurl;1
+0;0,
+(Q) is the solution of the variational formulation (10), and Ah ∈ S1
+0,(T t
+α) ⊗
+N 0
+I (T x
+ν ) is the solution of space-time finite element method (28) or (46). For the vectorial
+wave equation (3) and its variational formulation (10), we use the manufactured solution
+A(t, x1, x2) =
+�A1(t, x1, x2)
+A2(t, x1, x2)
+�
+=
+�−5t2x2(1 − x2)
+t2x1(1 − x1)
+�
++ t3
+�sin(πx1)x2(1 − x2)
+0
+�
+(51)
+21
+
+00.6
+0.40.20.82
+X0.4Level 20.60.8X
+10.2for (t, x1, x2) ∈ Q, which fulfills the homogeneous boundary condition
+γtA(t, x1, x2) = A|Σ(t, x1, x2) × nx(x1, x2) = 0
+for (t, x1, x2) ∈ Σ
+and the homogeneous initial conditions
+A(0, x1, x2) = ∂tA(0, x1, x2) = 0
+for (x1, x2) ∈ Ω.
+The related right-hand side j is given by
+j(t, x1, x2) =
+�−10(t2 − x2
+2 + x2)
+2(t2 − x2
+1 + x1)
+�
++
+�2t3 sin(πx1) + 6t sin(πx1)x2(1 − x2)
+πt3(1 − 2x2) cos(πx1)
+�
+for (t, x1, x2) ∈ Q with
+divx j(t, x1, x2) = −6πt(x2 − 1)x2 cos(πx1),
+(t, x1, x2) ∈ Q.
+The integrals for computing the projections Π0
+hj, ΠRT ,1
+h
+j in (38), (41) are calculated by using-
+high-order quadrature rules.
+The temporal matrices involving the modified Hilbert trans-
+formation HT , e.g.
+the matrices (48), are assembled as proposed in [44, Subsection 2.2],
+see also [45] for further assembling strategies. The calculation of all spatial matrices, e.g.
+the matrices (32), is done with the help of the finite element library Netgen/NGSolve, see
+www.ngsolve.org and [33]. The linear systems are solved by the sparse direct solver UMF-
+PACK 5.7.1 [10] in the standard configuration. All calculations, presented in this section,
+were performed on a PC with two Intel Xeon E5-2687W v4 CPUs 3.00 GHz, i.e. in sum 24
+cores and 512 GB main memory.
+hx
+ht
+0.2828
+0.1414
+0.0707
+0.0354
+0.0177
+0.1768
+6.64e-01
+6.53e-01
+6.50e-01
+6.49e-01
+6.49e-01
+0.0884
+3.49e-01
+3.27e-01
+3.21e-01
+3.20e-01
+3.20e-01
+0.0442
+2.12e-01
+1.74e-01
+1.63e-01
+1.60e-01
+1.59e-01
+0.0221
+1.61e-01
+1.06e-01
+8.68e-02
+8.13e-02
+7.99e-02
+0.0110
+1.46e-01
+8.04e-02
+5.29e-02
+4.34e-02
+4.07e-02
+Table 1: Interpolation errors in |·|Hcurl;1(Q) for the unit square Ω and T =
+√
+2 for the function
+A in (51) using a uniform refinement strategy.
+hx
+ht
+0.2828
+0.1414
+0.0707
+0.0354
+0.0177
+0.1768
+7.50e-02
+7.49e-02
+7.49e-02
+7.49e-02
+7.49e-02
+0.0884
+1.99e-02
+1.93e-02
+1.93e-02
+1.93e-02
+1.93e-02
+0.0442
+6.97e-03
+4.96e-03
+4.82e-03
+4.81e-03
+4.81e-03
+0.0221
+5.25e-03
+1.74e-03
+1.24e-03
+1.20e-03
+1.20e-03
+0.0110
+5.13e-03
+1.31e-03
+4.37e-04
+3.11e-04
+3.01e-04
+Table 2: Interpolation errors in ∥·∥L2(Q) for the unit square Ω and T =
+√
+2 for the function A
+in (51) using a uniform refinement strategy.
+Last, in Table 1, Table 2, we report the interpolation error in the norms |·|Hcurl;1(Q) and
+∥·∥L2(Q) for the unit square Ω and T =
+√
+2 for the function A defined in (51), where first-order
+convergence is observed in |·|Hcurl;1(Q) and second-order convergence is obtained in ∥·∥L2(Q).
+22
+
+7.1
+Galerkin–Petrov FEM
+In this subsection, we investigate numerical examples for the Galerkin–Petrov finite element
+method (28) in the situation described at the beginning of this section.
+hx
+ht
+0.2828
+0.1414
+0.0707
+0.0354
+0.0177
+0.1768
+6.68e-01
+6.55e-01
+6.52e-01
+6.52e-01
+6.51e-01
+0.0884
+3.56e-01
+9.03e-01
+3.27e-01
+3.26e-01
+3.26e-01
+0.0442
+2.18e-01
+7.22e-01
+4.37e+03
+1.64e-01
+1.63e-01
+0.0221
+1.66e-01
+2.19e-01
+2.02e+04
+8.75e+13
+8.19e-02
+0.0110
+1.50e-01
+9.82e-02
+8.64e+03
+5.02e+17
+8.43e+21
+Table 3: Errors in |·|Hcurl;1(Q) of the Galerkin–Petrov FEM (28) with the approximate right-
+hand side Π0
+hj ∈ S0(T t
+α) ⊗ S0(T x
+ν )2 for the unit square Ω and T =
+√
+2 and the solution A in
+(51) using a uniform refinement strategy.
+hx
+ht
+0.2828
+0.1414
+0.0707
+0.0354
+0.0177
+0.1768
+9.79e-02
+9.73e-02
+9.73e-02
+9.73e-02
+9.73e-02
+0.0884
+4.83e-02
+4.95e-02
+4.67e-02
+4.67e-02
+4.67e-02
+0.0442
+2.64e-02
+2.47e-02
+4.27e+01
+2.33e-02
+2.33e-02
+0.0221
+1.70e-02
+1.22e-02
+1.09e+02
+4.55e+11
+1.17e-02
+0.0110
+1.37e-02
+6.63e-03
+2.69e+01
+1.56e+15
+2.28e+19
+Table 4: Errors in ∥·∥L2(Q) of the Galerkin–Petrov FEM (28) with the approximate right-hand
+side Π0
+hj ∈ S0(T t
+α) ⊗ S0(T x
+ν )2 for the unit square Ω and T =
+√
+2 and the solution A in (51)
+using a uniform refinement strategy.
+hx
+ht
+0.2828
+0.1414
+0.0707
+0.0354
+0.0177
+0.1768
+6.38e-01
+6.26e-01
+6.23e-01
+6.22e-01
+6.22e-01
+0.0884
+3.42e-01
+8.26e-01
+3.10e-01
+3.09e-01
+3.08e-01
+0.0442
+2.16e-01
+9.97e-01
+3.70e+03
+1.55e-01
+1.54e-01
+0.0221
+1.72e-01
+1.06e+00
+1.81e+04
+6.10e+13
+7.73e-02
+0.0110
+1.59e-01
+1.24e+00
+1.91e+04
+5.71e+18
+3.25e+21
+Table 5: Errors in |·|Hcurl;1(Q) of the Galerkin–Petrov FEM (28) with the approximate right-
+hand side ΠRT ,1
+h
+j ∈ S1(T t
+α) ⊗ RT 0(T x
+ν ) for the unit square Ω and T =
+√
+2 and the solution A
+in (51) using a uniform refinement strategy.
+First, we consider the terminal time T =
+√
+2. In Table 3 and Table 4, we present the
+numerical results for the Galerkin–Petrov finite element method (28) with the approximate
+right-hand side Π0
+hj ∈ S0(T t
+α)⊗S0(T x
+ν )2 of Subsection 6.1.1. In Table 5 and Table 6, we report
+the results for ΠRT ,1
+h
+j ∈ S1(T t
+α) ⊗ RT 0(T x
+ν ) of Subsection 6.1.2. All tables show conditional
+stability, i.e. the CFL condition (37) is required for stability. Note that the ratio of the mesh
+sizes ht = 0.0177 and hx = 0.0221, i.e. the last column and second last row of Tables 3 to 6,
+23
+
+hx
+ht
+0.2828
+0.1414
+0.0707
+0.0354
+0.0177
+0.1768
+4.67e-02
+4.29e-02
+4.21e-02
+4.19e-02
+4.19e-02
+0.0884
+1.80e-02
+1.93e-02
+1.06e-02
+1.04e-02
+1.04e-02
+0.0442
+1.27e-02
+1.32e-02
+3.61e+01
+2.66e-03
+2.61e-03
+0.0221
+1.18e-02
+1.36e-02
+1.01e+02
+3.17e+11
+6.64e-04
+0.0110
+1.16e-02
+1.53e-02
+8.53e+01
+1.74e+16
+8.78e+18
+Table 6: Errors in ∥·∥L2(Q) of the Galerkin–Petrov FEM (28) with the approximate right-hand
+side ΠRT ,1
+h
+j ∈ S1(T t
+α) ⊗ RT 0(T x
+ν ) for the unit square Ω and T =
+√
+2 and the solution A in
+(51) using a uniform refinement strategy.
+is given by
+ht
+hx
+≈ 0.0177
+0.0221 ≈ 0.801
+and thus, fulfills the CFL condition (37) resulting in a stable method. In the case of stability,
+we observe first-order convergence in |·|Hcurl;1(Q), where the errors in Table 1, Table 3 and
+Table 5 are within the same range. When considering the errors in ∥·∥L2(Q), Table 4 reports
+only first-order convergence, whereas in Table 6, second-order convergence is observed as in
+Table 2 on the interpolation error.
+Next, we elaborate on this convergence behavior by investigating the difference |A(T, x) −
+Ah(T, x)| for x ∈ Ω. For the mesh sizes ht = 0.0707, hx = 0.0884, we plot the function Ω ∋
+x �→ |A(T, x) − Ah(T, x)| in Figure 3 for both projections Π0
+h and ΠRT ,1
+h
+. In the literature, we
+find the term spurious solutions, e.g. in [22], or spurious modes, e.g. in [5]. Spurious modes are
+parts of the numerical solution, which correspond to eigensolutions of the discrete differential
+operator. They are oscillations that should not be part of the computed solution. Spurious
+modes can be understood as numerically generated noise and have no physical meaning. An
+example can be found in [22, Figure 5.8]. In Figure 3, we obtain a similar noise behavior
+that occurs for the projection Π0
+hj of the right-hand side. Here, we observe some of the zero
+eigensolutions of the curl-curl operator added to the solution. Hence, we get a worse L2(Q)-
+error, see Table 4 and Table 6, but a similar error behavior in the |·|Hcurl;1(Q)-norm for both
+projections if the CFL condition is met, see Table 3 and Table 5.
+To summarize, the approximation Π0
+hj ∈ S0(T t
+α) ⊗ S0(T x
+ν )2 of j is good enough, when the
+error is measured in |·|Hcurl;1(Q), while the better approximation ΠRT ,1
+h
+j ∈ S1(T t
+α) ⊗ RT 0(T x
+ν )
+of j is needed for optimal convergence rates in ∥·∥L2(Q).
+Second, we examine the terminal time T = 3
+2. In Table 7, Table 8, the numerical re-
+sults for the Galerkin–Petrov finite element method (28) with the approximate right-hand
+side ΠRT ,1
+h
+j ∈ S1(T t
+α) ⊗ RT 0(T x
+ν ) of Subsection 6.1.2 show that slightly violating the CFL
+condition (37) leads to instability. More precisely, the ratio of the mesh sizes ht = 0.0187 and
+hx = 0.0221, i.e. the last column and second last row of Table 7, Table 8, is given by
+ht
+hx
+≈ 0.0187
+0.0221 ≈ 0.846
+and thus, violates the CFL condition (37) resulting in an unstable method. In other words,
+the CFL condition (37) seems to be sharp for this particular situation.
+24
+
+Figure 3: The magnitude of the difference |A(T, ·) − Ah(T, ·)| for T =
+√
+2, ν = 1, α = 2,
+displayed over the spatial domain Ω for Π0
+hj ∈ S0(T t
+α)⊗S0(T x
+ν )2 (left) and ΠRT ,1
+h
+j ∈ S1(T t
+α)⊗
+RT 0(T x
+ν ) (right).
+hx
+ht
+0.3000
+0.1500
+0.0750
+0.0375
+0.0187
+0.1768
+7.31e-01
+7.18e-01
+7.16e-01
+7.15e-01
+7.15e-01
+0.0884
+3.90e-01
+1.09e+00
+3.57e-01
+3.55e-01
+3.55e-01
+0.0442
+2.44e-01
+1.24e+00
+6.37e+03
+4.40e-01
+1.77e-01
+0.0221
+1.92e-01
+1.37e+00
+2.28e+04
+9.80e+14
+1.29e+04
+0.0110
+1.77e-01
+1.58e+00
+2.44e+04
+5.36e+17
+7.84e+21
+Table 7: Errors in |·|Hcurl;1(Q) of the Galerkin–Petrov FEM (28) with the approximate right-
+hand side ΠRT ,1
+h
+j ∈ S1(T t
+α) ⊗ RT 0(T x
+ν ) for the unit square Ω and T = 3
+2 and the solution A
+in (51) using a uniform refinement strategy.
+hx
+ht
+0.3000
+0.1500
+0.0750
+0.0375
+0.0187
+0.1768
+5.45e-02
+5.06e-02
+4.99e-02
+4.97e-02
+4.97e-02
+0.0884
+2.05e-02
+2.49e-02
+1.26e-02
+1.24e-02
+1.23e-02
+0.0442
+1.43e-02
+1.69e-02
+6.28e+01
+4.40e-03
+3.09e-03
+0.0221
+1.33e-02
+1.85e-02
+1.30e+02
+5.17e+12
+4.92e+01
+0.0110
+1.31e-02
+2.05e-02
+1.16e+02
+1.67e+15
+2.16e+19
+Table 8: Errors in ∥·∥L2(Q) of the Galerkin–Petrov FEM (28) with the approximate right-hand
+side ΠRT ,1
+h
+j ∈ S1(T t
+α) ⊗ RT 0(T x
+ν ) for the unit square Ω and T = 3
+2 and the solution A in (51)
+using a uniform refinement strategy.
+7.2
+Galerkin–Bubnov FEM
+In this subsection, we report on numerical results for the Galerkin–Bubnov finite element
+method (46) using the modified Hilbert transformation in the situation described at the be-
+ginning of this section. We only show numerical results for the terminal time T =
+√
+2, as
+25
+
+those for T = 3
+2 are similar.
+hx
+ht
+0.2828
+0.1414
+0.0707
+0.0354
+0.0177
+0.1768
+6.77e-01
+6.58e-01
+6.53e-01
+6.52e-01
+6.51e-01
+0.0884
+3.72e-01
+3.36e-01
+3.28e-01
+3.26e-01
+3.26e-01
+0.0442
+2.41e-01
+1.83e-01
+1.68e-01
+1.64e-01
+1.63e-01
+0.0221
+1.95e-01
+1.16e-01
+9.13e-02
+8.40e-02
+8.21e-02
+0.0110
+1.81e-01
+9.25e-02
+5.79e-02
+4.56e-02
+4.20e-02
+Table 9: Errors in |·|Hcurl;1(Q) of the Galerkin–Bubnov FEM (46) with the approximate right-
+hand side Π0
+hj ∈ S0(T t
+α) ⊗ S0(T x
+ν )2 for the unit square Ω and T =
+√
+2 and the solution A in
+(51) using a uniform refinement strategy.
+hx
+ht
+0.2828
+0.1414
+0.0707
+0.0354
+0.0177
+0.1768
+1.03e-01
+9.61e-02
+9.70e-02
+9.73e-02
+9.73e-02
+0.0884
+5.20e-02
+4.61e-02
+4.65e-02
+4.67e-02
+4.67e-02
+0.0442
+2.98e-02
+2.33e-02
+2.32e-02
+2.32e-02
+2.33e-02
+0.0221
+2.08e-02
+1.22e-02
+1.16e-02
+1.16e-02
+1.16e-02
+0.0110
+1.79e-02
+7.12e-03
+5.91e-03
+5.84e-03
+5.83e-03
+Table 10: Errors in ∥·∥L2(Q) of the Galerkin–Bubnov FEM (46) with the approximate right-
+hand side Π0
+hj ∈ S0(T t
+α) ⊗ S0(T x
+ν )2 for the unit square Ω and T =
+√
+2 and the solution A in
+(51) using a uniform refinement strategy.
+hx
+ht
+0.2828
+0.1414
+0.0707
+0.0354
+0.0177
+0.1768
+6.45e-01
+6.27e-01
+6.23e-01
+6.23e-01
+6.22e-01
+0.0884
+3.62e-01
+3.19e-01
+3.11e-01
+3.09e-01
+3.08e-01
+0.0442
+2.48e-01
+1.75e-01
+1.59e-01
+1.55e-01
+1.54e-01
+0.0221
+2.10e-01
+1.13e-01
+8.72e-02
+7.95e-02
+7.75e-02
+0.0110
+2.00e-01
+9.14e-02
+5.63e-02
+4.36e-02
+3.98e-02
+Table 11: Errors in |·|Hcurl;1(Q) of the Galerkin–Bubnov FEM (46) with the approximate right-
+hand side ΠRT ,1
+h
+j ∈ S1(T t
+α) ⊗ RT 0(T x
+ν ) for the unit square Ω and T =
+√
+2 and the solution A
+in (51) using a uniform refinement strategy.
+In Table 9, Table 10, Table 11, Table 12, we observe unconditional stability, i.e.
+no
+CFL condition is needed. This is the main difference between the results in this subsection
+and the results of Subsection 7.1, where for stability, a CFL condition is required. Besides
+the stability issue, the errors in Table 9, Table 10, Table 11, Table 12 are comparable with
+the errors of the previous Subsection 7.1 regarding the projection of the right-hand side.
+Thus, the approximation Π0
+hj ∈ S0(T t
+α) ⊗ S0(T x
+ν )2 of j is good enough to get first-order
+convergence in |·|Hcurl;1(Q), see Table 9. In Table 12, we again see that the better approximation
+ΠRT ,1
+h
+j ∈ S1(T t
+α) ⊗ RT 0(T x
+ν ) of j is needed for optimal convergence rates in ∥·∥L2(Q).
+26
+
+hx
+ht
+0.2828
+0.1414
+0.0707
+0.0354
+0.0177
+0.1768
+5.28e-02
+4.23e-02
+4.20e-02
+4.19e-02
+4.19e-02
+0.0884
+2.70e-02
+1.10e-02
+1.05e-02
+1.04e-02
+1.04e-02
+0.0442
+2.28e-02
+3.99e-03
+2.75e-03
+2.61e-03
+2.59e-03
+0.0221
+2.21e-02
+2.91e-03
+9.92e-04
+6.85e-04
+6.52e-04
+0.0110
+2.19e-02
+2.79e-03
+7.22e-04
+2.46e-04
+1.71e-04
+Table 12: Errors in ∥·∥L2(Q) of the Galerkin–Bubnov FEM (46) with the approximate right-
+hand side ΠRT ,1
+h
+j ∈ S1(T t
+α) ⊗ RT 0(T x
+ν ) for the unit square Ω and T =
+√
+2 and the solution A
+in (51) using a uniform refinement strategy.
+8
+Conclusion
+In this paper, we presented two conforming space-time finite element methods for the vectorial
+wave equation. First, we stated a variational formulation with different trial and test spaces.
+Its conforming discretization with piecewise multilinear functions leads to a Galerkin–Petrov
+method, which is only conditionally stable, i.e. a CFL condition is required. For a particular
+choice of spatial meshes, we stated the CFL condition, where numerical examples showed its
+sharpness. Further, we investigated the influence of projections of the right-hand side on the
+convergence. In numerical examples, we observed only first-order convergence in ∥·∥L2(Q) when
+the right-hand side was projected onto piecewise constants, whereas second-order convergence
+was obtained when the right-hand side was projected onto piecewise linear functions in time
+and lowest-order Raviart–Thomas functions in space.
+Second, to tackle the problem of a CFL condition, we introduced a variational formulation
+for the vectorial wave equation with equal trial and test spaces using the modified Hilbert
+transformation HT . We gave a rigorous derivation of this variational setting. A conforming
+discretization with piecewise multilinear functions of this new variational approach results in
+a space-time Galerkin–Bubnov method. All numerical examples showed the unconditional
+stability of this conforming space-time method. In addition, the influence of the projection of
+the right-hand side is similar to that of the Galerkin–Petrov approach.
+For both presented conforming space-time approaches, a generalization to piecewise poly-
+nomials of higher-order is possible, see [17].
+Further, the fast solvers [25, 41, 42] for the
+resulting linear systems, where some allow for time parallelization, are also applicable. In ad-
+dition, other time-dependent problems in electromagnetics can be handled with the presented
+variational frameworks, which are topics of future work, see e.g. [17].
+References
+[1] Abedi, R., and Mudaliar, S.
+An asynchronous spacetime discontinuous Galerkin
+finite element method for time domain electromagnetics. J. Comput. Phys. 351 (2017),
+121–144.
+[2] Arnold, D. N., Falk, R. S., and Winther, R. Finite element exterior calculus,
+homological techniques, and applications. Acta Numer. 15 (2006), 1–155.
+27
+
+[3] Assous, F., Ciarlet, P., and Labrunie, S. Mathematical foundations of compu-
+tational electromagnetism, vol. 198 of Applied Mathematical Sciences. Springer, Cham,
+2018.
+[4] Bai, X., and Rui, H. A second-order space-time accurate scheme for Maxwell’s equa-
+tions in a Cole–Cole dispersive medium. Engineering with Computers 38, 6 (2022), 5153–
+5172.
+[5] Bossavit, A. Solving Maxwell equations in a closed cavity, and the question of ’spurious
+modes’. IEEE Transactions on Magnetics 26, 2 (1990), 702–705.
+[6] Buffa, A., Costabel, M., and Sheen, D. On traces for H(curl, Ω) in Lipschitz
+domains. J. Math. Anal. Appl. 276, 2 (2002), 845–867.
+[7] Chari, M., Konrad, A., Palmo, M., and D’Angelo, J. Three-dimensional vector
+potential analysis for machine field problems. IEEE Transactions on Magnetics 18, 2
+(1982), 436–446.
+[8] Chauhan, A., Ferrouillat, P., Ramdane, B., Maréchal, Y., and Meunier,
+G. A review on methods to simulate three dimensional rotating electrical machine in
+magnetic vector potential formulation using edge finite element method under sliding
+surface principle. International Journal of Numerical Modelling: Electronic Networks,
+Devices and Fields 35, 1 (2021), e2925.
+[9] Crawford, Z. D., Li, J., Christlieb, A., and Shanker, B. Unconditionally stable
+time stepping method for mixed finite element Maxwell solvers. Progress In Electromag-
+netics Research C 103 (2020), 17 – 30.
+[10] Davis, T. A. Algorithm 832: UMFPACK V4.3 – an unsymmetric-pattern multifrontal
+method. ACM Trans. Math. Softw. 30, 2 (2004), 196–199.
+[11] Dörfler, W., Findeisen, S., and Wieners, C. Space-time discontinuous Galerkin
+discretizations for linear first-order hyperbolic evolution systems. Comput. Methods Appl.
+Math. 16, 3 (2016), 409–428.
+[12] Egger, H., Kretzschmar, F., Schnepp, S. M., and Weiland, T. A space-time
+discontinuous Galerkin Trefftz method for time dependent Maxwell’s equations. SIAM
+J. Sci. Comput. 37, 5 (2015), B689–B711.
+[13] Ern, A., and Guermond, J.-L. Finite elements I—Approximation and interpolation,
+vol. 72 of Texts in Applied Mathematics. Springer, Cham, 2021.
+[14] Ern, A., and Guermond, J.-L. Finite elements II—Galerkin Approximation, Elliptic
+and Mixed PDEs, vol. 73 of Texts in Applied Mathematics. Springer, Cham, 2021.
+[15] Girault, V., and Raviart, P.-A. Finite element methods for Navier-Stokes equations,
+vol. 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1986.
+[16] Hauser, J. I. M. Space-Time Methods for Maxwell’s Equations - Solving the Vectorial
+Wave Equation. PhD thesis, Graz University of Technology, 2021.
+28
+
+[17] Hauser, J. I. M. Space-Time FEM for the Vectorial Wave Equation under Consideration
+of Ohm’s Law. [math.NA] arXiv:2301.03381, arXiv.org, 2023.
+[18] Hauser, J. I. M., Kurz, S., and Steinbach, O. Space-time finite element methods
+for the vectorial wave equation. In preparation.
+[19] Hochbruck, M., and Sturm, A. Upwind discontinuous Galerkin space discretization
+and locally implicit time integration for linear Maxwell’s equations. Math. Comp. 88, 317
+(2019), 1121–1153.
+[20] Ionel, D. M., and Popescu, M. Finite-element surrogate model for electric machines
+with revolving field-application to IPM motors. IEEE Transactions on Industry Applica-
+tions 46, 6 (2010), 2424 – 2433.
+[21] Jackson, J. D., and Okun, L. B. Historical roots of gauge invariance. Rev. Modern
+Phys. 73, 3 (2001), 663–680.
+[22] Jin, J. The finite element method in electromagnetics, second ed. Wiley-Interscience,
+New York, 2002.
+[23] Ladyzhenskaya, O. A. The boundary value problems of mathematical physics, vol. 49
+of Applied Mathematical Sciences. Springer-Verlag, New York, 1985.
+[24] Lang, S. Differential and Riemannian manifolds, third ed., vol. 160 of Graduate Texts
+in Mathematics. Springer-Verlag, New York, 1995.
+[25] Langer, U., and Zank, M.
+Efficient direct space-time finite element solvers for
+parabolic initial-boundary value problems in anisotropic Sobolev spaces. SIAM J. Sci.
+Comput. 43, 4 (2021), A2714–A2736.
+[26] Lilienthal, M., Schnepp, S. M., and Weiland, T. Non-dissipative space-time hp-
+discontinuous Galerkin method for the time-dependent Maxwell equations. J. Comput.
+Phys. 275 (2014), 589–607.
+[27] Logg, A., Mardal, K.-A., and Wells, G., Eds. Automated solution of differential
+equations by the finite element method. The FEniCS book, vol. 84 of Lect. Notes Comput.
+Sci. Eng. Berlin: Springer, 2012.
+[28] Löscher, R., Steinbach, O., and Zank, M. An unconditionally stable space-time
+finite element method for the wave equation. In preparation.
+[29] Löscher, R., Steinbach, O., and Zank, M. Numerical results for an unconditionally
+stable space-time finite element method for the wave equation. In Domain Decomposition
+Methods in Science and Engineering XXVI, vol. 145 of Lect. Notes Comput. Sci. Eng.
+Springer, Cham, 2022, pp. 587–594.
+[30] Monk, P. Finite element methods for Maxwell’s equations. Numer. Math. Sci. Comput.
+Oxford: Oxford University Press, 2003.
+[31] Newmark, N. M. A method of computation for structural dynamics. Journal of the
+Engineering Mechanics Division 85, 3 (1959), 67–94.
+29
+
+[32] Pauly, D., and Schomburg, M. Hilbert complexes with mixed boundary conditions
+part 1: de Rham complex. Math. Methods Appl. Sci. 45, 5 (2022), 2465–2507.
+[33] Schöberl, J. NETGEN: An advancing front 2d/3d-mesh generator based on abstract
+rules. Comput. Vis. Sci. 1, 1 (1997), 41–52.
+[34] Steinbach, O., and Missoni, A. A note on a modified Hilbert transform. Applicable
+Analysis (2022), 1–8.
+[35] Steinbach, O., and Zank, M. Coercive space-time finite element methods for initial
+boundary value problems. Electron. Trans. Numer. Anal. 52 (2020), 154–194.
+[36] Steinbach, O., and Zank, M. A note on the efficient evaluation of a modified Hilbert
+transformation. J. Numer. Math. 29, 1 (2021), 47–61.
+[37] Stern, A., Tong, Y., Desbrun, M., and Marsden, J. E. Geometric computational
+electrodynamics with variational integrators and discrete differential forms. In Geometry,
+mechanics, and dynamics, vol. 73 of Fields Inst. Commun. Springer, New York, 2015,
+pp. 437–475.
+[38] Tiegna, H., Amara, Y., and Barakat, G. Overview of analytical models of per-
+manent magnet electrical machines for analysis and design purposes. Mathematics and
+Computers in Simulation 90 (2013), 162–177.
+[39] Xie, J., Liang, D., and Zhang, Z. Energy-preserving local mesh-refined splitting
+FDTD schemes for two dimensional Maxwell’s equations. J. Comput. Phys. 425 (2021),
+Paper No. 109896, 29.
+[40] Xie, Z., Wang, B., and Zhang, Z. Space-time discontinuous Galerkin method for
+Maxwell’s equations. Commun. Comput. Phys. 14, 4 (2013), 916–939.
+[41] Zank, M.
+Efficient direct space-time finite element solvers for the wave equation in
+second-order formulation. In preparation.
+[42] Zank, M. High-order discretisations and efficient direct space-time finite element solvers
+for parabolic initial-boundary value problems. To appear in Proceedings of Spectral and
+High Order Methods for Partial Differential Equations ICOSAHOM 2020+1.
+[43] Zank, M. Inf-Sup Stable Space-Time Methods for Time-Dependent Partial Differential
+Equations, vol. 36 of Monographic Series TU Graz: Computation in Engineering and
+Science. 2020.
+[44] Zank, M.
+An exact realization of a modified Hilbert transformation for space-time
+methods for parabolic evolution equations. Comput. Methods Appl. Math. 21, 2 (2021),
+479–496.
+[45] Zank, M.
+Integral representations and quadrature schemes for the modified hilbert
+transformation. Computational Methods in Applied Mathematics (2022).
+30
+

PNFOT4oBgHgl3EQf4TQo/content/tmp_files/2301.12949v1.pdf.txt

@@ -0,0 +1,2449 @@
+arXiv:2301.12949v1  [math.FA]  30 Jan 2023
+MOMENT PROBLEM FOR
+ALGEBRAS GENERATED BY A NUCLEAR SPACE
+MARIA INFUSINO, SALMA KUHLMANN, TOBIAS KUNA, PATRICK MICHALSKI
+Abstract. We establish a criterion for the existence of a representing Radon
+measure for linear functionals defined on a unital commutative real algebra
+A, which we assume to be generated by a vector space V endowed with a
+Hilbertian seminorm q. Such a general criterion provides representing measures
+with support contained in the space of characters of A whose restrictions to
+V are q−continuous. This allows us in turn to prove existence results for the
+case when V is endowed with a nuclear topology. In particular, we apply our
+findings to the symmetric tensor algebra of a nuclear space.
+Introduction
+Given a unital commutative (not necessarily finitely generated) R–algebra A and
+a linear subspace V of A, we say that A is generated by V if there exists a set of
+generators G of A such that V is the linear span of G, i.e. V = span(G). Equiva-
+lently, A is generated by V if V contains a set of generators of A. This article deals
+with the moment problem for A generated by a vector space V which is endowed
+with a topology τV compatible with the addition and the scalar multiplication,
+namely (V, τV ) is a topological vector space. Moreover, we always assume that the
+character space of A, i.e., the set X(A) of all R–algebras homomorphisms from A
+to R, is non-empty and we always endow X(A) with the weakest (Hausdorff) topol-
+ogy τX(A) on X(A) such that for each a ∈ A the function ˆa: X(A) → R, α �→ α(a)
+is continuous and consider on X(A) the Borel σ−algebra B(τX(A)) w.r.t. τX(A).
+Our main question is the following.
+Main Question.
+Let (V, τV ) a topological vector space and A be an algebra generated by V such
+that {α ∈ X(A) : α ↾V
+is τV − continuous} ̸= ∅. Given a linear functional L on A
+with L(1) = 1, does there exist a Radon measure ν on X(A) with support contained
+in {α ∈ X(A) : α ↾V
+is τV − continuous} such that
+(0.1)
+L(a) =
+�
+X(A)
+ˆa(α)dν(α)
+for all a ∈ A?
+If a Radon measure ν as in (0.1) does exist, then we call ν a representing Radon
+measure for L.
+We recall that a Radon measure ν on X(A) is a non-negative
+measure on the Borel σ–algebra w.r.t. τX(A) that is locally finite and inner regular
+w.r.t. compact subsets of X(A).
+The support of ν, denoted by supp(ν), is the
+smallest closed subset C of X(A) for which ν(X(A) \ C) = 0.
+The main difficulty is to understand how different choices of τV as well as different
+topological properties of L impact the solvability of Main Question and the support
+of the corresponding representing measures. In this article we first focus on the
+Date: January 31, 2023.
+2020 Mathematics Subject Classification. Primary 44A60, 46M40, 28C20.
+Key words and phrases. moment problem; infinite dimensional moment problem; nuclear
+space; projective limit; Prokhorov’s condition.
+1
+
+2
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+case when τV is the topology generated by a Hilbertian seminorm (i.e. a seminorm
+induced by a symmetric positive semidefinite bilinear form) and then consider the
+case when (V, τV ) is a nuclear space, as nuclear topologies are generated by a system
+of Hilbertian seminorms.
+An early study of Main Question for (V, τV ) nuclear can be found e.g. in [12],
+[2, Chapter 5, Section 2], [3], [7], [11], [21, Section 12.5 and 15.1], [1], where A is
+the symmetric (tensor) algebra S(V ) of V . This is a very natural choice as S(V )
+is isomorphic to the ring of polynomials having as variables the coordinate vectors
+with respect to a basis of V and the character space X(S(V )) of S(V ) can be
+identified with the algebraic dual V ∗ of V . In fact, in those works the nuclearity
+assumption on V on L allow to get the existence of representing measures with
+support contained in V ′, where V ′ is the topological dual of V . More recently, the
+role of the nuclearity assumption on V was discussed in [23, Sections 5 and 6], [9]
+and [14, Section 3] while in [15] and [16] a better localization of the support was
+obtained for a specific choice of the nuclear space, namely V = C∞
+c (Rn), i.e. the
+space of infinitely differentiable functions with compact support.
+Let us describe the two main results in this article.
+First, when τV is the topology generated by a Hilbertian seminorm q, we derive
+in Theorem 2.5 a criterion for the existence of a representing measure with support
+contained in the characters of A whose restrictions to V are q−continuous (see
+also Remark 2.9 and Theorem 2.8). The criterion is based on the projective limit
+approach to the moment problem introduced in [17], that is, we build the repre-
+senting measure for L on A from representing measures for L restricted to finitely
+generated subalgebras of A. In fact, we prove that a representing measure for L
+exists if and only if for any finitely generated subalgebra S of A the restriction L ↾S
+is represented by a Radon measure νS such that the family of all νS’s is concen-
+trated w.r.t. another q−continuous Hilbertian seminorm p, which has finite trace
+with respect to q. The concentration of a family of measures is a classical concept
+in measure theory and is crucial for us, because it ensures the applicability of our
+projective limit approach in [17] by implying a Prokhorov type condition.
+Second, in Theorem 2.10 we show that, when A itself is endowed with a Hilbertian
+seminorm q and there exists C > 0 such that L(a2) ≤ Cq(a)2 for all a ∈ A, it is
+enough to check the conditions in our criterion only on a dense subalgebra of A
+to get the existence of a representing measure for L with support contained in the
+q−continuous characters of A (see also Theorem 2.11).
+These two main results are based on two Hilbertian seminorms q and p. We inves-
+tigate different choices of them in terms of the functional L. For example, a natural
+choice for p is Hilbertian seminorm induced by L, i.e. sL(a) :=
+�
+L(a2) for all a ∈
+A. For this choice, the concentration of the νS’s holds automatically and so we get
+more concrete sufficient conditions for the existence of a representing measure for L
+in Corollary 2.12 and Corollary 2.13. We then exploit in Corollary 2.15 the choice
+of sL to demonstrate how one can give sufficient conditions only in terms of L to
+guarantee existence of a representing measures for L ↾S for all finite subalgebras S.
+Those corollaries all reveal the fundamental role played by the Hilbertian seminorm
+q. Thus, in the last part of Section 2.2, we explore the case when no Hilbertian
+seminorm q on V is pre-given. In particular, in Corollary 2.19 we give conditions
+under which one can construct a suitable q and derive a solution for Main Question
+in this case.
+Another setting in which it is always possible to obtain a suitable q is when
+(V, τV ) is a nuclear space. Therefore, in Section 2.3, we prove analogous results for
+Main Question when A is generated by a nuclear space (V, τV ) (see Corollary 2.20,
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+3
+Corollary 2.21 and Corollary 2.22). From those corollaries, some of the results in
+literature mentioned above can be retrieved.
+The structure of the paper is as follows.
+In Section 1, we present our general context, thereby providing definitions and
+notations. In particular, we review the notions of Hilbertian seminorm and nu-
+clear space in Subsection 1.1. In Subsection 1.2, we state and prove Lemma 1.6
+(about the support localization of a Radon probability measure on a finite dimen-
+sional space with a Hilbertian seminorm), which we need for the proof of our main
+theorem Theorem 2.5. Section 2 contains our main results, as described above.
+Subsection 2.1 is dedicated to the concept of p-concentration of a family of Radon
+measures for a given seminorm p, which is exploited in the subsequent Subsections
+2.2 and 2.3 when studying Main Question for V endowed with the topology τV
+induced by a Hilbertian seminorm (respectively, a nuclear topology). In Section 3
+we apply our main results to the case when A is the symmetric algebra S(V ) of
+a nuclear space (V, τV ), see Corollary 3.1 and Corollary 3.2. In Theorem 3.3, we
+consider the case when some of the sufficient conditions for the existence of the rep-
+resenting measure for L on S(V ) are only given on a total subset E of the nuclear
+space (V, τV ). Then the nuclearity allows us to obtain a Hilbertian norm q on V but,
+in order to apply our criterion Theorem 2.10 to the dense sub-algebra S(span(E)),
+we need a Hilbertian seminorm ˜q on S(V ), which we construct in Lemma 3.5. We
+note that Theorem 3.3 is a generalization of the classical solution to Main Question
+when A = S(V ) with (V, τV ) nuclear due to Berezansky and Kondratiev. Finally,
+in Subsection 4.1 of the Appendix 4, we explain the relation between the notion
+of trace of a Hilbertian seminorm w.r.t. to another and the classical definition of
+trace of a positive continuous operator on a Hilbert space. We then compare in
+Subsection 4.2 the definition of nuclear space used in this article (due to Yamasaki
+[27]) with that due to Grothendieck [10] and Mityagin [19], as well as with the
+definitions of this concept given by Berezansky and Kondratiev in [2, p. 14] and
+by Schmüdgen in [23, p. 445] (this comparision is needed in Section 3). We also
+provide in Subsection 4.3 a complete proof of the measure theoretical identity (2.7),
+which we exploited in the proof of Theorem 2.10.
+1. Preliminaries
+In this section we collect some fundamental concepts, notations, and results
+which we will repeatedly use in the following.
+Throughout this article A denotes a unital commutative R–algebra with non-empty
+character space X(A).
+A subset Q ⊆ A is a quadratic module (in A) if 1 ∈ Q, Q+Q ⊆ Q, and A2Q ⊆ Q.
+The set � A2 of all finite sums of squares of elements in A is the smallest quadratic
+module in A. The non-negativity set of a quadratic module Q is defined as
+KQ := {α ∈ X(A) : ˆa(α) ≥ 0 for all a ∈ Q} ⊆ X(A),
+which is closed. Given C ⊆ X(A) closed, the set
+Pos(C) := {a ∈ A : ˆa(α) ≥ 0 for all α ∈ C}
+is a quadratic module with KPos(C) = C (see, e.g. [17, Proposition 2.1-(i)]).
+Throughout this article each linear functional L: A → R is assumed to be nor-
+malized, that is, L(1) = 1.
+Given a a quadratic module Q in A, we say that a linear functional L: A → R
+is Q–positive if L(Q) ⊆ [0, ∞). In particular, each � A2–positive linear functional
+
+4
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+L: A → R satisfies the Cauchy–Bunyakovsky–Schwarz inequality, i.e.,
+(1.1)
+L(ab)2 ≤ L(a2)L(b2)
+for all a, b ∈ A.
+Throughout this article we consider A generated by an R–vector space V endowed
+with a locally convex topology, namely a topology induced by a family of seminorms.
+Therefore, let us recall that a function p: V → [0, ∞) is a seminorm if p(λv) =
+|λ| p(v) and p(v + w) ≤ p(v) + p(w) for all λ ∈ R and all v, w ∈ V . We denote
+by Br(p) the closed semi-ball of radius r > 0 centered at the origin in (V, p), i.e.
+Br(p) := {v ∈ V : p(v) ≤ r}.
+A linear functional l: V → R is continuous w.r.t. a seminorm p on V if there
+exists C > 0 such that |l(v)| ≤ Cp(v) for all v ∈ V . We denote by V ′
+p the topological
+dual of (V, p), i.e. the collection of all p−continuous linear functionals on V , while
+V ∗ denotes the algebraic dual of V . The operator seminorm p′ on V ′
+p is defined
+as p′(ℓ) := supv∈B1(p) |ℓ(v)| < ∞. The weak topology on the algebraic dual (resp.,
+topological dual) of (V, p) is the weakest topology on V ∗ (resp., on V ′
+p ) such that
+for each v ∈ V the evaluation function evv : V ∗ → R (resp., V ′
+p → R) is continuous.
+We will often use the restriction map φV : X(A) → V ∗ defined by φV (α) = α ↾V ,
+∀α ∈ X(A). Note that φV is continuous as X(A) is endowed with τX(A) and V ∗
+with the weak topology.
+We recall that the spectrum of a seminorm p is defined as
+sp(p) := {α ∈ X(A) : α is p–continuous}.
+More generally, for each C > 0 we define
+spC(p) := {α ∈ X(A) : |α(a)| ≤ Cp(a), ∀a ∈ A},
+which is compact in X(A), as it is closed and continuously embeds into the prod-
+uct �
+a∈A[−Cp(a), Cp(a)]. Note that the spectrum sp(p) = �
+n∈N spn(p), which
+provides that sp(p) is σ−compact in X(A) and so Borel measurable.
+1.1. Hilbertian seminorms and nuclear spaces.
+Throughout this section V will denote a real vector space.
+Definition 1.1. A seminorm p on V is called Hilbertian if it is induced by a
+symmetric positive semidefinite bilinear form ⟨·, ·⟩ on V , i.e., p(v) =
+�
+⟨v, v⟩ for all
+v ∈ V .
+Note that a seminorm p on V is Hilbertian if and only if p fulfills the parallelo-
+gram law, i.e. p(v+w)2+p(v−w)2 = 2p(v)2+2p(w)2 for all v, w ∈ V , in which case
+the bilinear form ⟨·, ·⟩p is uniquely determined by p via the polarization identity:
+(1.2)
+⟨v, w⟩ = 1
+2
+�
+p(v + w)2 − p(v)2 − p(w)2�
+for all v, w ∈ V.
+For this reason, in the following we denote the positive semidefinite bilinear form
+inducing p by ⟨·, ·⟩p.
+The term “Hilbertian seminorm”, used e.g. in [26] and [27], is also sometimes
+replaced by the term “prehilbertian seminorm” according to the Bourbaki’s tradi-
+tion [6, V.4, Definition 3]. Both terms hints to the fact that this type of seminorms
+can be aways used to construct a Hilbert space (see Remark 4.5).
+Let us also observe that there always exists an Hilbertian seminorm on every
+non-trivial vector space V . Indeed, if (ei)i∈I is an algebraic basis of V then for
+any x = �
+i∈I xiei ∈ V and y = �
+i∈I yiei ∈ V we can define ⟨x, y⟩ := �
+i∈I xiyi.
+As only finite many summands are unequal to zero, the sum is finite and p(x) :=
+�
+⟨x, x⟩ defines a Hilbertian seminorm on V .
+Let us now introduce the notion of trace of a Hilbertian seminorm w.r.t. to
+another one (see [6, V.58, No. 9]) which will be fundamental in the definition of
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+5
+a nuclear space used in this article. To this purpose, let us recall that given a
+Hilbertian seminorm p on V , a subset E of V is called:
+• p–orthogonal if ⟨e1, e2⟩p = 0 for all distinct elements e1, e2 ∈ E.
+• p–orthonormal if E is p–orthogonal and p(e) = 1 for all e ∈ E.
+In particular, a p–orthonormal set E is said to be a complete p−orthonormal system
+if E is total in E, i.e. span(E)
+p = V . Such a system is also known as orthonormal
+basis.
+Definition 1.2. Let p and q be two Hilbertian seminorms on V . The trace of p
+w.r.t. q is denoted by tr(p/q) and defined as
+tr(p/q) :=
+
+
+
+sup
+E∈FON(q)
+�
+e∈E
+p(e)2,
+if ker(q) ⊆ ker(p)
+∞,
+otherwise
+,
+where FON(q) denotes the collection of all finite q–orthonormal subsets of V .
+When there exists C > 0 such that p ≤ Cq the following characterization of the
+trace of p w.r.t. q holds (by combining Proposition 4.6 and (4.2) in Appendix 4.1):
+(1.3)
+∀E complete q−orthonormal system in V , tr(p/q) =
+�
+e∈E
+p(e)2.
+The following properties are immediate from the Definition 1.2.
+Lemma 1.3. Let p and q be two Hilbertian seminorms on V with tr(p/q) < ∞.
+Then:
+(i) p2 ≤ tr(p/q)q2.
+(ii) ∀ε, δ > 0, tr(εp/δq) =
+� ε
+δ
+�2 tr(p/q).
+(iii) ∀ W subspace of V, tr(p↾W /q↾W ) ≤ tr(p/q).
+We are equipped now with all notions needed to introduce the definition of a
+nuclear space due to Yamasaki (see [27, Definition 20.1]), which we are going to
+adopt in this article.
+Definition 1.4. A locally convex space (V, τ) is called nuclear if τ is induced by
+a directed family P of Hilbertian seminorms on V such that for each p ∈ P there
+exists q ∈ P with tr(p/q) < ∞.
+Definition 1.4 is equivalent to the more traditional ones in [10] and [19], which
+we report in Appendix 4.2 for the convenience of the reader (see Definition 4.10
+and Definition 4.11).
+Note that a nuclear topology can be always constructed on every vector space V .
+However, this nuclear topology has typically no relation with a pre-given topology
+τV on V . However, when (V, τV ) is a separable locally convex space with a Schauder
+basis, there exists a dense subspace U of V on which a nuclear topology stronger
+than τV ↾U can be constructed.
+1.2. Probabilities on finite dimensional Hilbertian seminormed spaces.
+In the following we introduce a fundamental result about the support localization
+of a Radon measure defined on the dual of a finite dimensional real vector space,
+namely Lemma 1.6, which is inspired by [26, Fundamental lemma (p. 24)] and will
+play a crucial role in the proof of our main theorem Theorem 2.5. For this, let us
+recall two properties of the Gaussian measure on a finite dimensional real vector
+space endowed with a Hilbertian seminorm (see [26, p. 26-28] for a proof).
+
+6
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+Proposition 1.5. Let q be a Hilbertian seminorm on an n−dimensional R–vector
+space V with ker(q) = {0} and E a complete q−orthonormal system of V . Let γ be
+the Gaussian measure on V , i.e.,
+dγ(v) := (2π)− n
+2 exp
+�
+− 1
+2q(v)2�
+dλ(v),
+where λ is the measure on V corresponding to the Lebesgue measure on Rn under
+the identification V → Rn, v �→ (⟨v, e⟩q)e∈E. Then the following properties hold
+(i)
+�
+⟨v, w⟩2
+qdγ(v) = 1 for all w ∈ V such that q(w) = 1.
+(ii) γ({v ∈ V : |ℓ(v)| ≥ 1}) ≥ 7−1 for all ℓ ∈ V ′ with q′(ℓ) ≥ 1, where q′ denotes
+the operator seminorm on V ′.
+Lemma 1.6. Let p and q Hilbertian seminorms on a finite dimensional R–vector
+space such that tr(p/q) < ∞ and let µ be a probability measure on V ′.
+If for any ε > 0 there exists δ > 0 such that µ({l ∈ V ′ : |l(v)| ≥ 1}) ≤ ε for all
+v ∈ Bδ(p), then
+µ(B1(q′)) ≥ 1 − 7(ε + tr(p/δq)),
+where q′ denotes the operator seminorm on V ′.
+Proof. Let µ be a probability measure on V ′. W.l.o.g. we can assume ker(q) = {0},
+because otherwise for each complement W of ker(q) in V we have µ–almost surely
+that B1(q′) = {l ∈ V ′ : |l(w)| ≤ q(w) ∀ w ∈ W} holds1. Consider
+D := (µ × γ)({(l, v) ∈ V ′ × V : |l(v)| ≥ 1}),
+where µ × γ denotes the product measure between the given measure µ on V ′ and
+the Gaussian measure γ on V . Now, let l ∈ V ′ \ B1(q′). Then Fubini’s theorem on
+the one hand, combined with Proposition 1.5-((ii)), provides that
+D =
+�
+V ′ γ({v ∈ V : |l(v)| ≥ 1})dµ(l) ≥ 7−1µ(V ′ \ B1(q′)) = 7−1 (1 − µ(B1(q′))) ,
+and, on the other hand, combined with the assumption yields that
+(1.4)
+D =
+�
+V
+µ({l ∈ V ′ : |l(v)| ≥ 1})dγ(v) ≤ εγ(Bδ(p)) + γ(V \ Bδ(p)).
+Moreover, by [6, V, §4.8, Theorem 2], there exists a complete q−orthonormal system
+E of V that is p−orthogonal. In particular, p(v)2 = �
+e∈E
+⟨v, e⟩2
+qp(e)2 holds for all
+v ∈ V, which combined with Proposition 1.5-((i)) and (1.3) gives
+γ(V \ Bδ(p)) ≤ δ−2
+�
+V
+p(v)2dγ(v) = δ−2 �
+e∈E
+p(e)2
+�
+V
+⟨v, e⟩2
+qdγ(v) = δ−2tr(p/q).
+The latter together with (1.4) and Lemma 1.3-((ii)) provides
+(1.5)
+D ≤ ε + δ−2tr(p/q) = ε + tr(p/δq).
+Combining (1.4) and (1.5) yields the assertion.
+□
+1On the one hand, it is immediate that B1(q′) = {l ∈ V ′ : q′(l) ≤ 1} = {l ∈ V ′ :
+|l(v)| ≤ q(v) for all v ∈ V } ⊆ {l ∈ V ′ : |l(w)| ≤ q(w) for all w ∈ W }.
+On the other
+hand, since p2 ≤ tr(p/q)q2, we have that ker(q) ⊆ ker(p) ⊂ Bδ(p) for all δ > 0 and so
+the assumption on µ in Lemma 1.6 ensures that µ({l ∈ V ′
+: |l(v)| ≥ 1}) = 0 for all
+v ∈ ker(q), i.e.
+µ({l ∈ V ′ : |l(v)| = 0, ∀v ∈ ker(q)}) = 1, which immediately provides that
+µ {B1(q′) \ {l ∈ V ′ : |l(w)| ≤ q(w) for all w ∈ W }} = 0.
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+7
+2. Main results
+In this section we are going to present our main results concerning Main Question
+for a unital commutative real algebra A generated by a vector space V first endowed
+with a Hilbertian seminorm q and then with a nuclear topology. More precisely,
+in Subsection 2.2 we first establish a criterion for the existence of a representing
+measure with support contained in the set of characters of A whose restrictions to
+V are q−continuous (see Theorem 2.5 and Remark 2.9, as well as Theorem 2.8).
+When the seminorm q is defined on the full algebra A, i.e. A = V , this result
+provides in particular a criterion for the existence of a representing measure on the
+Gelfand spectrum of q. We actually show that when L ≤ Cq on A for some C
+then it is enough to check the latter criterion just on a dense subalgebra of A (see
+Theorem 2.10). Moreover, in Lemma 2.17 we provide an explicit bound on L which
+guarantees the existence of a Hilbertian seminorm q on A satisfying our criteria.
+Exploiting our general criteria, in Corollary 2.15, we identify more concrete suf-
+ficient conditions on L and q for the existence of such a representing measure for L.
+Those allow us to clarify in Subsection 2.3 the relation between the solvability of
+Main Question and the presence of a nuclear topology on V . Our general criteria
+are based on the projective limit approach introduced in [17] which allows to reduce
+Main Question to a family of finite-dimensional moment problems whose solutions
+satisfy a concentration condition to which we dedicate Subsection 2.1.
+2.1. The concentration condition.
+Let A be a unital commutative R-algebra generated by a linear subspace V ⊆ A
+such that X(A) ̸= ∅, and L a normalized linear functional on A.
+As already mentioned, in proving our main results for Main Question we will
+exploit the projective limite approach we developed in [17]. This is based on the
+construction of (X(A), τX(A)) together with the maps {πS : S ∈ J} as the projective
+limit of the projective system of Hausdorff spaces {(X(S), τX(S)), πS,T , J}, where
+J := {⟨W⟩ : W finite dimensional subspace of V }
+is ordered by inclusion, ⟨W⟩ denotes by the subalgebra of A generated by W, τX(S)
+is the weak topology on X(S), for any S, T subalgebras of A with S ⊆ T the map
+πS,T : X(T ) → X(S) is the natural restriction and πS := πS,A. The correspond-
+ing projective system of measurable spaces is given by {(X(S), B(τX(S))), πS,T , J},
+where B(τX(S)) is the Borel σ−algebra w.r.t. τX(S). Recall that this means that
+πS,T is measurable for all S ⊆ T in J and that πS,T ◦ πT,R for all S ⊆ T ⊆ R in J.
+Roughly speaking, in [17], we establish that there exists a representing Radon
+measure for L on A supported in (X(A), B(τX(A))) if and only if for each S ∈
+J there exists a representing Radon measure νS supported in (X(S), B(τX(S)))
+such that {νS :
+S ∈ J} fulfills the so-called Prokhorov condition. In the next
+subsection we will exploit this result when studying Main Question for V endowed
+with the topology τV induced by a Hilbertian seminorm and we will exploit the
+given topological structure on V to prove that the Prokhorov condition (see [17,
+Section 1.2] and references therein) is satisfied whenever {νS : S ∈ J} fulfills the
+following concentration property.
+Definition 2.1. Given a seminorm p on V and for each S ∈ J a Radon measure νS
+on (X(S), B(τX(S))), we say that {νS : S ∈ J} is p−concentrated (or concentrated
+w.r.t. p) if
+(2.1)
+∀ε > 0 ∃δ > 0: ∀S ∈ J, ∀a ∈ Bδ(p) ∩ S, νS({α ∈ X(S) : |α(a)| ≥ 1}) ≤ ε.
+This definition is an adaptation to our setting of the notion of continuity for
+cylindrical measures introduced in [8, 16, Chapter IV, Section 1.4]. It also easily
+
+8
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+relates to the notion of concentrations of cylindrical measures in [24, Definition 1,
+p.192]. In fact, (2.1) is weaker than assuming that the cylindrical quasi-measure as-
+sociated to {νS : S ∈ J} is cylindrically concentrated on {spC(p) : C > 0}, namely
+∀ε > 0 ∃δ > 0: ∀S ∈ J, νS(πS(spδ(p))) ≥ 1 − ε.
+Let us now provide a useful characterization of the p−concentration of a collec-
+tion of Radon measures.
+Proposition 2.2. Given a seminorm p on V and for each S ∈ J a Radon measure
+νS on (X(S), B(τX(S))), we have that {νS : S ∈ J} is p−concentrated if and only
+if the following holds
+(2.2) ∀ε > 0 ∃γ > 0: ∀S ∈ J, ∀a ∈ S ∩V, νS({α ∈ X(S) : |α(a)| ≤ γp(a)}) ≥ 1 − ε.
+Proof.
+Suppose (2.1) holds and fix ε > 0. Taking 0 < δ′ < δ with δ as in (2.1), we have
+that (2.2) holds for γ =
+1
+δ′ . In fact, for any S ∈ J, let b ∈ S ∩ V and distinguish
+the following two cases.
+• If p(b) ̸= 0, then
+δ′b
+p(b) ∈ Bδ(p) ∩ S and so (2.1) provides that νS({α ∈ X(S) :
+|α(b)| ≥ p(b)
+δ }) ≤ ε, which implies νS({α ∈ X(S) : |α(b)| ≤ p(b)
+δ′ }) ≥ 1 − ε.
+• If p(b) = 0, then clearly span(b) ⊆ Bδ(p) ∩ V ∩ S and so (2.1) gives that
+∀λ > 0, ∀a ∈ span(b), νS ({α ∈ X(S) : |α(a)| ≥ 1}) ≤ λ, i.e. ∀a ∈ span(b),
+νS ({α ∈ X(S) : |α(a)| < 1}) = 1. Then
+∀r > 0, νS
+��
+α ∈ X(S) : |α(b)| < 1
+r
+��
+= 1,
+and so we get νS({α ∈ X(S) : |α(b)| = 0}) = 1, which in particular gives that
+νS({α ∈ X(S) : |α(b)| ≤ p(b)
+δ′ }) = 1 ≥ 1 − ε.
+Conversely, suppose (2.2) holds and fix ε > 0. Taking γ as in (2.2), we have that
+(2.1) holds for δ ≤ 1
+γ . In fact, for any S ∈ J, let b ∈ Bδ(p) ∩ S and distinguish the
+following two cases.
+• If p(b) ̸= 0, then (2.2) provides that νS({α ∈ X(S) : |α(b)| ≤ γp(b)}) ≥ 1 − ε
+which implies νS({α ∈ X(S) : |α(b)| < 1}) ≥ 1 − ε.
+• If p(b) = 0, then (2.2) provides that νS({α ∈ X(S) : |α(b)| = 0}) ≥ 1 − ε, i.e.
+νS({α ∈ X(S) : |α(b)| > 0}) ≤ ε, which implies νS({α ∈ X(S) : |α(b)| ≥ 1}) ≤ ε.
+□
+Remark 2.3.
+If ν is a Radon measure on X(A) s.t. {πS#ν : S ∈ J} is p−concentrated, then
+(2.3)
+ν({α ∈ X(A) : |α(b)| = 0 ∀b ∈ ker(p)}) = 1,
+where πS#ν denotes the pushforward measure of ν w.r.t. πS. Indeed, using the same
+argument as in the proof of Proposition 2.2, we can show that ∀b ∈ ker(p), ν({α ∈
+X(A) : |α(b)| = 0}) = π⟨b⟩#ν({α ∈ X(⟨b⟩) : |α(b)| = 0}) = 1. This together with
+the fact that {α ∈ X(A) : |α(b)| = 0} is a closed subset of X(A) and ν a Radon
+measure yields (2.3) by [24, Part I, Chapter I, 6.(a)].
+Let us establish now a sufficient condition for the p−concentration of a collection
+of representing measures, which we will often exploit in the rest of the article.
+Lemma 2.4. Let A be an algebra generated by a linear subspace V and p a semi-
+norm on V . Given a linear functional L on A such that L(� A2) ⊆ [0, ∞) and,
+for each S ∈ J, a representing measure νS for L ↾S, if
+(2.4)
+∃ C > 0 : L(a2) ≤ Cp(a)2
+for all a ∈ V
+holds, then {νS : S ∈ J} is p−concentrated.
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+9
+Proof. Let ε > 0 and take δ := � ε
+C . Then for all a ∈ Bδ(p) ∩ S
+νS({α ∈ X(S) : |α(a)| ≥ 1}) ≤
+�
+X(S)
+ˆa2dνS = L(a2) ≤ Cp(a)2 ≤ Cδ2 ≤ ε
+i.e. {νS : S ∈ J} fulfills (2.1).
+□
+With a similar proof one gets the following generalization of (2.4)
+(2.5)
+∀ε > 0 ∃ C > 0 : L(a2) ≤ Cp(a)2 + ε
+for all a ∈ V.
+2.2. The case when τV generated by a Hilbertian seminorm.
+Theorem 2.5. Let A be an algebra generated by a linear subspace V ⊆ A, q a
+Hilbertian seminorm on V such that {α ∈ X(A) : α ↾V is q–continuous} ̸= ∅ and
+J := {⟨W⟩ : W finite dimensional subspace of V }. Let L be a normalized linear
+functional on A.
+There exists a representing Radon measure ν for L with support contained in
+{α ∈ X(A) : α↾V is q–continuous} if and only if there exists a Hilbertian seminorm
+p on V with tr(p/q) < ∞ and for each S ∈ J there exists a representing Radon
+measure νS for L↾S with support contained in X(S) and such that {νS : S ∈ J} is
+p−concentrated.
+Proof. For each S ∈ J, let νS be a representing Radon measure for L↾S with support
+contained in X(S) and p be a Hilbertian seminorm p on V with tr(p/q) < ∞ such
+that {νS : S ∈ J} is p−concentrated. Let us first show that the family {νS : S ∈ J}
+fulfils the so-called Prokhorov condition by means of the characterization in [17,
+Proposition 1.18], that is, we aim to show that for all ε > 0 and for all S ∈ J, there
+exists K(S) ⊆ X(S) compact such that νS(K(S)) ≥ 1 − ε and πS,T (K(T )) ⊆ K(S)
+for all T ∈ J with S ⊆ T .
+Let ε > 0. Since {νS : S ∈ J} is p−concentrated and tr(p/q) < ∞, we can
+take δ > 0 as in (2.1) and set rε := q(δ√ε)−1�
+tr(p/q). For each S ∈ J, define
+K(S) := {α ∈ X(S) : |α(v)| ≤ rε(v) for all v ∈ S ∩ V }. Then K(S) is compact in
+X(S) as it is closed and embeds into the compact product �
+v∈S∩V [−rε(v), rε(v)]
+via the continuous map α �→ (α(v))v∈S∩V .
+Now for any S ⊆ T in J the in-
+clusion πS,T(K(T )) ⊆ K(S) holds by definition and for each S ∈ J the estimate
+νS(K(S)) ≥ 1 − 14ε holds by Lemma 2.6 below. Hence, the family {νS : S ∈ J}
+fulfils Prokhorov’s condition and so we can apply [17, Theorem 3.10-(ii)], which
+guaranteed the existence of a representing Radon measure ν for L with support
+contained in X(A).
+It remains to show that the support of ν is contained in
+{α ∈ X(A) : α↾V
+is q–continuous}. For this, set φV : X(A) → V ∗, α �→ α ↾V and
+Kε := {α ∈ X(A) : |α(v)| ≤ rε(v) for all v ∈ V } =
+�
+S∈J
+π−1
+S (K(S)).
+Then [5, Propositions 7.2.2-(i) and 7.2.5-(iii)] and Lemma 2.6 imply that
+ν(Kε) = lim
+S∈I νS(K(S)) ≥ 1 − 14ε.
+Since Kε ⊆ φV
+−1(V ′
+rε) = φV
+−1(V ′
+q) for all ε > 0 this yields that ν(φV
+−1(V ′
+q)) = 1,
+i.e. ν has support contained in φV
+−1(V ′
+q ) = {α ∈ X(A) : α↾V
+is q–continuous}.
+Conversely, let ν be a representing Radon measure for L with support contained
+in {α ∈ X(A) : α ↾V
+is q–continuous}. Then, for each S ∈ J, the push-forward
+νS := πS#ν is a representing Radon measure for L ↾S with support contained
+in X(S). For each n ∈ N, set Kn := φV
+−1(Bn(q′)) and define
+(2.6)
+p(v)2 :=
+∞
+�
+n=1
+1
+n4
+�
+Kn
+ˆv2dν
+for all v ∈ V.
+
+10
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+It is easy to verify that p defines a Hilbertian seminorm on V .
+Then for each
+E ∈ FON(q) we have that
+�
+e∈E
+p(e)2 (2.6)
+=
+∞
+�
+n=1
+1
+n4
+�
+Kn
+�
+e∈E
+ˆe2dν
+Lemma 2.7
+≤
+∞
+�
+n=1
+1
+n4
+�
+Kn
+n2dν ≤
+∞
+�
+n=1
+1
+n2 < 2,
+that is, tr(p/q) ≤ 2 < ∞. To show that {νS : S ∈ J} is p−concentrated, let ε > 0
+and take n ∈ N such that ν(X(A) \ Kn) ≤ 2−1ε. Then there exists δ > 0 such that
+n4δ2 ≤ 2−1ε and so, for each S ∈ J and each v ∈ Bδ(p) ∩ S, we obtain that
+νS({α ∈ X(S) : |α(v)| ≥ 1})
+≤
+ν({α ∈ X(A) : |α(v)| ≥ 1} ∩ Kn) + ν(X(A) \ Kn)
+≤
+�
+Kn
+ˆv2dνS + 2−1ε ≤ n4p(v)2 + 2−1ε ≤ ε,
+i.e. (2.1) holds.
+□
+Lemma 2.6. For each S ∈ J, the estimate νS(K(S)) ≥ 1 − 14ε holds.
+Proof. Let S ∈ J and set I := {W ⊆ S ∩ V : W fin. dim. subspace of S ∩ V }. Let
+W ∈ I and consider the continuous restriction map φW : X(S) → W ′. Then the
+push-forward µ′
+W := φW #νS is a probability measure on W ′ that satisfies
+µ′
+W ({l ∈ W ′ : |l(w)| ≥ 1}) = νS({α ∈ X(S) : |α(w)| ≥ 1}) ≤ ε
+for all w ∈ Bδ(p↾W ). Then Lemma 1.6 implies that
+νS(φW
+−1(B1(rε ↾′
+W ))) = µ′
+W (B1(rε ↾′
+W ))) ≥ 1 − 7(ε + tr(p↾W /δr↾W )) = 1 − 14ε
+as tr(p↾W /δrε ↾W ) ≤ tr(p/δrε) and, by Lemma 1.3-((ii)), tr(p/δrε) ≤ ε.
+Since K(S) = �
+W∈I φW
+−1(B1(rε ↾W ′)) by definition, [5, Propositions 7.2.2-(i)
+and 7.2.5-(iii)] imply that νS(K(S)) = limW∈I νS(φW
+−1(B1(r↾W ′))) ≥ 1 − 14ε.
+□
+Lemma 2.7. Let n ∈ N and E ∈ FON(q). Then �
+e∈E
+ˆe(α) ≤ n2 for all α ∈ Kn.
+Proof. Let α ∈ Kn and set H := span(E) (for convenience, set α = α ↾H and
+q = q ↾H). Since E ∈ FON(q) is finite, the space (H, q) is Hilbertian and E is a
+complete q−orthonormal system. In particular, α ↾H∈ Bn(q ↾′
+H) and by the Riesz
+representation theorem there exists a ∈ H such that α(x) = ⟨x, a⟩q for all x ∈ H
+and q(a) = q′(α) ≤ n. Therefore,
+�
+e∈E
+ˆe(α)2 =
+�
+e∈E
+α(e)2 =
+�
+e∈E
+⟨e, a⟩2
+q = q(a)2 ≤ n2
+yields the assertion.
+□
+Using exactly the same proof scheme but exploiting [17, Corollary 3.11-(ii)] in-
+stead of [17, Theorem 3.10-(ii)], it is easy to obtain the following more general
+version of Theorem 2.5 including the localization of the support of the representing
+measure.
+Theorem 2.8.
+Let A be an algebra generated by a linear subspace V ⊆ A, K ⊆ X(A) closed, q
+a Hilbertian seminorm on V such that {α ∈ K : α ↾V is q–continuous} ̸= ∅ and
+J := {⟨W⟩ : W finite dimensional subspace of V }. Let L be a normalized linear
+functional on A and Q a quadratic module such that K = KQ.
+There exists a representing Radon measure ν for L with support contained in
+{α ∈ K : α ↾V is q–continuous} if and only if there exists a Hilbertian seminorm
+p on V with tr(p/q) < ∞ and for each S ∈ J there exists a representing Radon
+measure νS for L↾S with support contained in KQ∩S and such that {νS : S ∈ J} is
+p−concentrated.
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+11
+Remark 2.9.
+(1) If in Theorem 2.5 (resp. Theorem 2.8) we assume for each S ∈ J the
+uniqueness of the representing measure for L↾S with support contained in
+X(S) (resp. in KQ∩S) , then by [17, Remark 3.12-(ii)] we get the uniqueness
+of the corresponding representing measure for L.
+(2) If in Theorem 2.5 (resp. Theorem 2.8) we take V = A, we obtain a criterion
+for the existence of a representing measure for L with support contained in
+sp(q) (resp. on KQ ∩ sp(q)).
+(3) Combining Theorem 2.5 (resp. Theorem 2.8) and Remark 2.3, it is easy to
+see that if there exists a representing measure for L with support contained
+in {α ∈ X(A) : α ↾V
+is q−continuous} then L vanishes on ker(p) and so
+on ker(q).
+When A is endowed with a Hilbertian seminorm q and there exists C > 0 such
+that L(a2) ≤ Cq(a)2 for all a ∈ A, we can characterize the representing measures
+for L with support contained in sp(q) only through conditions on a dense subalgebra
+of A.
+Theorem 2.10. Let A be an algebra, q a Hilbertian seminorm on A with sp(q) ̸= ∅,
+B a subalgebra of A which is dense in (A, q) and I := {⟨W⟩ : W finite dimensional
+subspace of B}. Let L be a normalized � A2–positive linear functional on A for
+which there exists C > 0 such that L(a2) ≤ Cq(a)2 for all a ∈ A.
+There exists a representing Radon measure ν for L with support contained in
+sp(q) if and only if there exists a Hilbertian seminorm p on B with tr(p/q) < ∞
+and for each S ∈ I there exists a representing Radon measure νS for L ↾S with
+support contained in X(S) such that {νS : S ∈ I} is p−concentrated.
+Proof. By the density of B in (A, q), there is a one-to-one correspondence between
+the set of all q−continuous characters of A and the set of all q−continuous characters
+of B, which will therefore both denote simply by sp(q). Moreover, let τsp(q)A (resp.
+τsp(q)B) be the weakest topology on sp(q) which makes ˆa : sp(q) → R, α �→ α(a)
+continuous for all a ∈ A (resp. for all a ∈ B) and by B(τsp(q)A) (resp. B(τsp(q)B))
+the associated Borel σ-algebra. We demand to Appendix 4.3 the proof that
+(2.7)
+B(τsp(q)A) = B(τsp(q)B).
+and so we will not distinguish between the measurable spaces (sp(q), B(τsp(q)A))
+and (sp(q), B(τsp(q)B)), which will be both simply denoted by (sp(q), B(τsp(q))).
+Suppose that there exists a representing Radon measure ν for L with support
+contained in sp(q). Then, applying Theorem 2.5 for V = A = B, we get that there
+exists a Hilbertian seminorm p on B with tr(p/q) < ∞ and for each S ∈ I there
+exists a representing Radon measure νS for L ↾S with support contained in X(S)
+such that {νS : S ∈ I} is p−concentrated.
+Conversely, suppose there exists a Hilbertian seminorm p on B with tr(p/q) < ∞
+and for each S ∈ I there exists a representing Radon measure νS for L ↾S with
+support contained in X(S) such that {νS : S ∈ I} is p−concentrated.
+Then,
+applying Theorem 2.5 for V = A = B (see also Remark 2.9-(2)), we obtain that
+there exists a representing measure ν for L ↾B with support contained in sp(q). We
+aim to prove that ν is actually a representing measure for L, so it remains to show
+that L(a) =
+�
+ˆa(α)dν(α), ∀a ∈ A \ B.
+Let a ∈ A\B. By the density of B in (A, q), there exists a sequence (bn)n∈N ⊆ B
+such that q(bn − a) → 0 as n → ∞. Hence, for any α ∈ sp(q) we get α(bn) → α(a)
+as n → ∞, i.e. limn→∞ ˆbn(α) = ˆa(α). Then, using Fatou’s lemma, we obtain that
+
+12
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+�
+sp(q)
+ˆa(α)2dν =
+�
+sp(q)
+lim
+n→∞
+ˆbn(α)2dν ≤ lim inf
+n→∞
+�
+sp(q)
+ˆbn(α)2dν = lim inf
+n→∞ L(b2
+n) = L(a2),
+where in the last equality we used the Cauchy–Bunyakovsky–Schwarz inequality
+and the existence of C > 0 such that L(a2) ≤ Cq(a)2 for all a ∈ A.
+Hence,
+ˆa ∈ L2(sp(q), B(τsp(q))) and so ˆa ∈ L1(sp(q), B(τsp(q))).
+Then for any M > 0 we have that
+�
+sp(q)
+|ˆa(α) − ˆbn(α)|dν(α) ≤
+�
+sp(q)
+���ˆa(α)1{β:|ˆa(β)|≤M}(α) − ˆbn(α)1{β:|ˆbn(β)|≤M}(α)
+��� dν(α)
++
+�
+sp(q)
+��ˆa(α)1{β:|ˆa(β)|≤M}(α) − ˆa(α)
+�� dν(α)
++
+�
+sp(q)
+��� ˆbn(α)1{β:|ˆbn(β)|≤M}(α) − ˆbn(α)
+��� dν(α)
+=
+�
+sp(q)
+���ˆa(α)1{β:|ˆa(β)|≤M}(α) − ˆbn(α)1{β:|ˆbn(β)|≤M}(α)
+��� dν(α)
++
+�
+sp(q)
+|ˆa(α)|
+1{β:|ˆa(β)|>M}(α)dν(α)
++
+�
+sp(q)
+���ˆbn(α)
+���
+1{β:|ˆbn(β)|>M}(α)dν(α)
+(2.8)
+Using that
+1{β:|ˆbn(β)|>M}(α) ≤
+1
+M |ˆbn(α)| and that ν is a representing measure
+for L ↾B, we easily see that:
+�
+sp(q)
+���ˆbn(α)
+���
+1{β:|ˆbn(β)|>M}(α) ≤ 1
+M
+�
+sp(q)
+ˆbn(α)2dν(α) = L(b2
+n)
+M
+→ L(a2)
+M
+, as n → ∞.
+Therefore, passing to the limit for n → ∞ in (2.8), we get that for any M > 0:
+(2.9)
+lim
+n→∞
+�
+sp(q)
+|ˆa(α)− ˆbn(α)|dν(α) ≤
+�
+sp(q)
+|ˆa(α)|
+1{β:|ˆa(β)|>M}(α)dν(α)+L(a2)
+M
+Since ˆa ∈ L1(sp(q), B(τsp(q))) and |ˆa(α)|
+1{β:|ˆa(β)|>M}(α) ≤ |ˆa(α)| for all M > 0,
+we can apply the dominated converge theorem, which ensures that:
+lim
+M→∞
+�
+sp(q)
+|ˆa(α)|
+1{β:|ˆa(β)|>M}(α)dν(α) =
+�
+sp(q)
+lim
+M→∞ |ˆa(α)|
+1{β:|ˆa(β)|>M}(α)dν(α) = 0
+Hence, passing to the limit for M → ∞ in (2.9), we obtain that:
+lim
+n→∞
+�
+sp(q)
+|ˆa(α) − ˆbn(α)|dν(α) = 0
+and so
+L(a) = lim
+n→∞ L(bn) = lim
+n→∞
+�
+sp(q)
+ˆbn(α)dν(α) =
+�
+sp(q)
+ˆa(α)dν(α).
+□
+With a similar proof, it is possible to show the following more general version of
+Theorem 2.10.
+Theorem 2.11. Let A be an algebra, K ⊆ X(A) closed, q a Hilbertian seminorm
+on A with sp(q) ∩ K ̸= ∅, B a subalgebra of A which is dense in (A, q) and I :=
+{⟨W⟩ : W finite dimensional subspace of B}. Let L be a normalized � A2–positive
+linear functional on A for which there exists C > 0 such that L(a2) ≤ Cq(a)2 for
+all a ∈ A and Q a quadratic module such that K = KQ.
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+13
+There exists a representing Radon measure ν for L with support contained in
+sp(q)∩K if and only if there exists a Hilbertian seminorm p on B with tr(p/q) < ∞
+and for each S ∈ I there exists a representing Radon measure νS for L ↾S with
+support contained in KQ∩S such that {νS : S ∈ I} is p−concentrated.
+Given a normalized � A2–positive linear functional L, the map (a, b) �→ L(ab)
+defines a symmetric positive semidefinite bilinear form and so the following is a
+natural Hilbertian seminorm on A
+(2.10)
+sL(a) :=
+�
+L(a2) for all a ∈ A.
+Then, combining Lemma 2.4 with our main results Theorem 2.8 and Theorem 2.11,
+we easily obtain the following two results.
+Corollary 2.12. Let A be an algebra generated by a linear subspace V ⊆ A, K ⊆
+X(A) closed and J := {⟨W⟩ : W finite dimensional subspace of V }. Let L be a
+normalized � A2–positive linear functional L on A and Q a quadratic module such
+that K = KQ.
+If there exists a Hilbertian seminorm q on V such that tr(sL ↾V /q) < ∞ and
+{α ∈ K : α↾V is q–continuous} ̸= ∅ and for each S ∈ J there exists a representing
+Radon measure νS for L↾S with support contained in KQ∩S, then there exists a rep-
+resenting measure for L with support contained in {α ∈ K : α↾V is q–continuous}.
+Proof. Since L(a2) = sL(a)2 for all a ∈ V , (2.4) holds for p = sL and C = 1. Hence,
+Lemma 2.4 ensures that {νS : S ∈ I} is sL−concentrated. This together with the
+assumption tr(sL ↾V /q) < ∞ allows us to apply Theorem 2.8 for p = sL, ensuring
+that there exists a representing Radon measure ν for L with support contained in
+{α ∈ K : α↾V is q–continuous}.
+□
+Corollary 2.13. Let (A, τ) be a locally convex topological algebra, B a sub-algebra
+of A which is dense in (A, τ) and I := {⟨W⟩ : W finite dimensional subspace of B}.
+Let L be a normalized � A2–positive linear functional on A and Q a quadratic
+module such that K = KQ.
+If there exists a τ−continuous Hilbertian seminorm q on A with tr(sL/q) < ∞
+and sp(q)∩K ̸= ∅ and if for each S ∈ I there exists a representing Radon measure νS
+for L↾S with support contained in KQ∩S, then there exists a representing measure
+for L with support contained in sp(q) ∩ K.
+Proof. The τ–continuity of q and the density of B in (A, τ) provide that B is dense
+in (A, q). Moreover, since L(a2) = sL(a)2 for all a ∈ A, (2.4) holds for p = sL,
+C = 1 and V = A, so Lemma 2.4 ensures that {νS : S ∈ I} is sL−concentrated.
+Then, by Theorem 2.11, there exists a representing measure for L with support
+contained in sp(q) ∩ K.
+□
+Remark 2.14. In Corollary 2.12 and Corollary 2.13 we could actually replace A
+with A/ ker(sL), because it is readily seen from the Cauchy-Schwartz inequality that
+L vanishes on ker(sL). Moreover, we have that V/ ker(sL ↾V ) = V/(ker(sL) ∩ V ) =
+V/ ker(sL) and, whenever tr(sL ↾V /q) < ∞ for some Hilbertian norm q, the space
+V/ ker(sL) endowed with the quotient seminorm induced by sL (and also denoted by
+sL with a slight abuse of notation) is separable. Hence, whenever these techniques
+work, (V, sL ↾V ) is essentially a separable space.
+Let us now exploit Corollary 2.12 to obtain more concrete sufficient conditions
+for the existence of a representing measure for L in presence of a fixed Hilbertian
+seminorm q on A.
+Corollary 2.15. Let A be an algebra generated by a linear subspace V ⊆ A, L a
+normalized linear functional on A, Q a quadratic module in A and q a Hilbertian
+seminorm on A with {α ∈ KQ : α↾V
+is q–continuous} ̸= ∅. If
+
+14
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+(a) L(Q) ⊆ [0, ∞),
+(b) for each v ∈ V ,
+∞
+�
+n=1
+1
+2n√
+L(v2n) = ∞,
+(c) tr(sL ↾V /q) < ∞, where sL(a) :=
+�
+L(a2) for all a ∈ A,
+then there exists a unique representing Radon measure ν for L with support con-
+tained in {α ∈ KQ : α↾V
+is q–continuous}.
+Proof. Let J := {⟨W⟩ : W finite dimensional subspace of V }. By [17, Theorem 3.17-
+(i)], the assumptions (a) and (b) guarantee that for each S ∈ J there exists a unique
+representing Radon measure νS for L↾S with support contained in KQ∩S. This to-
+gether with the assumption (c) allows us to apply Corollary 2.12, ensuring that
+there exists unique representing Radon measure ν for L with support contained in
+{α ∈ KQ : α↾V
+is q–continuous}.
+□
+Remark 2.16. Corollary 2.15 still holds if we replace the assumptions (a) and
+(b) with the assumption (a’) L(Pos(KQ)) ⊆ [0, +∞) and in the proof we use [17,
+Theorem 3.14] instead of [17, Theorem 3.17]. However, under this replacement, the
+uniqueness is not anymore ensured.
+The following lemma provides an explicit construction of a Hilbertian seminorm
+q as required in Corollary 2.12 and so in Corollary 2.15.
+Lemma 2.17. Let A be an algebra generated by a linear subspace V ⊆ A and p a
+Hilbertian seminorm on V such that there exists a complete p−orthonormal system
+{en : n ∈ N} in V . Choose (λn)n∈N ⊆ R>0 such that �∞
+n=1 λ2
+n < ∞. Then
+q(v) :=
+�
+�
+�
+�
+∞
+�
+n=1
+λ−2
+n ⟨v, en⟩2p,
+∀v ∈ V
+defines a Hilbertian seminorm on U :=
+�
+v ∈ V :
+∞
+�
+n=1
+λ−2
+n ⟨v, en⟩2
+p < ∞
+�
+such that
+tr(p ↾U /q) < ∞ and U is dense in (V, p).
+Proof. For each a, b ∈ U, let us define ⟨a, b⟩q :=
+∞
+�
+n=1
+λ−2
+n ⟨a, en⟩p⟨b, en⟩p (note that
+the Cauchy-Schwartz inequality provides that ⟨a, b⟩q < ∞, since ⟨v, v⟩q < ∞ for
+all v ∈ U by the definition of U). Then ⟨·, ·⟩q is a symmetric positive semidefinite
+bilinear form on U × U and thus, q(v) =
+�
+⟨v, v⟩q for all v ∈ U defines a Hilbertian
+seminorm on U.
+As {en : n ∈ N} is a complete p−orthonormal system in V , we have that Parse-
+val’s equality holds and so
+∀ v ∈ U,
+p(v)2 =
+∞
+�
+n=1
+⟨v, en⟩2
+p =
+∞
+�
+n=1
+λ2
+nλ−2
+n ⟨v, en⟩2
+p ≤
+�
+sup
+n∈N
+λ2
+n
+�
+q(v)2,
+i.e. ∀v ∈ U, p(v) ≤ Cq(v) where C := supn∈N λ2
+n is finite because of the assumption
+�∞
+n=1 λ2
+n < ∞.
+Moreover, since for all i, j ∈ N we have
+⟨λiei, λjej⟩q =
+∞
+�
+n=1
+λ−2
+n ⟨λiei, en⟩p⟨λjej, en⟩p =
+∞
+�
+n=1
+λ−2
+n (λiδi,n)(λjδj,n) = δi,j,
+the set {λnen : n ∈ N} is q−orthonormal. Moreover, for all v ∈ U and all n ∈ N we
+have that ⟨v, λnen⟩q = λ−1
+n ⟨v, en⟩p and so
+∀ v ∈ U,
+⟨v, v⟩2
+q =
+∞
+�
+n=1
+λ−2
+n ⟨v, en⟩2
+p =
+∞
+�
+n=1
+⟨v, λnen⟩2
+q,
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+15
+i.e. Parseval’s equality is satisfied, which is equivalent to say that {λnen : n ∈ N}
+is a complete q−orthonornal system in U by [6, Chapter 5, §2.3, Proposition 5].
+Hence, using (1.3), we get that
+(2.11)
+tr(p ↾U /q) =
+∞
+�
+n=1
+p(λnen)2 =
+∞
+�
+n=1
+λ2
+np(en)2 =
+∞
+�
+n=1
+λ2
+n < ∞
+As U contains {en : n ∈ N}, we get that U is dense in V .
+□
+Remark 2.18.
+(i) If (V, p) is Hausdorff and contains a countable total subset, then the ex-
+istence of a complete p−orthonormal system in V is guaranteed by [6,
+Chapter 5, §2.4, Corollary p. V.24]. In particular, such a system exists
+when (V, p) is Hausdorff and separable.
+(ii) If {fn : n ∈ N} is another complete p−orthonormal system in V , then we
+get that �∞
+n=1 λ−2
+n ⟨v, fn⟩2
+p < ∞ for all v ∈ T U where T is the linear oper-
+ator T : V → V given by en �→ fn for all n ∈ N. Indeed, since T maps a
+complete p−orthonormal system to another complete p−orthonormal sys-
+tem, T is orthogonal and so for all v ∈ U we get that �∞
+n=1 λ−2
+n ⟨T v, fn⟩2
+p =
+�∞
+n=1 λ−2
+n ⟨T v, T en⟩2
+p = �∞
+n=1 λ−2
+n ⟨v, en⟩2
+p < ∞.
+(iii) Iterating the construction in Lemma 2.17, we can show that there exists
+dense subset U of (V, p) such that U can be equipped with a nuclear topol-
+ogy stronger than the one inherited from p.
+Combining Lemma 2.17 with Corollary 2.12, we obtain the following corollary.
+Corollary 2.19. Let A be an algebra generated by a linear subspace V ⊆ A and L
+a normalized � A2–positive linear functional on A such that there exists a complete
+sL−orthonormal system {en : n ∈ N} in V . Choose U and q as in Lemma 2.17,
+and set J(U) := {⟨W⟩ : W finite dim. subspace of U}.
+If {α ∈ X(A) : α↾U is sL–continuous} ̸= ∅ and for each S ∈ J(U) there exists a
+representing Radon measure νS for L ↾S, then there exists a representing measure
+for L ↾⟨U⟩ with support contained in {α ∈ X(⟨U⟩) : α↾U is q–continuous}.
+Proof. The assumptions ensure that we can apply Lemma 2.17 to (V, sL ↾V ), which
+provides a Hilbertian seminorm q on a dense subset U of (V, sL ↾V ) such that
+tr(sL ↾U /q) < ∞. Then {α ∈ X(A) : α↾U is sL–continuous} ⊆ {α ∈ X(⟨U⟩) : α↾U
+is q–continuous} and so {α ∈ X(⟨U⟩) : α ↾U is q–continuous} ̸= ∅. Hence, we can
+apply Corollary 2.12 to L ↾⟨U⟩ and get the conclusion.
+□
+In the above corollary the Hilbertian seminorm q on V is not pre-given as in
+Theorem 2.5 (resp. Theorem 2.8), but explicitly constructed through Lemma 2.17.
+The price to pay for this is that we obtain an integral representation for the starting
+linear functional L not on the whole A but just on the sublagebra ⟨U⟩ of A. Note
+that the latter subalgebra is actually dense in (A, sL) (or more in general in (A, p)
+when p is defined on the whole of A, see Lemma 4.17 in the Appendix) and so
+Corollary 2.19 provides a representing measure for L restricted to a dense subalgebra
+of (A, sL). However, the representing measure is supported on characters whose
+restrictions to U lie in the topological dual of (U, q) and the density of ⟨U⟩ in (A, sL)
+does not allow us to show that is supported on characters whose restrictions to V
+are in the topological dual of (V, sL ↾V ) and so to get an integral representation
+for L on the full A.
+The latter effect is not an artefact of the techniques used
+here.
+Indeed, if V = ℓ2 is endowed with the usual norm ∥ · ∥ℓ2 which makes
+it a Hilbert space, then the associated Gaussian measure (which is the product
+of infinitely many one-dimensional standard Gaussian measures) gives rise to a
+functional L on S(V ) and the Gaussian measure is the only measure representing
+
+16
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+L. As sL ↾V = ∥·∥ℓ2, the Gaussian measure cannot be supported on {α ∈ X(S(V )) :
+α ↾V
+is sL ↾V −continuous} because it is well-known that this set has measure zero
+(see e.g. [2, Theorem 1.3].
+In the case when U = V , Corollary 2.19 provides a representing measure for the
+starting L on the whole A. This is for example the case when (V, τV ) is separa-
+ble nuclear and sL ↾V is τV −continuous, as analyzed in more details in the next
+subsection.
+2.3. Results on Main Question for τV nuclear.
+Corollary 2.12 and Corollary 2.15 nicely applies to the case when the generating
+subspace of the algebra is endowed with a nuclear topology.
+Corollary 2.20. Let (V, τV ) be a nuclear space, A an algebra generated by V ,
+J := {⟨W⟩ : W finite dimensional subspace of V } and K ⊆ X(A) closed. Let L
+be a normalized � A2–positive linear functional L on A and Q a quadratic module
+such that K = KQ.
+If for each S ∈ J there exists a representing Radon measure νS for L ↾S with
+support contained in KQ∩S and sL ↾V is τV –continuous, then for each Hilbertian
+seminorm q on V s.t. {α ∈ KQ : α↾V
+is q–continuous} ̸= ∅ and tr(sL ↾V /q) < ∞,
+there exists a representing measure for L with support contained in {α ∈ K : α↾V
+is q–continuous}.
+Proof. As (V, τV ) is nuclear and sL ↾V is τV –continuous, using Lemma 1.3 and the
+directnedness of the generating family for τ, we can easily derive that there exists
+a τV –continuous Hilbertian seminorm q on V such that tr(sL ↾V /q) < ∞. Thus,
+we can apply Corollary 2.12 and get the desired conclusion.
+□
+Using exactly the same argument but replacing Corollary 2.12 with Corollary 2.15,
+we obtain the following.
+Corollary 2.21. Let (V, τV ) be a nuclear space, A be generated by V , L a normal-
+ized linear functional on A such that L(� A2) ⊆ [0, ∞) and Q a quadratic module
+in A. If
+(a) L(Q) ⊆ [0, ∞),
+(b) for each v ∈ V ,
+∞
+�
+n=1
+1
+2n√
+L(v2n) = ∞,
+(c) sL ↾V is τV –continuous,
+then, for each Hilbertian seminorm q on V s.t. {α ∈ KQ : α↾V
+is q–continuous} ̸=
+∅ and tr(sL ↾V /q) < ∞, there exists a unique representing Radon measure ν for L
+such that ν({α ∈ KQ : α↾V
+is q–continuous}) = 1.
+We can retrieve [23, Theorem 13] from Corollary 2.21 applied to Q = � A2.
+Indeed, in [23, Theorem 13] the assumption (ii) exactly correspond to (a) and (b)
+of Corollary 2.21 for Q = � A2 (the alternative assumption (i) corresponds to (a’)
+in Remark 2.16), and the assumption of the existence of a τ−continuous seminorm
+q on V such that L(v2) ≤ q(v)2 for all v ∈ V guarantees that sL ↾V (a) ≤ q(v) for
+all v ∈ V , i.e. also (c) in Corollary 2.21 is satisfied.
+Note that if there exists a τ−continuous Hilbertian seminorm q on A such that
+L(a2) ≤ q(a)2 for all a ∈ A, then not only sL is τ−continuous but also L itself is
+τ–continuous, since by the Cauchy-Schwarz inequality we have that
+|L(a)|2 = |L(1 · a)|2 ≤ L(1)L(a2) ≤ q(a)2
+for all a ∈ A.
+Viceversa, the continuity of L on certain classes of nuclear topological algebra
+provides the continuity of sL, allowing us to establish the following result.
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+17
+Corollary 2.22. Let (A, τ) be a locally convex nuclear topological algebra which is
+also barrelled (respectively, has also jointly continuous multiplication), L a τ−continuous
+linear functional on A and Q a quadratic module in A. If
+(a) L(Q) ⊆ [0, ∞),
+(b) for each v ∈ V ,
+∞
+�
+n=1
+1
+2n√
+L(v2n) = ∞,
+then, for each Hilbertian seminorm q on V s.t. KQ ∩ sp(q) ̸= ∅ and tr(sL/q) < ∞,
+there exists a unique (KQ ∩ sp(q))–representing Radon measure ν for L.
+Proof. Let us first observe that the τ–continuity of sL is ensured both when (A, τ)
+is barrelled and when has jointly continuous multiplication. Indeed, in the first
+case the τ–continuity of L ensures the existence of a Hilbertian seminorm q on
+A such that L(a2) ≤ q(a)2 for all a ∈ A(see, e.g. [23, Lemma 14]) and so, as
+observed above, sL is τ–continuous. In the second case, the joint continuity of the
+multiplication provides the existence of a τ–continuous seminorm p on A such that
+L(ab) ≤ p(a)p(b) for all a, b ∈ A and so sL(a)2 = L(a2) ≤ p(a)2 for all a ∈ A, which
+shows that sL is τ–continuous.
+Hence, in both cases we can apply Corollary 2.21 for V = A and get the desired
+conclusion.
+□
+We can easily retrieve [23, Theorem 15] from Corollary 2.22 applied to Q = � A2.
+3. The case of the symmetric algebra of a nuclear space
+Let us apply the results of Section 2 to the case when A is the symmetric algebra
+S(V ) with (V, τV ) nuclear. Corollary 2.21 immediately gives the following result.
+Corollary 3.1. Let (V, τV ) be a nuclear space, L a normalized linear functional on
+S(V ) and Q a quadratic module in S(V ). If
+(a) L(Q) ⊆ [0, ∞),
+(b) for each v ∈ V ,
+∞
+�
+n=1
+1
+2n√
+L(v2n) = ∞,
+(c) sL ↾V is τ–continuous
+then, for each Hilbertian seminorm q on V such that tr(sL ↾V /q) < ∞ and {α ∈
+KQ : α↾V
+is q–continuous} ̸= ∅, there exists a unique representing Radon measure
+ν for L such that ν({α ∈ KQ : α↾V
+is q–continuous}) = 1.
+We can retrieve [23, Theorem 16] from Corollary 3.1 applied to Q = � A2.
+Indeed, the definition of nuclear space in [23, p. 445] is covered by Definition 1.4
+(for more details see Remark 4.16), [23, Theorem 16] the assumption (ii) exactly
+correspond to (a) and (b) of Corollary 3.1 for Q = � A2 (the alternative assumption
+(i) corresponds to (a’) in Remark 2.16), and the assumption of the existence of a
+τ−continuous seminorm q on V such that L(v2) ≤ q(v)2 for all v ∈ V guarantees
+that sL ↾V (a) ≤ q(v) for all v ∈ V , i.e. also (c) in Corollary 3.1 is satisfied.
+If to each v ∈ V we associate the operator Av(w) = vw for any w ∈ S(V ), then
+we can also retrieve [7, Theorem 4.3, (i) ↔ (iii)] for such operators from the version
+of Corollary 3.1 with (a) and (b) replaced by (a’) in Remark 2.16 by taking L = T
+and K = Z (see also [14, Theorem 3.11]).
+Corollary 3.1 also allows to easily prove the following result.
+Corollary 3.2. Let (V, τV ) be a nuclear space with τV induce by a directed family
+of seminorms P on V , L a normalized linear functional on S(V ) and Q a quadratic
+module in S(V ). If
+
+18
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+(a) L(Q) ⊆ [0, ∞),
+(b) for each v ∈ V ,
+∞
+�
+n=1
+1
+2n√
+L(v2n) = ∞,
+(c) for each d ∈ N, there exists p ∈ P such that the restriction L: S(V )d → R is
+pd–continuous, where pd is the quotient seminorm on the d−th homogeneous
+component S(V )d of S(V ) induced by the projective tensor seminorm p⊗d,
+then, for each Hilbertian seminorm q on V such that tr(sL ↾V /q) < ∞ and {α ∈
+KQ : α↾V
+is q–continuous} ̸= ∅, there exists a unique representing Radon measure
+ν for L such that ν({α ∈ KQ : α↾V
+is q–continuous}) = 1.
+Proof. Since L ↾S(V )2 is p2–continuous for some p ∈ P, there exists C > 0 such
+that L(v2) ≤ Cp2(v) for all v ∈ V . Moreover, as pd comes from the projective
+tensor seminorm p⊗d, we easily get that pd(vd) ≤ p(v)d holds for all v ∈ V and
+all d ∈ N, see e.g. [9, Lemma 3.1]. Using the latter for d = 2, we obtain that
+L(v2) ≤ Cp2(v) ≤ Cp(v)2 for all v ∈ V , namely that the Hilbertian seminorm
+sL ↾V is p–continuous and so τV –continuous. Hence, the conclusion follows at once
+from Corollary 3.1.
+□
+Using Theorem 2.10 instead of Corollary 2.21, we can prove a slightly generaliza-
+tion of the classical solution to Main Question for (V, τV ) nuclear in [2, Chapter 5,
+Theorem 2.1] (cf. [2, Chapter 5, Section 2.3] and [3]).
+Theorem 3.3. Let (V, τV ) be a Hausdorff separable nuclear space with τV induced
+by a directed family P of Hilbertian seminorms on V , L a normalized linear func-
+tional on S(V ) and K a closed subset of V ∗. For any n ∈ N and s ∈ P, let ˜s(n) the
+Hilbertian seminorm on S(V )n given by �s(n)(b) :=
+�
+N
+�
+i=1
+N
+�
+j=1
+⟨bi1, bj1⟩s · · · ⟨bin, bjn⟩s
+for any b:=
+N
+�
+i=1
+bi1· · ·bin ∈ S(V )n with N ∈ N and bik ∈ V for k = 1, . . . , n. If
+(1) L(Q) ⊆ [0, ∞), where Q is a quadratic module of S(V ) such that K = KQ
+(2) there exists a countable subset E of V whose linear span is dense in (V, τV )
+such that �∞
+k=1
+1
+2k√
+L(v2k) = ∞ for all v ∈ span(E)
+(3) For any d ∈ N, there exists p2d ∈ P such that the restriction L: S(V )2d → R
+is �
+p2d
+(2d)–continuous
+(4) K ∩ V ′
+q2 ̸= ∅, where q2 ∈ P is such that tr(p2/q2) < ∞,
+then there exists a representing Radon measure µ for L with support contained in
+K ∩ V ′
+q2.
+Remark 3.4.
+If for each d the map V → R, v �→ L(vd) is τV −continuous, then the assump-
+tion (3) in Theorem 3.3 holds. Indeed, the τV −continuity of the map V → R, v �→
+L(vd) implies that for any d there exists a rd ∈ P such that |L(vd)| ≤ 1 for all
+v ∈ V with rd(v) ≤ 1. Then
+����L
+��
+v
+rd(v)
+�d����� ≤ 1 for all v ∈ V and so
+��L(vd)
+�� ≤ rd(v)d, ∀v ∈ V.
+By using the multivariate polarization identity, this in turn provides that
+|L(v1 · · · vd)| ≤ dd
+d! rd(v1) · · · rd(vd), ∀v1, . . . , vd ∈ V.
+Then, since (V, τ) is nuclear, Lemma 3.5-(1) below ensures that for any pd ∈ P
+with tr(rd/pd) < ∞ we have
+|L(a)| ≤ (tr(rd/pd)d)d
+d!
+�p(d)
+d (a), ∀a ∈ S(V )d
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+19
+and hence, in particular, L ↾S(V )2d is �p(2d)
+2d −continuous.
+Lemma 3.5. Let (V, τV ) be a separable nuclear space with τV induced by a directed
+family of Hilbertian seminorms P on V and L a normalized linear functional.
+(1) If for some d ∈ N, there exists r ∈ P and �CL,d, such that
+|L(v1 . . . vd)| ≤ �CL,d r(v1) . . . r(vd)
+∀v1, . . . , vd ∈ V,
+then for any s ∈ P with tr(r/s) < ∞ we have that
+|L(a)| ≤ �CL,d (tr(r/s))d �s(d)(a)
+∀a ∈ S(V )d.
+(2) Let ℓ ∈ V ∗ for some r ∈ P and αℓ the character on S(V ) associated to ℓ,
+which is uniquely determined by defining αℓ(v1 . . . vd) := ℓ(v1) . . . ℓ(vd) for
+all d ∈ N and v1, . . . , vd ∈ V . If ℓ ∈ V ′
+r for some r ∈ P, then the associate
+character αℓ on S(V ) is such that for any s ∈ P with tr(r/s) < ∞ and any
+d ∈ N the following holds
+|αℓ(a)| ≤ (r′(ℓ)tr(r/s))d �s(d)(a)
+∀a ∈ S(V )d.
+(3) If the assumption (3) in Theorem 3.3 holds with continuity constant CL,2d
+and (λd)d∈N0 is a sequence of real numbers such that
+(3.1)
+∞
+�
+d=0
+λ−2
+d
+< ∞,
+then the seminorm defined by
+(3.2)
+˜p(a)2 := λ2
+0|a(0)|2 +
+∞
+�
+d=1
+λ2
+dCL,2d
+�
+�
+p2d
+(d)(a(d))
+�2
+,
+∀ a :=
+∞
+�
+d=0
+a(d) ∈ S(V )
+is Hilbertian and
+|L(a)|2 ≤ L(a2) ≤
+� ∞
+�
+d=0
+λ−2
+d
+�
+˜p(a)2
+for all a ∈ S(V ).
+(4) Let CL,d, (λd)d∈N0 and ˜p as in (3), and for each d ∈ N take a seminorm
+q2d ∈ P such that tr(p2d/q2d) < ∞ (such a seminorm always exists by
+nuclearity). If (ηd)d∈N0 is a sequence of real numbers such that
+(3.3)
+∞
+�
+d=1
+λ2
+d
+η2
+d
+CL,2dtr(p2d/q2d)d < ∞,
+then the seminorm defined by
+(3.4)
+˜q(a)2 := η2
+0|a(0)|2 +
+∞
+�
+d=1
+η2
+d
+�
+�
+q2d
+(d)(a(d))
+�2
+,
+∀ a :=
+∞
+�
+d=0
+a(d) ∈ S(V )
+is Hilbertian and such that tr(˜p/˜q) < ∞.
+(5) Let CL,d, (λd)d∈N0 and ˜p as in (3), and for each d ∈ N take a seminorm
+q2d ∈ P such that tr(p2d/q2d) < ∞ for all d ∈ N and also tr(q2/q2d) < ∞
+for all d ∈ N with d ≥ 2 (such a seminorm always exists by nuclearity). If
+(ηd)d∈N0 is a sequence of real numbers fulfilling (3.3) and
+(3.5)
+∞
+�
+d=1
+c2d
+η2
+d
+< ∞, ∀c > 0,
+then ℓ ∈ V ∗ is q2-continuous if and only if αℓ is ˜q continuous, i.e. V ′
+q2 ans
+sp(˜q) are isomorphic, where ˜q is as in (3.4).
+Proof.
+
+20
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+(1) This is a direct consequence of the multilinear Schwartz kernel theorem for
+nuclear spaces, see e.g. [4, Lemma 6.1 and Theorem 6.1].
+(2) By the r−continuity of ℓ, we obtain that |αℓ(v1 . . . vd)| ≤ r′(ℓ)dr(v1) . . . r(vd)
+for all v1, . . . , vd ∈ V . Hence, the result directly follows from (1) applied
+for L replaced with αℓ.
+(3) Let d ∈ N and b :=
+N
+�
+i=1
+bi1 · · · bid ∈ S(V )d with N ∈ N and bik ∈ V
+for k = 1, . . . , d.
+Since the assumption (3) of Theorem 3.3 holds and
+b2 ∈ S(V )2d, we have that L(b2) ≤ C2d �
+p2d
+(2d)(b2). Moreover, since b2 =
+N
+�
+i=1
+N
+�
+h=1
+bi1 · · · bidbh1 · · · bhd, we obtain that
+�
+p2d
+(2d)(b2)
+=
+�
+�
+�
+�
+N
+�
+i=1
+N
+�
+h=1
+⟨bi1, bj1⟩p2d · · · ⟨bid, bjd⟩p2d
+N
+�
+j=1
+N
+�
+k=1
+⟨bh1, bk1⟩p2d · · · ⟨bhd, bkd⟩p2d
+=
+�
+�
+�
+�
+� N
+�
+i=1
+N
+�
+h=1
+⟨bi1, bj1⟩p2d · · · ⟨bid, bjd⟩p2d
+�2
+=
+N
+�
+i=1
+N
+�
+h=1
+⟨bi1, bj1⟩p2d · · · ⟨bid, bjd⟩p2d = �
+p2d
+(d)(b)2
+Hence, we get that
+(3.6)
+L(b2) ≤ C2d �
+p2d
+(2d)(b2) ≤ C2d �
+p2d
+(d)(b)2,
+∀b ∈ S(V )d.
+Let (λd)d∈N0 as in (3.1) and a := �∞
+d=0 a(d) ∈ S(V ). Then there exists
+Da ∈ N such that a(d) = 0 for all d > Da and so we get that
+|L(a)|2 ≤ |L(a2)| ≤
+Da
+�
+d=0
+Da
+�
+j=0
+|L(a(d))L(a(j))| ≤
+Da
+�
+d=0
+Da
+�
+j=0
+�
+L
+��
+a(d)�2��
+L
+��
+a(j)�2�
+=
+� Da
+�
+d=0
+�
+L
+��
+a(d)�2��2
+≤
+� ∞
+�
+d=0
+λ−2
+d
+� � Da
+�
+d=0
+λ2
+dL
+��
+a(d)�2��
+(3.6)
+≤
+� ∞
+�
+d=0
+λ−2
+d
+� �
+λ2
+0|a(0)| +
+Da
+�
+d=1
+λ2
+dC2d �
+p2d
+(d) �
+a(d)�2
+�
+=
+� ∞
+�
+d=0
+λ−2
+d
+�
+˜p(a)2.
+(4) Since (V, τV ) is Hausdorff and separable, we have that for each d ∈ N there
+exists a complete q2d−orthonormal system Ed in V (see Remark 2.18).
+Then B :=
+�
+1
+ηn ei1 · · · ein : n ∈ N0, ei1, . . . , ein ∈ En
+�
+is a complete ˜q−orthonormal
+system in S(V ) and thus, for any (λd)d∈N0 as in (3.1) and (ηd)d∈N0 as
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+21
+in (3.3), we obtain that
+tr(˜p/˜q)
+=
+�
+e∈B
+˜p(e)2 = λ2
+0
+η2
+0
++
+∞
+�
+d=1
+�
+ei1 ,...,eid∈Ed
+λ2
+dC2d �
+p2d
+(2d)(ei1 · · · eid)2
+η2
+d
+=
+λ2
+0
+η2
+0
++
+∞
+�
+d=1
+�
+ei1 ,...,eid∈Ed
+λ2
+dC2dp2d(ei1)2 · · · p2d(eid)2
+η2
+d
+=
+λ2
+0
+η2
+0
++
+∞
+�
+d=1
+λ2
+d
+η2
+d
+C2d
+�
+ei1 ∈Ed
+p2d(ei1)2 · · ·
+�
+eid∈Ed
+p2d(eid)2
+=
+λ2
+0
+η2
+0
++
+∞
+�
+d=1
+λ2
+d
+η2
+d
+C2dtr(p2d/q2d)d < ∞.
+Hence, tr(˜p/˜q) < ∞.
+(5) Let us first show why the existence of a seminorm q2d with the properties
+as in the statement is guaranteed by the nuclearity of V . As P is directed,
+for each d ≥ 2 there exists a seminorm r2d ∈ P such that p2d ≤ r2d and
+q2 ≤ r2d. Then, by the nuclearity of V , we can choose a q2d ∈ P such that
+tr(r2d/q2d) < ∞ and hence, by definition of trace, tr(p2d/q2d) < ∞ and
+tr(q2/q2d) < ∞ for all d ≥ 2.
+Let ℓ be q2-continuous. Then, for any d ≥ 2, we get that
+|αℓ((a(d))2)|
+(2)
+≤ (q′
+2(ℓ)tr(q2/q2d))2d �
+q2d
+(2d)((a(d))2),
+∀a(d) ∈ S(V )d,
+while, for d = 1, we have that
+|αℓ((a(1))2)| = ℓ(a(1))2 ≤ q′
+2(ℓ)q2(a(1))2,
+∀ a(1) ∈ S(V )1 = V.
+Moreover, arguing as in (3), it is easy to see that for all d ∈ N
+�
+q2d
+(2d)((a(d))2) = �
+q2d
+(2d)(a(d))2,
+∀a(d) ∈ S(V )d.
+Now, for any a := �∞
+d=0 a(d) ∈ S(V ), there exists Da ∈ N such that
+a(d) = 0 for all d > Da. Thus, setting ˜ηd := ηdq′
+2(ℓ)−d (1 + tr(q2/q2d))−d
+for all d ∈ N0 and exploiting the previous three inequalities, we get that
+|αℓ(a)|2
+≤
+|αℓ(a2)| ≤
+� ∞
+�
+d=0
+˜η−2
+d
+� � Da
+�
+d=0
+˜η2
+d αℓ
+��
+a(d)�2��
+≤
+� ∞
+�
+d=0
+˜η−2
+d
+� �
+˜η2
+0|a(0)| + ˜η2
+1q′
+2(ℓ)2q2(a(1))2 +
+Da
+�
+d=2
+˜η2
+d (q′
+2(ℓ)tr(q2/q2d))2d �
+q2d
+(d)�
+a(d)�2
+�
+≤
+� ∞
+�
+d=0
+˜η−2
+d
+�
+˜q(a)2,
+which provides the ˜q−continuity of αℓ since
+��∞
+d=0 ˜η−2
+d
+�
+< ∞ by (3.5).
+Conversely, if αℓ is ˜q−continuous, then there exists C ≥ 0 such that
+|ℓ(v)| = |αℓ(v)| ≤ C˜q(v) = Cη1q2(v),
+∀v ∈ V.
+□
+Proof of Theorem 3.3.
+Let I := {⟨W⟩ : W finite dimensional subspace of span(E)}.
+Recall that (X(S(V )), τX(S(V ))) is isomorphic to V ∗ equipped with the weak
+topology. Then, by the generalization of the classical Nussbaum theorem to any
+finitely generated algebra (see e.g. [17, Theorem 3.16]), the assumptions (1) and
+(2) ensure that for each S ∈ I there exists a unique KQ∩S–representing measure
+
+22
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+νS for L↾S. Moreover, the separability and the nuclearity of (V, τV ) as well as the
+assumptions (1) and (3) ensure that we can apply Lemma 3.5 and get two Hilbertian
+seminorms ˜p and ˜q on S(V ) such that tr(˜p/˜q) < ∞ and L(a2) ≤
+��∞
+d=0 λ−2
+d
+�
+˜p(a)2
+for all a ∈ S(V ). Thus, by Lemma 2.4, {νS : S ∈ I} is ˜p-concentrated.
+Also the density of span(E) in (V, τV ) given by assumption (2) implies the den-
+sity of S(span(E)) in (S(V ), ˜q). Then, by Lemma 3.5-(5) we have that sp(˜q) is
+isomorphic to V ′
+q2. Thus, exploiting also the assumption (4), the conclusion follows
+by applying Theorem 2.11 to A := S(V ), q = ˜q, B := S(span(E)), and p = ˜p.
+□
+We can retrieve [2, Chapter 5, Theorem 2.1] from Theorem 3.3, because their
+definition of nuclear space (V, τ) is covered by Definition 1.4 (for more details see
+Remark 4.14), their regularity assumption on the starting sequence [2, Chapter
+5, Section 2.1, p.52] corresponds to Theorem 3.3-(3), their positivity assumption
+[2, Chapter 5, (2.1)] is equivalent Theorem 3.3-(1), and those together with their
+growth condition in [2, Chapter 5, (2.5)] imply that Theorem 3.3-(2) holds. For the
+convenience of the reader, we restate their growth condition in our setting
+∃ E ⊂ V countable s.t. span(E) is dense in (V, τ) and C{zk} is quasi-analytic,
+(3.7)
+where zk :=
+�
+sup
+v∈E
+p2k(v)
+�k
+�
+�
+�
+�
+sup
+v1,...,v2k∈V
+�
+|L(v1 · · · v2k)|
+�
+p2k
+(2k)(v1 · · · v2k)
+�
+∀k ∈ N,
+p2k and �
+p2k
+(2k) are as in Theorem 3.3-(3).
+and prove in detail the above mentioned implication.
+Proposition 3.6. Let (V, τV ) be a separable nuclear space with τV induced by a
+directed family of seminorms P on V , L: S(V ) → R linear such that (1) and (3)
+of Theorem 3.3 hold. If (3.7) is fulfilled, then Theorem 3.3-(2) holds.
+Proof. Let us preliminarily observe that for each k ∈ N we have
+(3.8)
+mk :=
+�
+sup
+v1,...,v2k∈E
+|L(v1 . . . v2k)| ≤ zk.
+Moreover, the following properties hold:
+(a) (mk)k∈N is log-convex as L(ab)2 ≤ L(a2)L(b2) for all a, b ∈ S(V ).
+(b) for each a ∈ S(V ) the sequence (
+�
+L(a2k))k∈N is increasing (by repeated
+applications of the Cauchy-Schwartz inequality).
+(c) for each a ∈ S(V ) and k ∈ N, the map a �→
+2k�
+L(a2k) defines a semi-norm
+on S(V ) (by [18, Remark 3.2-(ii)] )
+(d) for each f ∈ E and k ∈ N,
+�
+L(f 2k) ≤ mk (by definition of mk).
+Fix k ∈ N and v = �l
+i=1 λivi with vi ∈ E and λi ∈ R. Let us choose d ∈ N such
+that 2d ≤ 2k ≤ 2d+1. Then
+2k�
+L(v2k)
+(b)
+≤
+2d+1�
+L(v2d+1)
+(c)
+≤
+l
+�
+i=1
+|λi|
+2d+1�
+L(v2d+1
+i
+) ≤
+l
+�
+i=1
+|λi|
+2d
+��
+L(v2·2d
+i
+)
+(d)
+≤
+l
+�
+i=1
+|λi|
+2d√m2d ≤
+�
+1 +
+l
+�
+i=1
+|λi|
+�
+2d√m2d ≤ Kv
+2k√m2k.
+(3.9)
+where in the last inequality we used that ( k√mk)k∈N is increasing, see e.g. [13,
+Corollary 4.2] and Kv := 1 + �l
+i=1 |λi| > 0.
+Since the class C{zk} is quasi-analytic and (3.8) holds, also C{mk}. This together
+with the property (a) ensures that C{√m2k} is quasi-analytic, see e.g. [13, Lemma
+4.3].
+Hence, by the Denjoy-Carleman theorem, �∞
+k=1
+1
+2k√m2k = ∞ holds.
+This
+combined with (3.9) provides the conclusion.
+□
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+23
+4. Appendix
+In the following we first explain the relation between the notion of trace of a
+Hilbertian seminorm w.r.t.
+to another and the classical definition of trace of a
+positive continuous operator on a Hilbert space. Then we compare the definition
+of nuclear space used in this article due to Yamasaki [27] with the more traditional
+ones due to Grothendieck [10] and Mityagin [19], and with the definitions of this
+concept given by Berezansky and Kondratiev in [2, p. 14] and by Schmüdgen in
+[23, p. 445], whose results we compared to ours in Section 3. Finally, we provide a
+complete proof of the measure theoretical identity (2.7), which we exploited in the
+proof of Theorem 2.10.
+4.1. Trace of positive continuous operators on Hilbert spaces.
+Let us start by recalling the definition of trace of a positive continuous operator on
+a Hilbert space, which we also denote with the symbol tr.
+Definition 4.1 (cf. [6, V.50, (24’)]). Given a Hilbert space (H, ⟨·, ·⟩), the trace of
+a continuous and positive operator f : H → H is defined as
+(4.1)
+tr(f) :=
+sup
+e1,...,en
+n
+�
+i=1
+⟨ei, f(ei)⟩,
+where n ranges over N and e1, . . . , en ∈ H ranges over the set of all finite sequences
+that are orthonormal w.r.t. ⟨·, ·⟩.
+In fact, by [6, V.48, Lemma 2], we have that for every complete orthonormal
+system {ei : i ∈ Ω} in H the following holds
+(4.2)
+tr(f) =
+�
+i∈Ω
+⟨ei, f(ei)⟩.
+If D ⊆ H is dense, then there exists a complete orthonormal system in H that is
+contained in D. Therefore, in (4.1) it suffices to let e1, . . . , en ∈ H range over the
+set of all finite sequences in D that are orthonormal w.r.t. ⟨·, ·⟩.
+For the convenience of the reader, we also recall here some fundamental classes
+of operators that will be needed in showing the relation between traces mentioned
+above.
+Definition 4.2. Given a Hilbert space (H, ⟨·, ·⟩), we say that a bounded linear
+operator f : H → H is trace-class if tr(√f ∗f) < ∞, where f ∗ denotes the adjoint
+of f. The positive bounded operator √f ∗f is called absolute value of f.
+Definition 4.3. Given two Hilbert spaces (H1, p1) and (H2, p2), we say that a
+continuous operator f : H1 → H2 is
+(1) Hilbert-Schmidt (or quasi-nuclear) if tr(f ∗f) < ∞.
+(2) nuclear if there exist (vn)n∈N ⊆ H1 and (wn)n∈N ⊆ H2 such that
+∞
+�
+n=1
+p1(vn)p2(wn) < ∞
+and
+f(·) =
+∞
+�
+n=1
+⟨·, vn⟩p1wn.
+Note that (vn)n∈N ⊆ (H1, p1) and (wn)n∈N ⊆ (H2, p2) can be chosen to be
+orthogonal (see, e.g., [25, Corollary, p. 494]).
+Proposition 4.4. Let f : (H1, p1) → (H2, p2) be a nuclear operator. If H ⊆ H1
+closed, then f ↾H : H → f(H) is also nuclear.
+Proof. Since f is nuclear, there exists (vn)n∈N ⊆ (H1, p1) and (wn)n∈N ⊆ (H2, p2)
+orthogonal such that f(·) = �∞
+n=1⟨·, vn⟩p1wn and �∞
+n=1 p1(vn)p2(wn) < ∞. Then
+
+24
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+f(vn) = ⟨vn, vn⟩p1wn for all n ∈ N, since (vn)n∈N ⊆ (H1, p1) is orthogonal, and so
+(wn)n∈N ⊆ f(H). Furthermore, for each n ∈ N there exist xn ∈ H, yn ∈ H⊥ such
+that vn = xn + yn. Thus,
+f(x) =
+∞
+�
+n=1
+⟨x, xn + yn⟩p1wn =
+∞
+�
+n=1
+⟨x, xn⟩p1wn
+for all x ∈ H.
+Moreover, ⟨xn, yn⟩p1 = 0 implies that p1(xn) ≤ p1(xn + yn) = p1(vn) for all n ∈ N
+and hence, �∞
+n=1 p1(xn)p2(wn) ≤ �∞
+n=1 p1(vn)p2(wn) < ∞.
+□
+We are ready now to relate Definition 1.2 and Definition 4.1.
+Remark 4.5. A Hilbertian seminorm p on a real vector space V can be always
+used to construct a Hilbert space out of V .
+Indeed, p induces a seminorm on
+Vp := V/ ker(p) given by v + ker(p) �→ p(v) and denoted, with a slight abuse of
+notation, also by p. Thus, (Vp, p) is a pre-Hilbert space, as p clearly induces an
+inner product on Vp. Now, Vp is dense in the completion V p of (Vp, p) and so p
+extends to a norm p on V p which makes (V p, p) a Hilbert space.
+Proposition 4.6.
+Let p and q be two Hilbertian seminorms on a real vector space V .
+(1) If ker(q) ⊆ ker(p) then u: Vq → Vp, v + ker(q) �→ v + ker(p) is well-defined.
+Note that u is injective iff ker(q) = ker(p).
+(2) If there exists C > 0 such that p ≤ Cq, then u is continuous and uniquely
+continuously extends to u: (V q, q) → (V p, p). Moreover, u is injective iff
+for any Cauchy sequence (vn) in Vq s.t. u(vn) converges to 0 in p we have
+that vn converges to 0 in q.
+(3) tr(p/q) < ∞ if and only if u is Hilbert-Schmidt, i.e., tr(u∗u) < ∞, where
+u∗ denotes the adjoint of u.
+Proof.
+(1) For any v ∈ V , let us set for convenience [v]q := v + ker(q) and [v]p :=
+v + ker(p). Recalling the notation and the properties introduced in Remark 4.5,
+it is easy to see that ker(q) ⊆ ker(p) implies (1), because under this assuption
+[x]q = [y]q implies x − y ∈ ker(p) and so [x]p = [y]p, i.e. u([x]q) = u([y]q).
+Moreover, suppose that u is injective and that there exists v ∈ ker(p) \ ker(q).
+Then [v]q ̸= [0]q and [v]p = [0]p. Hence, on the one hand the injectivity of u ensures
+that u([v]q) ̸= [0]p, but on the other hand u([v]q) = [v]p = [0]p which leads to a
+contradiction. Conversely, if ker(p) = ker(q), then u is the identity which is clearly
+injective.
+(2) Suppose there exists C > 0 such that p ≤ Cq. Then ker(q) ⊆ ker(p) and
+so u is well-defined by (1). Also, for any [v]q ∈ Vq we have p(u([v]q)) = p([v]p) =
+p(v) ≤ Cq(v) = Cq([v]q), i.e. u is continuous and so can be uniquely extended to
+the completions giving the desired u.
+For proving the second part of (2), suppose that u is injective and let (vn) be
+a Cauchy sequence in Vq s.t. p(u(vn)) → 0. Then, by completeness, there exists
+w ∈ V q such that q(vn − w) → 0 and so, by continuity of u, p(u(vn) − u(w)) → 0.
+Therefore, p(u(w)) ≤ p(u(vn)−u(w))+p(u(vn)) = p(u(vn)−u(w))+p(u(vn)) → 0,
+i.e. p(u(w)) = 0 that is u(w) = 0. Hence, the injectivity of u implies that w = 0
+and so that q(vn) = q(vn) = q(vn − w) → 0.
+Conversely, suppose that for any Cauchy sequence (vn) in Vq s.t. p(u(vn)) → 0
+we have q(vn) → 0. If w ∈ ker(u) ⊆ V q, then there exists a Cauchy sequence
+(wn) in Vq converging to w, i.e. q(wn − w) → 0. By continuity of u, we have that
+p(u(wn) − u(w)) → 0 but u(w) = 0 and so p(u(wn)) = p(u(wn)) → 0. Hence, our
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+25
+assumption implies q(wn) → 0 and so q(w) ≤ q(wn − w) + q(wn) → 0, which is
+equivalent to w = 0 and so provides the injectivity of u.
+(3) directly follows from the following observation
+tr(u∗u)
+(4.1)
+=
+sup
+e1,...,en
+n
+�
+i=1
+⟨[ei]q, u∗u([ei]q)⟩q
+=
+sup
+e1,...,en
+n
+�
+i=1
+⟨u([ei]q), u([ei]q)⟩p
+=
+sup
+e1,...,en
+n
+�
+i=1
+⟨[ei]p, [ei]p⟩p =
+sup
+e1,...,en
+n
+�
+i=1
+⟨ei, ei⟩p
+Def.1.2
+=
+tr(p/q),
+where n ranges over N and e1, . . . , en ∈ V ranges over the set of all finite sequences
+that are orthonormal w.r.t. ⟨·, ·⟩q.
+□
+Proposition 4.7. Let p and q be two Hilbertian seminorms on V . If ker(p) =
+ker(q) and tr(p/q) < ∞, then Vq is separable.
+Proof. Suppose that Vq is not separable. Then there exists (ej)j ∈ J orthonormal
+basis of Vq with J uncountable.
+Since q(ej) = 1 for all j ∈ J and ker(p) =
+ker(q), we have that p(ej) > 0 for all j ∈ J. However, tr(p/q) < ∞ implies that
+supn∈N supj1,...,jn∈J
+�n
+k=1 p(ejk)2 < ∞ and so for all but countably many n−tuples
+in (ej)j ∈ J we have �n
+k=1 p(ejk)2 = 0, which contradicts the fact that p(ej) > 0
+for all j ∈ J.
+□
+Corollary 4.8. Let p and q be two Hilbertian seminorms on V s.t. there exists
+C > 0 such that p ≤ Cq then u injective and Hilbert-Schmidt implies that Vq is
+separable.
+Corollary 4.9. Let A be an algebra generated by a linear subspace V ⊆ A, and L a
+normalized linear functional on A such that L(� A2) ⊆ [0, ∞). If q is a Hilbertian
+seminorm q on V such that tr(sL ↾V /q) < ∞, then the space V/ ker(sL) endowed
+with the quotient seminorm induced by sL (and also denoted by sL with a slight
+abuse of notation) is separable.
+Proof. Let us endow Vq with the quotient seminorm induced by q, which we also
+denote by q with a slight abuse of notation. As tr(sL ↾V /q) < ∞, Lemma 1.3-
+(i) ensures the q−continuity of sL ↾V and so that ker(q) ⊂ ker(sL).
+Then the
+quotient seminorm induced on Vq by sL ↾V actually reduces to itself, i.e. ∀v ∈
+V, inf{sL(v + w) : w ∈ ker(q)} = sL(v), and can be continuously extended to a
+norm pL on the completion Vq of Vq. Hence, both (Vq, pL) and (Vq, q) are Hilbert
+spaces.
+Consider ker(sL) in (Vq, q) and denote by ˜q the quotient norm induced on
+Vq/ker(sL) by q (respectively by ˜sL the quotient norm induced on Vq/ker(sL) by sL).
+Then ˜q is a Hilbertian norm as (Vq, q) is a Hilbert space and we have the follow-
+ing orthogonal decomposition Vq = ker(sL) ⊕ ker(sL)
+⊥. Then (Vq/ker(sL), ˜q) is a
+Hilbert space. Hence, denoting by π the orthogonal projection Vq onto ker(sL)
+⊥,
+we get Vq/ ker(π) ∼= π(Vq), i.e. Vq/ker(sL) ∼= ker(sL)
+⊥. Exploiting this isomor-
+phism and the fact that ker(sL) is closed in (Vq, q), it is easy to see that any
+finite ˜q-orthonormal subset { ˜ej}j=1,...,J with J ∈ N in Vq/ker(sL) provides a fi-
+nite q−orthonormal subset {hj}j=1,...,J := π−1 ({ ˜ej}j=1,...,J) in Vq. By density,
+for each n ∈ N and each j ∈ {1, . . ., J}, we can choose h(n)
+j
+∈ V such that
+
+26
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+q(hj − h(n)
+j
+) ≤ 1/n. Orthogonalizing {h(n)
+j
+}j∈{1,...,J} via the Gram-Schmidt pro-
+cess, we obtain a q−orthogonal subset {e(n)
+j
+}j∈{1,...,J} in V defined inductively
+by e(n)
+1
+:= h(n)
+1
+and e(n)
+k
+:= h(n)
+k
+− �k−1
+j=1
+⟨h(n)
+k
+,e(n)
+j
+⟩q
+⟨e(n)
+j
+,e(n)
+j
+⟩q e(n)
+j
+for all k ≥ 2.
+Defin-
+ing ˜e(n)
+k
+:=
+e(n)
+k
+q(e(n)
+k
+) for all k ∈ N, we get a q−orthonormal subset in V. So for
+each k ∈ {1, . . ., J} as n → ∞ we get inductively that q(e(n)
+k
+− hk) → 0 and
+hence sL(e(n)
+k
+− hk) → 0. Thus, for each ε > 0 there exists N ∈ N such that
+sL(e(N)
+j
+− hj) ≤ ε/J for all j and so
+�
+j=1,...,J
+˜sL( ˜ej) =
+�
+j=1,...,J
+sL(hj) ≤
+�
+j=1,...,J
+sL(e(N)
+j
+) + ε ≤ tr(sL ↾V /q) + ε.
+As this holds for an arbitrary finite ˜q-orthonormal subset { ˜ej}j=1,...,J, we have by
+the definition of the trace that tr( ˜sL/˜q) ≤ tr(sL ↾V /q) + ε, which together with
+tr(sL ↾V /q) < ∞ implies tr( ˜sL/˜q) < ∞. Then, since the kernel of ˜sL in Vq/ker(sL)
+is clearly trivial, so it is the kernel of ˜q in Vq/ker(sL).
+Hence, Proposition 4.7
+provides that
+�
+Vq/ker(sL)
+�
+/ ker(˜q) is separable, i.e. Vq/ker(sL) is separable. As the
+latter space is also metric, we have that its subspace Vq/ker(sL) is also separable.
+Moreover, since ker(q) ⊆ ker(sL) ⊆ V , we get that Vq/ker(sL) ∼= V/ ker(sL) and so
+V/ ker(sL) is also separable.
+□
+4.2. Other definitions of nuclear space.
+The definition of nuclear space used in this article, namely Definition 1.4, is
+due to Yamasaki but it is equivalent to the more traditional definitions of nuclear
+space due to Grothendieck [10] and Mityagin [19], which we report here for the
+convenience of the reader (see e.g. [27, Theorems A.1, A.2] for a proof of these
+equivalences).
+Definition 4.10. A TVS (V, τ) is called nuclear if τ is induced by a family P
+of seminorms on V such that for each p ∈ P there exists q ∈ P and there exist
+sequences (vn)n∈N ⊆ V, (ln)n∈N ⊆ V ∗ with the following property
+∞
+�
+n=1
+p(vn)q′(ln) < ∞ for all v ∈ V with v =
+∞
+�
+n=1
+ln(v)vn w.r.t. p,
+here q′ denotes the dual norm of q.
+Definition 4.11. A TVS (A, τ) is called nuclear if τ is induced by a family P of
+seminorms on V such that for each p ∈ P there exists q ∈ P with dn(Uq, Up) ∈
+O(n−λ) for some λ > 0, where Up denotes the (closed) semiball of p and dn(Uq, Up)
+denotes n–dimensional width of Uq w.r.t. Up, that is dn(Uq, Up) is defined as
+inf{c > 0 : Uq ⊆ Vn−1 + cUp for some Vn−1 ⊆ V with dim(Vn−1) = n − 1}.
+Using Proposition 4.6, it is easy to establish that Definition 1.4 coincides with
+the following one.
+Definition 4.12. A TVS (V, τ) is called nuclear if τ is induced by a directed family
+P of Hilbertian seminorms on V such that for each p ∈ P there exists q ∈ P with
+ker(q) ⊆ ker(p) and the continuous extension u: (V q, q) → (V p, p) of the canonical
+map u : (Vq, q) → (Vp, p) is Hilbert-Schmidt, i.e. tr(u∗u) < ∞.
+This equivalent refomulation of Definition 1.4 allows more easily to see its relation
+with the definitions of this concept given by Berezansky and Kondratiev in [2, p. 14]
+and by Schmüdgen in [23, p. 445] whose results we compare to ours in Section 3.
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+27
+Definition 4.13 (cf. [2, p. 14]). Let I be a directed index set and (Hi, pi)i∈I
+a family of Hilbert spaces such that V := �
+i∈I Hi is dense in each (Hi, pi) and
+for all i, j ∈ I there exists k ∈ I with i, j ≤ k and (Hk, pk) ⊆ (Hi, pi) as well
+as (Hk, pk) ⊆ (Hj, pj).
+The space V endowed with the topology τ induced by
+P := {pi : i ∈ I} is called nuclear if for each i ∈ I there exists j ≥ i in I such that
+the embedding (Hj, pj) ⊆ (Hi, pi) is Hilbert-Schmidt.
+Remark 4.14. Berezansky and Kondratiev’s Definition 4.13 of nuclear space is
+covered by Definition 4.12 (and thus, by Definition 1.4). Indeed, let (V, τ) be a
+nuclear space with defining family (Hi, pi)i∈I in the sense of Definition 4.13. Since
+for each i ∈ I we have that ker(pi) = {o}, we get Vpi = V . This together with
+the fact that V is dense in each (Hi, pi) ensures that the completion (V pi, pi) is
+isomorphic to (Hi, pi) for all i ∈ I. Thus, for i ≤ j in I the embedding (Hj, pj) ⊆
+(Hi, pi), which is Hilbert-Schmidt by assumption, coincides with u: (V pj, pj) →
+(V pi, pi). Hence, (V, τ) is nuclear in the sense of Definition 4.12.
+Definition 4.15 (c.f.
+[23, p. 445]). Let (Hn, pn)i∈N be a sequence of Hilbert
+spaces such that (Hn, pn) ⊆ (Hm, pm) for all m ≤ n in N. The space V := �∞
+n=1 Hn
+endowed with the topology τ induced by P := {pn : n ∈ N} is called nuclear if for
+each m ∈ N there exists n ∈ N such that the embedding (Hn, pn) ⊆ (Hm, pm) is
+nuclear (see Definition 4.3-(2)).
+Remark 4.16. Schmüdgen’s Definition 4.15 of nuclear space is covered by Defini-
+tion 4.12 (and thus, by Definition 1.4). Indeed, let (V, τ) be a nuclear space with
+defining family (Hn, pn)n∈N in the sense of Definition 4.15. For each n ∈ N, since
+ker(pn) = {o}, we get that Vpn = V ⊆ Hn and so that the completion (V pn, pn)
+is isomorphic to a closed subspace of (Hn, pn).
+As the embedding (Hm, pm) ⊆
+(Hn, pn) is nuclear, also its restriction r to V pm is nuclear by Proposition 4.4. The
+continuity of r guarantees that r(V pm) ⊆ r(Vpm) = r(V ) = V = V pn and so the
+map r coincides with u: (V pm, pm) → (V pn, pn). Hence, u is nuclear. Then [25,
+Theorem 48.2] ensures that tr(
+√
+u∗u) < ∞, i.e. u is a trace-class operator (see
+Definition 4.2). Since the family of trace-class operators on a Hilbert space forms
+an ideal in the space of bounded operators on the same space (see e.g. [20, Theorem
+VI.19]), we have that tr(u∗u) < ∞, i.e. u is Hibert-Schmidt. Thus, (V, τ) is also
+nuclear in the sense of Definition 4.12.
+4.3. Two auxiliary results.
+We provide here a proof of (2.7), which we used in the proof of Theorem 2.10 as
+well as a result Lemma 4.17 about dense subalgebras of topological algebras which
+we exploited in the analysis of Corollary 2.19.
+Proof of (2.7). For notational convenience, let τ A := τsp(q)A and τB := τsp(q)B.
+We preliminarily observe that
+q′(α) :=
+sup
+a∈A : q(a)≤1
+|α(a)| =
+sup
+a∈B : q(a)≤1
+|α(a)|,
+∀α ∈ sp(q)
+and so q′ is lower semi-continuous w.r.t. both τA and τ B. Hence, all sublevel sets
+of q′ are closed in both (sp(q), τ A) and (sp(q), τ B), i.e. Bn(q′)c ∈ τ A ∩ τ B, for
+all n ∈ N, which gives in turn Bn(q′) ∈ B(τ A) ∩ B(τ B). This together with the
+following two properties
+(i) τ A ∩ Bn(q′) = τ B ∩ Bn(q′),
+∀ n ∈ N.
+(ii) B(τ C) ∩ Bn(q′) = B(τC ∩ Bn(q′)),
+∀ n ∈ N, C ∈ {A, B}.
+provide that
+B(τ A) ∩ Bn(q′)
+(ii)
+= B(τ A ∩ Bn(q′))
+(i)
+= B(τ B ∩ Bn(q′))
+(ii)
+= B(τ B) ∩ Bn(q′) ⊆ B(τ B).
+
+28
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+The latter ensures that if Y ∈ B(τA) then Y ∩ Bn(q′) ∈ B(τ B) for all n ∈ N and so
+that Y = �
+n∈N Y ∩ Bn(q′) ∈ B(τB), i.e. B(τ A) ⊆ B(τ B). The opposite inclusion
+easily follows from τ B ⊂ τA. Hence, B(τ A) = B(τ B).
+□
+It remains to show (i) and (ii).
+Proof of (i). Let n ∈ N. Since τ B ⊂ τ A, we have that τ A ∩ Bn(q′) ⊇ τ B ∩ Bn(q′).
+For the opposite inclusion, let α ∈ sp(q)∩Bn(q′) and recall that for any C ∈ {A, B}
+a basis of neighbourhoods of α in the topology τ C ∩ Bn(q′) is given by
+{Uc1,...,ck;λ : k ∈ N, c1, . . . , ck ∈ C, λ > 0} ,
+where
+Uc1,...,ck;λ(α) := {γ ∈ sp(q) : | ˆcj(γ) − ˆcj(α)| < λ for j = 1, . . . , k and q′(γ) < n}
+We need to show that for any k ∈ N, a1, . . . , ak ∈ A and ε > 0 there exist
+b1, . . . , bk ∈ B and δ > 0 such that Ub1,...,bk;δ(α) ⊆ Ua1,...,ak;ε(α).
+Fixed k ∈ N, a1, . . . , ak ∈ A and ε > 0, by the density of B in (A, q) we can
+always choose b1, . . . , bk ∈ B such that q(aj − bj) <
+ε
+3n for j = 1, . . . , k. Then
+taking δ < ε
+3 we have that for any β ∈ Ub1,...,bk;δ(α) and any j ∈ {1, . . ., k} the
+following holds:
+|β(aj) − α(aj)|
+≤
+|β(aj) − β(bj)| + |β(bj) − α(bj)| + |α(bj) − α(aj)|
+≤
+nq(aj − bj) + δ + nq(bj − aj) < ε
+i.e. β ∈ Ua1,...,ak;ε(α) and hence Ub1,...,bk;δ(α) ⊆ Ua1,...,ak;ε(α).
+□
+Proof of (ii). Let n ∈ N and C ∈ {A, B},
+We have already showed that Bn(q′) ∈ B(τ C) and so τ C ∩Bn(q′) ⊆ B(τC), which
+in turn implies that B(τ C) ∩ Bn(q′) ⊇ B(τ C ∩ Bn(q′)).
+Now let i : sp(q) ∩ Bn(q′)∩ → sp(q) be the identity map.
+On the hand,
+the continuity of i :
+�
+sp(q) ∩ Bn(q′), τ C ∩ Bn(q′)
+�
+∩ →
+�
+sp(q), τ C�
+provides that
+i :
+�
+sp(q) ∩ Bn(q′), B(τ C ∩ Bn(q′))
+�
+∩ →
+�
+sp(q), B(τ C)
+�
+is measurable.
+On the
+other hand, B(τ C) ∩ Bn(q′) is the smallest σ−algebra on sp(q) ∩ Bn(q′) making i
+measurable.Hence, we have that B(τ C) ∩ Bn(q′) ⊂ B(τC ∩ Bn(q′)).
+□
+Lemma 4.17. Let A be an algebra generated by a linear subspace V ⊆ A and τ
+a topology on A such that (A, τ) is a topological algebra. If U is a subspace of V
+which is dense in (V, τ ↾V ), then ⟨U⟩ is dense in (A, τ).
+Proof. Let w ∈ U and v ∈ V . Then there exists a net (vα)α∈I with vα ∈ U such that
+vα → v. Since the multiplication is separately continuous, we have that wvα → wv
+and hence wv ∈ ⟨U⟩.
+Now let us take also u ∈ V .
+We will show by induction on n that
+(4.3)
+v1, . . . , vn ∈ V ⇒ v1 · · · vn ∈ ⟨U⟩, ∀n ∈ N.
+which implies that ⟨V ⟩ = ⟨U⟩ and so the conclusion A = ⟨U⟩.
+Let us first show the base case n = 2. If v1, v2 ∈ V then Then there exist nets
+(uα)α∈I and (wβ)β∈J with uα, wβ ∈ U such that uα → v1 and wβ → v2. Since the
+multiplication is separately continuous, for each u ∈ U, we have that uwβ → uv2
+and hence uv2 ∈ ⟨U⟩. In particular each uαv2 ∈ ⟨U⟩ and, using again the separate
+continuity of the multiplication, uαv2 → v1v2. Hence, v1v2 ∈ ⟨U⟩.
+Suppose now that (4.3) holds for a fixed n and let v1, . . . , vn+1 ∈ V .
+Then
+there exists (gα)α∈I with gα ∈ U such that gα → vn+1. Moreover, by inductive
+
+MOMENT PROBLEM FOR ALGEBRAS GENERATED BY A NUCLEAR SPACE
+29
+assumption, v1 · · · vn ∈ ⟨U⟩ and so there exists (hβ)β∈J with hβ ∈ ⟨U⟩ such that
+hβ → v1 · · · vn. Then for any u ∈ U, by the separate continuity of the multiplication,
+we have that hβu → v1 · · · vn · u and hence v1 · · · vn · u ∈ ⟨U⟩.
+In particular
+for each α ∈ I we get that v1 · · · vn · gα ∈ ⟨U⟩. Thus, using again the separate
+continuity of the multiplication, we obtain that v1 · · · vn · gα → v1 · · · vn · vn+1 and
+so v1 · · · vn · vn+1 ∈ ⟨U⟩.
+□
+Acknowledgments
+We are indebted to the Baden–Württemberg Stiftung for the financial support
+to this work by the Eliteprogramme for Postdocs. This work was also partially
+supported by the Ausschuss für Forschungsfragen (AFF). Maria Infusino is member
+of GNAMPA group of INdAM. Tobias Kuna is member of GNFM group of INdAM.
+References
+[1] S. Albeverio and F. Herzberg: The moment problem on the Wiener space, Bull. Sci. Math.
+132(2008), 7–18
+[2] Yu. M. Berezansky and Yu. G. Kondratiev, Spectral methods in infinite-dimensional anal-
+ysis. Vol. II (Transl. from the 1988 Russian original), Mathematical Physics and Applied
+Mathematics 12/2, Kluwer Academic Publishers, Dordrecht 1995.
+[3] Yu. M. Berezansky and S. N. Šifrin, A generalized symmetric power moment problem (Rus-
+sian), Ukrain. Mat. Ž. 23 (1971), 291–306.
+[4] Yu. M. Berezansky, Z. G. Sheftel, G. F. Us, Functional Analysis. Vol. II, Birkháuser, Basel,
+1996.
+[5] V. I. Bogachev, Measure theory. Vol. II, Springer, Berlin, 2007.
+[6] N. Bourbaki, Elements of mathematics. Topological vector spaces. Chapters 1–5., Springer,
+Berlin; 2003.
+[7] H.J. Borchers, J. Yngvason. Integral representations for Schwinger functionals and the mo-
+ment problem over nuclear spaces. Comm. Math. Phys. 43(3): 255–271, 1975.
+[8] I. M. Gel’fand and N. Ya. Vilenkin, Generalized functions. Vol. 4: Applications of harmonic
+analysis, Academic Press, New York and London, 1964.
+[9] M. Ghasemi, M. Infusino, S. Kuhlmann and M. Marshall, Moment problem for symmetric
+algebras of locally convex spaces, Integr. Equat. Oper. Th. 90 (2018), no. 3, Art. 29, 19 pp.
+[10] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Am. Math.
+Soc. 16, Am. Math. Soc., Providence, RI 1955.
+[11] G. C. Hegerfeldt. Extremal decomposition of Wightman functions and of states on nuclear
+*-algebras by Choquet theory. Comm. Math. Phys. 45(2): 133–135, 1975.
+[12] A. G. Kostyuchenko, B.S. Mityagin. Positive definite functionals on nuclear spaces, Trudy
+Mosk. Mat. Obshch. 9 (1960): 283–316.
+[13] M. Infusino, Quasi-analyticity and determinacy of the full moment problem from finite to
+infinite dimensions, Stochastic and Infinite Dimensional Analysis, Chap. 9: 161–194, Trends
+in Mathematics, Birkhäuser, 2016.
+[14] M. Infusino and S. Kuhlmann, Infinite dimensional moment problem: open questions and
+applications, Ordered Algebraic Structures and Related Topics, Contemporary Mathematics
+697, 187–201, Amer. Math. Soc., Providence, RI 2017.
+[15] M. Infusino, T. Kuna, A. Rota. The full infinite dimensional moment problem on semi-
+algebraic sets of generalized functions. J. Funct. Analysis, 267(5):1382–1418, 2014.
+[16] M. Infusino, T. Kuna, The full moment problem on subsets of probabilities and point config-
+urations, J. Math. Anal. Appl., 483(1), 123551, 2020.
+[17] M. Infusino, S. Kuhlmann, T. Kuna, and P. Michalski, Projective limit techniques for the
+infinite dimensional moment problem, Integr. Equ. Oper. Theory 94, 12 (2022).
+[18] M. Infusino, S. Kuhlmann, T. Kuna, and P. Michalski, An intrinsic characterization of
+moment functionals in the compact case, International Mathematics Research Notices, 3, 1
+(2023).
+[19] B.S. Mityagin, Approximate dimension and bases in nuclear spaces, Uspekhi Mat. Nauk 16
+(1961), no. 4, 59–127.
+[20] M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol.I: Functional Analysis,
+Academic Press, New York, London, 1975.
+[21] K. Schmüdgen. Unbounded operator algebras and representation theory. Operator Theory:
+Advances and Applications 37, Birkhäuser Verlag, Basel, 1990.
+
+30
+INFUSINO, KUHLMANN, KUNA, MICHALSKI
+[22] K. Schmüdgen, The moment problem, Grad. Texts in Math. 277, Springer, Cham 2017.
+[23] K. Schmüdgen, On the infinite dimensional moment problem, Ark. Mat. 56 (2018), no. 2,
+441–459.
+[24] L. Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures, Tata
+Inst. Fund. Res. Stud. Math. 6, Oxford University Press, London 1973.
+[25] F. Trèves, Topological vector spaces, distributions and kernels. New York-London: Academic
+Press 1967.
+[26] Y. Umemura, Measures on infinite dimensional vector spaces, Publ. Res. Inst. Math. Sci.,
+Kyoto Univ., Ser. A 1, 1-47 (1965).
+[27] Y. Yamasaki, Measures on infinite-dimensional spaces, Series in Pure Mathematics 5, World
+Scientific Publishing Co., Singapore 1985.
+(M. Infusino)
+Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari,
+Via Ospedale 72, 09124 Cagliari, Italy.
+Email address: [email protected]
+(S. Kuhlmann, P. Michalski)
+Fachbereich Mathematik und Statistik, Universität Konstanz,
+Universitätstrasse 10, 78457 Konstanz, Germany.
+Email address: [email protected]
+Email address: [email protected]
+(T. Kuna)
+Dipartimento di Ingegneria, Scienze dell’ Informazione e Matematica,
+Università degli Studi dell’Aquila, Via Vetoio, 67100, L’Aquila, Italy
+Email address: [email protected]
+

QtAzT4oBgHgl3EQfW_yd/content/tmp_files/2301.01311v1.pdf.txt

@@ -0,0 +1,2689 @@
+Large N analytical functional bootstrap I:
+1D CFTs and total positivity
+Zhijin Li 1,2
+1 Shing-Tung Yau Center and School of Physics, Southeast University , Nanjing 210096,
+China
+2 Department of Physics, Yale University, New Haven, CT 06511, USA
+We initiate the analytical functional bootstrap study of conformal field theories with large
+N limits. In this first paper we particularly focus on the 1D O(N) vector bootstrap. We
+obtain a remarkably simple bootstrap equation from the O(N) vector crossing equations in
+the large N limit. The numerical bootstrap bound is saturated by the generalized free field
+theory. We study the analytical extremal functionals of this crossing equation, for which
+the total positivity of the SL(2, R) conformal block plays a critical role. We prove the
+total positivity of the SL(2, R) conformal block for large scaling dimension ∆ and verify
+it numerically for ∆’s related to the bootstrap results. We construct a series of analytical
+functionals {αM} which satisfy the bootstrap positive conditions up to a range ∆ ⩽ ΛM.
+The functionals {αM} have a trivial large M limit. Surprisingly, due to total positivity, they
+can approach the large M limit in a way consistent with the bootstrap positive conditions
+for arbitrarily high ΛM, therefore proving the bootstrap bound analytically.
+Our result
+provides a concrete example to illustrate how the analytical properties of the conformal
+block lead to nontrivial bootstrap bounds. We expect this work paves the way for large N
+analytical functional bootstrap in higher dimensions.
+arXiv:2301.01311v1  [hep-th]  3 Jan 2023
+
+Contents
+1. Introduction
+2
+2. Large N numerical conformal bootstrap in 1D
+5
+2.1. O(N) vector crossing equations in 1D . . . . . . . . . . . . . . . . . .
+5
+2.2. O(N) vector bootstrap bounds in the large N limit . . . . . . . . . . . .
+8
+2.3. Extremal solutions and the simplified bootstrap equation
+. . . . . . . . .
+11
+3. SL(2, R) conformal block and total positivity
+13
+3.1. Total positivity: definition and theorems . . . . . . . . . . . . . . . . .
+14
+3.2. Total positivity of the Gauss hypergeometric function . . . . . . . . . . .
+16
+3.3. Total positivity of the 1D SL(2, R) conformal block
+. . . . . . . . . . .
+18
+4. Analytical functionals for the 1D O(N) vector bootstrap bound
+23
+4.1. Analytical functional basis
+. . . . . . . . . . . . . . . . . . . . . . .
+23
+4.2. Analytical functional for Regge superbounded conformal correlator . . . . .
+28
+4.2.1.
+Inverse of the infinite equation group . . . . . . . . . . . . . . .
+29
+4.2.2.
+Inverse of the finite subset of equation group
+. . . . . . . . . . .
+32
+4.2.3.
+Positivity from total positivity
+. . . . . . . . . . . . . . . . . .
+35
+4.3. Analytical functional for general conformal correlators
+. . . . . . . . . .
+38
+5. Conclusion and Outlook
+39
+A. Examples of the totally positive functions
+42
+A.1. Example 1: f(∆, x) = x∆ . . . . . . . . . . . . . . . . . . . . . . . .
+42
+A.2. Example 2: f(x, y) =
+1
+x+y . . . . . . . . . . . . . . . . . . . . . . . .
+43
+1
+
+1. Introduction
+The conformal bootstrap [1,2] has been revived since the breakthrough work [3], which
+shows that strong constraints on the parameter space of general conformal field theories
+(CFTs) can be obtained merely from few consistency conditions. This approach has led
+to remarkable successes in studying the strongly coupled critical phenomena, see [4,5] for
+comprehensive reviews. It is followed by an immediate question: how can such strong results
+be obtained from such few inputs? The bootstrap results, or the “numerical experiments”
+indicate certain mysterious mathematical structures in conformal theories which can play
+key roles in determining the CFT landscape. Since the ingredients in conformal bootstrap
+are just unitarity and the conformal blocks of the conformal group SO(D + 1, 1), the
+unreasonable effectiveness of the bootstrap method is likely related to certain properties
+of the SO(D + 1, 1) conformal blocks. The goal of this work is to explore such presumed
+mathematical structures. We will focus on 1D large N conformal bootstrap for which the
+extremal bootstrap functional can be studied analytically. Moreover, we will clarify the
+key mathematical property which makes our construction possible.
+We will start with the 1D O(N) vector bootstrap in the large N limit. Under suitable
+conditions the O(N) vector crossing equations are reduced to one of the simplest bootstrap
+equations
+�
+O∈S
+λ2
+O z−2∆φ G∆(z) −
+�
+O∈T
+λ2
+O (1 − z)−2∆φ G∆(1 − z) = 0,
+where G∆(z) is the 1D SL(2, R) conformal block. The bootstrap bound on the scaling
+dimension of the lowest operator in the O(N) traceless symmetric sector (T) is saturated
+by the generalized free field theory. We find the key mathematical property responsible for
+the bootstrap constraints can be provided by the total positivity of the SL(2, R) conformal
+block:
+Total positivity of the SL(2, R) conformal block −→ Large N bootstrap bound.
+Based on the total positivity of the SL(2, R) conformal block, we construct a series of
+analytical functionals for above bootstrap equation which can satisfy the bootstrap positive
+conditions up to arbitrarily high scaling dimension.
+Our interests in the large N CFTs and their bootstrap studies are motivated by several
+reasons.
+The large N CFTs play fundamental roles in the AdS/CFT correspondence [6–8]. In
+the large N limit, the conformal correlation functions are dominated by the generalized free
+field theories, and they provide pivotal solutions to the conformal crossing equations [9–11].
+2
+
+Perturbative CFT data can be obtained by expanding the solutions to the crossing equa-
+tions near generalized free field theories [12–14]. The role of large N CFTs in holography
+has been extensively studied, e.g. [15,16]. The generalized free field theories also provide
+nice examples for the harmonic analysis of the Euclidean conformal group [17]. In this
+work, we will show that the generalized free field theories are not just pivotal solutions
+to the crossing equations, but can also saturate the bootstrap bounds. This indicates a
+special positive structure in the generalized free field theories which restricts any dynami-
+cal corrections to the bounded parameters are either vanishing or negative. Decoding the
+positive structure is an interesting problem for bootstrap studies.
+We use O(N) vector bootstrap to study the large N CFTs. The O(N) vector bootstrap
+plays a special role in conformal bootstrap with global symmetries. Due to novel algebraic
+relations between crossing equations with different global symmetries [18,19], the non-O(N)
+vector crossing equations can be mapped to those of O(N) vector’s, and their bootstrap
+bounds are identical or weaker than the O(N) vector bootstrap bounds. On the physics
+side, the O(N) vector bootstrap bounds have close relation to several interesting theories.
+For instance, the 3D O(N) vector bootstrap bounds have two types of kinks [20,21]. The
+type I kinks with an O(N) vector scalar φ near a free boson ∆φ =
+1
+2 are related to
+the critical O(N) vector model [20], while the type II kinks with ∆φ near free fermion
+bilinears also appear in general dimensions and show close relation with conformal gauge
+theories [18,21].
+The analytical construction of the extremal bootstrap functional [22] provides a sub-
+stantial approach to uncover the positive structure in conformal bootstrap.
+Analytical
+extremal functionals have been firstly constructed in [23–25] for a 1D conformal bootstrap
+problem, in which the bootstrap bound is known to be saturated by the generalized free
+fermion theory [26]. Analytical functionals for higher dimensional conformal bootstrap have
+been studied in [27–29]. In [28] a family of functional basis dual to the generalized free
+field spectrum has been constructed, which shows close relation to the conformal dispersion
+relation [30]. Nevertheless, it remains a puzzle to realize the positive conditions, which is
+the crucial ingredient for conformal bootstrap. In this work, we aim to answer this critical
+question for the large N analytical functional bootstrap in a simplified laboratory, the 1D
+O(N) vector bootstrap. In 1D CFTs, there is only one conformal invariant cross ratio (z)
+and the spectrum does not depend on spin. Interestingly, although the crossing equations
+have been simplified notably in 1D, the bootstrap bounds show similar patterns as their
+higher dimension analogies. Therefore we expect the 1D analytical functional bootstrap
+studies are instructive for similar studies in higher dimensions.
+3
+
+In addition, the 1D conformal bootstrap with global symmetries also corresponds to
+many interesting physics problems. A large set of 1D CFTs are given by the line defects
+of higher dimensional CFTs. Two typical examples are provided by the monodromy line
+defect in the 3D Ising model [31,26] and the Wilson lines in the 4D N = 4 SYM [32,33].
+The 1D CFTs can also be realized as boundary theories of quantum field theories in AdS2
+background [34].
+Recently, there are growing interests in the applications of 1D O(N)
+symmetric CFTs in the celestial holography [35,36]. Conformal bootstrap in 1D has been
+a powerful approach to extract dynamical information in above theories.
+Total positivity of the 1D SL(2, R) conformal block will play a key role in constructing
+the analytical functionals of the 1D large N bootstrap. In mathematics the total positivity
+has been extensively studied since the early of 20th century. It has deep connections to
+quantum field theories, see e.g. [37–40]. The possible role of total positivity in conformal
+bootstrap has been proposed in [41], in which the authors focused on the geometrical
+configuration supported by the SL(2, R) conformal blocks. Due to total positivity, the 1D
+bootstrap equation (without an O(N) global symmetry) admits a cyclic polytope structure
+which can lead to nontrivial constraints on the CFT data, see also [42,43]. In this work,
+we will focus on a new 1D bootstrap equation with a different approach, but we will reach
+a similar conclusion that the total positivity of the SL(2, R) conformal block can play a
+key role for the bootstrap constraints.
+This paper is organized as follows. In Section 2 we study the 1D O(N) vector nu-
+merical bootstrap with large N. We discuss similarities and differences between the 1D
+and higher dimensional O(N) vector bootstrap. We obtain a simplified crossing equation,
+which determines the first part of the O(∞) vector bootstrap bound and provides an ideal
+example for analytical functional bootstrap study. In Section 3 we study total positivity of
+the SL(2, R) conformal block, which will be important to construct the analytical function-
+als. In Section 4 we construct the analytical functionals for the 1D O(∞) vector bootstrap
+which is saturated by the generalized free field theory. We firstly review the functional
+basis for 1D conformal block obtained from the dispersion relation. Then we explain how
+the total positivity of the conformal block function can play a key role to construct the
+analytical functionals satisfying the bootstrap positivity conditions. This work initiates a
+series of analytical functional bootstrap studies of the large N CFTs and their holographic
+duals, for which we briefly discuss in Section 5.
+4
+
+2. Large N numerical conformal bootstrap in 1D
+In this section we study 1D O(N) vector numerical conformal bootstrap in the large
+N limit.
+The 1D numerical conformal bootstrap has been studied in [26, 44, 45, 32, 33,
+46]. Our interest in the 1D O(N) vector bootstrap is from the observation that the 1D
+bootstrap bounds share several key properties of the O(N) vector bootstrap bounds in
+higher dimensions, thus it can provide a drastically simplified while still representative
+example to study the underlying mathematical structures in conformal bootstrap.
+The
+numerical bootstrap results provide insightful bases for analytical functional bootstrap in
+Section 4.
+2.1. O(N) vector crossing equations in 1D
+Let us consider an operator φi which forms a vector representation of the O(N) global
+symmetry. Its four point correlation function is given by
+⟨φi(x1)φj(x2)φk(x3)φl(x4)⟩ =
+1
+�
+x2
+12x2
+34
+�∆φ Gijkl(z),
+(2.1)
+where the variables xi are the 1D coordinates, xij = xi − xj and the conformal invariant
+cross-ratio z is defined as
+z = x12x34
+x13x24
+.
+(2.2)
+When the external operators φi(xi) are in the ordered configuration x1 < x2 < x3 < x4, the
+cross-ratio stays in the range z ∈ (0, 1). The stripped correlation function Gijkl(z) in (2.1)
+can be analytically continued in the complex plane except the branch points at z = 0, 1, ∞,
+which correspond to coincidences of two operators. The Gijkl(z) is a holomorphic function
+with two branch cuts at (−∞, 0] and [1, +∞). In the s-channel (12)(34) limit with z → 0,
+the conformal correlation function Gijkl(z) can be expanded in terms of the four point
+invariant tensors of O(N) singlet (S), traceless symmetric (T) and anti-symmetric (A)
+representations
+Gijkl(z) = δijδklGS(z) +
+�
+δikδjl + δilδjk − 2
+N δijδkl
+�
+GT(z) + (δilδjk − δikδjl)GA(z),
+(2.3)
+in which GR denotes the series expansion
+GR(z) =
+�
+OR
+λ2
+ORG∆(z)
+(2.4)
+5
+
+of the s-channel SL(2, R) conformal block [47]
+G∆(z) = z∆
+2F1(∆, ∆, 2∆; z).
+(2.5)
+Alternatively, one can expand the same correlation functions (2.3) in the t-channel (23)(41)
+limit, which can be formally written as Gjkli(1−z). The crossing symmetry of the correlation
+function (2.1) identifies the s- and t-channel expansions
+z−2∆φGijkl(z) = (1 − z)−2∆φGjkli(1 − z).
+(2.6)
+Together with (2.3), above crossing equation leads to following independent equations
+z−2∆φ
+�
+GT(z) − GA(z)
+�
+=(1 − z)−2∆φ
+�
+GT(1 − z) − GA(1 − z)
+�
+,
+(2.7)
+z−2∆φ
+�
+GS(z) − 2
+N GT(z)
+�
+=(1 − z)−2∆φ
+�
+GT(1 − z) + GA(1 − z)
+�
+.
+(2.8)
+Note to derive above crossing equations, we do not assume the statistical property of the
+external operator φi, so they can be applied to both fermions and bosons in 1D.
+A family of unitary solutions to the O(N) vector crossing equations are provided by
+GS(z) = 1 + 2
+N GT(z),
+(2.9)
+GT(z) = 1
+2z2∆φ
+�
+(1 − z)−2∆φ − λ
+�
+,
+(2.10)
+GA(z) = 1
+2z2∆φ
+�
+(1 − z)−2∆φ + λ
+�
+,
+(2.11)
+in which λ = ∓1 give the O(N) symmetric generalized free fermion and boson theories.
+Above correlation functions can be decomposed into the 1D conformal blocks
+GR(z) =
+∞
+�
+n=0
+cR
+n G2∆φ+n(z),
+(2.12)
+where
+cT
+n =
+(2∆φ)2
+n
+2 n!(4∆φ + n − 1)n
+(1 − (−1)nλ),
+(2.13)
+cA
+n =
+(2∆φ)2
+n
+2 n!(4∆φ + n − 1)n
+(1 + (−1)nλ).
+(2.14)
+6
+
+For different λ’s (|λ| < 1), the correlation functions GS/T/A contain the same spectrum
+∆n = 2∆φ + n, n ∈ N. An interesting question in CFT studies is that given the whole
+spectrum of a CFT, can we determine the theory uniquely? The correlation function (2.9-
+2.11) provides a counter example for this question.
+The O(N) vector crossing equations (2.7,2.8) have the same algebraic structure as
+those in higher dimensions [48,20]. However, in higher dimensions, there are spin selection
+rules in different O(N) representations due to the boson symmetry of the external scalars.
+The correlation function is invariant under permutation (i, x1) ↔ (j, x2), which leads to
+λφφOS = cS(−1)ℓλφφOS,
+λφφOT = cT(−1)ℓλφφOT ,
+λφφOA = cA(−1)ℓλφφOA,
+(2.15)
+where cS = cT = 1, cA = −1 are the signs from O(N) indices when permuting two φ’s.
+Therefore only even (odd) spins can appear in the S/T (A) representations. While there
+is no spin in 1D, do we have similar selection rules in different O(N) representations? The
+answer is yes and it relates to the so-called S-parity symmetry [31,26].
+The action of the S-parity is
+S :
+x → −x,
+S O(x) S = (−1)SOO(−x).
+(2.16)
+In 1D, the continuous part of the conformal symmetry preserves the cyclic order of the
+three point function ⟨O1(x1)O2(x2)O3(x3)⟩ with x1 < x2 < x3. However, the cyclic order
+can be modified by the S transformation
+S :
+⟨O1(x1)O2(x2)O3(x3)⟩ → (−1)SO1+SO2+SO3⟨O3(−x3)O2(−x2)O1(−x1)⟩.
+(2.17)
+Therefore in an S-parity invariant theory, we have
+λO1O2O3 = (−1)SO1+SO2+SO3λO2O1O3.
+(2.18)
+In the 1D O(N) vector bootstrap, if the external operators φi are scalars, the boson sym-
+metry between the two φ’s requires
+λφφOS = (−1)SOS λφφOS,
+λφφOT = (−1)SOT λφφOT ,
+λφφOA = −(−1)SOAλφφOA,
+(2.19)
+which leads to
+SOS = 1,
+SOT = 1,
+SOA = −1.
+(2.20)
+7
+
+While for the external fermions, the S-parity charges in the O(N) representations are
+opposite
+SOS = −1,
+SOT = −1,
+SOA = 1.
+(2.21)
+In the generalized free boson theory, the S-parity of the double-trace operators On = φ∂nφ
+is Sn = (−1)n. According to the S-parity charges in (2.20), the generalized free boson
+theory has spectrum O2n with ∆ = 2∆φ + 2n in the S/T sectors and spectrum O2n+1 with
+∆ = 2∆φ + 2n + 1 in the A sector. The spectra in S/T and A sectors are switched in the
+generalized free fermion theory due to the S-parity charges (2.21).
+The O(N) vector bootstrap plays a special role in bootstrapping CFTs with general
+global symmetries. In [18, 19] it has been verified that for a large variety of symmetries
+G and representations R, the crossing equations of the four point correlator ⟨R ¯RR ¯R⟩
+and ⟨RR ¯R ¯R⟩ can be linearly mapped into the O(N) symmetric form (2.7,2.8) through a
+transformation TR which is consistent with positivity conditions in the bootstrap algorithm.
+The transformation TR is purely algebraic so can also be applied in 1D conformal bootstrap.
+In consequence, the bootstrap bound on the lowest G singlet scalar coincides with the
+bound on the O(N) singlet scalar, while the O(N) vector bootstrap bound on the lowest
+T scalar can be interpreted as the bound on the lowest G non-singlet scalar appearing in
+the bootstrap equations of R.
+2.2. O(N) vector bootstrap bounds in the large N limit
+In the large N limit, the O(N) vector crossing equations (2.7,2.8) become
+�
+O∈S
+λ2
+O
+�
+0
+E∆(z)
+�
++
+�
+O∈T
+λ2
+O
+�
+F∆(z)
+−E∆(1 − z)
+�
++
+�
+O∈A
+λ2
+O
+�
+−F∆(z)
+−E∆(1 − z)
+�
+= 0,
+(2.22)
+where
+E∆(z) = z−2∆φG∆(z),
+(2.23)
+F∆(z) = E∆(z) − E∆(1 − z).
+(2.24)
+Their bootstrap bound on the lowest non-unit O(N) singlet operator goes to infinity. A
+solution to such bound is given by the correlation function (2.9-2.11) with N = ∞, in which
+the only O(N) singlet operator is the unit operator, while all the double-trace singlets have
+vanishing OPE coefficients and are decoupled in the crossing equation. Moreover, without
+extra assumptions on the spectrum, there is no upper bound on the lowest operator in
+8
+
+the T or A representation.
+To show this, let us consider the bootstrap bound on the
+scaling dimension of the lowest operator in the T sector, denoted ∆∗
+T.
+A solution to
+the crossing equation (2.22) can be constructed as follows. Given a four point correlator
+⟨φ(x1)φ(x2)φ(x3)φ(x4)⟩ ∼ G∗ which satisfy:
+z−2∆φG∗(z) − (1 − z)−2∆φG∗(1 − z) = 0,
+(2.25)
+the O(N) vector correlation functions
+GA = GS = G∗,
+GT = 0
+(2.26)
+satisfy the crossing equation (2.22). In this solution the T sector is empty therefore cor-
+responding to an infinity high upper bound ∆∗
+T = ∞. Due to the same logic there is no
+upper bound on ∆∗
+A either. Note the unit operator is an indispensable ingredient in the
+OPEs of the correlation functions G∗ and GS/A.
+It seems the 1D large N bootstrap is
+too simplified to capture nontrivial dynamics and is not insightful for higher dimensional
+bootstrap. However, this is not the case.
+As discussed before, different O(N) representations carry different S-parity charges,
+similar to the spin selection rules in higher dimensional bootstrap. With different S-parity
+charges it is expected that the spectra in different sectors are notably different. Specifically,
+in the O(N) vector bootstrap, to bound ∆∗
+T, we may expect a non-trivial gap for the lowest
+operator in the A sector which has opposite S-parity. In the bootstrap studies of defect
+CFTs, such gaps can be justified by the physical spectrum [26,46].
+With a gap assumption on the A sector spectrum, the bootstrap bound on ∆∗
+T can
+be modified drastically. The result is shown in Fig. 1, in which we have introduced an
+assumption that the lowest operator in the A sector satisfies ∆A ⩾ ∆c = 1.1 In the range
+∆φ ∈ (0, ∆c/2) the bootstrap bound on ∆∗
+T is given by ∆∗
+T = 2∆φ. It is followed by a sharp
+kink at ∆φ = ∆c/2, where the bound on ∆∗
+T jumps to ∆∗
+T = 2∆φ + 1. The solution (2.26)
+requires a unit operator in the A sector, therefore is excluded by the gap assumption. The
+bootstrap bound on ∆∗
+T disappears near ∆φ = 0.744, which suggest an end of the scalar
+bootstrap constraints.2
+The 1D O(∞) vector bootstrap bound is remarkably similar to the higher dimensional
+O(∞) vector bootstrap bounds shown in Fig. 2. It has been known since [20] that in
+1Bootstrap bounds with different gaps ∆c are qualitatively similar to Fig. 1.
+2Note the bound on ∆∗
+T provides the strongest constraint among the scalar bootstrap with global sym-
+metries.
+9
+
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Δϕ
+0
+1
+2
+3
+4
+5
+ΔT
+Figure 1: 1D O(∞) vector bootstrap bound on the scaling dimension of the lowest operator
+in the T sector. Gap assumption ∆A > 1.0 in the A sector. Λ = 20.
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Δϕ
+0
+5
+10
+15
+ΔT
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+Δϕ
+5
+10
+15
+ΔT
+Figure 2: 2D (left) and 3D (right) O(∞) vector bootstrap bounds on the scaling dimensions
+of the lowest scalars in the T sector. No gaps. Λ = 31.
+10
+
+3D, the O(N) vector bootstrap bounds show sharp kinks (type I) which are saturated by
+the 3D critical O(N) vector models. Moreover, in [21] the author observed that besides
+the type I kinks, the 3D O(N) vector bootstrap bounds also show another family of kinks
+(type II) which approach the free fermion theory in the large N limit. The type II kinks
+appear in general dimensions [18], and the kink in Fig. 1 at ∆φ = ∆c/2 could be considered
+as their dimensional continuation in 1D. In higher dimensions the type II kinks at finite
+N are conjectured to be related to the conformal gauge theories, while mixed with the
+bootstrap bound coincidences due to a positive algebraic structure in the four point crossing
+equations [19]. The numerical bootstrap results of the type II kinks are affected by the
+numerical convergence issue and it is hard to evaluate the CFT data numerically.
+One of the motivations of this work is to develop an analytical functional bootstrap
+method to study the kinks in the O(N) vector bootstrap bounds and clarify their putative
+connections to the conformal gauge theories. The higher dimensional bootstrap equations
+relate to conformal blocks with two cross ratios z, ¯z and spins, which make the analytical
+functional bootstrap more intricate. Here our results suggest that similar bootstrap bounds
+can also be realized in 1D conformal bootstrap, with a drastically simplified bootstrap
+setup.
+Therefore the 1D large N bootstrap can provide a key to unlock the large N
+analytical functional bootstrap in higher dimensions.
+2.3. Extremal solutions and the simplified bootstrap equation
+We focus on the 1D large N bootstrap bound in the range ∆φ ∈ (0, ∆c/2). Spectrum
+of the theory saturating the bootstrap bound can be obtained from the extremal functionals
+[22], which are shown in Fig. 3 for ∆φ = 0.1, 0.3. In the S sector the spectrum is trivial
+with only one first order zero at ∆ = 0, corresponding to the unit operator. Surprisingly,
+the extremal functional in the T sector shows a first order zero at ∆ = 2∆φ, and double
+zeros at ∆ = 2∆φ + n, n ∈ N+. Therefore the extremal spectrum is not from generalized
+free boson or fermion alone, but is given by the correlation functions (2.9-2.11) with |λ| < 1.
+Furthermore, action of the extremal functional in the A sector is the same as that of T
+sector up to numerical errors! In the A sector, we only introduced the positivity constraint
+above the gap ∆A ⩾ ∆c, while the extremal solution automatically satisfies the positivity
+condition down to ∆ > 2∆φ!
+Let us go back to the O(∞) vector crossing equation (2.22) and check what does it
+mean by two “almost” identical actions in T and A sectors. Consider a linear functional
+11
+
+Δϕ=0.1
+Δϕ=0.3
+Δϕ=0.6
+2
+4
+6
+8
+10
+12 Δ-2Δϕ
+210
+215
+220
+225
+230
+235
+240
+Log[f]
+Δϕ=0.1
+Δϕ=0.3
+Δϕ=0.6
+2
+4
+6
+8
+10
+12 Δ
+220
+230
+240
+250
+260
+270
+280
+Log[f]
+Figure 3: Extremal functional spectra in the O(N) T sector (left) and S sector (right).
+⃗α for the O(∞) vector crossing equation (2.22)
+⃗α · ⃗VT ≡ α1 · F∆(z) − α2 · E∆(1 − z),
+(2.27)
+⃗α · ⃗VA ≡ −α1 · F∆(z) − α2 · E∆(1 − z).
+(2.28)
+The observation ⃗α· ⃗VT = ⃗α· ⃗VA suggests α1 → 0 ! That is to say, to get the upper bound in
+Fig. 1 for ∆φ < ∆c/2, the first row in the crossing equation (2.22) is not necessary! The
+extremely small α1 has been verified in our numerical bootstrap results.
+Without the first row of (2.22), the conformal blocks in the T and A sectors are the
+same and the positivity constraint in the A sector
+⃗α · ⃗VA ⩾ 0,
+∀∆ > ∆c,
+(2.29)
+is substituted by the positivity constraint in the T sector
+⃗α · ⃗VT ⩾ 0,
+∀∆ > ∆∗
+T,
+(2.30)
+given ∆∗
+T < ∆c. While for ∆∗
+T ⩾ ∆c, the positivity constraints between the two sectors
+are switched and the bootstrap bound in Fig. 1 suggests the first row of (2.22) becomes
+important. The bootstrap constraints have a transition at ∆∗
+T = ∆c, corresponding to the
+jump of the bootstrap bound at ∆φ = ∆c/2 in Fig. 1. We leave a detailed study of the
+bootstrap bound with ∆φ ⩾ ∆c/2 for future work.
+The correlation functions (2.9-2.11) with different |λ| < 1 have the same spectrum
+while different OPE coefficients (2.13,2.14). The extremal OPE coefficients c∗
+n are given by
+12
+
+the generalized free boson or fermion theories when |λ| → 1
+c∗
+n =
+(2∆φ)2
+n
+n!(4∆φ + n − 1)n
+.
+(2.31)
+We have checked that our bootstrap bounds on the OPE coefficients of low-lying spectrum
+∆ = 2∆φ + n are well consistent with (2.31) up to n = 9.
+To summarize, the O(∞) vector bootstrap leads to a rather simple crossing equation
+�
+O∈S
+λ2
+O z−2∆φ G∆(z) −
+�
+O∈T
+λ2
+O (1 − z)−2∆φG∆(1 − z) = 0.
+(2.32)
+Bootstrap bound on ∆∗
+T from above crossing equation is given by ∆∗
+T = 2∆φ for general
+∆φ, and its extremal spectrum is the same as those in Fig. 3. The O(∞) vector bootstrap
+equation (2.22) is reduced to (2.32) in the range ∆φ < ∆c/2. For ∆φ ⩾ ∆c/2, the bootstrap
+bound from (2.32) stays in the line ∆∗
+T = 2∆φ, see e.g. the extremal spectrum at ∆φ = 0.6
+in Fig. 3, while the bound from (2.22) goes differently as the first row in (2.22) starts
+to play a role. The rest part of this work aims to construct analytical functionals for the
+crossing equation (2.32).
+We would like to add comments on O(∞) vector bootstrap in higher dimensions [49]. In
+the range between free boson and free fermion bilinear: ∆φ ∈
+� D−2
+2 , D − 1
+�
+, the bootstrap
+bound on ∆∗
+T is also saturated by the generalized free theory and the O(∞) vector bootstrap
+equations are reduced to the higher dimensional form of (2.32)
+�
+O∈S
+λ2
+O(z¯z)−∆φ G∆,ℓ(z, ¯z) −
+�
+O∈T
+λ2
+O((1 − z)(1 − ¯z))−∆φ G∆,ℓ(1 − z, 1 − ¯z)
+−
+�
+O∈A
+λ2
+O((1 − z)(1 − ¯z))−∆φ G∆,ℓ(1 − z, 1 − ¯z) = 0,
+(2.33)
+where G∆,ℓ(z, ¯z) are the SO(D + 1, 1) conformal blocks [50, 47].
+Considering the close
+relation between the O(∞) vector bootstrap in 1D and higher D’s, the analytical functional
+for 1D O(∞) vector bootstrap constructed in this work will be instructive to construct
+analytical functionals in higher dimensions [49].
+3. SL(2, R) conformal block and total positivity
+The 1D large N bootstrap provides an ideal example to decode the underlying math-
+ematical structures of conformal bootstrap.
+Considering there are only few ingredients
+13
+
+in the bootstrap crossing equation (2.32), it is expected that the presumed mathematical
+structures should be certain properties of the SL(2, R) conformal block. In section 4 we
+will construct the analytical functionals for the crossing equation (2.32) and show that the
+answer to this riddle is total positivity. In this section we provide a brief explanation of
+total positivity and study its relation to the conformal block G∆(z).
+3.1. Total positivity: definition and theorems
+Definition. A two-variable function K(x, y) defined on I × J with I, J ⊂ R is totally
+positive of the order k, if for all 1 ⩽ m ⩽ k, and arbitrary ordered variables x1 < ... <
+xm, y1 < ... < ym, xi ∈ I, yj ∈ J, the following determinants are positive
+||K(x, y)||m ≡ K
+�
+x1,
+...
+xm
+y1,
+...
+ym
+�
+= det
+�
+���
+K(x1, y1)
+...
+K(x1, ym)
+...
+...
+K(xm, y1)
+...
+K(xm, ym)
+�
+��� > 0.
+(3.1)
+We are interested in the totally positive functions of the order infinity, which will be
+assumed implicitly in the following part. For the finite sets I, J, the two-variable functions
+K(x, y) are reduced to the matrices K(x, y) → Ki,j. In this case, the definition (3.1) for
+totally positive matrices becomes that all the minors of the matrix K are positive.
+From the definition (3.1), it is straightforward to show following rules for totally pos-
+itive functions:
+• If g(x) and h(y) are positive functions defined on I and J, respectively, and K(x, y)
+is totally positive, then so is the function g(x)K(x, y)h(y).
+• If g(x) ∈ I and h(y) ∈ J are defined on x ∈ U and y ∈ V , and monotone in the same
+direction, and if K(x, y) is totally positive on I × J, then the function K(g(x), h(y))
+is totally positive on U × V .
+An important tool to study total positivity is the so-called “basic composition formula”.
+It shows how to construct a new totally positive function from two such functions and
+provides a powerful method to prove total positivity of certain functions.
+Basic composition formula. Let K, L, M be two-variable functions which satisfy
+M(x, y) =
+�
+K(x, z)L(z, y)dσ(z),
+(3.2)
+where σ(z) is a σ-finite measure and the integral converges absolutely, then the basic
+14
+
+composition formula suggests
+M
+�
+x1,
+...
+xm
+y1,
+...
+ym
+�
+=
+�
+· · ·
+�
+z1<···<zm
+K
+�
+x1,
+...
+xm
+z1,
+...
+zm
+�
+L
+�
+z1,
+...
+zm
+y1,
+...
+ym
+�
+dσ(z1) . . . dσ(zm). (3.3)
+A proof of this formula is sketched in [51].
+The convolution (3.2) of two kernels K(x, z) and L(z, y) can be considered as a contin-
+uous version of the standard matrix product, then above basic composition formula (3.3) is
+an extension of the Cauchy-Binet formula in matrix multiplication which expands subde-
+terminants of Mij in terms of those of Kim and Lmj. The basic composition formula (3.3)
+directly leads to following theorem.
+Theorem. The convolution (3.2) of two totally positive kernels is also totally positive.
+Variation Diminishing Property. Consider a function f : I → R, where I ⊂ R. The
+number of sign changes of f on I, denoted S+
+I (f), is defined as the maximum number of
+sign changes in a finite sequence {f(x1), f(x2), . . . , f(xm)}, xi ∈ I, x1 < · · · < xm.3 An
+important property of the totally positive function is given by [51]:
+Theorem. For I, J ⊂ R, consider a totally positive kernel K : I × J → R which is Borel-
+measurable. Let σ(y) be a regular σ-finite measure on J and f : J → R be a bounded and
+Borel-measurable function on J, so that the convolution of f(y) converges absolutely
+g(x) =
+�
+J
+K(x, y)f(y)dσ(y).
+(3.4)
+Then the number of sign changes of g(x) on I is not larger than that of f(y) on J:
+S+
+I (g) ⩽ S+
+J (f).
+(3.5)
+Moreover, if S+
+I (g) = S+
+J (f), then the two functions f(y) and g(x) should have the same
+arrangement of signs.
+The variation diminishing property of totally positive functions will play a critical role
+3For the possible zeros terms in the sequences, they are simply discarded when counting the number of
+sign changes.
+15
+
+to construct analytical functionals of 1D large N bootstrap.
+3.2. Total positivity of the Gauss hypergeometric function
+The Gauss hypergeometric function 2F1(∆, ∆, 2∆, z) is a substantial ingredient of the
+SL(2, R) conformal block G∆(z). We are particularly interested in its connection to the
+total positivity. There is numerical evidence indicating that the function 2F1(∆, ∆, 2∆, z)
+is indeed totally positive in the region z ∈ (0, 1), ∆ > 0 [41]. We have also numerically
+verified the total positivity of this function using a large set of data. While it is hard to
+provide a complete prove for the total positivity of 2F1(∆, ∆, 2∆, z), we can get promising
+evidence for this observation beyond the numerical checks.
+Total positivity of 2F1(∆, ∆, 2∆, z) in the large ∆ limit
+In the large ∆ limit, the hypergeometric function 2F1(∆, ∆, 2∆, z) has a much simpler
+asymptotic form, for which the total positivity can be proved easily. Let us consider the
+integral formula of the hypergeometric function
+2F1(∆, ∆, 2∆, z) =
+1
+B(∆, ∆)
+� 1
+0
+x∆−1(1 − x)∆−1(1 − zx)−∆dx,
+(3.6)
+where B(∆, ∆) = Γ(2∆)
+Γ(∆)2 is the Euler Beta function. In the large ∆ limit above integration
+can be solved using the method of steepest descent:
+� 1
+0
+x∆−1(1 − x)∆−1(1 − zx)−∆dx =
+� 1
+0
+1
+x(1 − x)e−∆ log[ 1−xz
+x(1−x)]dx,
+(3.7)
+which has a single stationary point x = 1−√1−z
+z
+in the region x ∈ (0, 1). Then the integration
+(3.6) is approximately given by
+2F1(∆, ∆, 2∆, z)|∆→∞ ≈
+1
+B(∆, ∆)
+� π
+∆(1 − z)− 1
+4 �
+1 +
+√
+1 − z
+�1−2∆ .
+(3.8)
+We find above approximation is reasonably good even for ∆ = 5.
+It is straightforward to prove the total positivity of the right hand side of (3.8). Since
+the positive factors depending on z or ∆ have no effect on the total positivity, the only
+relevant factor in the approximated formula is
+�
+1 +
+√
+1 − z
+�−2∆ = ρ(z)2∆,
+(3.9)
+16
+
+where ρ(z) =
+�
+1 + √1 − z
+�−1 is a monotone increasing function in z ∈ (0, 1). Therefore the
+asymptotic formula (3.8) has the same total positivity as the function z∆, which has been
+proved in Appendix A.1.
+A sufficient condition for the total positivity of 2F1(∆, ∆, 2∆, z)
+Both the large ∆ approximation and numerical tests with small ∆’s suggest the hyper-
+geometric function 2F1(∆, ∆, 2∆, z) is totally positive. Here we discuss a sufficient condition
+which, if true, can prove the totally positivity of 2F1(∆, ∆, 2∆, z) for general ∆ > 0.
+The hypergeometric function has a series expansion
+2F1(∆, ∆, 2∆, z) =
+∞
+�
+i=0
+(∆)2
+i
+(2∆)i
+zi
+i! ,
+(3.10)
+where (a)i is the Pochhammer symbol. Above expansion can be considered as a convo-
+lution of K(∆, i) ≡ (∆)2
+i /(2∆)i and f(i, z) ≡ zi/i! with a discrete σ-measure in (3.2).
+Therefore according to the basic composition formula (3.3), the hypergeometric function
+2F1(∆, ∆, 2∆, z) is totally positive if both of the two functions K(∆, i) and f(i, z) are
+totally positive. The function zi has been shown to be totally positive.
+The total positivity of the function K(∆, i) requires
+||K(∆, i)||m = K
+�
+∆1,
+...
+∆m
+i1,
+...
+im
+�
+= det
+�
+����
+(∆1)2
+i1
+(2∆1)i1
+...
+(∆m)2
+i1
+(2∆m)i1
+...
+...
+(∆1)2
+im
+(2∆1)im
+...
+(∆m)2
+im
+(2∆m)im
+�
+���� > 0,
+(3.11)
+with 0 ⩽ ∆1 < · · · < ∆m, 0 ⩽ i1 < · · · < im, ∆k ∈ R, ik ∈ N for any integer m. A compact
+formula for above determinants with general m is not known. Here we show for small m,
+above determinants are indeed positive.
+Consider the determinant ||K(∆, i)||m=2 for general ∆k and ik in the domain of defi-
+nition
+||K(∆, i)||2 = det
+�
+��
+(∆1)2
+i1
+(2∆1)i1
+(∆2)2
+i1
+(2∆2)i1
+(∆1)2
+i2
+(2∆1)i2
+(∆2)2
+i2
+(2∆2)i2
+�
+��
+=
+(∆2)2
+i1(∆1)2
+i2
+(2∆2)i1(2∆1)i2
+� i2−i1−1
+�
+k=0
+(∆2 + i1 + k)2(2∆1 + i1 + k)
+(∆1 + i1 + k)2(2∆2 + i1 + k) − 1
+�
+.
+(3.12)
+17
+
+For each term in the product with k ⩾ 0, we have
+(∆2 + i1 + k) 2 (2∆1 + i1 + k) − (∆1 + i1 + k) 2 (2∆2 + i1 + k) =
+(∆2 − ∆1) (2∆2∆1 + ∆1i1 + ∆2i1 + ∆1k + ∆2k) > 0,
+(3.13)
+and consequently
+(∆2 + i1 + k)2(2∆1 + i1 + k)
+(∆1 + i1 + k)2(2∆2 + i1 + k) > 1.
+(3.14)
+Therefore the RHS of (3.12) is positive.
+With higher m’s the determinant formula ||K(∆, i)||m is too complicated for a general
+study. By choosing a specific set of ik’s one can evaluate the determinants explicitly. For
+instances, taking ik = k, the determinants ||K(∆, k)||m are given by
+||K(∆, k)||m=3 =
+(3.15)
+(∆2 − ∆1) (∆3 − ∆1) (∆3 − ∆2) ∆1∆2∆3
+(∆1∆2 + ∆3∆2 + ∆1∆3 + 2∆1∆2∆3)
+16 (2∆1 + 1) (2∆2 + 1) (2∆3 + 1)
+for m = 3 and
+||K(∆, k)||m=4 =
+(3.16)
+(∆2 − ∆1) (∆3 − ∆1) (∆3 − ∆2) (∆4 − ∆1) (∆4 − ∆2) (∆4 − ∆3) ∆1∆2∆3∆4
+× (3∆1∆2∆3 (9∆3 + ∆1 (2∆2 + 3) (2∆3 + 3) + ∆2 (6∆3 + 9) + 13) +
+(9∆3 + ∆1 (2∆2 + 3) (2∆3 + 3) + ∆2 (6∆3 + 9) + 13) (3∆2∆3 + ∆1 (3∆3 + ∆2 (8∆3 + 3))) ∆4
++ (2∆1 + 3) (2∆2 + 3) (2∆3 + 3) (∆2∆3 + ∆1 (∆3 + ∆2 (2∆3 + 1))) ∆2
+4
+�
+/64 (2∆1 + 1) (2∆1 + 3) (2∆2 + 1) (2∆2 + 3) (2∆3 + 1) (2∆3 + 3) (2∆4 + 1) (2∆4 + 3)
+for m = 4, both of which are obviously positive for ordered ∆i’s. In all similar checks we
+find the results are well consistent with the total positivity. We conjecture this function is
+totally positive at infinity order for general ∆ ⩾ 0.
+3.3. Total positivity of the 1D SL(2, R) conformal block
+Now we study the total positivity of the fundamental ingredient in 1D conformal
+bootstrap, the SL(2, R) conformal block G∆(z) = z∆
+2F1(∆, ∆, 2∆, z). Here G∆(z) is a
+product (but not convolution) of two totally positive factors. However, it is not guaranteed
+that the product of two totally positive functions is also totally positive, and indeed,
+18
+
+the function G∆(z) loses its total positivity in the region with multiple small ∆i’s. This
+surprising fact was firstly observed in [41].4 Here we study the total positivity of G∆(z)
+from different aspects.
+Total positivity of G∆(z) in the large ∆ limit
+Using the asymptotic formula (3.8) of the Gauss hypergeometric function, the large ∆
+limit of the 1D conformal block is given by
+G∆(z)|∆→∞ ≈
+1
+B(∆, ∆)
+� π
+∆(1 − z)− 1
+4 z∆ �
+1 +
+√
+1 − z
+�1−2∆ .
+(3.17)
+The total positivity of above formula is determined by the factors depending on both z
+and ∆:5
+z∆ �
+1 +
+√
+1 − z
+�−2∆ = ˜ρ(z)∆,
+(3.18)
+where ˜ρ(z) = z (1 + √1 − z)−2, like ρ(z) in (3.9), is a monotone increasing function in
+z ∈ (0, 1). Thus the 1D conformal block function is totally positive for sufficiently large ∆.
+However, for small ∆, the large ∆ approximation (3.17) fails and it cannot say anything
+about the total positivity of G∆(z).
+A “Fixed point” of the 1D conformal block G∆(z)
+We show an interesting property of the conformal block G∆(z), though its physical
+correspondence is not clear yet.
+The conformal blocks G∆(z) with different ∆’s are plotted in Fig. 4. A surprising
+fact is that all these functions intersect near ∆ ∈ (0.62, 0.64) with G∆(z) ≃ 1. This tiny
+intersection region looks like a “fixed point” of the conformal block G∆(z), besides another
+trivial “fixed point” at z = 0. Why?
+Let us first consider the large ∆ approximation of G∆(z) (3.17). The dominating part
+of G∆(z) in this limit is
+G∆(z)|∆→∞ ∼
+1
+B(∆, ∆)
+�
+z
+(1 + √1 − z)2
+�∆
+≈ 4∆
+�1 − √1 − z
+1 + √1 − z
+�∆
+.
+(3.19)
+Here we have used the Stirling’s formula for the Gamma function which gives B(∆, ∆) ∼
+4−∆.
+From (3.19) it is clear that in the large ∆ limit, the equation G∆(z) = 1, or
+4The author would like to thank Nima Arkani-Hamed for the inspiring discussion on this problem.
+5Interestingly, the variable ˜ρ in (3.18) is just the variable ρ(z) in [52] motivated by different reasons.
+19
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0z
+0.0
+0.5
+1.0
+1.5
+2.0
+GΔ(z)
+0.615
+0.620
+0.625
+0.630
+0.635
+0.640
+z
+0.98
+0.99
+1.00
+1.01
+1.02
+1.03
+1.04
+1.05
+GΔ(z)
+Figure 4: Plots for the conformal block functions G∆(z) with ∆ = 0.005, 0.05, 0.1, 0.3 (red
+curves), ∆ = 1 (blue curve) and ∆ = 2, 4, 10, 50 (Green curves).
+log(G∆(z)) = 0 has a ∆-independent solution at z = 0.64.
+Contributions from extra
+factors are exponentially suppressed.
+Then let us go to the small ∆ limit.
+With a small ∆ the Gauss hypergeometric
+function is simplified to
+2F1(∆, ∆, 2∆, z)|∆≪1 ≈ 1 − ∆
+2 log(1 − z) + O(∆3)
+(3.20)
+and the conformal block function G∆(z) becomes
+G∆(z)|∆≪1 ≈ 1 + ∆ log
+�
+z
+√1 − z
+�
++ O(∆2),
+(3.21)
+in which the equation G∆(z) = 1 is solved by z = (
+√
+5 − 1)/2 ≈ 0.618.
+So both in the large and small ∆ limits, the equation G∆(z) = 1 has a solution
+independent of ∆. The solution walks slowly from z ≈ 0.618 near ∆ = 0 to z = 0.64 near
+∆ = ∞. Such a “fixed point” shows an interesting interplay between the factors z∆ and
+the hypergeometric function 2F1(∆, ∆, 2∆, z) in G∆(z). As will be shown below, the factor
+z∆ also changes the total positivity of G∆(z) with multiple small ∆i’s.
+Loss of total positivity of G∆(z) with small ∆s
+The total positivity is violated by the the 1D conformal block G∆(z) with multiple
+small ∆i ≪ 1 at the order 3.
+For sufficiently small ∆ it is convenient to take the lower order expansion (3.21) of
+20
+
+G∆(z). Up to the order ∆2, it is given by
+G∆(z)||∆|≪1 ≈ 1 + ∆ log
+�
+z
+√1 − z
+�
+− ∆2 log(z) tanh−1(1 − 2z) + O(∆3),
+(3.22)
+Let us consider the determinant of function G∆(z) at the third order
+||G∆(z)||3 = det
+�
+��
+G∆1(z1)
+G∆1(z2)
+G∆1(z3)
+G∆2(z1)
+G∆2(z2)
+G∆2(z3)
+G∆3(z1)
+G∆3(z2)
+G∆3(z3)
+�
+��
+(3.23)
+= (∆2 − ∆1) (∆3 − ∆1) (∆3 − ∆2)
+�
+log (z1) log
+�
+z1
+1 − z1
+�
+log
+�z2
+3 (1 − z2)
+z2
+2 (1 − z3)
+�
++ log (z3) log
+�
+z3
+1 − z3
+�
+log
+�z2
+2 (1 − z1)
+z2
+1 (1 − z2)
+�
++ log (z2) log
+�
+z2
+1 − z2
+�
+log
+�z2
+1 (1 − z3)
+z2
+3 (1 − z1)
+��
+,
+in which the ∆ factors are positive for the ordered ∆i, like those in (3.15-3.16). However,
+the zi-dependent factor is not definitely positive. Considering zi = 1+(i−4)δ with a small
+variable δ, at the leading order the z-dependent factor in (3.23) is
+�
+log 4
+3 log δ + log 2 log 3
+�
+δ + O(δ2),
+(3.24)
+which is negative for 0 < δ < 2
+− log 3
+log 4
+3 . A nonperturbative picture of the whole z-dependent
+factor in (3.23) is shown in Fig. 5. This confirms that the total positivity is violated by
+the function G∆(z) with multiple small ∆i and 1 − z.
+Using the same approach one can compute the determinant at the forth order ||G∆(z)||4,
+and its limit with small ∆ and 1 − z is actually positive
+||G∆(z)||4 ≈ 0.024 δ2 �
+i<j
+�
+∆j − ∆i
+�
+.
+(3.25)
+Non-positive determinants appear again at the fifth order. The ∆-expansion of G∆(z) at
+the order O(∆4) is rather complicated and similar analytical results for ||G∆(z)||5 are not
+available yet.
+Numerically one can show that for ∆i = 0.1 ∗ i, zj = 1 + 0.001(j − 6),
+||G∆(z)||5 ≈ −8.7 × 10−26.
+An interesting fact in these examples is that usually the magnitude of the negative
+21
+
+0.02
+0.04
+0.06
+0.08
+0.10δ
+-0.01
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Figure 5: The z-dependent factor in (3.23) is negative in a small range 0 < δ < 0.047.
+factor in ||G∆(z)||m is rather small.6 In [41] it has been commented that such tiny negative
+factors obtained from functions with normal parameters are reminiscent to the well-known
+hierarchy problem in particle physics. Our perturbative results for the m = 3 determi-
+nant suggest that the negative factor originates from the logarithmic term log δ in (3.24),
+which further comes from a logarithmic term log(1−z) near the branch cut z ∈ [1, +∞) of
+the hypergeometric function (3.20) associated with a power-law factor z∆. Without such
+a power-law factor the hypergeometric function remains totally positive. It would be in-
+teresting to extend our analysis to higher orders m ⩾ 5 to clarify the origin of the tiny
+negative determinant further.
+Above violation of total positivity appears when all the ∆i’s are small. In conformal
+bootstrap studies, the interesting set of ∆i are the physical spectrum near the bootstrap
+bounds, e.g., ∆n = 2∆φ +n, n ∈ N, which contains at most one small ∆0. We have numer-
+ically checked that when there is only one small ∆ in the data set {∆i}, the determinant
+||G∆(z)||m is always positive.
+In Section 4, we will assume the bootstrap problems are
+in the parameter space where G∆(z) is totally positive. It would be important for future
+studies to obtain a thorough understanding of this problem beyond the numerical tests.
+6Note the negative factor of the determinant ||G∆(z)||m remains tiny after taking off the small positive
+factor �
+i<j(∆j − ∆i).
+22
+
+Total positivity of the linear functional action on G∆(z)
+In Section 4 we will study the functional α′
+i whose action is given by
+S(∆, i) ≡ α′
+i[G∆] =
+� 1
+0
+xi+a G∆(x)dx.
+(3.26)
+and it is important to know the total positivity of the function S(∆, i).
+With sufficiently large ∆, the total positivity of the function S(∆, i) can be proved
+using the basic composition formula (3.3). Since the function zi+a and G∆(z) with large
+∆ are totally positive, their convolution is also totally positive. Note the total positivity
+of G∆(z) is a sufficient but not necessary condition for S(∆, i) being totally positive, and
+it could be totally positive even with multiple small ∆i’s though this is not the case for
+G∆(z). The integration (3.26) can be evaluated using series expansion of G∆(z):
+S(∆, i) =
+∞
+�
+k=0
+1
+k!
+(∆)2
+k
+(2∆)k
+1
+k + i + a + 1.
+(3.27)
+In the above formula, the function
+1
+k+i+a+1 is the modified Cauchy’s matrix which is totally
+positive, see Appendix A.2. For another relevant factor
+(∆)2
+k
+(2∆)k , we have provided promising
+evidence for its total positivity before. Therefore the linear functional action S(∆, i) is also
+expected to be totally positive for ∆ ⩾ 0, i ∈ N.
+4. Analytical functionals for the 1D O(N) vector bootstrap bound
+In this section we construct the analytical functionals for the 1D large N bootstrap
+bound with ∆ ∈ (0, ∆c/2), which is saturated by the the generalized free field theory
+with spectrum ∆n = 2∆φ + n, n ∈ N in the T sector.
+By constructing the analytical
+functionals for this simple while representative bootstrap problem, we want to study the
+critical question in conformal bootstrap: what is the mathematical structure responsible
+for the nontrivial bootstrap constraints? To construct the analytical functionals, we utilize
+the functional basis dual to the spectrum of generalized free field theories [28], for which
+we review in the first part of this section.
+4.1. Analytical functional basis
+In Section 2, the linear functionals are constructed based on the derivatives of variable
+z at the crossing symmetric point z =
+1
+2: α = �
+i⩽Λ ci ∂i
+z · |z= 1
+2. These functionals are
+23
+
+convenient for numerical computations. Nevertheless, due to the singularities at z = 0, 1
+of the conformal block G∆(z), the series expansion of G∆(z) only converges in the range
+|z − 1
+2| < 1
+2. To construct functionals more effectively, it needs new basis which contains
+information of the singularities of G∆(z), namely the analytical functional basis [23–25].
+The analytical functional basis is dual to the function basis in terms of which the
+conformal correlation functions can be expanded. The function basis can be provided by
+the s- and t-channel conformal blocks
+Gs
+n ≡ z−2∆φ G∆n(z),
+Gt
+n ≡ (1 − z)−2∆φ G∆n(1 − z),
+(4.1)
+and their derivatives, associated with the spectrum of generalized free field theories, e.g.,
+the generalized free boson ∆n = 2∆φ + 2n or fermion ∆n = 2∆φ + 2n + 1 [24,25]. In this
+work, inspired by the extremal functional spectrum in Fig. 3, we adopt a different function
+basis for the conformal correlation function, which is given by the conformal blocks Gs
+n, Gt
+n
+without their derivatives, associated with the spectrum ∆n = 2∆φ + n, n ∈ N. Consider a
+correlation function G(z) which is superbounded in the u-channel Regge limit |z| → ∞:7
+|G(z)| < |z|−ϵ
+(4.2)
+with ϵ > 0, it admits a unique expansion in terms of the above function basis
+G(z) =
+∞
+�
+n=0
+λs
+n Gs
+n +
+∞
+�
+n=0
+λt
+n Gt
+n ≡ Gs(z) + Gt(z).
+(4.3)
+The basis Gs
+n is holomorphic away from z ∈ [1, +∞), so is Gs(z). Likewise, the function
+Gt(z) is holomorphic away from z ∈ (−∞, 0]. The functional basis αs,t
+m dual to the above
+function basis satisfies
+αs
+m · Gs
+n = δmn,
+αs
+m · Gt
+n = 0,
+(4.4)
+αt
+m · Gt
+n = δmn,
+αt
+m · Gs
+n = 0,
+(4.5)
+based on which the coefficients λs/t
+n
+in (4.3) can be extracted from the Regge superbounded
+conformal correlator
+λs
+n = αs
+n · G,
+λt
+n = αt
+n · G
+(4.6)
+7Here the correlation function G(z) is the correlation function G(z) in (2.1) dressed with a factor z−2∆φ.
+24
+
+and the expansion (4.3) can be formally rewritten as
+G(z) = Gs(z) + Gt(z) =
+∞
+�
+n=0
+(αs
+n · G) Gs
+n + (αt
+n · G) Gt
+n.
+(4.7)
+Above formula has close relation with the dispersion relation of conformal correlation func-
+tion [28,53]. Here we sketch the main idea. Consider the Cauchy’s integral formula for the
+conformal correlation function G(z):
+G(z) =
+�
+dw
+2πi
+1
+w − z G(w),
+(4.8)
+in which the contour encircles w = z but does not contact the branch cuts (−∞, 0] and
+[1, +∞). The contour can be deformed into contours wrapping the two branch cuts, denoted
+C∓ and the arcs at infinity. For the Regge superbounded correlation functions which satisfy
+G(w) = O(|w|−ϵ) in the Regge limit |w| → ∞, contributions from infinity vanishes and the
+integral of G(z) consists of two parts
+G(z) = −
+�
+C−
+dw
+2πi
+1
+w − z G(w) +
+�
+C+
+dw
+2πi
+1
+w − z G(w) ≡ Gt(z) + Gs(z),
+(4.9)
+in which the Gt(z) and Gs(z) are holomorphic away from z ∈ (−∞, 0] and z ∈ [(1, +∞),
+respectively. The holomorphicity of the two terms in (4.9) suggests they can be decomposed
+into the function basis of Gt
+n and Gs
+n, as in (4.3). Such decomposition can be alternatively
+fulfilled with the expansion of the integral kernel
+1
+w−z
+1
+w − z =
+∞
+�
+n=0
+Θn(w) z−2∆φG∆n(z)
+(4.10)
+for integral along the contour C+ and
+1
+w − z = −
+∞
+�
+n=0
+Θn(1 − w) (1 − z)−2∆φG∆n(1 − z)
+(4.11)
+for integral along the contour C−, in which
+Θn(w) =
+(−1)n(2∆φ)2
+n
+n!(4∆φ + n − 1)n
+1
+w 3F2(1, −n, 4∆φ + n − 1; 2∆φ, 2∆φ; 1
+w).
+(4.12)
+25
+
+The integrals in (4.9) turn into
+Gs(z) =
+�
+C+
+dw
+2πi
+1
+w − z G(w) =
+∞
+�
+n=0
+��
+C+
+dw
+2πiΘn(w) G(w)
+�
+Gs
+n ,
+(4.13)
+Gt(z) = −
+�
+C−
+dw
+2πi
+1
+w − z G(w) =
+∞
+�
+n=0
+��
+C−
+dw
+2πiΘn(1 − w) G(w)
+�
+Gt
+n .
+(4.14)
+Comparing with the expansion (4.7), it gives explicit formulas for the actions of the func-
+tional basis
+αs
+n · G =
+�
+C+
+dz
+2πiΘn(z) G(z),
+(4.15)
+αt
+n · G =
+�
+C−
+dz
+2πiΘn(1 − z) G(z).
+(4.16)
+By deforming the contours it is clear that the actions αs
+n · Gt
+n = αt
+n · Gs
+n = 0.
+There are constraints the analytical functionals need to satisfy [54,24,25]. Here Θn(z) ∼
+O(|z|−1) in the Regge limit |z| → ∞, therefore the integrals (4.15,4.16) only converge for the
+functions G(z) ∼ O(|z|−ϵ) in this limit. The most general conformal correlation functions
+have Regge limit G(z) → |z|0 for which the integrals (4.15,4.16) do not converge. In this
+case one can use the subtracted functionals [28]
+¯αr
+n = αr
+n −
+(−1)n(2∆φ)2
+n
+n!(4∆φ + n − 1)n
+αr
+0,
+r = s, t,
+(4.17)
+which correspond to new integral kernels with Regge behavior ¯Θn(z) ∼ O(z−2).
+Actions of the functional basis on conformal blocks
+The dual relations (4.4,4.5) show the actions of functional basis on the conformal
+blocks with ∆ = 2∆φ + n. For the actions on conformal blocks with general ∆’s, we need
+to evaluate the integrals (4.15) with G = z−2∆φG∆(z) and G = (1 − z)−2∆φG∆(1 − z)
+αs
+n ·
+�
+z−2∆φG∆(z)
+�
+≡ S(∆, n) =
+� ∞
+1
+dz
+2πiDisc[Θn(z)z−2∆φG∆(z)],
+(4.18)
+αs
+n ·
+�
+(1 − z)−2∆φG∆(1 − z)
+�
+≡ T(∆, n) =
+� ∞
+1
+dz
+2πiDisc[Θn(z)(1 − z)−2∆φG∆(1 − z)].
+(4.19)
+26
+
+The function Θn(z) is regular along z ∈ [1, +∞), while the conformal blocks acquire dis-
+continuities between the two sides of the branch cut [1, +∞)
+1
+2πiDisc[z−2∆φG∆(z)] = Γ(2∆)
+Γ(∆)2 z−2∆φ−∆+1
+2F1(1 − ∆, 1 − ∆, 1, 1 − z),
+(4.20)
+1
+2πiDisc[(1 − z)−2∆φG∆(1 − z)] = − 1
+π sin(π(∆ − 2∆φ))(1 − z)−2∆φG∆(1 − z).
+(4.21)
+Applying above two formulas in (4.18) and (4.19) it gives
+S(∆, n) =
+(−1)nΓ(2∆)Γ(n + 2∆φ)2
+Γ(n + 1)Γ(∆)2Γ(−∆ + 2∆φ + 1)Γ(∆ + 2∆φ)(n + 4∆φ − 1)n
+× 3F2(1, −n, n + 4∆φ − 1; −∆ + 2∆φ + 1, ∆ + 2∆φ; 1),
+(4.22)
+and
+T(∆, n) = sin(π(∆ − 2∆φ))
+π
+n
+�
+i=0
+(−1)i+n+1Γ(2∆)Γ(∆ − 2∆φ + 1)Γ(2∆φ + i)(2∆φ + i)2
+n−i
+Γ(n − i + 1)(4∆φ + n + i − 1)n−i
+× 3 ˜F2(∆, ∆, ∆ − 2∆φ + 1; 2∆, ∆ + i + 1; 1),
+(4.23)
+where 3 ˜F2 is the regularized generalized hypergeometric function. For ∆ = 2∆φ+m, m ∈ N,
+above formulas agree with the dual relation (4.4). Actions of the linear functionals αt
+n on
+the s- and t-channel conformal blocks can be obtained from (4.22) and (4.23) through
+z ↔ 1 − z transformation.
+For ∆φ = 1
+2 the kernel Θs
+n(z) in (4.4) is drastically simplified
+Θn(z) = 1
+z
+(−1)nΓ(n + 1)2
+Γ(2n + 1)
+2F1
+�
+−n, n + 1; 1; 1
+z
+�
+.
+(4.24)
+Its actions on conformal blocks are reduced to
+S(∆, n)|∆φ= 1
+2 =
+Γ(2∆)Γ(n + 1)2 sin(π(∆ − n − 1))
+πΓ(∆)2(∆ − n − 1)(∆ + n)Γ(2n + 1),
+(4.25)
+and
+T(∆, n)|∆φ= 1
+2 = (−1)n(n!)2 Γ(2∆) Γ(∆)2
+Γ(2n + 1)
+sin(π∆)
+π
+× 4 ˜F3(∆, ∆, ∆, ∆; 2∆, ∆ − n, ∆ + n + 1; 1).
+(4.26)
+27
+
+Above formulas provide necessary ingredients to construct analytical functionals. We firstly
+construct the analytical functionals for the Regge superbounded conformal correlators, then
+we extend the construction to general conformal correlators.
+4.2. Analytical functional for Regge superbounded conformal correlator
+In Section 2 we have shown numerically that in the range ∆ < ∆c, the bootstrap
+bound is determined by the crossing equation
+�
+O∈S
+λ2
+O z−2∆φ G∆(z) −
+�
+O∈T
+λ2
+O (1 − z)−2∆φ G∆(1 − z) = 0.
+(4.27)
+Actions of the extremal functional α∗ on above crossing equation should satisfy following
+positive conditions
+α∗ ·
+�
+z−2∆φG∆(z)
+�
+= 0,
+for ∆ = 0,
+(4.28)
+α∗ ·
+�
+z−2∆φG∆(z)
+�
+> 0,
+∀ ∆ > 0,
+(4.29)
+−α∗ ·
+�
+(1 − z)−2∆φG∆(1 − z)
+�
+= 01,
+for ∆ = 2∆φ,
+(4.30)
+−α∗ ·
+�
+(1 − z)−2∆φG∆(1 − z)
+�
+= 02,
+for ∆ = 2∆φ + n, n ∈ N+,
+(4.31)
+−α∗ ·
+�
+(1 − z)−2∆φG∆(1 − z)
+�
+> 0,
+∀ ∆ > 2∆φ & ∆ ̸= 2∆φ + n, n ∈ N+,
+(4.32)
+in which the notation 0i refers to the zeros of order i. In principle the zeros with even
+orders above 2 also satisfy the positive condition (4.31), however, we will show that there
+are only second order zeros in (4.31).
+For the Regge superbounded conformal correlators, e.g., correlation functions (2.9-2.11)
+with λ = 0, the bootstrap functional α can be expanded in terms of the functional basis
+α =
+∞
+�
+n=0
+cnαs
+n + dnαt
+n.
+(4.33)
+Considering the actions of the functional basis on Gs,t
+n
+(4.4,4.5), the positive condition
+(4.29) of the extremal functional α∗ requires
+cn > 0
+∀n ∈ N.
+(4.34)
+28
+
+Moreover, the conditions (4.30,4.31) suggest
+dn = 0
+∀n ∈ N.
+(4.35)
+The positive coefficients cn in α∗ =
+∞
+�
+n=0
+cnαs
+n should be arranged so that the extra positive
+conditions can be satisfied.
+It is easy to verify that the positive condition (4.28) is satisfied by the functional
+α∗ =
+∞
+�
+n=0
+cnαs
+n. For each αs
+n, its action is
+αs
+n · z−2∆φ =
+�
+C+
+dz
+2πiΘn(z) z−2∆φ,
+(4.36)
+in which the integrand has a single pole at z = 0 and is holomorphic for z ∈ [1, +∞)
+enclosed by the contour C+. Therefore the action vanishes, as required by (4.28).
+The critical constraints to solve cn are from (4.30,4.31). In the action (4.23) of func-
+tional basis αs
+n, the factor sin(π(∆ − 2∆φ)) generates single zeros at ∆ = 2∆φ + n with
+n ∈ N. To further form double zeros for n ∈ N+, the coefficients cn should satisfy
+∞
+�
+i=0
+T (2∆φ + n, i) · ˜ci = δn0,
+∀n ∈ N
+(4.37)
+in which the coefficients ˜ci is
+˜ci ≡ ci (−1)i Γ(i + 1)2
+Γ(2i + 1).
+(4.38)
+and T is the stripped action
+T(∆, i) = (−1)i+1 sin(π(∆ − 2∆φ))
+π
+T (∆, i) Γ(i + 1)2
+Γ(2i + 1).
+(4.39)
+One may expect the coefficients cn can be obtained by solving the whole infinite equation
+group (4.37), like the remarkable work [23]. However, the solutions to the whole equation
+group (4.37) lead to a trivial functional. We demonstrate this point using an example with
+∆φ = 1
+2.
+4.2.1. Inverse of the infinite equation group
+Solution to the linear equation group (4.37) is given by the inverse of the infinite
+matrix T (2∆φ + n, i). For general ∆φ the matrix T (2∆φ + n, i) is quite complicated and
+29
+
+hard to solve directly. The formulas are notably simplified with ∆φ = 1
+2, see (4.24-4.26). In
+this case it is convenient to take the variable transformation x = 1 − z−1 [23]. The kernel
+Θn(z) degenerates to the Legendre polynomials Pn(2x−1) and the stripped action T (∆, n)
+becomes
+T (∆, n) =
+� 1
+0
+dx x∆−1
+2F1(∆, ∆, 2∆, x)Pn(2x − 1).
+(4.40)
+Since the Legendre polynomials Pn(2x − 1) are orthogonal under the inner product
+� 1
+0
+dx Pn(2x − 1)Pm(2x − 1) =
+1
+2n + 1δn,m,
+(4.41)
+the matrix T (m, n) can be interpreted as the coefficients of the following expansion
+xm−1
+2F1(m, m, 2m, x) =
+∞
+�
+n=0
+(2n + 1)T (m, n)Pn(2x − 1).
+(4.42)
+Thus the inverse matrix T −1 is given by the inverse expansion
+Pn(2x − 1) =
+1
+2n + 1
+∞
+�
+m=1
+T −1(n, m) xm−1
+2F1(m, m, 2m, x).
+(4.43)
+Let us compare the constant term on both sides for each n. The Legendre polynomial
+satisfies Pn(2x − 1)|x=0 = (−1)n. While the function xm−1
+2F1(m, m, 2m, x) is equal to 1 at
+x = 0 for m = 1 and vanishes at x = 0 for m > 1. Therefore the elements T −1(n, 1), as
+well as the coefficients ˜cn in (4.37) can be solved from (4.43):
+T −1(n, 1) = ˜cn = (−1)n(2n + 1),
+(4.44)
+which gives
+cn = Γ(2n + 2)
+Γ(n + 1)2 .
+(4.45)
+A comment is that all the cn’s are positive, as needed to satisfy the positive condition
+(4.29).
+The whole extremal functional α∗ is given by
+α∗ · G(z) =
+�
+C+
+dz
+2πiΘ∗(z)G(z)
+(4.46)
+30
+
+with a kernel
+Θ∗(z) =
+∞
+�
+n=0
+cnΘn(z) ∝
+∞
+�
+n=0
+(−1)n(2n + 1)Pn(2x − 1)|x= z−1
+z
+≡ ¯Θ∗(x).
+(4.47)
+Using the generating function of Legendre polynomial
+1
+�
+t2 − 2tx + 1
+=
+∞
+�
+n=0
+Pn(x) tn,
+(4.48)
+one can show
+¯Θ∗(x) =
+1 − t2
+�
+4tx + (1 − t)2�3/2
+�����
+t→1
+,
+(4.49)
+In the limit t → 1, we have
+¯Θ∗(x)|x̸=0 = 0.
+(4.50)
+While with x = 0 the function ¯Θ∗(x) has a pole at t = 1. Such “extremal” kernel behaves
+like a Dirac δ-function δ(x). The action of functional α∗ on the t-channel conformal block
+−(1 − z)−2∆φG∆(1 − z) with ∆φ = 1
+2 becomes
+−α∗ ·
+�
+(1 − z)−1G∆(1 − z)
+�
+=
+sin(π(∆ − 1))
+π
+� 1
+0
+dx
+1 − t2
+�
+4tx + (1 − t)2�3/2 x∆−1
+2F1(∆, ∆, 2∆, x)
+�����
+t→1
+.
+(4.51)
+Due to the pole at x = 0 with t = 1, above integral only gives a nonzero value for
+∆ = 2∆φ = 1, while vanishes for ∆ > 1. Together with the factor sin(π(∆ − 1)), the whole
+action vanishes for all ∆ ⩾ 1. On the other hand, its action on the s-channel conformal
+block
+α∗ ·
+�
+z−1G∆(z)
+�
+= Γ(2∆)
+Γ(∆)2
+� 1
+0
+dx
+1 − t2
+�
+4tx + (1 − t)2�3/2
+2F1(∆, 1 − ∆, 1, x)
+�����
+t→1
+(4.52)
+vanishes at ∆ = 0 and is always positive for ∆ > 0.
+To summarize, the functional constructed from the whole infinity set of equations
+(4.37) is actually trivial due to its vanishing action on the t-channel conformal block, or
+the O(N) T sector in (4.27). We expect this is the case for general ∆φ.
+31
+
+1
+2
+3
+4
+5
+6 Δ
+-4
+-3
+-2
+-1
+1
+2
+f
+2
+4
+6
+8 Δ
+-15
+-10
+-5
+Log[f]
+Figure 6: Actions (denoted f) of the functionals α′
+N on t-channel conformal block with
+N = 1 (left) and N = 4 (right). Note in the right plot, the y-axis is log(f) which has no
+real value for negative f.
+4.2.2. Inverse of the finite subset of equation group
+Although the inverse of the whole equation group (4.37) leads to a degenerated func-
+tional, a key property is that the inverse of the finite subset of the equation group can
+produce functionals which satisfy all the positive conditions (4.28-4.32) within a range
+∆ ⩽ ΛN. This allows us to construct a series of functionals with arbitrarily high ΛN!
+Instead of constructing a functional whose action on t-channel conformal block has
+double zeros at ∆ = 2∆φ + n for all n ∈ N+, we would like to relax the restriction on the
+double zeros. Specifically, we consider the functional
+α′
+N =
+N
+�
+n=0
+cnαs
+n,
+(4.53)
+whose action has a single zero at ∆ = 2∆φ and double zeros at ∆ = 2∆φ + n for each
+integer 0 < n ⩽ N. This amounts to the following constraints
+N
+�
+i=0
+T (2∆φ + n, i) · ˜ci = δn0,
+0 ⩽ n ⩽ N.
+(4.54)
+It is straightforward to solve above equations for small N’s. Taking N = 4, the matrix
+32
+
+N=10
+N=15
+N=20
+N=30
+N=40
+10
+20
+30
+n
+10
+20
+30
+40
+50
+|c
+n|
+Figure 7: |˜cn| solved from (4.54) with finite N’s. The straight line is |˜cn| = 2n + 1.
+T (n + 1, i) are8
+�
+�
+�
+�
+�
+�
+�
+�
+π2
+6
+1
+6(12 − π2)
+1
+6(π2 − 9)
+1
+18(31 − 3π2)
+1
+72(12π2 − 115)
+12 − π2
+5π2 − 48
+129 − 13π2
+1
+3(75π2 − 739)
+1
+12(4859 − 492π2)
+5(π2 − 9)
+5(129 − 13π2)
+5(73π2 − 720)
+5
+3(7492 − 759π2)
+5
+12(7932π2 − 78283)
+70
+9 (31 − 3π2)
+70
+9 (75π2 − 739)
+70
+9 (7492 − 759π2)
+70
+9 (4335π2 − 42784)
+35
+18(679939 − 68892π2)
+35
+4 (12π2 − 115)
+35
+4 (4859 − 492π2)
+35
+4 (7932π2 − 78283)
+35
+4 (679939 − 68892π2)
+35(99003π2 − 977120)
+�
+�
+�
+�
+�
+�
+�
+�
+,
+which corresponds to the coefficients ˜ci
+˜ci ≈ (0.999753, −2.97534, 4.5689, −4.39935, 1.88092).
+(4.55)
+It is impressive that the first few elements are close to the limit ˜cn = (−1)−n(2n + 1) even
+for N = 4. In Fig. 7 we show more solutions of ˜cn with larger N’s.9 From these examples,
+the coefficients |˜cn| with n ⩽ N/2 are close to the limit |˜cn| = (2n + 1), while for larger
+n’s, |˜cn| deviates the straight line and decreases exponentially. Nevertheless, for all n’s the
+coefficients ˜cn have signs ˜cn ∝ (−1)n. This suggests the coefficients cn ∝ (−1)n˜cn are all
+positive, therefore satisfying the positive condition (4.29) up to ∆ < N + 1.
+In Figs. 8 and 9 we show the actions (denoted f) of the functional α′
+N on the t-channel
+8The cautious readers may notice that the 5 × 5 matrix is symmetric up to certain numerical factors.
+Actually it can be proved that for ∆φ = 1
+2, the matrix Mm,n ≡
+Γ(n)2
+Γ(2n−1)T (m, n − 1) is symmetric!
+9To solve ˜cn from (4.54) with large N, it is necessary to adopt high numerical precision, reminiscent of
+the numerical conformal bootstrap with SDPB [55,56].
+33
+
+N=6
+N=9
+N=12
+N=15
+0
+5
+10
+15
+20Δ
+10-19
+10-14
+10-9
+10-4
+10
+f
+Figure 8: Actions of the functionals α′
+N on the t-channel conformal block.
+5
+10
+15
+20
+Δ
+0.1
+100
+105
+108
+1011f
+Figure 9: Action of the functional α′
+20 on the s-channel conformal block.
+34
+
+and s-channel conformal blocks, respectively. The actions on the t-channel conformal block
+have a single zero at ∆ = 1 and double zeros at ∆ = n + 1 for integer 0 < n < N. For
+1 < ∆ < N + 1, the f decreases with larger N.
+This is consistent with our previous
+results that with N = ∞, the functional α∗ becomes trivial: f = 0 for ∆ ⩾ 2∆φ. The
+action on the s-channel conformal block (N = 20) shown in Fig. 9 has a single zero at
+∆ = 0, corresponding to the unit operator, and is positive for 0 < ∆ < N + 1. The fact
+f(n) |n>20 = 0, n ∈ N is expected since there is no αs
+n>20 in α′
+N=20. The numerical solutions
+of the equations (4.54) do satisfy all the positive conditions (4.28-4.32) up to ∆ < N + 1
+for finite N.
+4.2.3. Positivity from total positivity
+Our goal is to generalize previous results to arbitrarily large N ∈ N+ with general
+∆φ.
+However, it requires highly nontrivial conditions to guarantee the strong positive
+constraints (4.28-4.32) for large ∆. The reason why our previous examples can satisfy the
+positive conditions for ∆ ⩽ Λn is due to a simple fact: in the action
+f(∆) ∝ sin(π(∆ − 2∆φ)
+N
+�
+n=0
+˜cnT (∆, n),
+(4.56)
+the sign of the function �N
+n=0 ˜cnT (∆, n) oscillates in phase with sin(π(∆ − 2∆φ) in the
+range 2∆φ < ∆ < 2∆φ + N. This is equivalent to the following properties of the function
+�N
+n=0 ˜cnT (∆, n) with general N:
+• The matrix T (2∆φ + n, i)
+��
+0⩽n,i⩽N is non-degenerate and invertible.
+• All the zeros at n = 1, . . . , N are of order 1 or higher odd numbers.
+• There are no other zeros besides n = 1, . . . , N.
+Any violations of above properties will necessarily break the positive conditions and invalid
+the functionals constructed from (4.54).
+Surprisingly, above three properties are closely
+related to the total positivity of the SL(2, R) conformal block for which we have studied
+in Section 3.
+35
+
+Consider the equation (4.54) for general ∆φ. Its LHS corresponds to
+g(∆) ≡
+N
+�
+n=0
+T (∆, n)˜cn =
+� 1
+0
+dx(1 − x)2∆φ−1x−2∆φG∆(x)
+N
+�
+n=0
+˜cn 3F2(1, −n, 4∆φ + n − 1; 2∆φ, 2∆φ; 1 − x),
+(4.57)
+where
+˜cn = cn
+(−1)n(2∆φ)2
+n
+n!(n + 4∆φ − 1)n
+.
+(4.58)
+In (4.57), the total positivity of the factor (1−x)2∆φ−1x−2∆φG∆(x) follows the total positiv-
+ity of the conformal block G∆(x) for sufficiently large ∆. Therefore we have the Variation
+Diminishing Property (3.5) that the sign changes of the functions g(∆) and ¯Θ(x) = � ˜cn 3F2
+in (4.57) should satisfy the relation
+S+(g) ⩽ S+(¯Θ) ⩽ N,
+(4.59)
+i.e., the number of sign changes of g(∆) in ∆ ∈ (2∆φ, ∞) is not larger than the number
+of sign changes of ¯Θ(x) with x ∈ (0, 1)! The second inequality in (4.59) is due to the fact
+that ¯Θ(x) is a polynomial of order N
+¯Θ(x) =
+N
+�
+n=0
+anxn,
+(4.60)
+consequently including multiplicity, there are at most N zeros of ¯Θ(x). Only the odd order
+zeros relate to sign changes of ¯Θ(x), therefore the polynomial ¯Θ(x) can have at most N sign
+changes, corresponding to N first order zeros. Moreover, in the extremal case S+(¯Θ) = N,
+according to the Descartes’ rule of signs, the number of positive roots of ¯Θ(x) is at most
+the number of sign changes in the sequence of its coefficients {an}, therefore there has to
+be N sign changes in {an}, which requires an ∝ a0(−1)n.
+The inequality (4.59) provides substantial restrictions on the solutions of the equation
+group (4.54)! The function g(∆) has at most N zeros including multiplicity. Therefore the
+N zeros specified in the equations (4.54) at ∆ = 2∆φ + n, n = 1, 2, . . . , N are all the zeros
+allowed by the inequality (4.59). Moreover, they are single zeros!
+We also need to prove that there are indeed N zeros in g(∆), e.g., the equations
+(4.54) have non-trivial solutions.
+This is equivalent to the statement that the matrix
+36
+
+T (2∆φ + n, i)
+��
+0⩽n,i⩽N is invertible for any N, which can be proved within two steps based
+on the assumed total positivity of the function f(∆, i) = (∆)2
+i /(2∆)i. Firstly it can be
+shown that the integral
+P(∆, k) =
+� 1
+0
+dx(1 − x)2∆φ−1x−2∆φG∆(x)xk
+(4.61)
+is totally positive, similar to (3.27).
+Therefore the determinant of its sub-matrices are
+always nonzero. The determinant ||T (∆, k)||N is related to ||P(∆, k)||N through a non-
+degenerate basis transformation {xk} → {Θk} and is also nonzero.
+Therefore based on the total positivity of the SL(2, R) conformal block, for which
+we have provided promising evidence while a thorough understanding is not reached yet,
+previous three questions on the equation group (4.54) can be nicely addressed. It confirms
+that for general N, the equations (4.54) always have nontrivial solutions, and the related
+function g(∆) only has single zeros at ∆ = 2∆φ + n, n = 1, ..., N. This guarantees the
+positive conditions on the t-channel conformal block (4.30-4.32) for ∆ ⩽ 2∆φ + N with
+arbitrary positive integer N.
+An interesting question is whether the matrix T (∆, n) is also totally positive. If true,
+then it can prove that the coefficients cn solved from (4.54) are always positive.
+The
+Variation Diminishing Property
+can tell us the signs of an in (4.60) with basis {xn}. In
+future studies, it would be important to provide quantitative estimations of ˜cn’s and explain
+the non-monotonic shapes in Fig. 7.
+Here we summarize the properties of the analytical functional α′
+N = �N
+n=0 cnαs
+n con-
+structed through the equations (4.54): for a given positive integer N, the functional α′
+N can
+produce the spectrum consistent with the numerical bootstrap results up to ∆ ⩽ 2∆φ + N.
+It gives the unit operator in the O(N) singlet sector and double trace operators in the
+O(N) T sector below 2∆φ + N. The actions of the functional satisfy the positive condi-
+tions in both singlet and T sectors for ∆ < 2∆φ + N. Based on the total positivity of the
+SL(2, R) conformal block, such functionals exist for any finite N. The large N limit of the
+functional lim
+N→∞ α′
+N = α∗ is trivial, which produces zero action on the t-channel conformal
+block. In contrast, the truly nontrivial point here is the way how the series of functionals
+{α′
+N} approach the limit α∗: for any given large cutoff ΛN, one can construct α′
+N so that
+its action satisfies the required positive conditions for any ∆ < ΛN. This explains, for the
+Regge superbounded correlators, the numerical bootstrap bound of the crossing equation
+(2.32), or the bound with ∆φ < ∆c/2 in Fig. 1.
+37
+
+4.3. Analytical functional for general conformal correlators
+The functional constructed in the last section scales as O(|z|−1) in the Regge limit and
+only works for the superbounded conformal correlators, e.g. (2.9-2.11) with λ = 0. For
+more general correlators, such as (2.9-2.11) with λ ̸= 0, one can construct functionals using
+the subtracted basis ¯αs
+n (4.17):
+¯α′
+N =
+N
+�
+n=1
+cn¯αs
+n.
+(4.62)
+Above functional should satisfy the same positive conditions (4.28-4.32).
+Following the
+same reasons for (4.54) we can get similar equation group
+N
+�
+i=1
+¯T (2∆φ + n, i) · ˜ci = δn0,
+for n = 0, 1, . . . , N − 1,
+(4.63)
+in which
+¯T (∆, i) = T (∆, i) −
+(−1)i(2∆φ)2
+i
+i!(4∆φ + i − 1)i
+T (∆, 0)
+(4.64)
+is invertible. Solutions to (4.63) are related to the function ¯g
+¯g(∆) ≡
+N
+�
+n=1
+¯T (∆, n)˜cn =
+(4.65)
+� 1
+0
+dx(1 − x)2∆φ−1x−2∆φG∆(x)
+N
+�
+n=1
+˜cn
+�
+3F2(1, −n, 4∆φ + n − 1; 2∆φ, 2∆φ; 1 − x) − 1
+�
+.
+Again we want to bound the number of zeros of the function ¯g(∆) by the number of
+sign changes in the sequence of the coefficients of the polynomial ˜Θ(x) = � ˜cn( 3F2 − 1).
+However, the function ¯g(∆) is expected to have N − 1 zeros at ∆ = 2∆φ + n with n =
+1, ..., N −1, while ˜Θ(x) remains an order N polynomial of x, whose coefficients, in principle,
+could have N sign changes, indicating the function ¯g(∆) could have an extra zero besides
+the N − 1 zeros specified in (4.63). Solution to this puzzle is that the lowest term of ˜Θ(x),
+when expanded as a order N polynomial of (1 − x), is linear in (1 − x), while the constant
+term has been canceled in the subtraction (4.17) for better Regge behavior. Therefore this
+linear term can be factorized and ¯g(∆) becomes
+¯g(∆) =
+� 1
+0
+dx(1 − x)2∆φx−2∆φG∆(x)
+N−1
+�
+n=0
+λnxn.
+(4.66)
+38
+
+The function (1 − x)2∆φx−2∆φG∆(x) remains totally positive while the polynomial ˜Θ(x)
+is reduced to the order N − 1, which can have at most N − 1 zeros and sign changes.
+Therefore according to the Variation Diminishing Property
+(3.5), the function ¯g(∆) can
+have at most N − 1 zeros. This confirms the function ¯g(∆) has no other zeros besides
+2∆φ + n, n = 1, ..., N − 1. Moreover, they are single zeros. It can be verified numerically
+that the coefficients cn solved from (4.63) are all positive, corresponding to positive actions
+of ¯α′
+N on the s-channel conformal for ∆ ∈ (0, 2∆φ + N), similar to Fig. 9.
+The conclusion is that the subtraction (4.17) is consistent with Variation Diminishing
+Property and the functionals ¯α′
+N for the general conformal correlators have similar positive
+properties as the functionals α′
+N for the Regge superbounded correlators.
+5. Conclusion and Outlook
+We have studied the 1D O(N) vector bootstrap in the large N limit. We have shown
+that with suitable bootstrap implementations, the 1D large N bootstrap bound has several
+interesting properties which are reminiscent to its higher dimensional analogies. We par-
+ticularly focused on the bootstrap bound saturated by the generalized free field theory, for
+which we obtained a significantly simplified bootstrap equation. The most interesting part
+of this work is the construction of analytical functionals for the large N bootstrap bound.
+We proposed an approach to construct a series of bootstrap functionals {α′
+M} whose ac-
+tions on the crossing equations can satisfy the bootstrap positive conditions for ∆ ⩽ ΛM.
+A surprising fact is that although the large M limit of the functional α′
+M becomes triv-
+ial, the functionals {α′
+M} can approach the limit in a particular way so that the positive
+conditions of the bootstrap bound can be fulfilled at arbitrarily high ΛM, thus providing
+an analytical explanation for the bootstrap bound. In our construction of the analytical
+functionals, the total positivity of the SL(2, R) conformal block function plays a substan-
+tial role, which uncovers an interesting mathematical structure in conformal bootstrap and
+intriguing connections between mathematics and quantum field theories.
+We believe this work opens the door towards more systematical studies for many
+fascinating problems in quantum field theories and their connections to mathematics. Part
+of these problems are explained below.
+• The most fundamental question is the total positivity of the SL(2, R) conformal
+block G∆(z). Though it is the key ingredient for the analytical functional satisfying
+the positive conditions, we only proved the totally positivity of SL(2, R) conformal
+block in the large ∆ limit and verified it numerically for the parameters relevant to
+39
+
+the extremal spectrum. It has been observed in [41] that this function loses total
+positivity with small ∆. We also explicitly showed that negative minors appear with
+multiple small ∆i’s. A more comprehensive understanding on the total positivity of
+the SL(2, R) conformal block is of critical importance. On the mathematical side,
+it would be very helpful to develop more powerful techniques to determine total
+positivity of these functions.
+• In this work we only constructed the analytical functional for the first part of the
+1D large N bootstrap bound before the kink in Fig. 1, which is saturated by the
+generalized free field theory and the bootstrap equations are reduced to a simple form
+(2.32). It is tempting to know the theories saturating the second part of the bootstrap
+bound and construct the analytical functionals. Furthermore, the bootstrap bound
+almost disappears after ∆φ > 0.75 in Fig. 1. Similar phenomenon also appears in
+higher dimensions, see e.g., in Fig. 2. It would be interesting to know the reasons
+which dissolve the bootstrap restrictions.
+• Conformal field theories with large N limits have close relation to the quantum field
+theories in the AdS spacetime. Constraints on the CFT side can lead to nontrivial
+restrictions on the theories in AdS, see e.g. [34, 57–59]. It would be interesting to
+explore the constraints of the analytical functional constructed in this work on the
+S-matrices in AdS2. In particular, how does the total positivity affect the scattering
+process in AdS2? Do the AdS analogies of the conformal blocks, the Witten diagrams
+also satisfy total positivity? The role of total positivity in the 4D amplitude in flat
+spacetime has been extensively studied recently [37–40]. Our results on the 1D CFTs
+suggest that the AdS2 could provide another interesting and technically tractable
+laboratory to explore the role of (total) positivity in quantum field theories. We hope
+to report the applications of our analytical functional and total positivity on AdS
+physics in another work.
+• Total positivity is powerful to analyze positivity of the analytical functionals. In our
+construction, the positivity of bootstrap functionals can be established given the total
+positivity of the SL(2, R) conformal block while without solving the equation groups
+(4.54,4.63) explicitly. Nevertheless, it would be interesting to know more concrete
+information on the analytical functionals {α′
+M}.
+For instance, to understand the
+shape of the coefficients |˜c|n shown in Fig.
+7.
+One may wonder if the equation
+groups (4.54, 4.63) are easier to solve in Mellin space [60–62].
+40
+
+• The 1D large N vector bootstrap provides insights to study higher dimensional O(N)
+vector bootstrap. There are solid evidence for close relations between the two prob-
+lems. Firstly their bootstrap bounds have similar patterns, as shown in Figs. 1 and
+2. Moreover, for the bootstrap bounds saturated by the generalized free field theories,
+the O(N) vector bootstrap equations degenerate to similar forms in 1D and higher
+dimensions.
+The functional basis dual to higher dimensional generalized free field
+spectrum has been constructed in [28] and their relation to dispersion relation has
+been studied in [29], see also [63,27]. However, a crucial question is how to organize
+the functional basis in order to satisfy the positive conditions. The method developed
+in this work can be useful to construct analytical functionals with suitable positive
+properties in higher dimensions. We leave this problem for future work [49].
+• A more challenging problem along this direction is to construct the analytical func-
+tionals for the O(N) vector bootstrap bounds with large but finite N. This was one
+of the motivations for the author to start this work. In this case we need to go back
+to the whole O(N) vector crossing equations (2.7,2.8) and take the 1/N terms into
+account. These 1/N terms and the crossing equation (2.7) will necessarily introduce
+new ingredients responsible for the 1/N interactions in the underlying theories. The
+related analytical functionals could provide a new nonperturbative frame to study
+CFTs with large N limits, including the 3D critical O(N) vector models and the
+conformal gauge theories in general dimensions.
+• The series of analytical functionals {α′
+N} constructed in this work are sensitive to the
+large ∆ spectrum. Associated with total positivity, they can be employed to detect
+the non-unitarity in the large ∆ region, which relates to the high energy dynamics
+in AdS. We hope more systematical studies of the large N analytical functionals can
+provide solid conclusions for some widely interested questions on the large ∆ spectrum
+of large N unitary CFTs.
+Acknowledgements
+The author would like to thank Nima Arkani-Hamed, Greg Blekherman, Miguel Paulos
+and David Poland for discussions. The author is grateful to David Poland for the valuable
+support. The author thanks the organizers of the conferences “Bootstrapping Nature: Non-
+perturbative Approaches to Critical Phenomena” at Galileo Galilei Institute, “Positivity”
+at Princeton Center for Theoretical Science and Simons Collaboration on the Nonpertur-
+bative Bootstrap Annual Meeting for creating stimulating environments. This research was
+41
+
+supported by Shing-Tung Yau Center and Physics Department at Southeast University, Si-
+mons Foundation grant 488651 (Simons Collaboration on the Nonperturbative Bootstrap)
+and DOE grant DE-SC0017660. The bootstrap computations were carried out on the Yale
+Grace computing cluster, supported by the facilities and staff of the Yale University Faculty
+of Sciences High Performance Computing Center.
+A. Examples of the totally positive functions
+In this appendix we show some classical examples of the totally positive functions.
+Some of the results in this part have been applied in our study of the total positivity of the
+Gauss hypergeometric functions 2F1(∆, ∆, 2∆, z) and SL(2, R) conformal block functions
+G∆(z).
+A.1. Example 1: f(∆, x) = x∆
+The determinant formula (3.1) of the function f(∆, x) = x∆ is given by
+||f(∆, x)||m ≡ f
+�
+∆1,
+...
+∆m
+x1,
+...
+xm
+�
+= det
+�
+���
+x∆1
+1
+...
+x∆m
+1
+...
+...
+x∆1
+m
+...
+x∆m
+m
+�
+��� .
+(A.1)
+Taking ∆i = i − 1, above determinant goes back to the Vandermonde determinant, which
+is given by
+||f(∆, x)||m =
+�
+i>j
+(xi − xj)
+(A.2)
+and is positive for the ordered variables 0 < x1 < · · · < xm.
+Then to prove the total
+positivity of the function f(∆, x), one only needs to show that its determinant can never
+be zero, which can be done by induction [64,41].
+The statement ||f(∆, x)|| ̸= 0 is equivalent to the claim that for a given set of ci ∈ R,
+the equation
+hm(x) =
+m
+�
+i=1
+cix∆i
+(A.3)
+cannot have m solutions in the region x > 0. For n = 1, h1(x) = c1x∆1 and there is no
+positive solution for h. Assume above statement is true for hi(x) with i < n. If hn(x) has
+42
+
+n positive solutions, then according to Rolle’s theorem, the following function
+(x−∆1hn(x))′ =
+n
+�
+i=2
+(∆i − ∆1)ci x∆i−∆1 ∼ hn−1(x)
+(A.4)
+can have n−1 positive zeros, which is inconsistency with our previous induction assumption
+that hn−1(x) cannot have n−1 positive solutions. Therefore the function hn(x) should have
+positive solutions less than n. This completes the proof that the determinant ||f(∆, x)||m
+can never be zero.
+From the total positivity of the function x∆, one can show a family of totally positive
+functions. For instances, the function exy = (ex)y is also totally positive.
+A.2. Example 2: f(x, y) =
+1
+x+y
+The determinant formula (3.2) for the function f(x, y) is
+||f(∆, x)||m ≡ f
+�
+x1,
+...
+xm
+y1,
+...
+ym
+�
+= det
+�
+���
+1
+x1+y1
+...
+1
+x1+ym
+...
+...
+1
+xm+y1
+...
+1
+xm+ym
+�
+��� .
+(A.5)
+Above determinant can be solved in a compact form, i.e., the Cauchy formula
+||f(∆, x)||m =
+�
+i>k
+(xi − xk) �
+i>k
+(yi − yk)
+m�
+i,k=1
+(xi + yk)
+,
+(A.6)
+which is positive for the ordered variables x1 < · · · < xm, y1 < · · · < ym.
+The total positivity of f(x, y) can be alternatively proved using the basic composition
+formula (3.3). The function can be rewritten as
+1
+x + y =
+� ∞
+0
+e−(x+y)tdt =
+� 1
+0
+uxuyd(log(u)).
+(A.7)
+Due to the basic composition formula, the total positivity of above integral follows the
+total positivity of the function ux.
+43
+
+References
+[1] S. Ferrara, A. F. Grillo, and R. Gatto, “Tensor representations of conformal algebra
+and conformally covariant operator product expansion,” Annals Phys. 76 (1973)
+161–188.
+[2] A. M. Polyakov, “Nonhamiltonian approach to conformal quantum field theory,” Zh.
+Eksp. Teor. Fiz. 66 (1974) 23–42.
+[3] R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, “Bounding scalar operator
+dimensions in 4D CFT,” JHEP 12 (2008) 031, arXiv:0807.0004 [hep-th].
+[4] D. Poland, S. Rychkov, and A. Vichi, “The Conformal Bootstrap: Theory, Numerical
+Techniques, and Applications,” Rev. Mod. Phys. 91 (2019) 015002,
+arXiv:1805.04405 [hep-th].
+[5] D. Poland and D. Simmons-Duffin, “Snowmass White Paper: The Numerical
+Conformal Bootstrap,” in 2022 Snowmass Summer Study. 3, 2022.
+arXiv:2203.08117 [hep-th].
+[6] J. M. Maldacena, “The Large N limit of superconformal field theories and
+supergravity,” Adv. Theor. Math. Phys. 2 (1998) 231–252, arXiv:hep-th/9711200.
+[7] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2 (1998)
+253–291, arXiv:hep-th/9802150.
+[8] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from
+noncritical string theory,” Phys. Lett. B 428 (1998) 105–114, arXiv:hep-th/9802109.
+[9] A. L. Fitzpatrick, J. Kaplan, D. Poland, and D. Simmons-Duffin, “The Analytic
+Bootstrap and AdS Superhorizon Locality,” JHEP 12 (2013) 004, arXiv:1212.3616
+[hep-th].
+[10] Z. Komargodski and A. Zhiboedov, “Convexity and Liberation at Large Spin,” JHEP
+11 (2013) 140, arXiv:1212.4103 [hep-th].
+[11] S. Pal, J. Qiao, and S. Rychkov, “Twist accumulation in conformal field theory. A
+rigorous approach to the lightcone bootstrap,” arXiv:2212.04893 [hep-th].
+[12] R. Gopakumar, A. Kaviraj, K. Sen, and A. Sinha, “Conformal Bootstrap in Mellin
+Space,” Phys. Rev. Lett. 118 no. 8, (2017) 081601, arXiv:1609.00572 [hep-th].
+44
+
+[13] L. F. Alday, “Large Spin Perturbation Theory for Conformal Field Theories,” Phys.
+Rev. Lett. 119 no. 11, (2017) 111601, arXiv:1611.01500 [hep-th].
+[14] J. Penedones, J. A. Silva, and A. Zhiboedov, “Nonperturbative Mellin Amplitudes:
+Existence, Properties, Applications,” JHEP 08 (2020) 031, arXiv:1912.11100
+[hep-th].
+[15] I. Heemskerk, J. Penedones, J. Polchinski, and J. Sully, “Holography from Conformal
+Field Theory,” JHEP 10 (2009) 079, arXiv:0907.0151 [hep-th].
+[16] A. L. Fitzpatrick and J. Kaplan, “Unitarity and the Holographic S-Matrix,” JHEP
+10 (2012) 032, arXiv:1112.4845 [hep-th].
+[17] D. Karateev, P. Kravchuk, and D. Simmons-Duffin, “Harmonic Analysis and Mean
+Field Theory,” JHEP 10 (2019) 217, arXiv:1809.05111 [hep-th].
+[18] Z. Li and D. Poland, “Searching for gauge theories with the conformal bootstrap,”
+JHEP 03 (2021) 172, arXiv:2005.01721 [hep-th].
+[19] Z. Li, “Symmetries of conformal correlation functions,” Phys. Rev. D 105 no. 8,
+(2022) 085018, arXiv:2006.05119 [hep-th].
+[20] F. Kos, D. Poland, and D. Simmons-Duffin, “Bootstrapping the O(N) vector
+models,” JHEP 06 (2014) 091, arXiv:1307.6856 [hep-th].
+[21] Z. Li, “Bootstrapping conformal QED3 and deconfined quantum critical point,”
+JHEP 11 (2022) 005, arXiv:1812.09281 [hep-th].
+[22] S. El-Showk and M. F. Paulos, “Bootstrapping Conformal Field Theories with the
+Extremal Functional Method,” Phys. Rev. Lett. 111 no. 24, (2013) 241601,
+arXiv:1211.2810 [hep-th].
+[23] D. Mazac, “Analytic bounds and emergence of AdS2 physics from the conformal
+bootstrap,” JHEP 04 (2017) 146, arXiv:1611.10060 [hep-th].
+[24] D. Mazac and M. F. Paulos, “The analytic functional bootstrap. Part I: 1D CFTs
+and 2D S-matrices,” JHEP 02 (2019) 162, arXiv:1803.10233 [hep-th].
+[25] D. Mazac and M. F. Paulos, “The analytic functional bootstrap. Part II. Natural
+bases for the crossing equation,” JHEP 02 (2019) 163, arXiv:1811.10646 [hep-th].
+45
+
+[26] D. Gaiotto, D. Mazac, and M. F. Paulos, “Bootstrapping the 3d Ising twist defect,”
+JHEP 03 (2014) 100, arXiv:1310.5078 [hep-th].
+[27] M. F. Paulos, “Analytic functional bootstrap for CFTs in d > 1,” JHEP 04 (2020)
+093, arXiv:1910.08563 [hep-th].
+[28] D. Maz´aˇc, L. Rastelli, and X. Zhou, “A basis of analytic functionals for CFTs in
+general dimension,” JHEP 08 (2021) 140, arXiv:1910.12855 [hep-th].
+[29] S. Caron-Huot, D. Mazac, L. Rastelli, and D. Simmons-Duffin, “Dispersive CFT Sum
+Rules,” JHEP 05 (2021) 243, arXiv:2008.04931 [hep-th].
+[30] D. Carmi and S. Caron-Huot, “A Conformal Dispersion Relation: Correlations from
+Absorption,” JHEP 09 (2020) 009, arXiv:1910.12123 [hep-th].
+[31] M. Bill´o, M. Caselle, D. Gaiotto, F. Gliozzi, M. Meineri, and R. Pellegrini, “Line
+defects in the 3d Ising model,” JHEP 07 (2013) 055, arXiv:1304.4110 [hep-th].
+[32] A. Cavagli`a, N. Gromov, J. Julius, and M. Preti, “Integrability and conformal
+bootstrap: One dimensional defect conformal field theory,” Phys. Rev. D 105 no. 2,
+(2022) L021902, arXiv:2107.08510 [hep-th].
+[33] A. Cavagli`a, N. Gromov, J. Julius, and M. Preti, “Bootstrability in defect CFT:
+integrated correlators and sharper bounds,” JHEP 05 (2022) 164, arXiv:2203.09556
+[hep-th].
+[34] M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, “The S-matrix
+bootstrap. Part I: QFT in AdS,” JHEP 11 (2017) 133, arXiv:1607.06109 [hep-th].
+[35] D. Garc´ıa-Sep´ulveda, A. Guevara, J. Kulp, and J. Wu, “Notes on resonances and
+unitarity from celestial amplitudes,” JHEP 09 (2022) 245, arXiv:2205.14633
+[hep-th].
+[36] H. Jiang, “Celestial Mellin amplitude,” JHEP 10 (2022) 042, arXiv:2208.01576
+[hep-th].
+[37] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov, and
+J. Trnka, Grassmannian Geometry of Scattering Amplitudes. Cambridge University
+Press, 4, 2016. arXiv:1212.5605 [hep-th].
+46
+
+[38] N. Arkani-Hamed and J. Trnka, “The Amplituhedron,” JHEP 10 (2014) 030,
+arXiv:1312.2007 [hep-th].
+[39] N. Arkani-Hamed, T.-C. Huang, and Y.-T. Huang, “The EFT-Hedron,” JHEP 05
+(2021) 259, arXiv:2012.15849 [hep-th].
+[40] E. Herrmann and J. Trnka, “Chapter 7: Positive geometry of scattering amplitudes,”
+J. Phys. A 55 no. 44, (2022) 443008, arXiv:2203.13018 [hep-th].
+[41] N. Arkani-Hamed, Y.-T. Huang, and S.-H. Shao, “On the Positive Geometry of
+Conformal Field Theory,” JHEP 06 (2019) 124, arXiv:1812.07739 [hep-th].
+[42] K. Sen, A. Sinha, and A. Zahed, “Positive geometry in the diagonal limit of the
+conformal bootstrap,” JHEP 11 (2019) 059, arXiv:1906.07202 [hep-th].
+[43] Y.-T. Huang, W. Li, and G.-L. Lin, “The geometry of optimal functionals,”
+arXiv:1912.01273 [hep-th].
+[44] M. F. Paulos and B. Zan, “A functional approach to the numerical conformal
+bootstrap,” JHEP 09 (2020) 006, arXiv:1904.03193 [hep-th].
+[45] K. Ghosh, A. Kaviraj, and M. F. Paulos, “Charging up the functional bootstrap,”
+JHEP 10 (2021) 116, arXiv:2107.00041 [hep-th].
+[46] A. Gimenez-Grau, E. Lauria, P. Liendo, and P. van Vliet, “Bootstrapping line defects
+with O(2) global symmetry,” JHEP 11 (2022) 018, arXiv:2208.11715 [hep-th].
+[47] F. A. Dolan and H. Osborn, “Conformal Partial Waves: Further Mathematical
+Results,” arXiv:1108.6194 [hep-th].
+[48] R. Rattazzi, S. Rychkov, and A. Vichi, “Bounds in 4D Conformal Field Theories
+with Global Symmetry,” J. Phys. A 44 (2011) 035402, arXiv:1009.5985 [hep-th].
+[49] Z. Li, “Large N analytical functional bootstrap II,” Work in progress.
+[50] F. A. Dolan and H. Osborn, “Conformal partial waves and the operator product
+expansion,” Nucl. Phys. B 678 (2004) 491–507, arXiv:hep-th/0309180.
+[51] A. Erd´elyi, “S. karlin, total positivity, vol. i (stanford university press; london:
+Oxford university press, 1968), xi 576 pp., 166s. 6d.” Proceedings of the Edinburgh
+Mathematical Society 17 no. 1, (1970) 110–110.
+47
+
+[52] M. Hogervorst and S. Rychkov, “Radial Coordinates for Conformal Blocks,” Phys.
+Rev. D 87 (2013) 106004, arXiv:1303.1111 [hep-th].
+[53] A. Bissi, A. Sinha, and X. Zhou, “Selected topics in analytic conformal bootstrap: A
+guided journey,” Phys. Rept. 991 (2022) 1–89, arXiv:2202.08475 [hep-th].
+[54] J. Qiao and S. Rychkov, “Cut-touching linear functionals in the conformal
+bootstrap,” JHEP 06 (2017) 076, arXiv:1705.01357 [hep-th].
+[55] D. Simmons-Duffin, “A Semidefinite Program Solver for the Conformal Bootstrap,”
+JHEP 06 (2015) 174, arXiv:1502.02033 [hep-th].
+[56] W. Landry and D. Simmons-Duffin, “Scaling the semidefinite program solver SDPB,”
+arXiv:1909.09745 [hep-th].
+[57] S. Caron-Huot, D. Mazac, L. Rastelli, and D. Simmons-Duffin, “AdS bulk locality
+from sharp CFT bounds,” JHEP 11 (2021) 164, arXiv:2106.10274 [hep-th].
+[58] L. C´ordova, Y. He, and M. F. Paulos, “From conformal correlators to analytic
+S-matrices: CFT1/QFT2,” JHEP 08 (2022) 186, arXiv:2203.10840 [hep-th].
+[59] W. Knop and D. Mazac, “Dispersive sum rules in AdS2,” JHEP 10 (2022) 038,
+arXiv:2203.11170 [hep-th].
+[60] J. Penedones, “Writing CFT correlation functions as AdS scattering amplitudes,”
+JHEP 03 (2011) 025, arXiv:1011.1485 [hep-th].
+[61] A. L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju, and B. C. van Rees, “A Natural
+Language for AdS/CFT Correlators,” JHEP 11 (2011) 095, arXiv:1107.1499
+[hep-th].
+[62] M. F. Paulos, “Towards Feynman rules for Mellin amplitudes,” JHEP 10 (2011) 074,
+arXiv:1107.1504 [hep-th].
+[63] A. Bissi, P. Dey, and T. Hansen, “Dispersion Relation for CFT Four-Point
+Functions,” JHEP 04 (2020) 092, arXiv:1910.04661 [hep-th].
+[64] F. R. Gantmacher and M. Krein, “Oscillation matrices and kernels and small
+vibrations of mechanical systems,” 1961.
+48
+

UdE0T4oBgHgl3EQflgEz/content/tmp_files/2301.02486v1.pdf.txt

@@ -0,0 +1,1544 @@
+DRAFT VERSION JANUARY 9, 2023
+Typeset using LATEX twocolumn style in AASTeX61
+LONGTERM STABILITY OF PLANETARY SYSTEMS FORMED FROM A TRANSITIONAL DISK
+RORY BOWENS,1, 2, 3, ∗ ANDREW SHANNON,4, 1, 2 REBEKAH DAWSON,1, 2 AND JIAYIN DONG1, 5, 6, †
+1Department of Astronomy & Astrophysics, The Pennsylvania State University, State College, PA, USA
+2Center for Exoplanets and Habitable Worlds, The Pennsylvania State University, State College, PA, USA
+3Department of Astronomy, The University of Michigan, Ann Arbor, MI, USA
+4LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon, France
+5Center for Exoplanets and Habitable Worlds, The Pennsylvania State University, State College, PA, USA
+6Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA
+ABSTRACT
+Transitional disks are protoplanetary disks with large and deep central holes in the gas, possibly carved by young planets.
+Dong, R., & Dawson, R. 2016, ApJ, 825, 7 simulated systems with multiple giant planets that were capable of carving and
+maintaining such gaps during the disk stage. Here we continue their simulations by evolving the systems for 10 Gyr after disk
+dissipation and compare the resulting system architecture to observed giant planet properties, such as their orbital eccentricities
+and resonances. We find that the simulated systems contain a disproportionately large number of circular orbits compared to
+observed giant exoplanets. Large eccentricities are generated in simulated systems that go unstable, but too few of our systems
+go unstable, likely due to our demand that they remain stable during the gas disk stage to maintain cavities. We also explore
+whether transitional disk inspired initial conditions can account for the observed younger ages of 2:1 resonant systems orbiting
+mature host stars. Many simulated planet pairs lock into a 2:1 resonance during the gas disk stage, but those that are disrupted tend
+to be disrupted early, within the first 10 Myr. Our results suggest that systems of giant planets capable of carving and maintaining
+transitional disks are not the direct predecessors of observed giant planets, either because the transitional disk cavities have a
+different origin or another process is involved, such as convergent migration that pack planets close together at the end of the
+transitional disk stage.
+Keywords: Transitional Disks — Protoplanets — Resonance
+∗ [email protected]
+† Flatiron Research Fellow
+arXiv:2301.02486v1  [astro-ph.EP]  6 Jan 2023
+
+2
+BOWENS ET AL.
+1. INTRODUCTION
+Planets form in disks of gas and dust that surround young
+stars, known as protoplanetary disks. Protoplanetary disks
+are sometimes observed with a deep and wide gap in the
+dust distribution (e.g., Strom et al. 1989). These disks are
+called transitional disks (Piétu et al. 2005). About 10% of
+protoplanetary disks are transitional disks (Luhman et al.
+2010), but it remains unclear whether this fraction indicates
+that ∼ 10% of protoplanetary disks spend most of their lives
+as transitional disks or that most protoplanetary disks spend
+∼ 10% of their lives as transitional disks (Owen 2016). Nu-
+merous theories have been proposed to explain the origins
+of the gaps (e.g., Dullemond & Dominik 2005; Chiang &
+Murray-Clay 2007; Krauss et al. 2007; Suzuki & Inutsuka
+2009; Vorobyov et al. 2015); the leading theories are pho-
+toevaporation (Clarke et al. 2001; Alexander et al. 2006a,b;
+Owen et al. 2011, 2012) and planetary sculpting (Calvet et al.
+2005; Dodson-Robinson & Salyk 2011; Zhu et al. 2011,
+2012; Dong et al. 2015). Photoevaporative winds can de-
+plete the inner disk when the photoevaporative mass loss rate
+exceeds the accretion rate. Although early photoevaporation
+models (e.g., Owen et al. 2011, 2012) produced photoevapo-
+rative mass loss rates too low to be consistent with the high
+observed accretion rates in transitional disks (Ercolano &
+Pascucci 2017a), recent models (e.g., Ercolano et al. 2021;
+Picogna et al. 2021) that more accurately model the tempera-
+ture structure of the disk can meet or exceed half of observed
+accretion rates. Transitional disks with higher accretion rates
+may be the consequence of stronger photoevaporative winds
+in carbon-depleted disks (Ercolano et al. 2018; Wölfer et al.
+2019).
+Alternatively, they may be the result of magneti-
+cally driven supersonic accretion flows (Wang & Goodman
+2017; but see Ercolano et al. 2018 for observational argu-
+ments against this hypothesis). More in-depth reviews of the
+observational properties of transitional disks and theory of
+their origins can be found in Espaillat et al. (2014); Owen
+(2016); Ercolano & Pascucci (2017b); van der Marel (2017).
+In this work, we focus on the question of whether gaps
+in transitional disks form by planetary sculpting (Papaloizou
+& Lin 1984; Marsh & Mahoney 1993; Paardekooper &
+Mellema 2004). Planetary sculpting is a process in which
+planets or brown dwarfs formed in the protoplanetary disk
+clear a gap (Dodson-Robinson & Salyk 2011; Zhu et al. 2011,
+Rosenthal et al. 2020; see Paardekooper et al. 2022 for a re-
+cent review). This theory is supported by the discovery of
+protoplanets forming in the gap of the transitional disk PDS
+70 (Keppler et al. 2018; Haffert et al. 2019; Isella et al. 2019;
+Muley et al. 2019), possibly LkCa 15 (Kraus & Ireland 2012;
+Sallum et al. 2015; but see also Currie et al. 2019), and a
+few other candidates (HD 100546, HD 142527, HD 169142;
+Quanz et al. 2013; Biller et al. 2012; Reggiani et al. 2014).
+Further supporting this hypothesis, sub-structures in proto-
+planetary disks have also been attributed to planetary sculpt-
+ing (e.g., Huang et al. 2018; Long et al. 2018; Zhang et al.
+2018; Choksi & Chiang 2021). Although the sculpting plan-
+ets creating the deep and wide gaps in transitional disks are
+typically assumed to be Jovian – and we will make that as-
+sumption here – Fung & Chiang (2017) showed that com-
+pact configurations of super-Earths can clear inner cavities in
+disks with very low viscosity. See Ginzburg & Sari (2018)
+and Garrido-Deutelmoser et al. (2022) for more on the prop-
+erties of such gaps.
+To account for the gaps seen in transitional disks, planetary
+systems must remain stable over the observed disk timescale
+(Tamayo et al. 2015). To assess which planetary system ar-
+chitectures are capable of producing transitional disks, Dong
+& Dawson (2016) (hereafter DD16) performed N-body sim-
+ulations of such planetary systems, using three to six giant
+planets spaced from 3 to 30 AU. The depletion of the gaps the
+planets produced was based on different assumptions for the
+disk viscosity and scale height, and these depletions dictated
+the spacings of the planets necessary to produce a continu-
+ous gap. DD16 included an eccentricity damping force from
+the gas in the gap. They found that a subset of planetary sys-
+tem configurations remained stable over a typical one million
+year disk lifetime. Thus they concluded it was plausible that
+the subset of planetary systems that contain Jovians planets
+in the 3 to 30 AU range all appear as transitional disks dur-
+ing the protoplanetary stage. Under this hypothesis, the phe-
+nomenon of transitional disks is not pervasive, affecting all
+protoplanetary disks for ∼10% of their lifetime, but instead
+is restricted to the ∼10% of disks that happen to harbor giant
+planets.
+Because DD16 conducted simulations only during the gas
+disk stage, we cannot directly compare those systems to ma-
+ture planetary systems observed around main sequence field
+stars. After the protoplanetary disk dissipates in ≲ 107 years
+(Haisch et al. 2001; Pfalzner et al. 2014), the systems may
+go unstable, as systems of massive planets in such compact
+configurations often do (Chambers et al. 1996a; Smith & Lis-
+sauer 2009; Morrison & Kratter 2016). However, the damp-
+ing during the protoplanetary disk state may allow these sys-
+tems to find long-term stable compact dynamical states (e.g.,
+Melita & Woolfson 1996; Lee & Peale 2002; Dawson et al.
+2016; Morrison et al. 2020). The four tightly packed jovian
+planets around HR 8799 (Marois et al. 2008, 2010) may be
+in such a dynamical state (Fabrycky & Murray-Clay 2010;
+Go´zdziewski & Migaszewski 2014; Wang et al. 2018), and
+thus perhaps an example of the post-gas evolution of such
+systems.
+Here, we seek to understand the longer-term dynamical
+evolution of the giant planet planetary systems capable of
+sculpting deep and wide gaps in transitional disks. Simbu-
+lan et al. (2017) performed a case study of HL Tau along
+
+POST TRANSITIONAL DISK STABILITY
+3
+these lines, finding ejection of one or more planets to be the
+most common outcome. How this result can be generalized
+remains an open question. Previous works simulating the
+longterm evolution of planetary systems compared simulated
+systems’ eccentricity and semimajor axis distributions (e.g.,
+Chatterjee et al. 2008; Juri´c & Tremaine 2008; Malmberg &
+Davies 2009; Petrovich et al. 2014; Carrera et al. 2019) to
+those of observed planets (e.g., Mayor et al. 2011; Dawson
+& Johnson 2012; Winn & Fabrycky 2015; Xie et al. 2016).
+However, those works typically began with ad hoc tightly
+packed initial configurations of orbits to quickly induce insta-
+bilities. Here, we use initial conditions physically motivated
+from DD16, as our interest is in how well those hypothe-
+sized transitional-disk sculpting systems match observed sys-
+tems. We also explore the prevalence and evolution of mean
+motion resonances in these simulated systems, which were
+common during the stage simulated by DD16. Motivated
+by Koriski & Zucker (2011)’s finding that 2:1 mean motion
+resonance (MMR) systems are younger on average, we will
+assess whether the systems that begin in MMR will break
+within observable timescales.
+We describe our simulations in Section 2. We identify the
+presence and behavior of orbital resonances Section 3. We
+assess longterm stability and which characteristics affect it
+in Section 4. We investigate the planets’ eccentricities and
+compare to those of observed exoplanets in Section 5. We
+present our conclusions in Section 6.
+2. SIMULATIONS
+We simulate the long-term evolution of the multi-planet
+systems that are capable of opening the gaps observed in tran-
+sitional disks. Most of our simulations start where DD16’s
+left off, at the end of transitional disk stage, with gas damp-
+ing forces turned off, which is an approximation that the gas
+disk is immediately removed. Their final planet masses, po-
+sitions, and velocities from the gas stage are our initial post-
+gas conditions. However, since we do run some additional
+gas stage simulations, we describe the gas stage simulations
+in detail in Appendix A.
+For our post-gas simulations use the mercury6 Bulirsch-
+Stoer integrator with a 1000 AU ejection distance, a solar
+mass and solar radius central body, an accuracy parameter
+of 10−12, and medium precision outputs every million years.
+Our approximation of the instantaneous removal of the gas
+disk does not induce problems for several reasons: 1) as we
+will show the majority of systems remain stable after the re-
+moval, 2) the gas surface density was low to begin with due
+to depletion in the gas, and 3) simulations show that instanta-
+neously removing a depleted gas disk does not significantly
+alter the resonant dynamics (Morrison et al. 2020).
+The configurations are summarized in Table 1. Configura-
+tion names are defined as planet number followed by planet
+mass in Jupiters (with letters used for configurations with the
+same planet numbers and masses). Being continuations of
+DD16, the systems had between 3 and 6 equal mass planets,
+where the number of planets required was dictated by the gap
+widths. We did not include configurations that DD16 found
+to be unsuitable for creating proto-planetary disk cavities be-
+cause the amount of gas expected to be present in the cavity
+would drive the planets too far apart via resonant repulsion
+to create overlapping gaps (their 4-10acd, 5-5ac) or would be
+insufficient to stabilize the configuration during the gas disk
+stage (their 6-2). For each configuration, DD16 used various
+assumed gas surface densities since ALMA gas observations
+and chemical disk modeling are unable to constrain gas de-
+pletion in the inner few AU of a disk. For each configuration,
+we use the simulations with the gas surface density consistent
+with the expected depletion in the cavity (reported in Dong &
+Dawson 2016 based on Fung et al. 2014). For their gas stage
+simulations, DD16 ran 10 random realizations for each con-
+figuration. We supplement with an additional 10 gas stage
+realizations for each configuration, that we then continue in
+the post-gas stage. See Appendix A for more details about
+the gas stage simulations. In summary, we ran eighteen re-
+alizations of twenty variations of the disk initial conditions
+for 10 Gyr, for a total of 360 simulations. We found that
+14 simulations had gone unstable during the gas disk stage
+(i.e., lost a planet via ejections or collisions); those simula-
+tions are considered to have instability events at 0 Gyr in all
+future analysis. We also found that two simulations ended
+the 10 Gyr simulations with zero planets due to central body
+collisions.
+To better compare giant planets discovered by the radial
+velocity method – which are commonly observed at ∼ 1–3
+AU – we ran an additional set of simulations that added an
+interior planet. We refer to the original set as 3–30 AU sim-
+ulations and additional set as 1–30 AU simulations. The ad-
+ditional planet is placed interior to the others by the average
+Hill spacing of original configuration. First the average Hill
+spacing of the original configuration is determined by aver-
+aging every pair’s separation in all ~20 simulations for the
+configuration, excluding those that went unstable during the
+gas disk stage. Then the position for the innermost planet is
+determined relative to the initial location of the 3 AU planet
+according to the average Hill spacing. The newly created
+1 AU planet is appended to the corresponding original 3–30
+AU simulation during the gas stage, in order to make the final
+results more comparable. Similar to the 3–30 AU gas stage
+simulations (Appendix A), these systems are simulated for
+1 Myr with gas damping present using the hybrid integrator
+(timestep of 3 days, accuracy parameter of 10−12). However,
+they are then simulated for only 1 Gyr with gas damping shut
+off using the Bulirsch-Stoer integrator. This reduced simu-
+
+4
+BOWENS ET AL.
+Table 1. Summary of 3 – 30 AU Systems
+Name
+Σ30
+∆0
+GDI
+2:1 MMR %
+10 Gyr
+g cm−2
+(RH)
+Sims.
+(Near %)
+Stability %
+3-5
+0.001
+6.2
+0
+15, (20)
+100
+3-10a
+0.0001
+5.2
+0
+40, (40)
+100
+3-10b
+0.1
+4.8
+0
+75, (25)
+100
+3-10c
+0.0001
+4.4
+0
+40, (25)
+95
+3-10d
+0.1
+4.4
+0
+75, (25)
+100
+3-10e
+0.0001
+4.0
+1
+0, (25)
+80
+4-2
+0.001
+6.1
+0
+85, (15)
+95
+4-5a
+0.001
+4.7
+0
+85, (15)
+95
+4-5b
+0.1
+4.6
+0
+20, (80)
+100
+4-5c
+0.1
+4.6
+0
+20, (75)
+100
+4-5d
+0.001
+4.0
+1
+45, (45)
+20
+4-10b
+0.01
+3.9
+2
+40, (35)
+40
+5-1
+0.1
+6.2
+0
+60, (40)
+90
+5-2a
+0.1
+4.6
+0
+95, (5)
+30
+5-2b
+0.1
+4.2
+7
+65, (10)
+30
+5-5b
+0.01
+3.6
+1
+70, (25)
+40
+6-0.5
+1
+5.6
+0
+50, (40)
+5
+6-1
+1
+4.4
+2
+45, (25)
+5
+NOTE—The parameter Σ30 indicates normalization the gas surface
+density inside the depleted gap for the gas stage simulation
+(Appendix A), where Σgas = Σ30
+�
+a
+30AU
+�−3/2, and a normalization of
+Σ30 =10 g cm−2 corresponds to the minimum mass solar nebula.
+GDI Sims. gives the number of simulations (out of the twenty in a
+set) that experienced instabilities during the gas phase integration.
+These simulations were still run for the following 10 Gyr
+integration with their surviving planets.
+lation time is necessary for the efficient completion of the
+simulations.
+3. RESONANT BEHAVIOR
+Here we identify systems containing planets in two-body
+and three-body resonances. We assess resonance only for ad-
+jacent planet pairs (or triplets in the case of three body reso-
+nances) and by looking for librating resonant angles. For ex-
+ample, for the 2:1 resonance, the resonant angles are φin and
+φout:
+φin = 2λin −λout −ϖin
+(1)
+and
+φout = 2λin −λout −ϖout
+(2)
+where λin and λout are the mean longitude of the inner and
+outer planet, respectively, and ϖin and ϖout are the longitude
+of pericenter of the inner and outer planets, respectively. To
+ensure the resonant angles are well sampled, we run simu-
+lations with more frequent output (every 10 years) for 105
+years, using the same starting point as the standard simula-
+tions (i.e., immediately after gas disk stage). The 3–30 AU
+systems are run with a Bulirsch-Stoer integrator (10−12 ac-
+curacy parameter) while the 1–30 AU are run with a hybrid
+integrator (3 day timestep, 10−12 accuracy parameter).
+The libration centers for the massive planets in our simu-
+lations are 0 or 180◦ (Dong & Dawson 2016). We consider
+a planet pair resonant with libration about 0 if the resonant
+angle does not come within ±0.5◦ of 180◦ during the simu-
+lation (Fig. 1). Likewise, we consider a planet pair resonant
+with libration about 180◦ if the resonant angle does not come
+within ±0.5◦ of 0 during the simulation.
+Furthermore, we label systems with at least one planet pair
+whose resonant angle remained within ±170◦ of 0 or 180 for
+97.5% of the time as near resonance. These angles linger
+near a specific value as they circulate (Fig. 2). If the resonant
+angles circulate without lingering for all the planet pairs in a
+system, we classify the system as non-resonant. We inspect
+a subset of 54 of the resonant angle plots by eye to ensure
+the classification was reliable and that 0.5◦ worked well as a
+cut-off with our sampling frequency. We also considered a
+stricter resonant angle range, requiring it to stay ±50◦ of 0◦
+or 180 ◦ during the simulation. This did reduce the number
+of resonant systems by 12% but had no significant impacts
+on other results.
+Our resonance identification approach can fail for systems
+that go unstable quickly. To solve this problem, we trun-
+cate the resonance data at 75% of the instability timescale,
+defined as the timescale when a collision or ejection occurs
+(e.g., for a system that went unstable after four hundred thou-
+sand years, we use the resonant angle during the first three
+hundred thousand years). We label all planets that collided or
+were ejected before 60,000 years as having undergone "rapid
+instability." Less then ten systems had rapid instability so this
+classification is only a minor part of the results. Some sys-
+tems that went unstable during the gas disk stage only had
+a single planet remaining at the start of the gas-free stage.
+These systems are likewise given a unique label apart from
+resonance or no resonance.
+We find that it is more likely for the inner planets to be in
+2:1 MMR than the outer planets. Approximately 50% of the
+innermost planet pairs are in resonance in all planet systems.
+For outermost pairs, 14% of those in the five planet systems
+are in 2:1 MMR, 3% of the outer pairs in the six planet sys-
+tems, and none of those in the three or four planet systems.
+The higher fraction of true resonance among inner pairs vs.
+outer may be because the higher surface gas surface density
+and shorter orbital timescales in the inner disk facilitate res-
+onance capture during the gas disk stage.
+In our four, five, and six planet systems, outer pairs are near
+2:1 MMR (“lingering” systems with at least one planet pair
+whose resonant angle remained within ±170◦ of 0 or 180 for
+
+POST TRANSITIONAL DISK STABILITY
+5
+-180
+-90
+0
+90
+180
+2λ1 - λ2 - ϖ2 ( ° )
+-180
+-90
+0
+90
+180
+2λ1 - λ2 - ϖ1 ( ° )
+-180
+-90
+0
+90
+180
+2λ2 - λ3 - ϖ3 ( ° )
+0
+1
+2
+3
+4
+Time (kyr)
+-180
+-90
+0
+90
+180
+2λ2 - λ3 - ϖ2 ( ° )
+Figure 1. The resonance angle for planet pairs in a simulation from
+Config 3-10b, starting in the post-gas stage. Planets 2 and 3 (the
+inner two of the three; panel 3) happened to begin the gas disk stage
+in resonance and maintained this configuration in the post-gas stage.
+In other cases, our simulated planets get captured into resonance
+during the gas disk stage.
+97.5% of the time), in approximately 70% pairs while inner
+pairs experience closer to 30% with near 2:1 MMR. Interme-
+diate planet pairs show intermediate rates of near 2:1 MMR.
+However, in the three planet systems, 30% of the inner pairs
+are near resonance but none of the outer pairs are near reso-
+nance.
+We also examine the period ratios of 2:1 MMR systems
+(Fig. 3). Consistent with DD16’s findings, planets with pe-
+riod ratios far outside of their nominal resonance can have
+librating resonant angles. There are some systems that are
+librating in the 2:1 that have a period ratio greater than 2.5,
+even up to 6 (i.e., their 2:1 resonant angle is librating). Li-
+bration of the 2:1 resonant angle at such large period ratios
+is possible when the longitude of periapse precesses quickly
+and is caused in our simulations by the eccentricity damping
+during the gas disk stage. This phenomenon, which is more
+generally established by a dissipative change to the eccentric-
+-180
+-90
+0
+90
+180
+2λ1 - λ2 - ϖ2 ( ° )
+-180
+-90
+0
+90
+180
+2λ1 - λ2 - ϖ1 ( ° )
+-180
+-90
+0
+90
+180
+2λ2 - λ3 - ϖ3 ( ° )
+0
+1
+2
+3
+4
+Time (kyr)
+-180
+-90
+0
+90
+180
+2λ2 - λ3 - ϖ2 ( ° )
+Figure 2. The resonance angle among planet pairs in a simula-
+tion from Config 3-10a, starting in the post-gas stage. Each res-
+onance angle circulates but we consider planets 2 and 3 (the inner
+two of the three; panel 3) to be “near resonance” because the angle
+lingers within ±90◦ of 0.The pair reached this configuration part
+way through the earlier gas stage.
+ity (which may or may not be accompanied by a change in
+semi-major axis) is known as resonant repulsion (e.g., Lith-
+wick & Wu 2012). We find 47% of systems that lie within
+10% of a period ratio of 2 are in 2:1 MMR.
+In later sections, we will focus on the 2:1 MMR because
+other two body resonant angles rarely librate in our simula-
+tions (e.g., 3:2, 3:1, 4:3). Among the 360 simulations, we
+find four systems contain one or more pairs that librate in the
+3:2 resonance (twenty-five near resonance), one system in the
+3:1 resonance (one-hundred sixty-nine near-resonance), and
+zero systems in the 4:3 resonance. We also examined vari-
+ous three body resonances. First, we define the three body
+librating angle equation:
+φ3b/p,q(1,2,3) = pλ1 −(p+q)λ2 +qλ3
+(3)
+In the above, 1, 2, and 3 refer to the outer, middle, and inner
+planet, respectively. We examined multiple potential reso-
+
+6
+BOWENS ET AL.
+nances and found that only 9:6:4, 15:12:8, 4:2:1, and particu-
+larly 3:2:1 had a significant number of three body resonance
+cases. 22% of systems displayed at least one type of three
+body resonance (1% for three planet systems, 23% for four
+planet systems, 45% for five planet systems, and 35% for six
+planet systems). In all cases, there were less than 5 near res-
+onance simulations.
+0
+2
+4
+6
+8
+Period Ratio
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Fraction of Pairs
+All Pairs
+Pairs in 2:1 MMR
+Figure 3. The ratio of the periods of adjacent planets for all 3–30
+AU simulated systems (blue line) and systems in 2:1 MMR (red-
+dotted/dashed), as defined by libration of the resonant angle. Sys-
+tems in 2:1 MMR do not all have period ratios near 2. Many have
+ratios much larger than 2, even up to 6. Similar trends were ob-
+served for the 1–30 AU simulations.
+4. STABILITY
+In approximately 31% of the simulations, an ejection or
+collision occurs over 10 Gyr. We define a system as stable if
+none of the member planets were ejected or collided during
+either the gas or post-gas stage.
+4.1. Impact of Planet Number and 2:1 MMR
+Our data consists of 360 3–30 AU systems. Each set of 20
+systems is defined by a planet number and initial gas surface
+density profile (Table 1). In total there are 6 three planet sets,
+6 four planet sets, 4 five planet sets, and 2 six planet sets.
+We find that systems with more planets experience more
+collisions and/or ejections. Examining systems that remained
+stable during the gas disk stage, approximately 13% of the
+outermost planets are lost (ejected or collided) during the 10
+Gyr gas-free phase in contrast to 20% of inner planets lost.
+The difference is mostly due to more collisions among in-
+ner planets. In six planet systems, approximately 33% of
+planets were ejected and 15% collided. These values are sig-
+nificantly higher than the averages for the four planet sys-
+tems (9% ejected and 3% collided). We conclude that ad-
+ditional planets reduce a system’s stability but that origin of
+the ejected planets is fairly uniform across the semimajor axis
+range.
+We plot the initial fraction of 3–30 AU systems in 2:1
+MMR against the stability fraction, color coded by planet
+count (Fig. 4). Recall that for a given planet number, differ-
+ent sets have different planet masses and spacings. The initial
+fraction of systems in resonance alone does not appear to im-
+pact the stability but the initial planet count does. Moreover,
+the fraction of systems in resonance is not strongly correlated
+with the initial planet count, so it appears that planet count
+is independently driving the stability. The 3 planet systems
+remain stable ~90% of the time. The four and five planet
+systems display a range of stability from low (~20% of a set
+stable) to high (~95% of a set stable) values. Both sets of six
+planet systems have a low stability fraction (5%).
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+2:1 MMR Fraction
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Stability Fraction
+Three Planets
+Four Planets
+Five Planets
+Six Planets
+Figure 4. The fraction of the 3–30 AU simulations with at least one
+2:1 MMR in a configuration versus the overall stability of the con-
+figuration. Each configuration contains 20 simulations. For read-
+ability, the three pairs of identical sets (3-10b and 3-10d, 4-2 and
+4-5a, 4-5b and 4-5c) have had their 2:1 MMR % (75%, 85%, and
+20%, respectively) split slightly.
+We define the stability timescale as the time until any ejec-
+tion or collision occurs within the system. We find some
+systematic differences in stability timescale for systems with
+2:1 MMR and without (see Figure 5).
+The timescales at
+which systems went unstable differs slightly between the
+three MMR categories (resonance, near resonance, and no
+resonance). However, the overall rate of instabilities by 10
+Gyrs between the three categories was similar: 73% stable,
+68% stable, and 67% stable for resonance, near resonance,
+and no resonance; respectively. 10 systems could not have
+their resonance status determined due to rapid instability and
+thus were not used in the above assessments. From this data,
+we conclude that 2:1 MMR can improve the stability of a sys-
+tem in the short term (Myr timescale) but that this advantage
+is nullified with time. Based on only small differences in the
+
+POST TRANSITIONAL DISK STABILITY
+7
+final stability rates, we conclude the presence of one or more
+2:1 MMR pairs within a system did not significantly impact
+the final number of planets that collided or were ejected of
+10 Gyr.
+The above analysis of how resonances affect system stabil-
+ity looks at a system level: if at least one pair is in resonance,
+we consider it a resonant system. However, some trends may
+only be observable at the individual pair level. In Figure 6,
+we report stability timescales in a fashion identical to Figure
+5 but for adjacent pairs rather than systems. A pair is con-
+sidered stable until one member experiences an instability
+event. The pairs follow a similar trend to the systems, though
+we find more significant evidence (compared to the system-
+level analysis) that near 2:1 MMR pairs are less stable over
+the long term. The higher instability of near systems may be
+the result of close spacing without the stabilizing influence
+of resonance or chaotic behavior at the resonance separatrix.
+We also plot a calculation of resonance pairs based on pe-
+riod ratios rather than angle libration (Figure 7). Identifying
+systems near integer period ratios allows us to more directly
+compare to observational data where the orbital parameters
+are typically not well-constrained enough to determine if the
+resonance angle is librating. We determine if a period ratio is
+resonant according to the criteria in Koriski & Zucker (2011):
+δ = 2|r −rc|
+r +rc
+≤ 0.1
+(4)
+where r is the measured period ratio and rc the resonance
+period ratio (in this case 2). The output timesteps are every
+Myr and once a system leaves resonance we do not consider
+it capable of reentering. About 13% of our pairs begin in res-
+onance post-gas stage according to this criterion, compared
+to 24% using the angle libration criterion. Most of these pairs
+are in five or six planet configurations, with some sets having
+up to 40% of pairs near the 2:1. Although the fraction of pairs
+near the 2:1 period ratio declines over time, most disruptions
+occur in the first 50 Myr. Therefore, this behavior is not a
+good explanation for the trend of younger ages for resonant
+pairs found by Koriski & Zucker (2011), which requires a
+typical disruption timescale ∼1–10 Gy and near 100% initial
+fraction (Dong & Dawson 2016).
+4.2. Impact of Three-Body MMR
+We consider the impact of three-body MMR on the stabil-
+ity of systems. These classifications are once again carried
+out at the system level. The most common type of three-body
+MMR in our systems is 3:2:1, occurring in 11% of systems.
+We found that ~51% systems with at least one of the three-
+body resonances present had a final stability rate, similar to
+the rate of 55% of all systems with four or more planets (our
+three planet systems have a higher stability rate but no three
+body resonances). The other three body MMRs have small
+5
+6
+7
+8
+9
+10
+Time (log10(yrs))
+0.4
+0.6
+0.8
+1.0
+Stable Systems Fraction
+2:1 MMR
+Near 2:1 MMR
+No 2:1 MMR
+Figure 5. A CDF comparing instability times based on whether
+the system contains resonant, near-resonant, or no resonant planets.
+There are 185 resonant systems (of which 135 or 73% were stable),
+111 near resonant systems (of which 75 or 68% were stable), and
+54 no resonant systems (of which 36 or 67% were stable). Non-
+resonant systems start below 100% because some simulations went
+unstable during the gas disk stage and are marked as unstable at 0
+Gyr. Notably, there is little discrepancy between the final stability
+rates, suggesting the presence of 2:1 MMR does not significantly
+impact overall stability.
+5
+6
+7
+8
+9
+10
+Time (log10(yrs))
+0.4
+0.6
+0.8
+1.0
+Stable Pairs Fraction
+2:1 MMR
+Near 2:1 MMR
+No 2:1 MMR
+Figure 6. A CDF comparing instability times for the three cate-
+gories of resonance, now plotting per pair rather than per system.
+There are 262 resonant pairs (of which 67% were stable), 400 near
+resonant pairs (of which 57% were stable), and 411 non- resonant
+pairs (of which 73% were stable).
+
+8
+BOWENS ET AL.
+5
+6
+7
+8
+9
+10
+Time (log10(yrs))
+0.00
+0.05
+0.10
+0.15
+2:1 Period Fraction
+Figure 7. A CDF showing the fraction of pairs with 2:1 MMR
+based on the period ratio. About 13% of pairs begin in resonance
+according to this metric.
+occurrence rates and thus limited statistics, precluding any
+conclusions on their significance, if any.
+4.3. Impact of Mutual Hill radii
+We assess the impact of the initial post-gas spacing in mu-
+tual Hill radii and its relationship to stability of the 10 Gyr
+simulations. We present the starting (i.e., right after the gas
+disk phase) and final (i.e., after 10 Gyr) mutual Hill radii sep-
+arations for adjacent planet pairs in Figure 8. At the start of
+the lifetimes all the planets have roughly equal separations
+of mutual Hill radii. Some sets display average mutual Hill
+radii values that differ from the majority of sets (see the blue
+diamond clump at 3,2 and the purple square clump at 2,1).
+These pairs tend to remain stable at these wider separations.
+By comparing the two plots, we find that the planets fur-
+thest from the star (i.e., planets on the right) experience the
+greatest change in mutual Hill radii separation (especially
+amongst systems with four or more planets). Those pairs
+with Hill radii above 10 (right-side of Figure 8) are exclu-
+sively survivors of instability events (or pairs that began the
+post-gas stage with such wide separations, as mentioned in
+the previous paragraph).
+We find that if a system had an individual pair with a ini-
+tial mutual Hill radii values smaller than 3.5, it had a greater
+likelihood of instability. However, these represented a small
+fraction of the total pairs. We conclude that initial mutual Hill
+radius– within the narrow range encompassed in our starting
+conditions – was not a primary driver of the stability rate for
+a system, although its final value can be used to assess which
+systems experienced instabilities.
+4.4. 1 AU Planets
+The simulations by DD16 only included planets in a range
+from 3 to 30 AU (as based upon the protoplanetary disk ob-
+servations, it is unclear if the deep and wide gaps extend to
+within 3 AU). For simulation set, we run 10 new simulations
+with an additional planet near 1 AU (see Section 2 for de-
+tails). The 1 AU planets still fulfill the transitional disk crite-
+ria presented by DD16: i.e., a planet massive enough to carve
+out a deep gap; packed closely enough with other planets to
+create a continuous gap; and close enough to the star to clear
+the disk at 1 AU. We run the simulations for 1 Myr the same
+gas damping parameters as the corresponding 3–30 AU sys-
+tems. Then we run the simulations for 1 Gyr post-gas. The
+simulation timescale is shorter than for the 3–30 AU systems
+to keep the run time feasible. We run 200 simulations in total,
+each with 4 to 7 planets.
+The increased number of planets reduces the stability of
+the system. Several 1 – 30 AU sets had stability rates similar
+to their 3–30 AU counterparts but a majority of the 1–30 AU
+sets saw substantial decreases in stability compared to their
+3–30 AU counterparts. The reduction in stability is expected
+from past studies. For example, Chambers et al. (1996b)
+found that adding planets to a system led to shorter instability
+timescales. However, the impact was more modest for sys-
+tems with higher multiplicity: for example, they found that a
+four planet system would go unstable significantly faster than
+a three planet system, but a seven planet system would only
+go unstable a little faster than a six planet system. We find
+that the 1–30 AU systems have shorter instability timescales,
+particularly for sets that originally contained 3 or 4 planets.
+The 5 and 6 planet sets were already frequently unstable by 1
+Gyr: thus the additional instability from the extra planet has
+less impact on the final stability fraction.
+We present a CDF for pair-by-pair resonance status based
+on period ratio for the 1 – 30 AU systems (Figure 9; in a
+fashion identical to the 3–30 AU systems in Figure 7). The
+observed trends are mostly identical between the two types
+of systems, barring a slightly lower initial resonance fraction
+for the 1–30 AU systems (approximately 10% compared to
+the prior 13%). The fraction of 2:1 period pairs at 9 Gyrs is
+likewise approximately 2% lower for the 1–30 AU systems.
+Adding a planet at 1 AU is unable to replicate the trend of
+younger ages for resonant pairs found by Koriski & Zucker
+(2011), which requires a typical disruption timescale ∼ 1 −
+−10 Gyr and near 100% initial fraction (Dong & Dawson
+2016). Instead, our systems go unstable usually within ∼
+10−−100 Myr.
+5. ECCENTRICITY
+The eccentricity distributions of our final systems is a relic
+of how the planets evolved with time. In Figure 10, we plot
+the 1 Gyr eccentricity versus semimajor axis distribution for
+
+POST TRANSITIONAL DISK STABILITY
+9
+ 
+6,5
+5,4
+4,3
+3,2
+2,1
+ 
+Planet Pairs
+0
+5
+10
+15
+20
+25
+Pair Separation in Mutual Hill Radii
+Post Gas Age = 0 Gyr
+Three Planets
+Four Planets
+Five Planets
+Six Planets
+Outermost Planets
+ 
+6,5
+5,4
+4,3
+3,2
+2,1
+ 
+Planet Pairs
+0
+5
+10
+15
+20
+25
+Pair Separation in Mutual Hill Radii
+Post Gas Age = 10 Gyr
+Three Planets
+Four Planets
+Five Planets
+Six Planets
+Outermost Planets
+Figure 8. The mutual Hill radii separation of adjacent planet pairs for the 3–30 AU simulations after the gas disk simulations are complete
+(left) and after 10 Gyr (right). X position within a column is slightly randomized for readability. Planet 1 is the farthest from the star. Planets
+are numbered upwards so a system with four planets has planets 1, 2, 3, and 4. After the 10 Gyr integration, many systems have far higher
+hill separations then they initially had, owing to the tendency of instabilities to greatly alter a system. If a planet is ejected, the number
+corresponding to a planet is updated an new comparisons are made. That is, if planet 1 is ejected, planet 2 is renamed to planet 1 and a pair is
+then determined between the new planet 1 and planet 0.
+5
+6
+7
+8
+9
+Time (log10(yrs))
+0.00
+0.05
+0.10
+0.15
+2:1 Period Fraction
+Figure 9. A CDF showing the fraction of pairs with 2:1 MMR
+based on the period ratio for 1–30 AU systems. About 10% of pairs
+begin in resonance according to this metric, a slight decrease from
+the same analysis for 3–30 AU systems (Figure 7). The two types
+of systems otherwise should identical trends.
+the 3–30 AU systems (left) and 1–30 AU systems (center).
+Furthermore, in Figure 11, we plot a CDF for the 1 Gyr ec-
+centricity of several categories. Of particular importance are
+the "unstable" categories for both the 3–30 AU and 1–30 AU
+systems: the unstable category shows the CDF of 1 Gyr ec-
+centricities for surviving planets in systems that experienced
+instability events. Since extending the 3–30 AU systems to
+10 Gyr only only decreased the fraction of circular orbits
+(<0.05 eccentricity) from 80% to 70%, we show the results at
+1 Gyr to compare to the 1–30 AU systems, which were only
+simulated for 1 Gyr (Section 4.4).
+We studied the influence of ejections and collisions on the
+eccentricity of surviving planets. In four and five planet sys-
+tems (which had 80% and 50% stability, respectively), the
+mean eccentricity of all survivors was 0.16 ± 0.25 while the
+eccentricity of survivors from unstable systems was 0.47 ±
+0.23. As anticipated instability events increased the eccen-
+tricity of the survivors. Further trends were found in the type
+of instability events. Those systems which only experienced
+collisions had a final mean eccentricity of 0.33 ± 0.35 while
+those survivors that only experienced ejections had a final
+mean eccentricity of 0.50 ± 0.20. Although both types of
+instability event correlated with higher eccentricity, systems
+that experienced ejections saw a larger increase in the eccen-
+tricity of survivors.
+The 3–30 AU systems and the 1–30 AU systems exhibit
+similar trends: nearly all planets with an eccentricity greater
+than 0.2 are in systems that experienced some instability
+event (primarily ejections, which occurred about 2.5 times
+more frequently than collisions) during the simulation. Both
+sets of simulations display a high concentration of low eccen-
+tricity planets: trending below 0.1 within 30 AU and below
+0.2 beyond 30 AU. Including the additional planet near ∼ 1au
+slightly increased the typical eccentricities of surviving plan-
+ets, though the additional planet itself typically remains at
+very low eccentricity if it survives.
+
+10
+BOWENS ET AL.
+We compare the eccentricities between our simulations
+and observed exoplanets in both Figure 10 (right) and Fig-
+ure 11.
+We use known Jovian mass planets (0.3 to 10
+Jupiter M*sin(i)) taken from the Exoplanet Archive on Aug.
+4th, 2022. For our comparison sample, we select from the
+database those planets with eccentricities > 0. Planets dis-
+covered by various methods including radial velocity surveys
+and direct imaging are included in the comparison sample.
+Although there can be some biases in the measured values
+(Pan et al. 2010), observers can often measure eccentricity
+for giant planets detected by radial velocity, which is how
+the vast majority of planets suitable for comparison (i.e., a
+similar mass and semimajor axis range) to our sample are
+detected.
+The observed planets show eccentricity values with a con-
+centration towards low eccentricity (e < 0.15).
+However,
+the simulations show a much more extreme concentration
+towards very low eccentricity (e < 0.05). Unstable config-
+urations can reach high eccentricities (Figure 11), but our
+configurations (even if we limit to certain sets) apparently
+do not produce the correct mix of stable and unstable sys-
+tems. Based on the large discrepancy between the simula-
+tions (even when considering only unstable simulations or
+only 5 and 6 planet systems) and the observed exoplanet ec-
+centricities, we conclude that the systems created in DD16,
+while capable of creating transitional disks, can not repro-
+duce evolved systems.
+In contrast, other studies of planet-planet scattering that
+did not include a gas disk stage and/or require stability dur-
+ing the gas disk stage (e.g., Chatterjee et al. 2008; Juri´c
+& Tremaine 2008) produced eccentricity distributions that
+matched the observations well. Our use of equal-mass plan-
+ets likely does not account for the difference, as such configu-
+rations are just as likely to lead to elliptical orbits, albeit with
+a narrow distribution (Ford et al. 2001). Instead, our require-
+ment that configurations remain stable during the gas disk
+stage to maintain a cavity (Dong & Dawson 2016) appar-
+ently ensures too much stability to excite eccentricities. Our
+configurations do not achieve the “dynamically active” state
+identified by Juri´c & Tremaine (2008) that erases the memory
+of initial conditions and are reminiscent of the “dynamically
+cold” configurations explored by Dawson et al. (2016) that
+remain stable after the gas disk stage.
+6. CONCLUSIONS
+We simulated the long-term evolution of planetary systems
+capable of carving out and maintaining a transitional disk
+during the gas stage, using initial conditions from DD16. We
+subjected the planets to eccentricity damping during the disk
+stage and simulated the systems another 10 Gyr post-disk.
+Our main finding (Section 5) was that our systems tend to
+remain on stable, circular orbits. The typically very low ec-
+centricities are at odds with those observed in real systems of
+giant exoplanets discovered via the radial velocity method.
+For example, fraction of planets with eccentricities between
+0 and 0.2 was approximately 25% higher in our simulated
+systems compared to real systems. Among our simulated
+systems that experience an instability, the eccentricities can
+reach the observed high values, but too few of our systems
+go unstable. The stability of systems showed a dependence
+on multiplicity: three and four planet systems were more sta-
+ble (i.e., experienced no collisions or ejections) compared to
+five and six planet systems by a significant margin (86% sta-
+ble versus 33% stable, respectively). This trend is consis-
+tent with prior studies showing that higher planet multiplicity
+decreases stability (Chambers et al. 1996b). However, even
+when considering only high multiplicity systems and adding
+planets interior to those needed to carve observed gaps, our
+systems were too stable and circular.
+We also found that the presence of a 2:1 MMR in a sys-
+tem – which were commonly established in our simulations
+during the gas disk stage – did not significantly impact the
+overall likelihood of going unstable over 10 Gyr. Koriski &
+Zucker (2011) found that the presence of a 2:1 period ratio
+for a planetary pair indicates a system is younger on aver-
+age. However, in contrast to Koriski & Zucker (2011) who
+found that the typical lifetime of a 2:1 MMR is near 4 Gyr,
+we found that half of our pairs broke their 2:1 period ratio
+by 10 Myr. We noted a slightly higher instability rate for the
+near 2:1 MMR systems (those systems where the resonance
+angle still circled through all degrees but with a preference
+for a certain value) when considering resonance on a pair-
+by-pair basis (57% stable compared to approximately 70%
+stable for the resonance or no resonance systems).
+In future work, we could explore configurations of unequal
+mass planets, though past work has shown this is unlikely
+to significantly boost the resulting eccentricities (Ford et al.
+2001). It may be possible to further fine-tune our gas disk
+stage initial conditions (i.e., gas disk properties and planet
+spacing) to produce a higher fraction of post-gas unstable
+systems. However, more likely stability during the transi-
+tional disk stage precludes wide-spread instabilities after the
+gas disk disappears. If so, transitional disks may be typically
+caused by other processes besides giant planets, such as pho-
+toevaporation (e.g., Picogna et al. 2021) or compact config-
+urations of super-Earths and mini-Neptunes in low viscosity
+disk (Fung & Chiang 2017). Another possibility is that giant
+planets undergo convergent widescale migration at the very
+end of the transitional disk stage that packs them much closer
+together. As proposed by van der Marel & Mulders (2021),
+this explanation would also help account for the large size
+of transitional disks cavities, compared to the peak in giant
+planet occurrence at smaller semi-major axes. However, it
+
+POST TRANSITIONAL DISK STABILITY
+11
+0.6
+3
+6
+30
+60
+300 600
+Semimajor Axis (AU)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Eccentricity
+3
+4
+5
+6
+3--30 AU
+0.6
+3
+6
+30
+60
+300 600
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0.6
+3
+6
+30
+60
+300 600
+Semimajor Axis (AU)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+4
+5
+6
+7
+1--30 AU
+0.6
+3
+6
+30
+60
+300 600
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0.6
+3
+6
+30
+60
+300 600
+Semimajor Axis (AU)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Obs. Planets
+0.6
+3
+6
+30
+60
+300 600
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 10. Eccentricity versus semimajor axis for the 3–30 AU systems at 1 Gyr (left), the 1-30 AU systems at 1 Gyr (middle), and a selection
+of observed exoplanets (right). Simulated systems are color coded by the initial number of planets. Most simulated systems remain at low
+eccentricity e ≲ 0.1 for the entire evolution, although systems that began with more planets are more likely to undergo instabilities and end up
+on elliptical orbits.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Eccentricity
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fraction of Planets
+3-30 All Sim.
+1-30 All Sim.
+3-30 Unstable Sim.
+1-30 Unstable Sim.
+Observed Planets
+Figure 11. A CDF comparing the final eccentricities of five dif-
+ferent system pools. All 3-30 AU results are computed at 1 Gyr
+to make them directly comparable to the 1-30 AU results. Unsta-
+ble systems are systems where at least one planet experienced an
+instability event.
+would need to be explored what could trigger this just in-time
+migration.
+Computations for this research were performed on the
+Pennsylvania State University’s Institute for Computational
+& Data Sciences Advanced CyberInfrastructure (ICS-ACI).
+This content is solely the responsibility of the authors and
+does not necessarily represent the views of the Institute for
+CyberScience.
+The Center for Exoplanets and Habitable
+Worlds is supported by the Pennsylvania State University
+and the Eberly College of Science. This project was sup-
+ported in part by NASA XRP NNX16AB50G and NASA
+XRP 80NSSC18K0355, the National Science Foundation un-
+der Grant No. NSF PHY-1748958, and the Alfred P. Sloan
+Foundation’s Sloan Research Fellowship. This research has
+made use of the NASA Exoplanet Archive, which is operated
+by the California Institute of Technology, under contract with
+the National Aeronautics and Space Administration under the
+Exoplanet Exploration Program.
+Facility: Exoplanet Archive
+Software: Numpy
+REFERENCES
+Alexander, R. D., Clarke, C. J., & Pringle, J. E. 2006a, MNRAS,
+369, 216
+—. 2006b, MNRAS, 369, 229
+
+12
+BOWENS ET AL.
+Biller, B., Lacour, S., Juhász, A., et al. 2012, ApJ, 753, L38
+Calvet, N., D’Alessio, P., Watson, D. M., et al. 2005, ApJ, 630,
+L185
+Carrera, D., Raymond, S. N., & Davies, M. B. 2019, A&A, 629, L7
+Chambers, J. E. 1999, MNRAS, 304, 793
+Chambers, J. E., Wetherill, G. W., & Boss, A. P. 1996a, Icarus,
+119, 261
+—. 1996b, Icarus, 119, 261
+Chatterjee, S., Ford, E. B., Matsumura, S., & Rasio, F. A. 2008,
+ApJ, 686, 580
+Chiang, E., & Murray-Clay, R. 2007, Nature Physics, 3, 604
+Choksi, N., & Chiang, E. 2021, Monthly Notices of the Royal
+Astronomical Society, 510, 1657
+Clarke, C. J., Gendrin, A., & Sotomayor, M. 2001, MNRAS, 328,
+485
+Currie, T., Marois, C., Cieza, L., et al. 2019, ApJ, 877, L3
+Dawson, R. I., Chiang, E., & Lee, E. J. 2015, ArXiv:1506.06867,
+arXiv:1506.06867 [astro-ph.EP]
+Dawson, R. I., & Johnson, J. A. 2012, ApJ, 756, 122
+Dawson, R. I., Lee, E. J., & Chiang, E. 2016, ApJ, 822, 54
+Dodson-Robinson, S. E., & Salyk, C. 2011, ApJ, 738, 131
+Dong, R., & Dawson, R. 2016, ApJ, 825, 77
+Dong, R., Zhu, Z., & Whitney, B. 2015, ApJ, 809, 93
+Dullemond, C. P., & Dominik, C. 2005, A&A, 434, 971
+Ercolano, B., & Pascucci, I. 2017a, Royal Society Open Science, 4,
+170114
+—. 2017b, Royal Society Open Science, 4, 170114
+Ercolano, B., Picogna, G., Monsch, K., Drake, J. J., & Preibisch, T.
+2021, MNRAS, 508, 1675
+Ercolano, B., Weber, M. L., & Owen, J. E. 2018, MNRAS, 473,
+L64
+Espaillat, C., Muzerolle, J., Najita, J., et al. 2014, in Protostars and
+Planets VI, ed. H. Beuther, R. S. Klessen, C. P. Dullemond, &
+T. Henning, 497
+Fabrycky, D. C., & Murray-Clay, R. A. 2010, ApJ, 710, 1408
+Ford, E. B., & Chiang, E. I. 2007, ApJ, 661, 602
+Ford, E. B., Havlickova, M., & Rasio, F. A. 2001, Icarus, 150, 303
+Fung, J., & Chiang, E. 2017, ApJ, 839, 100
+Fung, J., Shi, J.-M., & Chiang, E. 2014, ApJ, 782, 88
+Garrido-Deutelmoser, J., Petrovich, C., Krapp, L., Kratter, K. M.,
+& Dong, R. 2022, ApJ, 932, 41
+Ginzburg, S., & Sari, R. 2018, MNRAS, 479, 1986
+Go´zdziewski, K., & Migaszewski, C. 2014, MNRAS, 440, 3140
+Haffert, S. Y., Bohn, A. J., de Boer, J., et al. 2019, Nature
+Astronomy, 329
+Haisch, Karl E., J., Lada, E. A., & Lada, C. J. 2001, ApJL, 553,
+L153
+Huang, J., Andrews, S. M., Dullemond, C. P., et al. 2018, ApJL,
+869, L42
+Isella, A., Benisty, M., Teague, R., et al. 2019, arXiv e-prints,
+arXiv:1906.06308
+Juri´c, M., & Tremaine, S. 2008, ApJ, 686, 603
+Keppler, M., Benisty, M., Müller, A., et al. 2018, A&A, 617, A44
+Kominami, J., & Ida, S. 2002, Icarus, 157, 43
+Koriski, S., & Zucker, S. 2011, ApJL, 741, L23
+Kraus, A. L., & Ireland, M. J. 2012, ApJ, 745, 5
+Krauss, O., Wurm, G., Mousis, O., et al. 2007, A&A, 462, 977
+Lee, M. H., & Peale, S. J. 2002, ApJ, 567, 596
+Lithwick, Y., & Wu, Y. 2012, ApJL, 756, L11
+Long, F., Pinilla, P., Herczeg, G. J., et al. 2018, ApJ, 869, 17
+Luhman, K. L., Allen, P. R., Espaillat, C., Hartmann, L., & Calvet,
+N. 2010, ApJS, 186, 111
+Malmberg, D., & Davies, M. B. 2009, MNRAS, 394, L26
+Marois, C., Macintosh, B., Barman, T., et al. 2008, Science, 322,
+1348
+Marois, C., Zuckerman, B., Konopacky, Q. M., Macintosh, B., &
+Barman, T. 2010, Nature, 468, 1080
+Marsh, K. A., & Mahoney, M. J. 1993, ApJL, 405, L71
+Mayor, M., Marmier, M., Lovis, C., et al. 2011, arXiv e-prints,
+arXiv:1109.2497
+Melita, M. D., & Woolfson, M. M. 1996, MNRAS, 280, 854
+Morrison, S. J., Dawson, R. I., & MacDonald, M. 2020, ApJ, 904,
+157
+Morrison, S. J., & Kratter, K. M. 2016, ApJ, 823, 118
+Muley, D., Fung, J., & van der Marel, N. 2019, ApJL, 879, L2
+Owen, J. E. 2016, PASA, 33, e005
+Owen, J. E., Clarke, C. J., & Ercolano, B. 2012, MNRAS, 422,
+1880
+Owen, J. E., Ercolano, B., & Clarke, C. J. 2011, MNRAS, 412, 13
+Paardekooper, S.-J., Dong, R., Duffell, P., et al. 2022, arXiv
+e-prints, arXiv:2203.09595
+Paardekooper, S. J., & Mellema, G. 2004, A&A, 425, L9
+Pan, M., Zakamska, N. L., & Ford, E. B. 2010, in EAS
+Publications Series, Vol. 42, EAS Publications Series, ed.
+K. Go˙zdziewski, A. Niedzielski, & J. Schneider, 169
+Papaloizou, J., & Lin, D. N. C. 1984, ApJ, 285, 818
+Papaloizou, J. C. B., & Larwood, J. D. 2000, MNRAS, 315, 823
+Petrovich, C., Tremaine, S., & Rafikov, R. 2014, ApJ, 786, 101
+Pfalzner, S., Steinhausen, M., & Menten, K. 2014, ApJL, 793, L34
+Picogna, G., Ercolano, B., & Espaillat, C. C. 2021, MNRAS, 508,
+3611
+Piétu, V., Guilloteau, S., & Dutrey, A. 2005, A&A, 443, 945
+Quanz, S. P., Amara, A., Meyer, M. R., et al. 2013, ApJ, 766, L1
+Reggiani, M., Quanz, S. P., Meyer, M. R., et al. 2014, ApJ, 792,
+L23
+Rein, H. 2012, MNRAS, 422, 3611
+Rosenthal, M. M., Chiang, E. I., Ginzburg, S., & Murray-Clay,
+R. A. 2020, MNRAS, 498, 2054
+
+POST TRANSITIONAL DISK STABILITY
+13
+Sallum, S., Follette, K. B., Eisner, J. A., et al. 2015, Nature, 527,
+342
+Simbulan, C., Rein, H., Tamayo, D., Petrovich, C., & Murray, N.
+2017, Monthly Notices of the Royal Astronomical Society, 469,
+3337
+Smith, A. W., & Lissauer, J. J. 2009, Icarus, 201, 381
+Strom, K. M., Strom, S. E., Edwards, S., Cabrit, S., & Skrutskie,
+M. F. 1989, AJ, 97, 1451
+Suzuki, T. K., & Inutsuka, S.-I. 2009, in American Institute of
+Physics Conference Series, ed. T. Usuda, M. Tamura, & M. Ishii,
+Vol. 1158, 161
+Tamayo, D., Triaud, A. H. M. J., Menou, K., & Rein, H. 2015, The
+Astrophysical Journal, 805, 100
+van der Marel, N. 2017, in Astrophysics and Space Science
+Library, ed. M. Pessah & O. Gressel, Vol. 445, 39
+van der Marel, N., & Mulders, G. D. 2021, AJ, 162, 28
+Vorobyov, E. I., Lin, D. N. C., & Guedel, M. 2015, A&A, 573, A5
+Wang, J. J., Graham, J. R., Dawson, R., et al. 2018, AJ, 156, 192
+Wang, L., & Goodman, J. J. 2017, ApJ, 835, 59
+Winn, J. N., & Fabrycky, D. C. 2015, ARA&A, 53, 409
+Wölfer, L., Picogna, G., Ercolano, B., & van Dishoeck, E. F. 2019,
+MNRAS, 490, 5596
+Xie, J.-W., Dong, S., Zhu, Z., et al. 2016, Proceedings of the
+National Academy of Science, 113, 11431
+Zhang, S., Zhu, Z., Huang, J., et al. 2018, ApJL, 869, L47
+Zhu, Z., Nelson, R. P., Dong, R., Espaillat, C., & Hartmann, L.
+2012, ApJ, 755, 6
+Zhu, Z., Nelson, R. P., Hartmann, L., Espaillat, C., & Calvet, N.
+2011, ApJ, 729, 47
+
+14
+BOWENS ET AL.
+APPENDIX
+A. GAS STAGE SIMULATIONS
+Some of the gas stage simulations are taking directly from DD16 but, as described in Section 2, we run supplemental gas stage
+simulations as part of this course. Gas stage simulations use the mercury6 Bulirsch-Stoer hybrid integrator with a timestep of
+and Bulirsch-Stoer accuracy parameter of 10−12) (Chambers 1999). Each configuration features three to six equal mass planets,
+with initial semimajor axes between 2 and 20 AU. The initial semi-major axes were selected so that the planets just barely open
+a common gap in the disk spanning 3–30 AU. Each configuration (Table 1) meets this condition for an assumed disk viscosity,
+disk scale height, and planet mass (0.5–10 Jupiter masses).
+For each configuration, we run 20 random realizations; 10 were run as part of DD16 and 10 are new to this paper. The semi-
+major was randomized from the default values listed in Table 2 by about 5%, drawing from a normal distribution centered on the
+default value. Initial eccentricities were 0. Initial inclinations were random set to ∼ 0.01◦ to avoid perfectly coplanar planets.
+The initial mean longitude, longitude of periapse, and longitude of ascending node of each planet are randomized between 0 and
+2π.
+The effects of the gas within the gap were implemented using prescriptions from Papaloizou & Larwood (2000), Kominami &
+Ida (2002), Ford & Chiang (2007), and Rein (2012) as user-defined forces in mercury6, following Dawson et al. (2015) and
+DD16. The timescales for gas damping are:
+τ = 0.029g cm−2
+Σ30
+� a
+AU
+�2 M⊙
+Mp
+yr×
+1
+,v < cs
+�
+v
+cs
+�3
+,v > cs,i < cs/vkep
+�
+v
+cs
+�4
+,i > cs/vkep
+(A1)
+where v =
+√
+e2 +i2vkep and vkep is the Keplerian velocity, and the sound speed cs = 1.29km/s
+� a
+AU
+�−1/4. We impose ˙e/e = −1/τ
+and ˙i/i = −2/τ (Kominami & Ida 2002). The value of Σ30 for each configuration is listed in Table 1. Following DD16, we do not
+impose migration (˙a) or precession (˙ϖ, ˙Ω) because DD16 found that these effects were negligible except for fine-tuned values.
+Furthermore, the direction and magnitude of migration is sensitive to uncertain disk conditions; the magnitude is typically small
+in a depleted cavity. See Section 4.2.3 of DD16 for more detail.
+
+POST TRANSITIONAL DISK STABILITY
+15
+Table 2. 3 – 30 AU Systems Default Semimajor Axis
+Name
+P1 a
+P2 a
+P3 a
+P4 a
+P5 a
+P6 a
+(AU)
+(AU)
+(AU)
+(AU)
+(AU)
+(AU)
+3-5
+2.6
+7.1
+19.4
+-
+-
+-
+3-10a
+2.2
+6.2
+18.1
+-
+-
+-
+3-10b
+2.6
+6.9
+18.5
+-
+-
+-
+3-10c
+3.3
+8.1
+19.6
+-
+-
+-
+3-10d
+3.3
+8.0
+19.0
+-
+-
+-
+3-10e
+4.3
+9.4
+20.5
+-
+-
+-
+4-2
+2.8
+5.7
+11.5
+23.1
+-
+-
+4-5a
+2.3
+4.8
+10.1
+21.0
+-
+-
+4-5b
+2.4
+4.9
+10.8
+20.7
+-
+-
+4-5c
+2.5
+5.0
+10.4
+21.3
+-
+-
+4-5d
+3.4
+6.3
+11.8
+22.1
+-
+-
+4-10b
+2.2
+4.6
+9.8
+20.8
+-
+-
+5-1
+2.5
+4.3
+7.6
+13.2
+23.1
+-
+5-2a
+2.9
+4.8
+8.2
+13.8
+23.3
+-
+5-2b
+3.7
+5.8
+9.3
+14.9
+23.8
+-
+5-5b
+2.5
+4.3
+7.5
+13.1
+22.9
+-
+6-0.5
+3.5
+5.1
+7.6
+11.3
+16.7
+24.8
+6-1
+3.6
+5.3
+7.8
+11.4
+16.8
+24.8
+NOTE—The default semimajor axis (a) for each planet in a set, rounded to 0.1 AU.
+

UdE5T4oBgHgl3EQfAw7h/content/tmp_files/2301.05382v1.pdf.txt

@@ -0,0 +1,914 @@
+Draft version January 16, 2023
+Typeset using LATEX manuscript style in AASTeX631
+Excitation of Multi-periodic Kink Motions in Solar Flare Loops: Possible Application
+to Quasi-periodic Pulsations
+Mijie Shi,1 Bo Li,1 Shao-Xia Chen,1 Mingzhe Guo,1, 2 and Shengju Yuan3
+1Shandong Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, School of Space Science and
+Physics, Institute of Space Sciences, Shandong University, Weihai, Shandong, 264209, China
+2Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, 3001 Leuven, Belgium
+3Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao, Shandong, 266237, China
+(Received; Revised; Accepted)
+ABSTRACT
+Magnetohydrodynamic (MHD) waves are often invoked to interpret quasi-periodic
+pulsations (QPPs) in solar flares. We study the response of a straight flare loop to a
+kink-like velocity perturbation using three-dimensional MHD simulations and forward
+model the microwave emissions using the fast gyrosynchrotron code.
+Kink motions
+with two periodicities are simultaneously generated, with the long-period component
+(PL = 57 s) being attributed to the radial fundamental kink mode and the short-period
+component (PS = 5.8 s) to the first leaky kink mode. Forward modeling results show
+that the two-periodic oscillations are detectable in the microwave intensities for some
+lines of sight. Increasing the beam size to (1′′)2 does not wipe out the microwave oscil-
+lations. We propose that the first leaky kink mode is a promising candidate mechanism
+to account for short-period QPPs. Radio telescopes with high spatial resolutions can
+Corresponding author: Mijie Shi
+[email protected]
+arXiv:2301.05382v1  [astro-ph.SR]  13 Jan 2023
+
+2
+Shi et al.
+help distinguish between this new mechanism with such customary interpretations as
+sausage modes.
+1. INTRODUCTION
+Quasi-periodic pulsations (QPPs) are oscillatory intensity variations commonly observed in solar
+flare emissions across a broad range of passbands (e.g., Tan et al. 2010; Kupriyanova et al. 2010;
+Van Doorsselaere et al. 2011; Kolotkov et al. 2015; Li et al. 2021). The typical periods of QPPs
+range from a fraction of a second to several minutes. A variety of candidate mechanisms have been
+proposed to explain QPPs (see reviews, e.g., Nakariakov & Melnikov 2009; Van Doorsselaere et al.
+2016; McLaughlin et al. 2018; Kupriyanova et al. 2020; Zimovets et al. 2021). However, the physical
+mechanisms responsible for generating QPPs remain uncertain.
+Rapid QPPs, namely those QPPs with periodicities on the order of seconds, are customarily at-
+tributed to such mechanisms as oscillatory reconnection (e.g., Craig & McClymont 1991; Hassam
+1992) or sausage modes (e.g., Roberts et al. 1983; Nakariakov et al. 2003). Oscillatory reconnec-
+tion can occur at a two-dimensional X-point as a consequence of, say, the impingement by velocity
+pulses (McLaughlin et al. 2009). This mechanism was shown to operate in the hot corona with the
+oscillatory periods independent of the initial velocity pulse (Karampelas et al. 2022). Sausage modes
+can cause periodic compression and rarefaction and thus modulate the microwave emissions of flare
+loops (e.g., Mossessian & Fleishman 2012; Reznikova et al. 2014; Kuznetsov et al. 2015; Guo et al.
+2021; Kupriyanova et al. 2022; Shi et al. 2022). Candidate sausage modes have been reported in
+observations of QPPs in various passbands (e.g., Melnikov et al. 2005; Zimovets & Struminsky 2010;
+Su et al. 2012; Kolotkov et al. 2015; Tian et al. 2016; see the recent review by Li et al. 2020).
+QPPs with multiple periodicities are often observed (Tan et al. 2010).
+Magnetohydrodynamic
+(MHD) waves are accepted to play an important role for the formation of these multi-periodic sig-
+nals. Inglis & Nakariakov (2009) interpreted their multi-periodic event as the result of kink-mode-
+triggered magnetic reconnection. Multiple periodic QPPs can also be attributed to the superposition
+of different modes, for example, fast and slow sausage modes (Van Doorsselaere et al. 2011), axial
+
+3
+fundamental kink mode and its axial overtones (Kupriyanova et al. 2013), kink mode and sausage
+mode (Kolotkov et al. 2015). In these studies, the long-period (several tens of seconds) signals are
+often attributed to kink or slow sausage modes, whereas the short-period (several seconds) signals to
+fast sausage modes.
+In this work, we simulate kink motions in flare loops and forward model their microwave signatures,
+examining the possibility that both short- and long-period kink motions can be excited simultane-
+ously.
+We present the MHD setup and results in Section 2, moving on to describe the forward
+modeling setup and results in Section 3. Section 4 summarizes this study.
+2. MHD SIMULATION
+2.1. Numerical Setup
+We model flare loops as field aligned, z-directed, axially uniform cylinders. The equilibrium density
+distribution is prescribed by
+ρ = ρe + (ρi − ρe)f(r),
+(1)
+where [ρi, ρe]=[4 × 1010, 1 × 109]mp cm−3 represent the internal and external mass densities.
+In
+addition, mp is the proton mass. By ‘internal’ (subscript i) and ‘external’ (subscript e), we refer to
+the equilibrium quantities at the loop axis and far from the loop, respectively. A continuous profile
+f(r) = exp[−(r/R)α] is used to connect ρi and ρe, with r =
+�
+x2 + y2, α = 15, and the nominal
+loop radius R = 3 Mm. The temperature distribution follows the same spatial dependence as the
+density, with [Ti, Te] = [10, 2] MK. The magnetic field B is z-directed, its magnitude varying from
+Bi = 60 G to Be = 80 G to maintain transverse force balance. The pertinent Alfv´en speeds are
+[vAi, vAe] = [654, 5510] km s−1. The loop length is L0 = 30 Mm. The specification of our equilibrium
+is largely in accordance with typical measurements (e.g., Tian et al. 2016). Figure 1 shows the initial
+density distributions in the z = L0/2 and x = 0 cuts. The equilibrium is perturbed by a kink-like
+initial velocity perturbation in the x-direction
+vx(x, y, z; t = 0) = v0exp
+�
+− r2
+2σ2
+�
+sin
+�πz
+L0
+�
+,
+(2)
+
+4
+Shi et al.
+where v0 = 20 km s−1 is the amplitude, and σ = R characterizes the spatial extent 1.
+The set of ideal MHD equations is evolved using the finite-volume code PLUTO (Mignone et al.
+2007).
+We use the piecewise parabolic method for spatial reconstruction, and the second-order
+Runge–Kutta method for time-stepping. The HLLD approximate Riemann solver is used for com-
+puting inter-cell fluxes, and a hyperbolic divergence cleaning method is used to keep the magnetic
+field divergence-free. The simulation domain is [−15, 15] Mm×[−15, 15] Mm×[0, 30] Mm. A uniform
+grids with cell numbers of [Nx, Ny, Nz] = [1000, 1000, 100] is adopted. We use the outflow boundary
+condition for all primitive variables at the lateral boundaries (x and y). For the top and bottom
+boundaries (z), the transverse velocities (vx and vy) are fixed at zero. The density, pressure, and
+Bz are fixed at their initial values. The other variables are set as outflow. Our numerical results
+vary little when, say, a finer grid or a larger domain is experimented with. In particular, no spurious
+reflection is discernible at the lateral boundaries.
+2.2. Numerical Results
+Figure 2 displays the temporal evolution of vx sampled at the loop center (x, y, z) = (0, 0, L0/2).
+One sees that two periodicities co-exist. We decompose the time sequence into a long-period (the red
+curve) and a short-period (green) component. This decomposition is performed using the low-(high-)
+pass filter of the filtfilt function in the SciPy package, with the threshold period chosen to be 10 s.
+The periods of the two components are PL = 57 s and PS = 5.8 s, respectively. Both components are
+seen to experience temporal damping.
+The snapshot at t = 116 s of the density at the loop apex is displayed in Figure 3(a). Some rolled-
+up vortices can be readily seen at the loop boundary close to the y-axis, suggesting the development
+of the Kelvin-Helmholtz instability as a result of localized shearing motions (e.g., Terradas et al.
+2008; Soler et al. 2010; Antolin et al. 2015). Figure 3(b) shows the velocity field at the loop apex,
+together with its low-pass (Figure 3(c)) and high-pass (Figure 3(d)) components. Figure 3(e) further
+displays the temporal evolution of vx(x, y = 0, z = L0/2; t). The multi-periodic oscillation is clearly
+1 The effect of σ is also worth examining. The results are collected in Appendix B to streamline the main text.
+
+5
+shown in the loop interior. The period of PL = 57 s is consistent with that of the radial fundamental
+kink mode. The low-pass component of the velocity field (Figure 3(c) and the related animation) is
+typical of a radial fundamental kink mode as well (e.g., Goossens et al. 2014). Therefore, the long-
+period oscillation is identified as the radial fundamental kink mode. This identification is further
+corroborated by a comparison with independent theoretical expectations from an eigenvalue problem
+perspective in Appendix A.1, where we show that the associated temporal damping is attributable
+to resonant absorption (Goossens et al. 2011).
+The short-period component (PS = 5.8 s) shows up as ridge-like structures crossing each other in
+the loop interior shown in Figure 3(e). The slope of each ridge matches the fast speed in the loop
+interior, indicating a fast wave nature of these ridges. The high-pass component of the velocity field
+(Figure 3(d) and the related animation) shows some rapid variation of a dipole-like velocity field at
+the loop interior. These features are consistent with the first leaky kink mode, i.e., the extension
+to the leaky regime of the first radial overtone with the damping attributable to lateral leakage
+(e.g., Spruit 1982; Cally 1986, 2003). The short-period oscillation associated with a first leaky kink
+mode is of particular interest as it has not been invoked for interpreting QPPs to our knowledge.
+Our numerical results show that the first leaky kink mode can be generated along with the radial
+fundamental kink mode and this scenario is likely to happen in flare loops. On this aspect we stress
+that the associated velocity field has not been demonstrated in initial value problem studies (see
+Appendix A.2 for further discussions).
+3. FORWARD MODELING THE GYROSYNCHROTRON EMISSION
+3.1. Method
+We compute gyrosynchrotron (GS) emissions using the fast GS code (Kuznetsov et al. 2011; Fleish-
+man & Kuznetsov 2010). The fast GS code computes the local values of the absorption coefficient
+and emissivity, thereby accounting for inhomogeneous sources by integrating the radiative transfer
+equation. We assume that non-thermal electrons occupy a time-varying volume that initially corre-
+sponds to r ≤ 2 Mm (the yellow volume in Figure 1). This assumption is based on the consideration
+
+6
+Shi et al.
+that the non-thermal electrons determined by some acceleration mechanism may fill only part of the
+loop, similar to the model used in Kupriyanova et al. (2022). This volume moves back and forth
+due to the kink motions. We trace this volume using a passive scaler as the simulation goes on.
+Within this volume, we further assume that the number density (Nb) of the non-thermal electrons
+is proportional to the thermal one (Ne), and specifically takes the form Nb = 0.005Ne. The spectral
+index of the non-thermal electrons is δ = 3, with the energy ranging from 0.1 to 10 MeV. The pitch
+angle distribution of the non-thermal electrons is taken to be isotropic. We assume that any line of
+sight (LoS) is located in the y − z plane, making an angle of 45◦ with both the y- and z-axis. The
+LoS intersects the apex plane z = L0/2 at (x0, y0) and threads a beam with a cross-sectional area of
+60 km×60 km. We select three beams with different x0 (i.e., x0 = 0, ±1 Mm) by adopting a fixed
+y0 = 0. The white line in Figure 1 illustrates the LoS for the case x0 = 0 Mm.
+3.2. Results
+We forward model the GS emission and examine the microwave signature of the kink motions,
+taking the 17 GHz emission as an example. Figure 4(a) shows the intensity variations for the three
+beams (x0 = 0, ±1 Mm). One sees that the intensity variation is negligible for the beam that passes
+through the loop axis (x0 = 0). However, the intensity shows obvious variations with two different
+periodicities for the rest of the beams (x0 = ±1 Mm). We decompose the intensity variations into
+a long-period (Figure 4(b)) and a short-period (Figure 4(c)) component for the cases x0 = ±1 Mm.
+The periods of the two components are consistent with PL and PS, meaning that the two periodicities
+of the microwave intensity are the manifestations of the simultaneously excited radial fundamental
+kink mode and the first leaky kink mode. We also find that the intensity variations are anti-correlated
+between the cases x0 = 1 Mm and x0 = −1 Mm. This behavior is caused by the asymmetric variations
+between the left and right sides of the loop for the two modes.
+Figure 5(a) shows the intensity variations for the case x0 = 1 Mm when different beam sizes are
+adopted. The intensity of a larger beam is achieved by adding the intensity of all single beams in the
+corresponding area projected onto the plane of sky. Figures 5(b) and 5(c) display the low-pass and
+high-pass component. One sees that increasing the beam size to as large as (1′′)2 does not change the
+
+7
+two-periodicity behavior of the microwave intensities. For the short-period component, the intensity
+variations are almost the same for different beam sizes.
+The kink motions with two periodicities are likely to be detectable using modern radio telescopes.
+The Mingantu Spectral Radioheliograph (MUSER, Yan et al. (2021)) observes the Sun with a time ca-
+dence of 0.2 s and a spatial resolution of 1.3′′ at 15 GHz. The Atacama Large Millimeter/submillimeter
+Array (ALMA, Wedemeyer et al. (2016)) can achieve unprecedented high resolutions at 85 GHz.
+4. SUMMARY
+We examined the response of a straight flare loop to a kink-like initial velocity perturbation using 3D
+MHD simulations. We found that kink motions with two periodicities are simultaneously generated
+in the loop interior. The long-period component (PL = 57 s) is attributed to the radial fundamental
+kink mode, and the short-period component (PS = 5.8 s) to the first leaky kink mode. We then
+examined the modulation of the microwave intensity by the kink motions via forward modeling the
+GS emissions.
+We found that the two-periodic signals are detectable in the 17 GHz microwave
+emission at some LoS directions. Increasing the beam size to as large as (1′′)2 does not wipe out
+these oscillatory signals.
+Leaky kink modes are a promising candidate mechanism to account for short-period QPPs in
+flare loops. Kolotkov et al. (2015) detected multi-periodic QPPs and interpreted the long-period
+component (100 s) as a radial fundamental kink mode and the short-period component (15 s) as a
+sausage mode. We argue that the short-period component in their observations can be alternatively
+interpreted as the first leaky kink mode. However, we cannot distinguish between the two mechanisms
+based purely on the oscillation period, because the timescales of the first leaky kink modes and
+sausage modes are close. Radio telescopes with high spatial resolutions would be very helpful to
+identify the first leaky kink modes. For sausage modes, the intensity variations are expected to be
+in-phase across the entire loop, whereas for the first leaky kink modes the intensity variations are
+anti-correlated between the left and right sides of the loop.
+
+8
+Shi et al.
+We thank the referee for constructive comments. This work is supported by the National Natural
+Science Foundation of China (41904150, 12273019, 41974200, 11761141002, 12203030). We gratefully
+acknowledge ISSI-BJ for supporting the international team “Magnetohydrodynamic wavetrains as a
+tool for probing the solar corona ”.
+1
+2
+3
+4
+APPENDIX
+A. COMPARISON BETWEEN 3D TIME-DEPENDENT RESULTS AND THEORETICAL
+EXPECTATIONS
+The temporal evolution of vx at three different locations in the apex plane (z = L0/2) is displayed
+by the blue curves in the left column of Figure 6. Two periodicities can be readily seen in any time
+sequence, which is therefore decomposed into a long-period (the red curves) and a short-period (green)
+component. We then fit both components using an exponentially damping sinusoid A0 sin(2πt/P +
+φ0)exp(−t/τ), printing the best-fit periods (PL,S) and damping times (τL,S) on each plot. The best-fit
+curves are additionally displayed by the black dotted lines.
+A.1. the Long-period Component
+The radial fundamental kink mode is well known to be resonantly absorbed in the Alfv´en continuum
+in a radially continuous equilibrium with our ordering of the characteristic speeds (e.g., Goossens
+et al. 2011). However, one complication is that our radial profile (see Equation (1)) is not readily
+amenable to analytical treatment. We therefore proceed with the numerical approach in Chen et al.
+(2021), computing the radial fundamental kink mode as a resistive eigenmode (see Goossens et al.
+2011, for conceptual clarifications). The code outputs a period of P EVP
+L
+= 57.3 s and a damping
+time of τ EVP
+L
+= 107 s, where the superscript indicates that these expectations are derived from an
+eigenvalue problem (EVP) perspective 2. One sees that the expected period and damping time agree
+2 The period of the radial fundamental kink mode reads 2L0/ck ≈ 55.8 s in the thin-tube (TT) thin-boundary (TB) limit,
+with ck being the kink speed (e.g., Goossens et al. 2011). This is not far from P EVP
+L
+. We refrain from comparing τ EVP
+L
+with the TTTB expectation, because an expression is not available for our radial profile. The detailed formulation of
+a radial profile, however, is known to impact the TTTB expectations for the damping time (e.g., Soler et al. 2014).
+
+9
+remarkably well with the best-fit values that we derive with the 3D time-dependent simulation (the
+left column of Figure 6).
+A.2. the Short-period Component
+This subsection compares our short-period component with the EVP expectations for the first
+leaky kink mode.
+We restrict ourselves to the piecewise constant version (i.e., α → ∞) of our
+equilibrium. The pertinent flow field is emphasized, despite that the expressions we offer are not
+new per se (e.g., Cally 1986, 2003). We work in a cylindrical coordinate system (r, θ, z), denoting
+the equilibrium quantities with the subscript 0. The interior and exterior are further discriminated
+by the subscripts i and e. There appears a set of primitive quantities {ρi,e, pi,e, Bi,e}, where ρ, p, and
+B represent the mass density, thermal pressure, and magnetic field strength, respectively. We define
+the Alfv´en (vA), adiabatic sound (cs), and tube speeds (cT) as
+v2
+Ai,e = B2
+i,e
+µ0ρi,e
+,
+c2
+si,e = γpi,e
+ρi,e
+,
+c2
+Ti,e =
+v2
+Ai,ec2
+si,e
+v2
+Ai,e + c2
+si,e
+,
+(A1)
+with µ0 the magnetic permeability of free space and γ = 5/3 the ratio of specific heats.
+Our EVP expectations are as follows. With kink motions in mind, we write any small-amplitude
+perturbation δg as
+δg(r, θ, z; t) = ℜ {˜g(r) exp [−i(ωt − kz − θ)]} ,
+(A2)
+where k denotes the real-valued axial wavenumber, and ω = ωR+iωI represents the angular frequency.
+Only temporally non-growing solutions are sought (ωI ≤ 0). Defining
+µ2
+i,e = (ω2 − k2v2
+Ai,e)(ω2 − k2c2
+si,e)
+(v2
+Ai,e + c2
+si,e)(ω2 − k2c2
+Ti,e) ,
+(A3)
+we further take ωR > 0 and −π/2 < arg µi,e ≤ π/2. A dispersion relation (DR) then writes
+µiR
+ω2 − k2v2
+Ai
+J′
+1(µiR)
+J1(µiR) = ρi
+ρe
+µeR
+ω2 − k2v2
+Ae
+(H(1)
+1 )′(µeR)
+H(1)
+1 (µeR)
+,
+(A4)
+which accounts for both the trapped and leaky regimes. Here Jn (H(1)
+n ) denotes the Bessel (Hankel)
+function of the first kind (with n = 1). The prime ′ represents, say, dJ1(z)/dz with z evaluated at
+
+10
+Shi et al.
+µiR. Standard procedure further yields that
+˜pT(r) = B2
+i
+µ0
+×
+�
+�
+�
+�
+�
+�
+�
+J1(µir)/J1(µiR),
+0 ≤ r ≤ R,
+H(1)
+1 (µer)/H(1)
+1 (µeR), r ≥ R,
+(A5)
+where the arbitrarily scaled ˜pT denotes the Fourier amplitude of the total pressure perturbation. For
+both the interior and exterior, the Fourier amplitudes for the radial and azimuthal speeds write
+˜vr(r)=
+−iω
+ρ0(ω2 − k2v2
+A)
+d˜pT
+dr ,
+(A6)
+˜vθ(r)=
+ω
+ρ0(ω2 − k2v2
+A)
+˜pT
+r .
+(A7)
+Our EVP expectations amount to time-dependent perturbations that are standing in both axial and
+azimuthal directions, the net results being
+vr(r, θ, z; t)=A sin(kz) cos θℜ
+�
+i˜vr(r)e−iωt�
+=
+�
+A sin(kz)eωIt� �
+cos θℜ
+�
+i˜vr(r)e−iωRt��
+,
+(A8)
+vθ(r, θ, z; t)=A sin(kz) sin θℜ
+�
+−˜vθ(r)e−iωt�
+=
+�
+A sin(kz)eωIt� �
+sin θℜ
+�
+−˜vθ(r)e−iωRt��
+.
+(A9)
+Here A is some arbitrary, real-valued, dimensionless magnitude.
+We now compare our short-period component with the EVP expectations. To start, plugging our
+equilibrium quantities into Equation (A4) yields a period of P EVP
+S
+= 5.96 s and a damping time of
+τ EVP
+S
+= 150 s. This pertains to the first leaky mode, namely the one that possesses the lowest ωR
+among all modes with ωI < 0 3. The expected period is in good agreement with the best-fit values
+printed in the left column of Figure 6. The best-fit damping times, on the other hand, are somehow
+shorter than expected, which is intuitively understandable because a radially discontinuous cylinder
+leads to more efficient wave trapping. Figure 7a further displays the velocity fields in the apex plane
+of our short-period component at some representative instant. Plotted in Figure 7b is the pertinent
+EVP expectation, namely the flow field constructed with the terms in the braces of Equations (A8)
+and (A9) at ωRt = 0. The two sets of flow fields are remarkably similar. Overall, we conclude that
+the short-period component can be confidently identified as the first leaky kink mode.
+3 Some analytical expressions are available that approximately solve Equation (A4) in some appropriate limits (e.g.,
+Cally 2003; Spruit 1982). We choose to solve Equation (A4) exactly, because the eigenfunctions are also of interest.
+
+11
+B. EFFECT OF VARYING σ
+This section examines how the relative importance of the long- and short-period components varies
+when we vary the spatial extent of the initial perturbation (i.e., σ in Equation 2). Two additional
+simulations are performed, one with σ = 0.5R and the other with σ = 2R.
+These additional
+computations are shown in the middle and right columns of Figure 6 in the same format as our
+reference results (with σ = R).
+Two pronounced periodicites can be readily told apart in any
+sampled vx. The following features stand out for the short-period components. Firstly, their best-fit
+periods and damping times vary little from one case to another, indicating the robust excitation of
+the first leaky kink mode. Secondly, the short-period component tends to be stronger for smaller
+values of σ, suggesting that leaky modes can receive a larger fraction of the energy imparted by the
+initial perturbation when it is more localized.
+Some subtlety exists for the long-period component when σ = 0.5R. Overall, the temporal behavior
+for vx deviates from an exponentially damping sinusoid, in contrast to what happens for other values
+of σ that we examine. We refrain from explaining why at this time of writing. Rather, we note
+that this deviation is not a numerical artifact but persists even when we adopt a considerably larger
+computational domain.
+REFERENCES
+Antolin, P., Okamoto, T. J., De Pontieu, B., et al.
+2015, ApJ, 809, 72
+Cally, P. S. 1986, SoPh, 103, 277
+—. 2003, SoPh, 217, 95
+Chen, S.-X., Li, B., Van Doorsselaere, T., et al.
+2021, ApJ, 908, 230
+Craig, I. J. D., & McClymont, A. N. 1991, ApJL,
+371, L41
+Fleishman, G. D., & Kuznetsov, A. A. 2010, ApJ,
+721, 1127
+Goossens, M., Erd´elyi, R., & Ruderman, M. S.
+2011, SSRv, 158, 289
+Goossens, M., Soler, R., Terradas, J., Van
+Doorsselaere, T., & Verth, G. 2014, ApJ, 788, 9
+Guo, M., Li, B., & Shi, M. 2021, ApJL, 921, L17
+Hassam, A. B. 1992, ApJ, 399, 159
+Inglis, A. R., & Nakariakov, V. M. 2009, A&A,
+493, 259
+Karampelas, K., McLaughlin, J. A., Botha, G.
+J. J., & R´egnier, S. 2022, ApJ, 933, 142
+
+12
+Shi et al.
+Kolotkov, D. Y., Nakariakov, V. M., Kupriyanova,
+E. G., Ratcliffe, H., & Shibasaki, K. 2015,
+A&A, 574, A53
+Kupriyanova, E., Kolotkov, D., Nakariakov, V., &
+Kaufman, A. 2020, Solar-Terrestrial Physics, 6,
+3
+Kupriyanova, E. G., Kaltman, T. I., & Kuznetsov,
+A. A. 2022, MNRAS, 516, 2292
+Kupriyanova, E. G., Melnikov, V. F., Nakariakov,
+V. M., & Shibasaki, K. 2010, SoPh, 267, 329
+Kupriyanova, E. G., Melnikov, V. F., & Shibasaki,
+K. 2013, SoPh, 284, 559
+Kuznetsov, A. A., Nita, G. M., & Fleishman,
+G. D. 2011, ApJ, 742, 87
+Kuznetsov, A. A., Van Doorsselaere, T., &
+Reznikova, V. E. 2015, SoPh, 290, 1173
+Li, B., Antolin, P., Guo, M. Z., et al. 2020, SSRv,
+216, 136
+Li, D., Ge, M., Dominique, M., et al. 2021, ApJ,
+921, 179
+McLaughlin, J. A., De Moortel, I., Hood, A. W.,
+& Brady, C. S. 2009, A&A, 493, 227
+McLaughlin, J. A., Nakariakov, V. M.,
+Dominique, M., Jel´ınek, P., & Takasao, S. 2018,
+SSRv, 214, 45
+Melnikov, V. F., Reznikova, V. E., Shibasaki, K.,
+& Nakariakov, V. M. 2005, A&A, 439, 727
+Mignone, A., Bodo, G., Massaglia, S., et al. 2007,
+ApJS, 170, 228
+Mossessian, G., & Fleishman, G. D. 2012, ApJ,
+748, 140
+Nakariakov, V. M., & Melnikov, V. F. 2009, SSRv,
+149, 119
+Nakariakov, V. M., Melnikov, V. F., & Reznikova,
+V. E. 2003, A&A, 412, L7
+Reznikova, V. E., Antolin, P., & Van Doorsselaere,
+T. 2014, ApJ, 785, 86
+Roberts, B., Edwin, P. M., & Benz, A. O. 1983,
+Nature, 305, 688
+Shi, M., Li, B., & Guo, M. 2022, ApJL, 937, L25
+Soler, R., Goossens, M., Terradas, J., & Oliver, R.
+2014, ApJ, 781, 111
+Soler, R., Terradas, J., Oliver, R., Ballester, J. L.,
+& Goossens, M. 2010, ApJ, 712, 875
+Spruit, H. C. 1982, SoPh, 75, 3
+Su, J. T., Shen, Y. D., Liu, Y., Liu, Y., & Mao,
+X. J. 2012, ApJ, 755, 113
+Tan, B., Zhang, Y., Tan, C., & Liu, Y. 2010, ApJ,
+723, 25
+Terradas, J., Andries, J., Goossens, M., et al.
+2008, ApJL, 687, L115
+Tian, H., Young, P. R., Reeves, K. K., et al. 2016,
+ApJL, 823, L16
+Van Doorsselaere, T., De Groof, A., Zender, J.,
+Berghmans, D., & Goossens, M. 2011, ApJ, 740,
+90
+Van Doorsselaere, T., Kupriyanova, E. G., &
+Yuan, D. 2016, SoPh, 291, 3143
+Wedemeyer, S., Bastian, T., Brajˇsa, R., et al.
+2016, SSRv, 200, 1
+Yan, Y., Chen, Z., Wang, W., et al. 2021, Frontiers
+in Astronomy and Space Sciences, 8, 20
+
+13
+Zimovets, I. V., & Struminsky, A. B. 2010, SoPh,
+263, 163
+Zimovets, I. V., McLaughlin, J. A., Srivastava,
+A. K., et al. 2021, SSRv, 217, 66
+
+14
+Shi et al.
+Figure 1. Initial density distributions in the z = L0/2 and x = 0 cuts. The yellow volume shows the region
+where the non-thermal electrons occupy at t = 0. The white line marks the line of sight (LoS) for the case
+x0 = 0.
+
+0-5
+X (Mm)
+Z (Mm)
+5
+10
+0
+5
+15
+20
+25
+30
++2
+4+
+0 Y (Mm)
+2
++-2
+-4
+Y (Mm) 0
+-2 -
+5
+10
+-4 +
+15z (Mm)
+20
+-5
+25
+5
+3015
+Figure 2. Temporal evolution of vx sampled at (x, y, z) = (0, 0, L0/2) (blue), together with its low-pass
+(red) and high-pass (green) components.
+
+20
+original
+low-pass
+15
+ high-pass
+10
+5
+[km/s]
+0
+-5
+-10
+-15
+0
+25
+50
+75
+100
+125
+150
+175
+200
+t [s]16
+Shi et al.
+Figure 3. (a) Density distribution at the loop apex. (b) Velocity field along with its (c) long-period and
+(d) short-period components at the loop apex. (e) Temporal evolution of vx(x, y = 0, z = L0/2). The black
+dashed line marks the instant (116 s) where the figures in the left column are produced. An animated version
+of this figure is available that has the same layout as the static figure, and runs from 0–200s.
+
+Vx [km/s]
+p[10°mpcm-3]
+20
+-10
+0
+10
+20
+10
+20
+30
+40
+200
+(a)
+4 -
+(e)
+2 -
+[Mm]
+0 -
+175
+y
+2 
+-4
+-4
+-2 
+0
+2
+4
+20 km/s
+-150
+10
+(b) original
+5 
+[Mm]
+0
+125
+y
+-5 
+-10
+10
+5
+0
+5
+10
+100日
+20 km/s
+t
+10
+(c) low-pass
+[Mm]
+-75
+0
+y
+5 
+-10
+50
+-10
+-5
+0
+5
+10
+2 km/s
+10
+5
+[Mm]
+25
+0
+y
+10-
+0
+-10
+-5
+0
+10
+-10
+5
+0
+10
+x [Mm]
+X [Mm]17
+Figure 4. (a) Intensity variations of the 17 GHz emission for three LoS beams. (b) Low-pass and (c)
+high-pass components of the intensity variations for x0 = ±1 Mm.
+
+1.75
+(a) original
+Xo=0 Mm
+Xo=1 Mm
+Xo=-1 Mm
+1.50
+25
+50
+75
+100
+125
+150
+175
+200
+0
+(b) low-pass
+1.6
+1.4
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0.02
+(c) high-pass
+0.00
+Q88888888888888888888888888888
+-0.02
+0
+25
+50
+75
+100
+125
+150
+175
+200
+t [s]18
+Shi et al.
+Figure 5. (a) Intensity variations of the 17 GHz emission for different beam sizes for the case x0 = 1 Mm.
+(b) Low-pass and (c) high-pass components of (a).
+
+2.4
+(60 km)2
+(0.5 arcsec)2
+(1.0 arcsec)2
+(a) original
+2.2
+2.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+(b) low-pass
+2.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0.02
+(c) high-pass
+0.00
+-0.02
+0
+25
+50
+75
+100
+125
+150
+175
+200
+t [s]19
+Figure 6. Left column: Temporal evolution of vx sampled in the apex plane at (a) (x, y) = (0, 0)Mm, (b)
+(x, y) = (1, 0)Mm, and (c) (x, y) = (2, 0)Mm for the case σ = R. Any blue curve represents the original
+signal, while the red and green curves give the low-pass and high-pass components, respectively. The black
+dotted lines display the fitting curves of the high-(low-) pass components by an exponentially damping
+sinusoid, with the best-fit periods and damping times shown in each panel. Middle column: same as the left
+but for σ = 0.5R. Right column: same as the left but for σ = 2R.
+
+O-R
+20
+(a) x=0Mm
+PL = 56.9s TL = 115.9s
+Ps = 5.8s
+Ts = 102.9s
+[km/s]
+0
+0
+50
+100
+150
+200
+20
+PL = 56.9s
+(b) x=1Mm
+T = 116.8s
+Ts = 92.3s
+10
+[km/s]
+-10
+0
+50
+100
+150
+200
+(c) x=2Mm
+PL = 57.1s TL = 118.8s
+Ps = 5.8s
+Ts = 109.7s
+10 
+[km/s] 
+-0
+-10
+0
+50
+100
+150
+200
+t [s]O=0.5R
+20
+(a1) x=0Mm
+Pr = 56.1s TL = 78.4s
+Ps = 5.8s 
+Ts = 99.0s
+10
+[km/s] 
+0
+-10
+0
+50
+100
+150
+200
+P, = 56.2s TL = 87.4s
+(b1) x=1Mm
+Ps = 5.8s
+Ts = 92.9s
+10
+[km/s]
+0
+-10
+0
+50
+100
+150
+200
+(c1) x=2Mm
+PL = 56.7s TL = 106.0s
+10
+Ps = 5.8s
+Ts = 102.6s
+[km/s]
+-10
+0
+50
+100
+150
+200
+t [s]O-2R
+20
+(a2) x=0Mm
+Pt = 57.1s T = 119.7s
+Ps = 5.8s 
+Ts = 110.3s
+10
+[km/s] 
+i
+F0
+M
+-10 
+-20
+0
+50
+100
+150
+200
+20
+PL = 57.1s
+(b2) x=1Mm
+ TL = 120.1s
+Ps = 5.8s
+Ts = 96.5s
+10
+[km/s]
+10
+-10
+0
+50
+100
+150
+200
+20
+(c2) x=2Mm
+P = 57.2s 
+T = 121.1s
+Ps = 5.8s
+Ts = 120.5s
+10
+[km/s]
+M
+-10
+0
+50
+100
+150
+200
+t [s]20
+Shi et al.
+Figure 7. Velocity fields in the apex plane of (a) the simulated short-period component at some represen-
+tative instant and (b) the eigenvalue problem expectation for the first leaky kink mode.
+
+(a) 3D t-dependent simulation (α = R,t = 52s)
+(b) EVP expectation (Wrt = O)
+6
+2
+3
+1
+[Mm]
+0
+0
+y
+-3
+-1 
+-6
+-2
+-6
+-3
+0
+3
+6
+-1
+0
+1
+2
+2
+X [Mm]
+X/R

WNAzT4oBgHgl3EQfmP1C/content/tmp_files/2301.01559v1.pdf.txt

@@ -0,0 +1,1096 @@
+Limits of single-photon storage in a single Λ-type atom
+Zhi-Lei Zhang1, 2 and Li-Ping Yang1, ∗
+1Center for Quantum Sciences and School of Physics, Northeast Normal University, Changchun 130024, China
+2Graduate School of China Academy of Engineering Physics, Beijing 100193, China
+We theoretically investigate the limits of single-photon storage in a single Λ-type atom, specifically the trade-
+off between storage efficiency and storage speed. A control field can be exploited to accelerate the storage
+process without degrading efficiency too much. We show that the storage speed is ultimately limited by the total
+decay rate of the involved excited state. For a single-photon pulse propagating in a regular one-dimensional
+waveguide, the storage efficiency has an upper limit of 50%. Perfect single-photon storage can be achieved
+by using a chiral waveguide or the Sagnac interferometry. By comparing the storage efficiencies of Fock-state
+and coherent-state pulses, we reveal the influence of quantum statistics of light on photon storage at the single-
+photon level. Our results could pave a new way for the optimization of single-photon storage.
+I.
+INTRODUCTION
+Quantum memories for photon pulses are crucial for quan-
+tum communications [1–3] and quantum computing [4, 5].
+Via the photon echo technique or electromagnetically in-
+duced transparency (EIT) effect, storage of weak coherent-
+state pulse with efficiency ∼ 90% has been achieved [6–11].
+Recently, storage of Fock-state single-photon (FSSP) pulse
+with efficiency > 85% has also been realized in laser-cooled
+rubidium atoms [12].
+However, in these experiments, the
+length of the target pulse (τp) is around tens of microseconds
+and it is almost three orders larger than the lifetime (1/γ)
+of the involved excited state of atoms. High-speed optical
+quantum memories for short pulses (∼ 1 ns) has also been
+demonstrated [13], but the storage efficiency is relatively low
+(< 30%) [14]. Storage of single-photon pulses with high effi-
+ciency and high speed remains a challenge.
+Compared to an atomic ensemble [15–19], single-atom sys-
+tem [20–22] provides a novel platform to explore the funda-
+mental limits of single-photon storage, specifically, the trade-
+off between storage efficiency and storage speed. A closely
+related problem, i.e., single- or few-photon scattering by an
+atom, has been extensively studied [23–29]. Recently, the
+time-delay induced interference effect attracts new interests
+about photon scattering by a giant atom
+[30–37].
+How-
+ever, these research works focus more on the reflection and
+transmission coefficients, not the storage properties. On the
+other hand, the impact of photon number quantum fluctua-
+tions, which play a crucial role in light-atom interaction at the
+single-photon level, has not been adequately explored.
+In this work, we investigate the limits of single-atom-based
+single-photon storage without and with a control field. For a
+∗ [email protected]
+three-level atom placed in a regular one-dimensional waveg-
+uide, there exists an upper limit (0.5) on the single-photon
+storage efficiency. A chiral waveguide [38–40] or Sagnac in-
+terference technique [41–43] could be used to improve the ef-
+ficiency and to realize perfect storage. In the absence of a
+control field, we find high storage efficiency could be obtained
+only for long single-photon pulses (τp ≫ 1/γ). Thus, there is
+a trade-off between storage efficiency and storage speed. A
+control field could be applied to enhance the storage speed
+and improve the storage efficiency for single photon pulses
+with length τp = 1/γ. However, the storage speed is ulti-
+mately limited by the total decay rate of the involved excited
+state. Different from an atomic ensemble, a single multi-level
+atom exhibits high non-linearity. We show that the storage
+efficiency of a coherent-state single-photon (CSSP) pulse is
+much lower than that of an FSSP pulse, since nonlinear multi-
+photon processes have been suppressed.
+This article is structured as follows. In Sec. II, we begin
+by introducing the master equation for a single Λ-type atom
+driven by a quantum pulse. In Sec. III, we investigate the stor-
+age of single-photon pulses without a control field. In Sec. III,
+we show the storage speed could be accelerated via a control
+field. In Sec. V, we show a chiral waveguide and the Sagnac
+interferometer could be exploited to realize perfect storage of
+FSSP pulses. We briefly summarize in Sec. VI. Some details
+about the master equation are given in Appendix A.
+II.
+MASTER EQUATIONS FOR A Λ-TYPE ATOM DRIVEN
+BY A QUANTUM PULSE
+Recently, substantial efforts have been devoted to investi-
+gating the scattering of propagating quantum pulses by a local
+quantum system [22, 44–47]. A systematic master-equation
+approach has been developed to handle the dynamics the lo-
+arXiv:2301.01559v1  [quant-ph]  4 Jan 2023
+
+2
+!
+"
+#
+$!
+%"#
+2
+'((*)
+$$
+Ω%(*)
+Δ
+FIG. 1. Scattering of a single-photon pulse with center frequency ω0
+and wave-packet function ˜ξ(t) by a three-level Λ-type atom placed
+in a one-dimensional waveguide. A control pulse with frequency ωc
+and strength Ωc(t) is applied to assist the storage process. The decay
+rates of the excited state |e⟩ to the two ground states are γeg and γes
+and ∆ is the two-photon detuning.
+cal quantum scatter [48–50]. The input-output relation has
+also been incorporated to give the information of the outgoing
+temporal mode [51, 52]. Here, we follow the approach given
+in [48] to handle the storage of both FSSP and CSSP pulses in
+a single Λ-type atom. We show that the quantum statistics of
+the quantum pulse affect the storage efficiency significantly.
+The basic elements of the storage process are illustrated
+in Fig. 1. The Λ-type atom, which is described by Hamil-
+tonian Ha = ωe|e⟩⟨e| + ωs|s⟩⟨s|, contained two stable ground
+states |g⟩ and |s⟩ and one excited state |e⟩. For a regular one-
+dimensional waveguide, both the forward-propagating modes
+a(ω) and backward-propagating modes b(ω) have to be con-
+sidered. The Hamiltonian for the waveguide photons is given
+by Hp =
+�
+dω(ω0 + ω)[a†(ω)a(ω) + b†(ω)b(ω)], where the
+frequency of the waveguide photons has been expanded to the
+first-order of the wave-vector along the propagating direction
+around the near-resonant mode ω0 [53, 54]. The interaction
+between the atom and waveguide photons is described by
+Hint =
+�
+dω
+�
+geg(ω)σ†
+ge+ges(ω)σ†
+se
+�
+[a(ω)+b(ω)]+h.c.,
+(1)
+where σge = |g⟩⟨e|, and σse = |s⟩⟨e|. In addition, an extra
+control laser pulse could be applied to assist and accelerate
+the storage process. The interaction to the control field is de-
+scribed by Hamiltonian Hc = [Ωc(t) exp(−iωct)σse + h.c.]. To
+enhance the storage efficiency, the two-photon-resonance con-
+dition is required ,i.e., ωe − ω0 = ωe − ωs − ωc = ∆.
+Both CSSP pulses and FSSP pulses have been com-
+monly used in storage experiments [55, 56].
+The dynam-
+ics of a nonlinear scatter exhibit very different features un-
+der these two types of quantum pulses [48, 57, 58]. Usu-
+ally, the single-photon wave-packet creation operator aξ =
+�
+dωξ(ω)a†(ω) is used to generate quantum photon pulse
+wave function [59]. The pulse shape is determined by the
+normalized spectral amplitude function
+�
+dω|ξ(ω)|2 = 1. A
+forward-propagating CSSP and FSSP pulses are described by
+|1CS⟩ = exp
+�
+a†
+ξ − 1/2
+�
+|0⟩ and |1FS⟩ = a†
+ξ|0⟩, respectively. Ini-
+tially, the atom is prepared in the ground state |g⟩. The incident
+single-photon quantum pulse excites the atom and transfers it
+to state |s⟩ to realize the storage.
+A CSSP pulse can be treated as a classical driving field.
+The dynamics of the atom density matrix are governed by a
+Lindblad master equation ˙ρ(t) = [Lac + Lp(t)]ρ(t), where
+Lacρ(t) = − i[Ha + Hc, ρ(t)] − γeg + γes
+2
+{|e⟩⟨e|, ρ(t)}
++ γegσgeρ(t)σ†
+ge + γesσgeρ(t)σ†
+ge,
+(2)
+describes the spontaneous decay of the excited state |e⟩ with a
+classical control on the storage channel. The pumping of the
+atom by the CSSP pulse is described by the Liouville opera-
+tor [48]
+Lp(t)ρ(t) = −i
+�γeg
+2
+��˜ξ(t)σ†
+ge, ρ(t)
+�
++
+�˜ξ∗(t)σge, ρ†(t)
+��
+, (3)
+where ˜ξ(t) is the wave-packet function of the CSSP pulse de-
+termined by the Fourier transform of ξ(ω) [58]. We emphasize
+that there is a factor 1/
+√
+2 in Lp, because both the forward
+and backward waveguide modes will contribute to the decay
+of the excited state |e⟩, but the target pulse only contains for-
+ward modes. Here, we see that a CSSP pulse functions as a
+classical driving, since ρ†(t) = ρ(t).
+The traditional Lindblad master equation cannot be used to
+describe the interaction between a FSSP pulse and a localized
+quantum system [48]. A generalized Fock-state master equa-
+tion has been developed [48, 49],
+˙ρ(t) = Lacρ(t) + Lp(t)ρ01(t)
+(4)
+˙ρ01(t) = Lacρ01(t) − i
+�γeg
+2
+˜ξ∗(t)[σge, ρ00(t)],
+(5)
+˙ρ00(t) = Lacρ00(t),
+(6)
+where
+ρ(t) = TrR[U(t)ρ(0) ⊗ |1FS⟩⟨1FS| ⊗ |0b⟩⟨0b|U†(t)],
+ρ01(t) = TrR[U(t)ρ(0) ⊗ |0a⟩⟨1FS| ⊗ |0b⟩⟨0b|U†(t)],
+ρ00(t) = TrR[U(t)ρ(0) ⊗ |0a⟩⟨0a| ⊗ |0b⟩⟨0b|U†(t)],
+(7)
+and U(t) = T exp
+�
+−i
+� t
+0 (Ha + Hp + Hc + Hint)dt
+�
+is the time
+evolution operator of the whole system.
+The initial state
+
+3
+(a)
+(b)
+FIG. 2. Optimization of the storage efficiency for a Fock-state single-
+photon pulse [panel (a)] and a coherent-state single-photon [panel
+(b)] by varying pulse length τp and decay rate γeg. No control field is
+applied (i.e., Ωc = 0) and the two-photon detuning ∆ is set as zero.
+waveguide modes is |1FS⟩ ⊗ |0b⟩. We note that significantly
+different from a CSSP pule, the pumping by an FSSP pulse
+[i.e., Lp(t)ρ01(t)] can not be regarded as a classical driving
+since ρ†
+01(t) � ρ01(t).
+(b)
+(a)
+FIG. 3. Contrast between the storage efficiency of a Fock-state sin-
+gle pulse (FSSP) and a coherent-state single-photon (CSSP) pulse.
+The two-photon detuning ∆ is set as zero.
+(a) Optimized stor-
+age efficiency in the absence of control field with Ωc = 0 and
+γeg = γes = γ/2. (b) Optimized storage efficiency in the presence
+of a control field with strength Ω = 0.7γ, length a = 0.9τp, and
+relative delay b = 0.6τp. The other parameters have been taken as
+γeg = 0.9γ, and γes = 0.1γ.
+III.
+STORAGE OF A SINGLE-PHOTON PULSE WITHOUT
+CONTROL FIELD
+In this section, we study the storage of a single-photon
+pulse in the absence of a control pulse, i.e., Ωc = 0. The ad-
+vantage of this storage scheme is that no information about the
+arrival time of the target pulse is needed. The atom initially
+prepared in state |g⟩ will be excited to state |e⟩ and sponta-
+(b)
+(a)
+FIG. 4. (a) Sketch map of the relative delay b between the target
+pulse (blue solid line) and the control pulse (orange dashed line). (b)
+Storage efficiency of a Fock-state pulse varies with the magnitude Ω
+and width a of a control pulse. γeg = 0.9γ, γes = 0.1γ, and b =
+0.6τp. The fitting white dashed line 2aΩ √π = 2.26 characterizes the
+constant area under the envelope function Ωc(t).
+neously decays to state |g⟩ or the storage state |s⟩. The storage
+efficiency of a single-photon pulse is defined as the steady-
+state probability Ps of state |s⟩. We show that the decay rates
+of the storage channel and the pumping channel must be care-
+fully matched to optimize storage efficiency. We also show
+that the storage efficiency of a CSSP pulse will be much lower
+than that of an FSSP pulse.
+There are three parameters to optimize the storage effi-
+ciency, i.e., the two decay rates γeg and γes and the length
+of the target pulse. Without loss of generality, we assume the
+target pulse is of the Gaussian shape
+˜ξ(t) =
+������
+1
+2πτ2p
+������
+1
+4
+exp
+������−(t − t0)2
+4τ2p
+������ ,
+(8)
+where t0 is the time of the pulse arriving at the atom and τp is
+the half-length of the pulse. In the following, we fix the total
+decay rate γ = γeg + γes of state |e⟩ and take it as the unit of
+frequency, i.e., γ = 1.
+Maximum storage efficiency will be obtained if the decay
+rates of the pumping and storage channels are equal to each
+other, i.e., γeg = γes = γ/2. In Fig. 2, we plot the storage
+efficiencies for an FSSP pulse [panel (a)] and a CSSP pulse
+[panel (b)] as a function of γeg and pulse length τp. For a
+given pulse length, the maximum storage efficiency locates
+at γeg = γ/2 for both FSSP and CSSP pulses. On the other
+hand, the storage efficiency of a longer pulse is larger. This
+can be seen more clearly in Fig. 3(a). To obtain higher storage
+efficiency, one needs to sacrifice the storage speed.
+The storage efficiency is strongly affected by the quantum
+statistics of the target pulse. As shown in Fig. 3(a), the stor-
+age efficiency of an FSSP pulse is much higher than that of a
+CSSP pulse. The few-level atom functions as a nonlinear sys-
+
+101
+100
+10-1
+0
+0.25
+0.5
+0.75101
+100
+0.5
+10-1
+0
+0
+0.25
+0.5
+0.75
+eq一FSSP
+---CSSP
+100
+10-1
+101
+102
+10-2
+YTp0.5
+-FSSP
+0.4
+---CSSP
+0.3
+S
+P
+0.2
+0.1
+0
+100
+101
+102
+Tp0.5
+3
+0.25
+2
+0.1
+2
+3
+1
+5
+4
+2+es Amplitude
+b>0
+Target pulse
+2ia
+p
+Time
+ Amplitude
+<0
+Control pulse
+2T
+Time
+p4
+(a)
+(b)
+(c)
+FIG. 5. Optimization of storage efficiency of a Fock-state single-photon pulse with γeg = 0.9γ, γes = 0.1γ, and Ω = 0.7γ. (a) Optimization
+with fixed pulse length τp = 1/γ and two-photon detuning ∆ = 0. The fitting white dashed line is given by b+2a = 1.2×2τp. (b) Optimization
+with fixed delay b = 0.6τp and ∆ = 0. (c) Optimization with half-length a = 0.9τp and delay b = 0.6τp of the control pulse.
+tem [58, 60], and multi-photon processes are suppressed in the
+extremely weak (single photon) pumping case. We also note
+that there exists an upper limit in the storage efficiency. When
+τp ≫ 1/γ, the storage efficiency of the FSSP (CSSP) pulse
+CSSP reaches the upper limit 0.5 (0.4). This low storage ef-
+ficiency fundamentally results from the fact that the pumping
+rate is half of the decay rate of the |g⟩ ↔ |e⟩ channel [46, 58].
+Perfect storage of single-photon pulses can be realized by en-
+hancing the pumping rate as shown in Sec. V.
+IV.
+STORAGE OF A SINGLE-PHOTON PULSE WITH A
+CONTROL FIELD
+In Sec. III, we show that the storage efficiency for short
+single-photon pulses (τp ≤ 1/γ) is relatively low. To assist
+and accelerate the single-photon storage, an extra control field
+could be applied to |e⟩ → |s⟩ channel [15, 19, 20, 61]. In the
+absence of a control pulse, maximum storage efficiency is ob-
+tained under the decay-rate matching condition γeg = γes. The
+control pulse provides new parameters, which can be much
+more easily controlled in experiments, to optimize storage ef-
+ficiency. We show that the total decay rate γ = γeg + γes plays
+an essential role in the storage process. Specifically, it limits
+the maximum storage speed. This marks a significant differ-
+ence from the storage of a single-photon pulse in an atomic
+ensemble, where more attention was paid to the
+√
+N-enhanced
+(N is the atom number) coupling strength between the target
+photon and the collective atomic states [62–64].
+In the following, we take a Gaussian control pulse as an
+example. Our main results are also valid for other types of
+control pules. The envelope of the control pulse is given by
+Ωc(t) = Ω exp
+�������−
+�t − t0 − b
+2a
+�2�������,
+(9)
+where Ω characterizes the effective strength of the control
+pulse, a is its half-width, and t0 is the time of the pulse center
+arriving at the atom. As shown in Fig. 4 (a), b is the relative
+delay between the target single-photon pulse and the control
+pulse. In addition to γeg and γes, we now have three more eas-
+ily controlled parameters to optimize single-photon storage.
+Similar to the Ωc
+=
+0 case, transition rates between
+the pumping channel and the storage channel also need to
+be balanced to obtain larger storage efficiency.
+As shown
+in Fig. 4 (b), maximum storage efficiency locates around
+(Ω + γes)/γeg ≈ 1 when the length of the control pulse is
+long enough. For a short control pulse (a < τp), a larger
+strength Ω is required to guarantee that the energy of the con-
+trol pulse is enough to transfer the population from state |e⟩
+to state |s⟩. The white dashed line denotes the fitting curve
+2aΩ √π = 2.26, i.e., the area under the envelope function
+Ωc(t) is a constant. To investigate the benefit of the control
+pulse, we will take γeg = 0.9γ and γes = 0.1γ when a control
+pulse is applied.
+The relative delay b and half-length a of the control pulse
+need to be matched to obtain larger storage efficiency. In Fig.5
+(a), we plot the storage probability Ps of an FSSP pulse as a
+function of a and b. The largest storage efficiency locates at
+b = 0.6τp and a = 0.9τp, i.e., a positive delay and a length
+comparable to the length of the target pulse. A similar de-
+lay was also required for an atomic ensemble optical mem-
+ory [19]. For a negative delay b, higher storage efficiency
+could also be obtained around the line b+2a = 1.2×2τp. This
+
+4
+2
+0
+-2
+1
+2
+3
+4
+5
+a5
+4
+3
+a
+2
+0.1
+10-1
+100
+101
+Tp101
+0.5
+0.25
+10-1
+0
+-3
+-2
+-1
+0
+1
+2
+35
+(a)
+(b)
+FIG. 6. Sketch map of two possible approaches to improving stor-
+age efficiency: (a) The atom only couples to the forward propagating
+photons in a perfect chiral waveguide. (b) For the Sagnac interferom-
+etry method, the target pulse is split into two smaller pulses, which
+enter the waveguide at different ends.
+(a)
+(b)
+FIG. 7. Comparison of improved storage efficiency of a Fock-state
+single-photon pulse (a) and a coherent-state single-photon pulse (b)
+without a control field.
+guarantees that the control pulse and the target single-photon
+pulse always have sufficient overlap.
+There exists a favorable length τp of the target pulse in stor-
+age efficiency optimization with fixed delay b and strength Ω
+of the control pulse. A larger storage efficiency could be ob-
+tained for τp = 1/γ as shown in Fig. 3 (b). This marks a signif-
+icant difference from the case in the absence of a control pulse,
+in which longer single-photon pulses (τp ≫ 1/γ) always have
+higher storage efficiency [see Fig. 3 (a)]. In Fig.5 (b), we plot
+the storage probability Ps of an FSSP pulse as a function of
+τp and a = 0.9τp with b = 0.6τp and Ω = 0.7γ. We show
+that larger storage efficiency is obtained around τp = 1/γ.
+Thus, the control pulse could be used to improve the stor-
+age speed. Previously, off-resonant Raman technique [19] has
+been explored to store a single broadband (short) photon in an
+atomic ensemble beyond the adiabatic storage frame based on
+EIT [15]. However, in the single-atom case, the two-photon
+detuning ∆ will reduce the storage efficiency greatly as shown
+in Fig. 5 (c). Moreover, large storage efficiency is still ob-
+tained around τp = 1/γ for fixed a and b. No extra acceler-
+ation is obtained with non-zero detuning ∆. The storage of a
+CSSP pulse is similar to that of an FSSP pulse, but with lower
+efficiency.
+(a)
+(b)
+FIG. 8. Comparison of the improved storage efficiency of Fock-
+state single-photon (FSSP) and coherent-state single-photon (CSSP)
+pulses without (a) and with (b) a control pulse. The two-photon de-
+tuning is set as ∆ = 0. (a) γeg = γes = γ/2. (b) γeg = 0.9γ, γes = 0.1γ,
+Ω = 0.7γ, a = 0.9τp, and b = 0.6τp.
+(a)
+(b)
+FIG. 9.
+Optimization of storage efficiency of a Fock-state single-
+photon pulse in presence of control field. γeg = 0.9γ, γes = 0.1γ,
+Ω = 0.7γ, b = 0.6τp. (a) Two-photon resonance case with ∆ = 0. (b)
+Off-resonance case witha = 0.9τp.
+V.
+EFFICIENT STORAGE VIA EXPLOITING A CHIRAL
+WAVEGUIDE OR A SAGNAC INTERFEROMETER
+In previous sections, we show that the storage of an FSSP
+pulse in a single three-level atom is limited to 0.5 with or
+without a control pulse. The storage efficiency for a CSSP is
+even lower. This low efficiency strongly hampers the practical
+application of the single-atom storage scheme. In this sec-
+tion, we show that perfect storage of single-photon pulse in a
+three-level atom can be realized by exploiting a chiral waveg-
+uide [65–68] or a Sagnac interferometer [41, 42]. Previously,
+these two methods have been applied successfully to enhance
+the frequency conversion efficiency [43, 69, 70] and to con-
+trol single-photon transport [38, 39, 71–73]. The underlying
+mechanism of both approaches is the same, i.e., increasing the
+coupling efficiency between the atom and the pulse modes.
+For a perfect chiral waveguide, the atom only interacts
+
+101
+100
+10-1
+0
+0.25
+0.5
+0.75
+Yeg101
+100
+0.5
+10-1
+0
+0
+0.25
+0.5
+0.75
+eg0.8
+一FSSP
+---CSSP
+0.6
+S
+0.4
+I
+0.2
+1
+100
+10-1
+101
+102
+10-2
+TpFSSP
+---CSSP
+100
+101
+102
+10-1
+10-2
+Tp5
+4
+3
+P
+a
+2
+1
+0.1
+100
+10-1
+101
+Tp101
+0.5
+100
+10-1
+0
+-3
+-2
+0
+2
+3
+-1
+V6
+(a)
+(b)
+with control
+without control
+without control
+%"# = 0.9
+%"# = 0.9
+%"# = 0.5
+FIG. 10.
+Global optimization of storage efficiency of a Fock-
+state single-photon pulse in high-dimensional parameter space. (a)
+The shift of the favorable length of the target pulse.
+The three
+lines {blue solid, orange dotted, yellow dashed} are obtained with
+parameters b = {1.3τp, 0.6τp, −0.4τp}, a = {0.7τp, 0.9τp, 1.2τp},
+Ω = {1.5τp, 0.7τp, 0.4τp}, and ∆ = 0. (b) Comparison of the stor-
+age efficiency with and without a control pulse. The blue solid line
+gives the global maximum storage efficiency with γeg = 0.9γ. The
+orange dashed line describes the optimal storage efficiency without
+control pulse (γeg = γes = 0.5γ). The yellow dashed line denotes the
+case without control pulse and γeg = 0.9γ.
+with photons propagating in one direction [see Fig. 6 (a)],
+such as the forward-propagating modes a(ω). The backward-
+propagating modes will not contribute to the scattering and
+storage of the target single-photon pulse. The spontaneous
+decay of the excited state comes solely from the interaction
+with forward-propagating modes. In this case, the pumping
+rate of the single-photon pulse does not change, but the decay
+rates of state |e⟩ are halved. Thus, the 1/
+√
+2-factor in Eqs. (3)
+and (5) will be removed.
+For a Sagnac interferometer case, the incident single-
+photon pulse will be split into two identical small pulses via
+a 50 : 50 beam splitter. These two small pulses enter the
+waveguide at two different ends [see Fig. 6 (b)].
+Mathe-
+matically, the wave-guide modes can always be re-expanded
+with even and odd modes a±(ω) = [a(ω) ± b(ω)] /
+√
+2. From
+Eq. (1), we see that the atom is only coupled to even modes.
+Thus, only even modes will contribute to the spontaneous de-
+cay of the atomic excited state. By carefully tuning the rel-
+ative phase between the two small pulses, one can guaran-
+tee that the target pulse (i.e., the superposition of two small
+pulses) only contains even modes. The target pulse is now
+described by a new single-photon wave-packet creation oper-
+ator aξ =
+�
+dωξ(ω)[a†(ω) + b†(ω)]/
+√
+2 =
+�
+dωξ(ω)a†
++(ω). In
+this case, the decay rates of state |e⟩ do not change, but the
+pumping rate of the single-photon pulse gets doubled. Thus,
+the 1/
+√
+2-factor in Eqs. (3) and (5) will be removed.
+We now show that the perfect storage of single-photon
+pulses in a single three-level atom can be realized with a chi-
+ral waveguide or Sagnac interferometer. In Fig.7, we plot the
+storage probability versus τp and γeg for an FSSP [panel (a)]
+and a CSSP pulse [panel (b)] in the absence of control pulse.
+Similar to the regular waveguide case (see Fig. 2), larger stor-
+age efficiency is obtained under the decay-rate matching con-
+dition γeg = γes. However, the maximum storage efficiency
+of an FSSP pule can now reach 1 at the long-pulse limit
+τp ≫ 1/γ as shown in Fig. 8 (a). The upper limit of the
+storage efficiency of a CSSP pulse has also been raised from
+0.4 to be larger than 0.6.
+The storage process can be accelerated by a control pulse
+without sacrificing the storage efficiency too much. Similar
+to Sec. IV, there exists an favorable pulse length τp in stor-
+age efficiency optimization with fixed b and Ω as shown in
+Fig. 8 and Fig. 9. We emphasize that the maximum storage ef-
+ficiency in Fig. 9 is a local one, not the global maximum in the
+high-dimensional parameter space {a, b, Ω, ∆, τp}. As shown
+in Fig. 10 (a), the favorable τp moves toward longer pulses by
+varying the control pulse parameters, specifically the relative
+delay b. We give the global maximum storage efficiency via
+brute-force numerical simulations as shown by the blue solid
+line in Fig. 10 (d). Compared to cases without a control pulse
+(the orange solid and yellow dotted lines), much larger storage
+efficiency for relatively short pulses τp ∼ 1/γ can be obtained
+under a control pulse. The storage efficiency of an FSSP pulse
+with τp = 1/γ can reach ∼ 0.9 [see Fig. 8 (b)]. However, the
+storage speed is still limited by the total spontaneous decay
+rate γ of the excited state |e⟩.
+VI.
+CONCLUSION
+We use a simple model, which is composed of a single Λ-
+type atom placed in a 1D waveguide, to explore the limits of
+single-photon storage.
+We show that for a regular waveg-
+uide, the storage efficiency of an FSSP pulse is limited to
+0.5 and the efficiency of a CSSP pulse is even lower. Perfect
+single-photon storage could be achieved by exploiting a chiral
+waveguide or a Sagnac interferometer. We find that there is
+a trade-off between storage efficiency and storage speed. A
+control pulse can be applied to accelerate the storage process.
+However, the storage speed is ultimately limited by the total
+decay rate of the involved excited state.
+One of the authors (L.P.Y) showed that the absorption speed
+of a single-photon pulse is limited by the width of the atom-
+light interaction spectrum [58]. For an atom interacting with
+1D wave-guide modes, the interaction spectrum is almost flat.
+Thus, the storage speed is mainly limited by de-excitation pro-
+
+10-
+100
+101
+102
+10-2
+YTp0.8
+0.6
+S
+0.4
+0.2
+10-1
+100
+101
+102
+10-2
+Tp7
+cesses. In most experiments, an atomic ensemble instead of a
+single atom was used as the storage media. In addition to
+the pumping strength, the decay rate of the pumping chan-
+nel is also enhanced by a factor of N (N is an effective atom
+number). Single-photon storage with high efficiency and high
+speed could be achieved in the atomic ensemble.
+ACKNOWLEDGEMENTS
+The authors thank Xin Yue for the helpful discussion.
+This work is supported by National Key R&D Program
+of China (Grant No.
+2021YFE0193500) and NSFC Grant
+No.12275048.
+Appendix A: Deduction of master equations
+We give some details of deriving the master equation for an
+atom driven by a quantum pulse. The Heisenberg equation of
+a waveguide mode is given by
+˙a(ω, t) = −iωa(ω) − igegσge(t) − igesσse(t)e−i(ωc−ω0)t,
+(A1)
+where we have set ℏ = 1. We integrate the formal solution of
+a(ω, t) over ω to obtain [48]
+�
+a(ω, t)dω =
+√
+2πain(t) − iπgegσge(t) − iπgesσse(t)e−i(ωc−ω0)t,
+(A2)
+where the so-called input-field ain(t) is an explicitly time-
+dependent operator
+ain(t) =
+1
+√
+2π
+� ∞
+−∞
+dωe−iωta(ω).
+(A3)
+Note that the two-time commutator of the input field yields a
+δ− function
+�
+ain(t), a†
+in(t′)
+�
+= δ(t − t′).
+(A4)
+In obtaining Eq.(A2), we have used the Wigner-Weisskopf
+approximation by treating the coupling coefficients as
+frequency-independent constants
+geg(ω) ≈ geg(ω0)=
+�γeg
+4π , ges(ω) ≈ ges(ω0)=
+�γes
+4π , (A5)
+where γeg and γes are the decay rates of the excited state |e⟩ to
+ground state |g⟩ and storage state |s⟩, respectively. We can get
+the similar expression of bin(t) similarly.
+The pumping effect from an incident FSSP pulse is charac-
+terized by the following relations
+ain(t) |1FS⟩ ⊗ |0b⟩=
+1
+√
+2π
+�
+dωξ(ω)e−iωt |0⟩= ˜ξ(t) |0⟩ ,
+(A6)
+bin(t) |1FS⟩ ⊗ |0b⟩ = 0.
+(A7)
+We can obtain the motion equation an arbitrary operator of the
+system [48, 74]
+˙X(t) =i[Hs, X(t)] + iΩc(t)[σ†
+se(t) + σse(t), X(t)]
++ [σ†
+ge(t), X(t)]
+������i
+�γeg
+2 ain(t) + i
+�γeg
+2 bin(t) + γeg
+2 σge(t) +
+√γegγes
+2
+σse(t)e−i(ωc−ω0)t
+������
++ ei(ωc−ω0)t[σ†
+se(t), X(t)]
+�
+i
+�
+γes
+2 ain(t) + i
+�
+γes
+2 bin(t) +
+√γegγes
+2
+σge(t) + γes
+2 σse(t)e−i(ωc−ω0)t
+�
++
+������i
+�γeg
+2 a†
+in(t) + i
+�γeg
+2 b†
+in(t) − γeg
+2 σ†
+ge(t) −
+√γegγes
+2
+σ†
+se(t)ei(ωc−ω0)t
+������ [σge(t), X(t)]
++ e−i(ωc−ω0)t
+�
+i
+�
+γes
+2 a†
+in(t) + i
+�
+γes
+2 b†
+in(t) −
+√γegγes
+2
+σ†
+ge(t) − γes
+2 σ†
+se(t)ei(ωc−ω0)t
+�
+[σse(t), X(t)].
+(A8)
+Using the relations ((A7)-(A8)), we obtain the motion equa-
+tions for ρ, ρ01, and ρ00 in the main text [i.e., Eqs. (4-6)]. Note
+that the fast-oscillating terms have been neglected. Different
+from Eq. (A6) for an FSSP pulse, the action of the input-field
+operator on a CSSP pulse is given by
+ain(t) |1CS⟩ ⊗ |0b⟩ = ˜ξ(t) |1CS⟩ ⊗ |0b⟩ .
+(A9)
+In this case, we obtain a single master equation as given in
+Eq. (3), where the CSSP pulse functions as a classical pump.
+
+8
+[1] L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Long-
+distance quantum communication with atomic ensembles and
+linear optics, Nature 414, 413 (2001).
+[2] H. J. Kimble, The quantum internet, Nature 453, 1023 (2008).
+[3] C. Simon, Towards a global quantum network, Nature Photon-
+ics 11, 678 (2017).
+[4] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and
+G. J. Milburn, Linear optical quantum computing with photonic
+qubits, Rev. Mod. Phys. 79, 135 (2007).
+[5] J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, Dis-
+tributed quantum computation over noisy channels, Phys. Rev.
+A 59, 4249 (1999).
+[6] M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, Efficient
+quantum memory for light, Nature 465, 1052 (2010).
+[7] Y.-W. Cho, G. Campbell, J. Everett, J. Bernu, D. Higginbottom,
+M. Cao, J. Geng, N. Robins, P. Lam, and B. Buchler, Highly
+efficient optical quantum memory with long coherence time in
+cold atoms, Optica 3, 100 (2016).
+[8] J. Geng, G. Campbell, J. Bernu, D. Higginbottom, B. Sparkes,
+S. Assad, W. Zhang, N. Robins, P. K. Lam, and B. Buchler,
+Electromagnetically induced transparency and four-wave mix-
+ing in a cold atomic ensemble with large optical depth, New
+Journal of Physics 16, 113053 (2014).
+[9] Y.-H. Chen, M.-J. Lee, I.-C. Wang, S. Du, Y.-F. Chen, Y.-C.
+Chen, and I. A. Yu, Coherent optical memory with high stor-
+age efficiency and large fractional delay, Phys. Rev. Lett. 110,
+083601 (2013).
+[10] P. Vernaz-Gris, K. Huang, M. Cao, A. S. Sheremet, and J. Lau-
+rat, Highly-efficient quantum memory for polarization qubits in
+a spatially-multiplexed cold atomic ensemble, Nature commu-
+nications 9, 1 (2018).
+[11] Y.-F. Hsiao, P.-J. Tsai, H.-S. Chen, S.-X. Lin, C.-C. Hung, C.-H.
+Lee, Y.-H. Chen, Y.-F. Chen, I. A. Yu, and Y.-C. Chen, Highly
+efficient coherent optical memory based on electromagnetically
+induced transparency, Phys. Rev. Lett. 120, 183602 (2018).
+[12] Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du,
+H. Yan, and S.-L. Zhu, Efficient quantum memory for single-
+photon polarization qubits, Nature Photonics 13, 346 (2019).
+[13] K. Reim, J. Nunn, V. Lorenz, B. Sussman, K. Lee, N. Lang-
+ford, D. Jaksch, and I. Walmsley, Towards high-speed optical
+quantum memories, Nature Photonics 4, 218 (2010).
+[14] N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet,
+B. Gouraud, and J. Laurat, Waveguide-coupled single collec-
+tive excitation of atomic arrays, Nature 566, 359 (2019).
+[15] M. Fleischhauer and M. D. Lukin, Dark-state polaritons in elec-
+tromagnetically induced transparency, Phys. Rev. Lett. 84, 5094
+(2000).
+[16] M. Fleischhauer and M. D. Lukin, Quantum memory for pho-
+tons: Dark-state polaritons, Phys. Rev. A 65, 022314 (2002).
+[17] A. E. Kozhekin, K. Mølmer, and E. Polzik, Quantum memory
+for light, Phys. Rev. A 62, 033809 (2000).
+[18] A. V. Gorshkov, A. Andr´e, M. Fleischhauer, A. S. Sørensen,
+and M. D. Lukin, Universal approach to optimal photon storage
+in atomic media, Phys. Rev. Lett. 98, 123601 (2007).
+[19] J. Nunn, I. A. Walmsley, M. G. Raymer, K. Surmacz, F. C. Wal-
+dermann, Z. Wang, and D. Jaksch, Mapping broadband single-
+photon wave packets into an atomic memory, Phys. Rev. A 75,
+011401 (2007).
+[20] A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kim-
+ble, Reversible state transfer between light and a single trapped
+atom, Phys. Rev. Lett. 98, 193601 (2007).
+[21] H. P. Specht, C. N¨olleke, A. Reiserer, M. Uphoff, E. Figueroa,
+S. Ritter, and G. Rempe, A single-atom quantum memory, Na-
+ture 473, 190 (2011).
+[22] L. Giannelli, T. Schmit, T. Calarco, C. P. Koch, S. Ritter, and
+G. Morigi, Optimal storage of a single photon by a single intra-
+cavity atom, New Journal of Physics 20, 105009 (2018).
+[23] J.-T. Shen and S. Fan, Coherent single photon transport in a one-
+dimensional waveguide coupled with superconducting quantum
+bits, Phys. Rev. Lett. 95, 213001 (2005).
+[24] L. Zhou, Z. R. Gong, Y.-x. Liu, C. P. Sun, and F. Nori, Con-
+trollable scattering of a single photon inside a one-dimensional
+resonator waveguide, Phys. Rev. Lett. 101, 100501 (2008).
+[25] T. Shi and C. P. Sun, Lehmann-symanzik-zimmermann reduc-
+tion approach to multiphoton scattering in coupled-resonator ar-
+rays, Phys. Rev. B 79, 205111 (2009).
+[26] D. Witthaut and A. S. Sørensen, Photon scattering by a three-
+level emitter in a one-dimensional waveguide, New Journal of
+Physics 12, 043052 (2010).
+[27] D. Roy, Two-photon scattering by a driven three-level emitter in
+a one-dimensional waveguide and electromagnetically induced
+transparency, Phys. Rev. Lett. 106, 053601 (2011).
+[28] H. Zheng and H. U. Baranger, Persistent quantum beats and
+long-distance entanglement from waveguide-mediated interac-
+tions, Phys. Rev. Lett. 110, 113601 (2013).
+[29] L. Zhou, L.-P. Yang, Y. Li, and C. P. Sun, Quantum routing of
+single photons with a cyclic three-level system, Phys. Rev. Lett.
+111, 103604 (2013).
+[30] L. Guo, A. Grimsmo, A. F. Kockum, M. Pletyukhov, and G. Jo-
+hansson, Giant acoustic atom: A single quantum system with a
+deterministic time delay, Phys. Rev. A 95, 053821 (2017).
+[31] W. Zhao and Z. Wang, Single-photon scattering and bound
+states in an atom-waveguide system with two or multiple cou-
+pling points, Phys. Rev. A 101, 053855 (2020).
+[32] X. Gu, A. F. Kockum, A. Miranowicz, Y.-x. Liu, and F. Nori,
+Microwave photonics with superconducting quantum circuits,
+Physics Reports 718, 1 (2017).
+[33] A. F. Kockum, G. Johansson, and F. Nori, Decoherence-free
+interaction between giant atoms in waveguide quantum electro-
+dynamics, Phys. Rev. Lett. 120, 140404 (2018).
+[34] L. Guo, A. F. Kockum, F. Marquardt, and G. Johansson, Os-
+cillating bound states for a giant atom, Phys. Rev. Research 2,
+
+9
+043014 (2020).
+[35] X. Wang, T. Liu, A. F. Kockum, H.-R. Li, and F. Nori, Tun-
+able chiral bound states with giant atoms, Phys. Rev. Lett. 126,
+043602 (2021).
+[36] J.-P. Zou, R.-Y. Gong, and Z.-L. Xiang, Tunable single-photon
+scattering of a giant λ-type atom in a squid-chain waveguide,
+Frontiers in Physics , 361 (2022).
+[37] X.-L. Yin, Y.-H. Liu, J.-F. Huang, and J.-Q. Liao, Single-photon
+scattering in a giant-molecule waveguide-qed system, Phys.
+Rev. A 106, 013715 (2022).
+[38] C. Gonzalez-Ballestero,
+E. Moreno,
+F. J. Garcia-Vidal,
+and A. Gonzalez-Tudela, Nonreciprocal few-photon routing
+schemes based on chiral waveguide-emitter couplings, Phys.
+Rev. A 94, 063817 (2016).
+[39] L. Du, Y.-T. Chen, and Y. Li, Nonreciprocal frequency conver-
+sion with chiral Λ-type atoms, Phys. Rev. Research 3, 043226
+(2021).
+[40] X. Wang, Z.-M. Gao, J.-Q. Li, H.-B. Zhu, and H.-R. Li, Un-
+conventional quantum electrodynamics with a hofstadter-ladder
+waveguide, Phys. Rev. A 106, 043703 (2022).
+[41] M. Bradford, K. C. Obi, and J.-T. Shen, Efficient single-photon
+frequency conversion using a sagnac interferometer, Phys. Rev.
+Lett. 108, 103902 (2012).
+[42] M. Bradford and J.-T. Shen, Single-photon frequency conver-
+sion by exploiting quantum interference, Phys. Rev. A 85,
+043814 (2012).
+[43] L. Du and Y. Li, Single-photon frequency conversion via a giant
+Λ-type atom, Phys. Rev. A 104, 023712 (2021).
+[44] T. Macha, E. Uru˜nuela, W. Alt, M. Ammenwerth, D. Pandey,
+H. Pfeifer, and D. Meschede, Nonadiabatic storage of short light
+pulses in an atom-cavity system, Phys. Rev. A 101, 053406
+(2020).
+[45] T. Shi, D. E. Chang, and J. I. Cirac, Multiphoton-scattering the-
+ory and generalized master equations, Phys. Rev. A 92, 053834
+(2015).
+[46] Z. Liao, X. Zeng, S.-Y. Zhu, and M. S. Zubairy, Single-photon
+transport through an atomic chain coupled to a one-dimensional
+nanophotonic waveguide, Phys. Rev. A 92, 023806 (2015).
+[47] Z. Liao, H. Nha, and M. S. Zubairy, Dynamical theory of single-
+photon transport in a one-dimensional waveguide coupled to
+identical and nonidentical emitters, Phys. Rev. A 94, 053842
+(2016).
+[48] K. M. Gheri, K. Ellinger, T. Pellizzari, and P. Zoller, Photon-
+wavepackets as flying quantum bits, Fortschritte der Physik:
+Progress of Physics 46, 401 (1998).
+[49] B. Q. Baragiola, R. L. Cook, A. M. Bra´nczyk, and J. Combes,
+n-photon wave packets interacting with an arbitrary quantum
+system, Phys. Rev. A 86, 013811 (2012).
+[50] J. Combes, J. Kerckhoff, and M. Sarovar, The slh frame-
+work for modeling quantum input-output networks, Advances
+in Physics: X 2, 784 (2017).
+[51] A. H. Kiilerich and K. Mølmer, Input-output theory with quan-
+tum pulses, Phys. Rev. Lett. 123, 123604 (2019).
+[52] A. H. Kiilerich and K. Mølmer, Quantum interactions with
+pulses of radiation, Phys. Rev. A 102, 023717 (2020).
+[53] J.-T. Shen and S. Fan, Theory of single-photon transport in a
+single-mode waveguide. i. coupling to a cavity containing a
+two-level atom, Phys. Rev. A 79, 023837 (2009).
+[54] J.-F. Huang, T. Shi, C. P. Sun, and F. Nori, Controlling single-
+photon transport in waveguides with finite cross section, Phys.
+Rev. A 88, 013836 (2013).
+[55] D.-C. Liu, P.-Y. Li, T.-X. Zhu, L. Zheng, J.-Y. Huang, Z.-Q.
+Zhou, C.-F. Li, and G.-C. Guo, On-demand storage of photonic
+qubits at telecom wavelengths, Phys. Rev. Lett. 129, 210501
+(2022).
+[56] J. V. Rakonjac,
+G. Corrielli,
+D. Lago-Rivera,
+A. Seri,
+M. Mazzera, S. Grandi, R. Osellame, and H. de Riedmatten,
+Storage and analysis of light-matter entanglement in a fiber-
+integrated system, Science Advances 8, eabn3919 (2022).
+[57] Y. Wang, J. c. v. Min´aˇr, L. Sheridan, and V. Scarani, Efficient
+excitation of a two-level atom by a single photon in a propagat-
+ing mode, Phys. Rev. A 83, 063842 (2011).
+[58] L.-P. Yang, H. X. Tang, and Z. Jacob, Concept of quantum tim-
+ing jitter and non-markovian limits in single-photon detection,
+Phys. Rev. A 97, 013833 (2018).
+[59] R. Loudon, The quantum theory of light (OUP Oxford, 2000) ,
+Chap. 6.
+[60] H.-Y. Yao and S.-W. Li, Enhancing photoelectric current by
+nonclassical light, New Journal of Physics 22, 123011 (2020).
+[61] A. E. Kozhekin, K. Mølmer, and E. Polzik, Quantum memory
+for light, Phys. Rev. A 62, 033809 (2000).
+[62] A. V. Gorshkov, A. Andr´e, M. D. Lukin, and A. S. Sørensen,
+Photon storage in Λ-type optically dense atomic media. i. cavity
+model, Phys. Rev. A 76, 033804 (2007).
+[63] A. V. Gorshkov, A. Andr´e, M. D. Lukin, and A. S. Sørensen,
+Photon storage in Λ-type optically dense atomic media. ii. free-
+space model, Phys. Rev. A 76, 033805 (2007).
+[64] A. V. Gorshkov, T. Calarco, M. D. Lukin, and A. S. Sørensen,
+Photon storage in Λ-type optically dense atomic media. iv. op-
+timal control using gradient ascent, Phys. Rev. A 77, 043806
+(2008).
+[65] R. Mitsch, C. Sayrin, B. Albrecht, P. Schneeweiss, and
+A. Rauschenbeutel, Quantum state-controlled directional spon-
+taneous emission of photons into a nanophotonic waveguide,
+Nature Communications 5, 5713 (2014).
+[66] J. Petersen, J. Volz, and A. Rauschenbeutel, Chiral nanopho-
+tonic waveguide interface based on spin-orbit interaction of
+light, Science 346, 67 (2014).
+[67] I. S¨ollner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi,
+G. Kirˇsansk˙e, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song,
+S. Stobbe, and P. Lodahl, Deterministic photon–emitter cou-
+pling in chiral photonic circuits, Nature Nanotechnology 10,
+775 (2015).
+[68] C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea,
+P. Schneeweiss, J. Volz, and A. Rauschenbeutel, Nanophotonic
+optical isolator controlled by the internal state of cold atoms,
+
+10
+Phys. Rev. X 5, 041036 (2015).
+[69] W.-B. Yan, J.-F. Huang, and H. Fan, Tunable single-photon fre-
+quency conversion in a sagnac interferometer, Scientific Re-
+ports 3, 1 (2013).
+[70] W. Z. Jia, Y. W. Wang, and Y.-x. Liu, Efficient single-photon
+frequency conversion in the microwave domain using supercon-
+ducting quantum circuits, Phys. Rev. A 96, 053832 (2017).
+[71] Y.-T. Chen, L. Du, L. Guo, Z. Wang, Y. Zhang, Y. Li, and J.-H.
+Wu, Nonreciprocal and chiral single-photon scattering for giant
+atoms, Communications Physics 5, 215 (2022).
+[72] Y. Lu, S. Gao, A. Fang, P. Li, F. Li, and M. S. Zubairy, Coherent
+frequency down-conversions and entanglement generation in a
+sagnac interferometer, Optics Express 25, 16151 (2017).
+[73] W.-B. Yan, W.-Y. Ni, J. Zhang, F.-Y. Zhang, and H. Fan, Tun-
+able single-photon diode by chiral quantum physics, Phys. Rev.
+A 98, 043852 (2018).
+[74] C. W. Gardiner and A. S. Parkins, Driving atoms with light of
+arbitrary statistics, Phys. Rev. A 50, 1792 (1994).
+

X9E5T4oBgHgl3EQfdQ8b/content/tmp_files/2301.05609v1.pdf.txt

@@ -0,0 +1,1202 @@
+Springer Nature 2021 LATEX template
+Co-manipulation of soft-materials estimating
+deformation from depth images
+Giorgio Nicola1*, Enrico Villagrossi1 and Nicola Pedrocchi1
+1Institute of Intelligent Industrial Technologies and Systems for
+Advanced Manufacturing, National Research Council of Italy, Via
+A. Corti 12, Milan, 20133, Italy.
+*Corresponding author(s). E-mail(s): [email protected];
+Abstract
+Human-robot co-manipulation of soft materials, such as fabrics, com-
+posites, and sheets of paper/cardboard, is a challenging operation
+that presents several relevant industrial applications. Estimating the
+deformation state of the co-manipulated material is one of the main
+challenges. Viable methods provide the indirect measure by calculating
+the human-robot relative distance. In this paper, we develop a data-
+driven model to estimate the deformation state of the material from
+a depth image through a Convolutional Neural Network (CNN). First,
+we define the deformation state of the material as the relative roto-
+translation from the current robot pose and a human grasping position.
+The model estimates the current deformation state through a Con-
+volutional Neural Network, specifically a DenseNet-121 pretrained on
+ImageNet. The delta between the current and the desired deforma-
+tion state is fed to the robot controller that outputs twist commands.
+The paper describes the developed approach to acquire, preprocess the
+dataset and train the model. The model is compared with the current
+state-of-the-art method based on a skeletal tracker from cameras. Results
+show that our approach achieves better performances and avoids the
+various drawbacks caused by using a skeletal tracker. Finally, we also
+studied the model performance according to different architectures and
+dataset dimensions to minimize the time required for dataset acquisition.
+Keywords: human-robot collaborative transportation, soft materials
+co-manipulation, vision-based robot manual guidance
+1
+arXiv:2301.05609v1  [cs.RO]  13 Jan 2023
+
+Springer Nature 2021 LATEX template
+2
+Co-manipulation of soft-materials estimating deformation from depth images
+Fig. 1: The studied problem of human-robot collaborative transportation of a
+soft material. The human holds the co-manipulated object from one side and,
+through an RGB-D camera, estimates the object’s deformation.
+1 Introduction
+Robots are increasingly assuming the role of assistants to humans in work-
+places. However, robots still lack the intelligence to behave as helpful assistants.
+A paradigmatic application that highlights such limits is collaborative manip-
+ulation and transport, which is impactful in multiple fields of application, from
+industrial logistics and construction to households.
+Human-Robot collaborative transportation offers multiple challenges.
+First, the robot might need to learn the path or the object’s final position.
+The robot infers them based on clear signals from the human operator or via
+haptic communication through the co-manipulated object. However, in the
+co-manipulation of non-rigid objects, haptic communication is limited since
+not all forces and torques applied by the human can be read from the robot
+end-effector. Second, the robot should share the workload with the human.
+Still, the human should be able to gain control over the robot, i.e., the robot
+and human should be able to exchange leader and follower roles continu-
+ously Franceschi, Pedrocchi, and Beschi (2022). Finally, a well-known problem
+of manipulating large objects is the rotation-translation ambiguity in human-
+robot Dumora, Geffard, Bidard, Brouillet, and Fraisse (2012) co-manipulation,
+but it can arise even during human-human Jensen, Salmon, and Killpack
+(2021) co-manipulation.
+In this paper, we study the problem of human-robot collaborative trans-
+portation of soft materials like fabric, shown in Figure 1. Unlike rigid objects,
+soft materials do not transfer compression forces since the absence of flexural
+
+sort
+CYBER
+SORTSpringer Nature 2021 LATEX template
+Co-manipulation of soft-materials estimating deformation from depth images
+3
+rigidity characterizes them. Furthermore, the rigidity along the other direc-
+tions is also very low, and the amount of force/torques they can sustain without
+damage is minimal. Thus, estimating the deformation state only by force and
+torque measurements is challenging. Consequently, standard control architec-
+tures based on impedance or admittance control cannot be used directly De
+Schepper, Moyaers, Schouterden, Kellens, and Demeester (2021); Sirintuna,
+Giammarino, and Ajoudani (2022).
+We propose to use a data-driven approach to develop a black-box model
+based on a Convolutional Neural Network (CNN), to estimate the deformation
+state of the co-manipulated soft material from depth images acquired by an
+RGB-D camera rigidly attached to the robot end-effector. Given a predefined
+rest configuration state, any estimated variation from such rest state caused by
+the human can be translated into a movement intention towards the direction
+that minimizes such difference. The proposed approach is applied to a case
+of human-robot collaborative transportation of a piece of carbon fiber fabric,
+proving the method’s effectiveness.
+The paper is structured as follows: in Section Introduction, related works
+and paper contributions are presented; in Section Method, the human-robot
+collaborative transportation problem is formalized, and the proposed solution,
+including the dataset acquisition strategy, is detailed; In Section Experiments,
+experimental evaluation of the approach is presented including comparison
+with state of the art, analysis of different neural network architectures and the
+dataset dimension; In section Conclusion, conclusions and future works are
+highlighted.
+1.1 Related Work
+The manipulation of deformable objects has been deeply investigated Sanchez,
+Corrales, Bouzgarrou, and Mezouar (2018), and multiple strategies have
+been developed, sometimes depending also on the geometric characteris-
+tics of the objects. Specifically, two main classes of deformable objects are
+studied: cables She et al. (2020); W. Wang and Balkcom (2018) and cloth-
+like Mcconachie, Dobson, Ruan, and Berenson (2020); Miller et al. (2012);
+Verleysen, Biondina, and Wyffels (2020). Manipulation of cloth-like objects is
+typically focused on the cloth folding task. However, few works study the prob-
+lem of the collaborative human-robot manipulation of such objects; instead,
+they typically focus on cloth folding Lee, Lu, Gupta, Levine, and Abbeel
+(2015); Li, Yue, Xu, Grinspun, and Allen (2015). One of the main difficulties
+in manipulating deformable materials is tracking the object and estimating
+its current shape, that is, its deformation Tang and Tomizuka (2022). Two
+main approaches have been developed to estimate the material deformation
+in human-robot manipulation of deformable materials: human motion capture
+and direct deformation estimation of the material. In the first case, the defor-
+mation is estimated by tracking the position of the material grasped points
+by the robot and human by implementing a motion capture system based on
+either IMU sensors Sirintuna et al. (2022) or camera Andronas, Kampourakis,
+
+Springer Nature 2021 LATEX template
+4
+Co-manipulation of soft-materials estimating deformation from depth images
+et al. (2021). Tracking the grasping point in the space assumes that the posi-
+tion grasping point on the manipulated object is known as a priori or is
+detectable. Therefore, it is possible to completely reconstruct the shape of the
+manipulated material thanks to physics simulation software Andronas, Kam-
+pourakis, et al. (2021); Kruse, Radke, and Wen (2017) or directly use the
+human-robot relative distance as input to the controller Sirintuna et al. (2022).
+In the second case, the deformation is estimated using a camera to detect visual
+features on the manipulated material converted into robot commands. Differ-
+ent types of visual features have been developed in the literature. In Kruse,
+Radke, and Wen (2015), material folds are detected and combined with force
+measurements to compute robot speed commands directly. In Jia, Hu, Pan,
+and Manocha (2018); Jia, Pan, Hu, Pan, and Manocha (2019), Histograms
+of Oriented Wrinkles (HOWs) are implemented as visual features to detect
+deformations. HOWs are computed by applying Gabor filters and extracting
+the high-frequency and low-frequency components on the RGB image. Sub-
+sequently, in Jia et al. (2018), a controller converts the desired speed in the
+feature to the robot end effector speed. Meanwhile, in Jia et al. (2019), a Ran-
+dom Forest controller is trained to compute the desired robot grasping point
+position from the HOW feature space.
+Finally, in De Schepper et al. (2021), a different approach is used. Instead of
+estimating the material deformation, motion tracking is used to detect hand-
+crafted coded gestures that, combined with torque force measures, are used
+directly to compute robot end effector speed.
+Recently, Deep Neural Networks have been used as feature extractors via
+autoencoder networks Tanaka, Arnold, and Yamazaki (2018); Tsurumine and
+Matsubara (2022); Yang et al. (2017). The extracted feature is subsequently
+used to plan robot movement to achieve the desired deformation. However,
+none of those methods based on DNN feature extraction has been currently
+applied to human-robot manipulation.
+The EU H2020 projects DrapeBot DrapeBot Consortium (2021) and Merg-
+ing MERGING Consortium (2019) are pioneering actions coping with these
+challenges in the industrial scenario. The DrapeBot project focuses on the
+robotic manipulation of carbon fiber and fiberglass plies during the draping
+process; in particular, Eitzinger, Frommel, Ghidoni, and Villagrossi (2021)
+highlights the importance of the human-robot co-manipulation when the
+dimension of the ply is such as to require more than one robot. The project
+Merging looks at the manipulation of flexible and fragile objects exploit-
+ing multiple industrial robots, designing new Electro-Adhesive (EA) grasping
+devices, using information from perception systems fused with the data from
+a digital twin to estimate the deformations of flexible elements Andronas,
+Kokotinis, and Makris (2021); Makris, Kampourakis, and Andronas (2022).
+1.2 Contribution
+This paper proposes to combine the two main approaches in the literature to
+estimate the material deformation by calculating the relative distance between
+
+Springer Nature 2021 LATEX template
+Co-manipulation of soft-materials estimating deformation from depth images
+5
+Fig. 2: Example of two different human grasping configurations on a carbon
+fiber ply. As can be noticed in both grasping configurations, the human hands
+do not introduce any deformation on the material between them. Thus, the por-
+tion of material between them can be considered rigid, and a single arbitrary
+point on the human side of the ply is sufficient to describe its position
+(a)
+(b)
+Fig. 3: Proposed formalization of the human-robot collaborative transporta-
+tion problem. a) Shows the top view highlighting the definition of the human
+grasping point Hgp. b) Shows a lateral view highlighting the parameters that
+compose T H
+R , T H
+R,des, and ∆T. For the sake of simplicity, rotations have been
+neglected.
+the robot and human grasping position directly from the material rather than
+from a motion tracking system. The human-robot relative pose is estimated
+through a convolutional deep neural network fed with the depth image of the
+handled material. The robot controller is designed to minimize the distance
+between the estimated relative pose and a rest relative pose.
+Using depth images has multiple advantages compared to state-of-the-art
+solutions. First, motion tracking-based methods assume to know exactly how
+
+Desired ply
+Current ply
+shape
+shape
+H
+gp
+R
+R,des
+Rdes
+△TPly reference
+Current
+shape
+Ply shape
+Ads
+gdes
+a
+2
+Adz
+gdes
+a
+2
+desSpringer Nature 2021 LATEX template
+6
+Co-manipulation of soft-materials estimating deformation from depth images
+the human will grasp the object because either defined a priori or detected.
+However, such an assumption is only sometimes realistic, especially in unstruc-
+tured environments such as factories and households. Our method does not
+track the hands, but it estimates the robot pose w.r.t. the human hands by
+looking at the deformation of the material.
+Second, motion tracking-based methods strongly rely on the sensor’s per-
+formance. On the one hand, IMU-based motion tracking is affected by a drift
+that can only be corrected with other motion capture techniques based on
+cameras or re-calibrating the setup. IMU, in addition, must be worn by the
+operator either with straps or specific suits that are uncomfortable for the
+operator, they can slide, and their position on the body is not repeatable. On
+the other hand, camera-based motion capture has the disadvantage that it has
+a limited working range both in the distance and in the field of view, and it is
+sensible for occlusions unless multiple cameras are used.
+Third, the proposed approach does not rely on handcrafted features as
+usually the methods that look at the material estimation do. On the one hand,
+the features might only partially describe the studied problem. On the other
+hand, transforming the feature space into robot commands takes a lot of work.
+Indeed, specific controllers are used, and in some cases, imitation learning on
+expert user samples is adopted.
+Finally, our methodology avoids any use of force/torque feedback. Such
+sensors are very noisy, limited bandwidth, and have high oscillations due
+to the object-handled dynamics. Furthermore, we avoid translation/rotation
+ambiguity.
+2 Method
+2.1 Problem formulation
+The problem of human-robot collaborative manipulation of soft materials is
+composed of two agents handling the soft material simultaneously, as shown
+in Figure 1. One agent is the human that leads the activity, and the second
+one is the robot that should follow the human movement. The objective is to
+manipulate the desired object while minimizing the deformations from a rest
+configuration that guarantees no damage to the material.
+Soft materials, like fabric, can be approximated as membranes Vasiliev and
+Morozov (2018) characterized by the absence of flexural rigidity. Therefore,
+they cannot sustain compressive loads, and deformations can be caused only
+by displacements or traction forces.
+Let us consider a soft material from now on called ply handled by two
+agents, a human and a robot. Denote hr and hl as the human’s hands grasping
+positions, and Rcur as the robot’s current grasping point. Given the assump-
+tion that the gripper geometry is constant, it is easy to see that a bi-unique
+proxy for the ply shape is the tuple of the relative roto-translations between
+the robot grasping position and the human hands holding positions, except
+for the local deformation around the hands that may change from person to
+
+Springer Nature 2021 LATEX template
+Co-manipulation of soft-materials estimating deformation from depth images
+7
+person. This model, however, is unnecessarily accurate since it also models the
+deformation of the ply in the corners, i.e., the roto-translation might change
+not only according to the distance human-robot but also to where the humans
+are grasping.
+A further likely assumption is that the human tenses the fabric between the
+hands, achieving a minimal internal tension of the ply, not enough to deform
+it but compensating for the gravity deformation effect. The material between
+the two hand-grasping points can be considered rigid in such a condition.
+Given this assumption, it is possible to define an arbitrary point on the ply
+between the hands invariant to the different grasping positions, Figure 2. The
+relative roto-translations between the robot grasping position and this point
+is a robust proxy for the ply deformation, except for the local deformation
+around the hands and the geometry of the corners, which are irrelevant to the
+control objective.
+Mathematically, denote Hgp as the proxy for the single human grasping
+point (Figure 3a). Denoting T H
+R
+and T H
+R,des as the actual roto-translation
+matrix between Rcur and Hgp and a target desired ply shape. Given this formu-
+lation, the problem consists of (i) imposing the target T H
+R,des and (ii) estimating
+T H
+R during the execution movement. The robot should be controlled based on
+such estimations to minimize the distance from the target’s desired pose.
+As a final note, given the frame reference in Figure 3b, a further simplifying
+assumption is that the x-axis rotation is always zero since it does not cause
+macroscopic deformations of the ply but only local deformations.
+2.2 Proposed solution
+We estimate the roto-translation T H
+R through an ensemble of CNNs that takes
+as input a depth image of the material from a camera rigidly attached to the
+robot end-effector. Specifically, the ensemble of CNNs outputs the parameters
+that fully describe T H
+R , i.e., three translations and two rotations following the
+xyz Euler conventions. For this purpose, an RGB-D camera is rigidly attached
+to the robot end-effector that looks at the top of the co-manipulated element,
+allowing the easy detection of the material shape and deflections due to the
+forces/displacement applied by the human on the material (see Figure 1).
+The 3D camera provides the depth map, and after proper preprocessing, a
+depth map is obtained composed only of carbon fiber ply segmented from the
+background and the human partner. The segmented depth map is fed to an
+ensemble of CNNs trained to estimate the deformation of the material, in other
+words, the distance between the robot gripper and the human grasping point
+Hgp. Applying Deep Learning techniques allows for defining a black-box model
+describing the relation between visual deformation and mechanical status.
+2.3 Dataset acquisition
+The training of the dataset should be done by acquiring a large number of
+depth images of the deformed material with different human-robot relative
+
+Springer Nature 2021 LATEX template
+8
+Co-manipulation of soft-materials estimating deformation from depth images
+Fig. 4: The setup developed to acquire the dataset of a carbon fiber ply. The
+setup comprises an RGB-D camera, Azure Kinect, a robot Universal Robot
+UR5, an aluminum frame to mimic the human, and a pair of fiducial markers,
+Apriltags, to localize the frame.
+distances and human grasping positions. To avoid a bothering effort to a human
+operator, we substituted the human with a frame that holds the soft material to
+achieve higher accuracy and repeatability of the dataset, as shown in Figure 4,
+and the relative distance is got by moving the robot and maintaining the frame
+fix.
+Precisely, a pair of metallic clips mimic the effect of the hands holding the
+ply. The frame allows the simulation of different human grasping positions on
+the material from now on, called human grasping configurations (Figure 2).
+The frame’s position is estimated using a pair of Apriltags J. Wang and Olson
+(2016), and the robot is moved relative to the frame. At each robot position,
+a set of RGB-D images are taken to account for the camera’s noisy output.
+The manipulated material is segmented from the background. First, the
+depth image is thresholded to a maximum and minimum value, and all pixels
+whose value is not within this range are set to zero to remove far and close
+objects quickly. Second, to remove the frame, Apriltags are used, the RGB
+image is converted into grayscale, and the tags’ positions in the 2D picture are
+detected. Subsequently, a parallel line to the line passing by the tags is drawn,
+and all the pixels above this line are set to zero. Finally, each image is cropped
+and resized to the resolution of 224×224. Each image is autonomously labeled
+with 3 Cartesian translations and the two rotations that fully describe T H
+R as
+in Section Problem formulation.
+
+AzureKinect
+EFFORTLESS
+Frame
+AprilTags
+Carbon Fiber Ply
+UR5Springer Nature 2021 LATEX template
+Co-manipulation of soft-materials estimating deformation from depth images
+9
+3 Experiments
+3.1 Experimental setup
+We tested the method with a piece of carbon fiber fabric with dimensions
+90 × 60 cm. Given a desired rest configuration of xref = 0.0 m, yref =
+0.6 m, zref = 0.0 m, θref = 0.0◦, γref = 0.0◦ the dataset of the deformed
+material was acquired in the range of ±0.105 m with step 0.03 m on the
+x − y − z axes and ±20◦ with step 5◦ on both y and z axis rotations for a
+total of 41472 different poses for each human grasping configuration of the
+material. The dataset resolution is the maximum permissible estimation error
+of the roto-translation matrix parameters. Furthermore, the study considered
+9 different human grasping configurations and 746496 depth images, i.e., two
+depth images were taken for each robot position. The setup for acquiring the
+dataset included an RGB-D camera, an Azure Kinect, and a Universal Robot
+UR5 as a robotic platform.
+3.2 Human Robot co-manipulation evaluation
+3 CNN models with Densenet-121 Huang, Liu, Van Der Maaten, and
+Weinberger (2017) compose of the system. The optimization software
+Optuna Akiba, Sano, Yanase, Ohta, and Koyama (2019) has computed the
+hyperparameters. The three input channels of the first convolutional layer of
+the Denesenet-121 were substituted with a convolutional layer with one input
+channel. This design choice is because the proposed method uses only depth
+images, while Densenet-121 is pretrained on the ImageNet dataset Deng et al.
+(2009) composed of RGB images. The weights of the new layer are equal to the
+sum of the weights along the three original channels. Figure 5 shows the error
+distributions of the estimation of the various parameters of T H
+R as boxplots.
+Subsequently, we compared our approach against the commonly used
+method in literature based on tracking the hands’ positions with a camera-
+based skeletal tracker and computing the human-robot distance. Therefore, the
+same Azure Kinect is used to acquire depth images of the human and perform
+skeletal tracking. The methods comparison exploited the same setup used for
+the dataset acquisition, the only difference being that a human was pretend-
+ing to grasp the deformable material. The depth image segmentation does not
+use the Apriltags to simulate actual operating conditions for the model. In
+contrast, it uses the hands’ key points positions in the 2D depth image, com-
+puted with the skeletal tracker from Azure Kinect. It is essential to note that
+the skeletal tracker is used for segmenting the depth image only for conve-
+nience since developing a method for segmenting the ply is out of the scope of
+this work. Any method to segment the co-manipulated material can be imple-
+mented based or not on a skeletal tracker. Finally, the robot was positioned
+in 20 known relative poses to the frame, i.e., set the material with a known
+deformation state, and we compared both methods on 20 frames for each pose.
+Results are shown in Figure 6.
+
+Springer Nature 2021 LATEX template
+10
+Co-manipulation of soft-materials estimating deformation from depth images
+Fig. 5: Comparison between the results of the single models and the model
+ensemble. Top left Cartesian estimation error. Top right estimation error on
+the x-axis. Center left estimation error on the y-axis. Center right estima-
+tion error on the z-axis. Bottom left estimation error on the y-axis rotation.
+Bottom right estimation error on the y-axis rotation.
+First, the CNN model estimation error on the x and z axis is sensibly lower
+and similar on y. The estimation error on the rotations is slightly higher but
+similar to the skeletal tracking error.
+Second, the error of the CNN model compared to the test dataset’s error
+(Figure 5) is significantly higher. The authors believe that this difference
+depends on the various sources of inaccuracies in the experimental setup rather
+than overfitting on the ply segmentation. Indeed, the robot grasping position
+and the frame grasping position are hard to reproduce.
+The same preprocessing pipeline of the dataset acquisition leads to similar
+results for the model. Even a tiny error in the grasping position, less than 1 cm,
+can significantly change the material shape when highly stretched. Indeed, note
+that the error in the deformation estimation is concentrated in those poses
+
+0.010
+wl
+0.005
+error
+0.008
+[m]
+0.006
+error
+lan
+0.000
+rtesi
+0.004
+X
+Cal
+0.002
+0.005
+0.000
+Model 1
+Model 2
+Model 3
+Model
+Model 1
+Model 2
+Model 3
+Model
+Ensemble
+Ensemble
+0.0075
+0.002
+0.0050
+0.000
+[m]
+-0.002
+error
+0.0025
+error
+0.004
+0.0000
+>
+N
+-0.006
+-0.0025
+-0.008
+Model 1
+Model 2
+Model 3
+Model
+Model 1
+Model 2
+Model 3
+Model
+Ensemble
+Ensemble
+0.4
+[deg]
+[deg]
+0.5
+0.2
+error
+error
+0.0
+0.0
+y
+-0.2
+N
+rot
+rot
+-0.5
+-0.4
+Model 1
+Model 2
+Model 3
+Model
+Model 1
+Model 2
+Model 3
+Model
+Ensemble
+EnsembleSpringer Nature 2021 LATEX template
+Co-manipulation of soft-materials estimating deformation from depth images
+11
+Fig. 6: Comparison of the error distributions in estimating the deformation
+state between the method used in literature based on skeletal tracking and
+our method based on a CNN model. Top left estimation error on the x-axis.
+Top right estimation error on the y-axis. Center left estimation error on
+the z-axis. Center right estimation error on y-axis rotation. Bottom left
+estimation error on the z-axis rotation.
+with high values of θ and γ rotations; nevertheless, the rotation was usually
+underestimated when the ply was highly stretched.
+In conclusion, the proposed approach proved to have a better estimation
+in the x, y, and z positions while slightly worse in the rotation. However,
+the skeleton tracker method was tested only in optimal conditions, i.e., when
+the distance camera key points are below 2.5 m when using the broad field
+of view. Indeed, as described in T¨olgyessy, Dekan, and Chovanec (2021), the
+skeletal tracking accuracy of the Azure Kinect drastically deteriorates above a
+threshold distance depending on the used field of view, thus, directly limiting
+the maximum size of the co-manipulated material.
+
+0.05
+0.05
+Error x [m]
+0.00
+Error y [m]
+0.00
+中
+0.05
+0.05
+0.10
+-0.10
+Skeletal
+CNN Mode
+Skeletal
+CNN Model
+tracking
+tracking
+0.05
+Error orientation y [rad]
+0.3
+中
+0.2
+Error z [m]
+0.00
+0.1
+-0.05
+0.0
+0.10
+0.1
+-0.2
+Skeletal
+CNN Model
+Skeletal
+CNN Model
+tracking
+tracking
+Error orientation  [rad]
+0.3
+0.2
+0.1
+0.0
+0.1
+0.2
+Skeletal
+CNN Model
+trackingSpringer Nature 2021 LATEX template
+12
+Co-manipulation of soft-materials estimating deformation from depth images
+Fig. 7: Training results of VGG not pretrained, VGG pretrained, and
+Densenet121 on the different percentage of the dataset. Top left shows the
+average loss for each architecture on the test dataset and the unused dataset.
+Top right shows the Cartesian distance error for each architecture as a
+boxplot. Bottom left shows the orientation error on the y-axis for each archi-
+tecture as a boxplot. Bottom right shows the orientation error on the z-axis
+for each model as a boxplot.
+3.3 Network architecture analysis
+In this section, we studied the model accuracy using different network archi-
+tectures. The network architectures studied were the following: a VGG11
+without initialized parameters as in Nicola, Villagrossi, and Pedrocchi (2022),
+VGG11 Simonyan and Zisserman (2015) with batch normalization provided by
+PyTorch and Densenet-121 Huang et al. (2017) provided by PyTorch Paszke
+et al. (2019).
+The learning rate is equal once to 1 × 10−4, and batch size is 256 on three
+different validation sets. The same conditions were applied to each combination
+of architecture and dataset dimension to evaluate them in the same situation.
+
+Graspconfigurations=9
+0.0035
+0.07
+Cartesian distance error [m]
+0.06
+0.0030
+0.05
+0.0025
+0.04
+SSOT
+0.0020
+0'0
+0.0015
+0.02
+0.0010
+0.01
+0.0005
+0.00
+0.0000
+25%
+50%
+75%
+100%
+25%
+50%
+75%
+100%
+10
+[deg]
+Orientation errory[deg]
+5
+Orientationerror
+2
+0
+0
+-2
+5
+-4
+-10
+25%
+50%
+75%
+100%
+25%
+50%
+75%
+100%
+Posespercentage
+VGG not pretrained
+Test set
+Densenet Test set
+VGG Unused labels set
+VGG not pretrained
+VGG Test set
+Densenet Unused labels
+Unused labels setSpringer Nature 2021 LATEX template
+Co-manipulation of soft-materials estimating deformation from depth images
+13
+After the first 5 epochs without improvements, the learning rate was divided
+by 10; otherwise, the training was stopped. The maximum number of epochs
+was 45.
+Figure 7 shows the training results of each architecture on different percent-
+ages of the dataset, including loss and distributions of the Cartesian distance
+error and orientation error over the y and z axis. It is worth noting that VGG11
+and Densenet121 achieve far better results on the test dataset than the VGG
+not pretrained. This result confirms our hypothesis that the feature learned
+on the ImageNet dataset are still beneficial even though the input image type
+is different. Furthermore, similarly to the results on ImageNet, DenseNet121
+achieved better results than VGG11.
+Subsequently, each architecture is trained on a different percentage of the
+dataset to verify whether a reduced-dimension dataset can still generalize on
+unseen data. In particular, results are compared between the test dataset
+and the unused dataset (set of labels removed from the dataset before divid-
+ing between training and test dataset). Not surprisingly, for every network
+architecture, the model performances decrease with the dataset size, with a
+significant drop at 25%. However, the loss values and the error distributions
+are almost identical between the test set and the unused set. It is possible to
+deduce that 1/4 of the acquired dataset with the deformation range is enough
+to train a model to generalize over unseen data and specifically unseen defor-
+mation combinations. Indeed, in Densenet121 on 25% of the data points in
+both the test and the unused datasets, the error is far below the dataset reso-
+lution (0.03 m on the x-y-z axis and 5° on θ and γ rotations). Increasing the
+dataset size, however, is still beneficial to improve the model’s accuracy.
+3.4 Dataset dimension analysis
+The dataset dimension plays an essential role in the training time, which is still
+a relevant obstacle to implementing Deep Learning methods in the industry.
+Three main factors influence the dataset dimension for the presented
+use case: (i) the number of photos for each pose; (ii) the number of poses
+for each human grasping configuration; (iii) the number of human grasping
+configurations.
+However, the first point’s relevance is minimal since most of the time during
+the dataset acquisition is spent moving the robot from one pose to another.
+The photo acquisition takes 0.03s while the time point-to-point motion takes
+0.2 s.
+The design of the experiments for study for the dataset dimension analysis
+foresees investigations with [100%, 75%, 50%, 25%] over the 373.248 poses
+randomly chosen before mentioned and [6, 4] human grasping configurations.
+Only Densenet121 was studied since it achieved the best performances in
+all previous training conditions. Each experiment provides the same training
+conditions of the network architecture analysis.
+We compared the results among the test dataset, the “unused labels set”,
+and the “unused grasp set”. Specifically, the ’unused label set’ includes all
+
+Springer Nature 2021 LATEX template
+14
+Co-manipulation of soft-materials estimating deformation from depth images
+Fig. 8: Results training of the DenseNet121 on the different percentages of
+the dataset and including only six grasp configurations over 9. Top left shows
+the loss for each model on the test dataset and the unused dataset. Top right
+shows the Cartesian distance error for each model as a boxplot. Bottom left
+shows the orientation error on the y-axis for each model as a boxplot. Bottom
+right shows the orientation error on the z-axis for each model as a boxplot.
+the datapoints in the unused set with the grasp configurations on which it is
+trained. The “unused grasp set” groups all the datapoints in the unused set
+with a discarded grasp configuration.
+Figure 8 and Figure 9 reports the models result with 6, and 4 grasp con-
+figurations. Similarly to the case with all nine grasp configurations, results
+between the test set and the unused label set are almost identical. On the
+contrary, there is a significant difference in the results on the unused grasp
+set. Such results prove that generalizing over unseen grasping configurations is
+much more challenging than generalizing over unseen poses. Thus, the dataset
+
+Graspconfigurations=6
+0.05
+0.005
+Cartesian distance error [m]
+0.04
+0.004
+0.03
+SSOT
+0.003
+0.02
+0.002
+0.01
+0.001
+0.00
+0.000
+25%
+50%
+75%
+100%
+25%
+50%
+75%
+100%
+4
+4
+Orientation error  [deg]
+3
+Orientation error y[deg]
+3
+2
+1
+0
+0
+-1
+-2
+-2
+3
+-3
+4
+25%
+50%
+75%
+100%
+25%
+50%
+75%
+100%
+Poses percentage
+Test set
+Unused labels set
+Unused grasp setSpringer Nature 2021 LATEX template
+Co-manipulation of soft-materials estimating deformation from depth images
+15
+Fig. 9: Results training of the DenseNet121 on the different percentage of the
+dataset and including only four grasp configurations over 9. Top left shows
+the loss for each model on the test dataset and the unused dataset. Top right
+shows the Cartesian distance error for each model as a boxplot. Bottom left
+shows the orientation error on the y-axis for each model as a boxplot. Bottom
+right shows the orientation error on the z-axis for each model as a boxplot.
+acquisition should focus on acquiring data with many grasping configurations
+rather than all the poses. Nevertheless, note that the worst performing model,
+trained on only 4 grasp configurations and 25% of training poses, estimates
+the Cartesian position with an error below the dataset resolution in more than
+75% of the datapoints.
+
+Graspconfigurations=4
+0.06
+0.014
+Cartesian distance error [m]
+0.05
+0.012
+0.010
+0.04
+SSOT
+0.008
+EO'0
+0.006
+0.02
+0.004
+0.01
+0.002
+0.00
+0.000
+25%
+50%
+75%
+100%
+25%
+50%
+75%
+100%
+7.5
+7
+5.0
+Orientation error  [deg]
+Orientation error y[deg]
+2.5
+2
+0.0
+0
+2.5
+-2
+-5.0
+-5
+7.5
+7
+10.0
+-10
+25%
+50%
+75%
+100%
+25%
+50%
+75%
+100%
+Posespercentage
+Test set
+Unused labels set
+Unused grasp setSpringer Nature 2021 LATEX template
+16
+Co-manipulation of soft-materials estimating deformation from depth images
+Fig. 10: Description of the developed pipeline to perform human-robot
+collaborative transportation.
+3.5 Human-robot collaborative transportation
+Refer to a scenario of human-robot collaborative transportation of carbon fiber
+fabric, as shown in Figure 1. The developed model is used to estimate the cur-
+rent deformation state of the carbon fiber fabric, and the difference concerning
+a predefined rest deformation state is converted into a twist command through
+a proportional controller as shown in Figure 10. A single PC runs the model
+and the robot controller, with a CPU, Intel Core i7-8700, and a GPU, NVIDIA
+GeForce RTX 3080Ti. The preprocessing run at approximately 30 Hz, and
+the robot controller with the model runs at 20 Hz. The user could smoothly
+move the co-manipulated material avoiding excessive material deformations as
+shown in the video in Nicola (2022).
+4 Conclusions
+This
+work
+presented
+a
+data-driven
+approach
+to
+co-manipulating
+soft
+deformable materials based on estimating the material deformation from depth
+images from an RGB-D camera rigidly attached to the robot end-effector
+through a CNN model. First, we formalized the problem of human-robot
+co-manipulation, and we defined the material deformation state based on
+the roto-translation between the robot grasping point and the human grasp-
+ing point. Second, we developed a Deep Learning model to estimate the
+roto-translation matrix parameters from a depth image of the deformed
+material.
+The proposed approach was compared with a well-known method from lit-
+erature based on computing the human-robot distance as the distance between
+the robot and the human hand keypoints obtained via a camera-based skele-
+ton tracker. Our approach proved more accurate and is not affected by the two
+main drawbacks of skeletal trackers. First, the human must always be in the
+camera’s field of view, and second, the accuracy of the skeletal tracker tends
+
+3Dcamera
+Depth image
+Pre-processing
+Co-manipulatedsoftmaterial
+Segmented ply
+Robot velocity
+depth image
+Robot
+set-point
+Deformation
+estimation
+Deformation set-point
+Control
+CNN Model
+algorithmSpringer Nature 2021 LATEX template
+Co-manipulation of soft-materials estimating deformation from depth images
+17
+to decrease as the human-camera distance increases. Those two drawbacks can
+have an extremely negative effect when the co-manipulated material is signifi-
+cant. Meanwhile, our approach can estimate the deformation state of material
+with any dimension since it only looks at the co-manipulated material.
+We evaluated multiple network architectures from the literature and stud-
+ied the model performances according to dataset dimension. Indeed, one of the
+main limitations to applying Deep Learning models in the industry is the neces-
+sity of acquiring large datasets, which is time-consuming. Results showed that
+the dataset could be acquired with a much lower resolution than the desired
+maximum estimation error. On the other hand, results also showed that the
+number of grasping configurations in the dataset is critical since reducing them
+causes a significant drop in model performances.
+Finally, the approach was tested in a real-world application of human-robot
+collaborative transportation of a carbon fiber fabric. The robot was able to
+follow human movements smoothly, avoiding excessive deformations.
+The main drawback of the proposed approach is that it needs quite a large
+dataset for every co-manipulated object that, even if it can be acquired chiefly
+autonomously, is still time-consuming. In future works, the authors will inves-
+tigate the usage of synthetic datasets to train the model. Furthermore, we will
+train multiple models simultaneously, one for each ply shape, sharing a stan-
+dard backbone for the CNN part of the network. Thus, the models should learn
+standard and possibly more general features that allow transfer learning by
+retraining the last fully connected layers. Finally, the deformation estimation
+was applied to the case of collaborative transportation with an anthropomor-
+phic manipulator. The authors plan to apply the proposed approach to a case
+with an anthropomorphic manipulator mounted on top of a mobile platform
+to increase the technological fallout of the method.
+Acknowledgments.
+This project has received funding from the European
+Union’s Horizon 2020 research and innovation program under grant agreement
+No 101006732, ”DrapeBot – A European Project developing collaborative
+draping of carbon fiber parts.”
+References
+Akiba, T., Sano, S., Yanase, T., Ohta, T., Koyama, M. (2019). Optuna: A next-
+generation hyperparameter optimization framework. Proceedings of the
+25th acm sigkdd international conference on knowledge discovery & data
+mining (p. 2623–2631). New York, NY, USA: Association for Comput-
+ing Machinery. Retrieved from https://doi.org/10.1145/3292500.3330701
+10.1145/3292500.3330701
+Andronas, D., Kampourakis, E., Bakopoulou, K., Gkournelos, C., Angelakis,
+P., Makris, S. (2021). Model-based robot control for human-robot flexible
+material co-manipulation.
+2021 26th ieee international conference on
+
+Springer Nature 2021 LATEX template
+18
+Co-manipulation of soft-materials estimating deformation from depth images
+emerging technologies and factory automation (etfa ) (p. 1-8). 10.1109/
+ETFA45728.2021.9613235
+Andronas, D., Kokotinis, G., Makris, S. (2021). On modelling and handling of
+flexible materials: A review on digital twins and planning systems. Proce-
+dia CIRP, 97, 447-452. (8th CIRP Conference of Assembly Technology
+and Systems)
+https://doi.org/10.1016/j.procir.2020.08.005
+De
+Schepper,
+D.,
+Moyaers,
+B.,
+Schouterden,
+G.,
+Kellens,
+K.,
+Demeester,
+E.
+(2021).
+Towards
+robust
+human-robot
+mobile
+co-manipulation
+for
+tasks
+involving
+the
+handling
+of
+non-rigid
+materials
+using
+sensor-fused
+force-torque,
+and
+skeleton
+track-
+ing
+data.
+Procedia
+CIRP,
+97,
+325-330.
+Retrieved
+from
+https://www.sciencedirect.com/science/article/pii/S2212827120314670
+(8th CIRP Conference of Assembly Technology and Systems)
+https://doi.org/10.1016/j.procir.2020.05.245
+Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., Fei-Fei, L. (2009). Ima-
+genet: A large-scale hierarchical image database. 2009 ieee conference
+on computer vision and pattern recognition (p. 248-255).
+10.1109/
+CVPR.2009.5206848
+DrapeBot Consortium
+(2021).
+Drapebot – a european project devel-
+oping collaborative draping of carbon fiber parts.
+Retrieved from
+https://www.drapebot.eu/
+Dumora, J., Geffard, F., Bidard, C., Brouillet, T., Fraisse, P. (2012). Experi-
+mental study on haptic communication of a human in a shared human-
+robot collaborative task. 2012 IEEE/RSJ International Conference on
+Intelligent Robots and Systems, 5137-5144.
+Eitzinger, C., Frommel, C., Ghidoni, S., Villagrossi, E. (2021). System con-
+cept for human-robot collaborative draping. Sampe europe conference
+(p. 7542-7549).
+Franceschi, P., Pedrocchi, N., Beschi, M. (2022). Adaptive impedance con-
+troller for human-robot arbitration based on cooperative differential
+game theory. 2022 international conference on robotics and automation
+(icra) (p. 7881-7887). 10.1109/ICRA46639.2022.9811853
+Huang, G., Liu, Z., Van Der Maaten, L., Weinberger, K.Q. (2017). Densely
+connected convolutional networks.
+2017 ieee conference on computer
+vision and pattern recognition (cvpr) (p. 2261-2269).
+10.1109/CVPR
+
+Springer Nature 2021 LATEX template
+Co-manipulation of soft-materials estimating deformation from depth images
+19
+.2017.243
+Jensen, S.W., Salmon, J.L., Killpack, M.D.
+(2021).
+Trends in haptic
+communication of human-human dyads: Toward natural human-robot
+co-manipulation.
+Frontiers in Neurorobotics, 15.
+Retrieved from
+https://www.frontiersin.org/articles/10.3389/fnbot.2021.626074
+10.3389/fnbot.2021.626074
+Jia, B., Hu, Z., Pan, J., Manocha, D. (2018). Manipulating highly deformable
+materials using a visual feedback dictionary.
+2018 ieee international
+conference on robotics and automation (icra) (p. 239-246).
+10.1109/
+ICRA.2018.8461264
+Jia, B., Pan, Z., Hu, Z., Pan, J., Manocha, D. (2019). Cloth manipulation using
+random-forest-based imitation learning. IEEE Robotics and Automation
+Letters, 4(2), 2086-2093.
+10.1109/LRA.2019.2897370
+Kruse, D., Radke, R.J., Wen, J.T.
+(2015).
+Collaborative human-robot
+manipulation of highly deformable materials.
+2015 ieee international
+conference on robotics and automation (icra) (p. 3782-3787).
+10.1109/
+ICRA.2015.7139725
+Kruse, D., Radke, R.J., Wen, J.T. (2017). Human-robot collaborative handling
+of highly deformable materials. 2017 american control conference (acc)
+(p. 1511-1516). 10.23919/ACC.2017.7963167
+Lee, A.X., Lu, H., Gupta, A., Levine, S., Abbeel, P. (2015). Learning force-
+based manipulation of deformable objects from multiple demonstrations.
+2015 ieee international conference on robotics and automation (icra)
+(p. 177-184). 10.1109/ICRA.2015.7138997
+Li, Y., Yue, Y., Xu, D., Grinspun, E., Allen, P.K. (2015). Folding deformable
+objects using predictive simulation and trajectory optimization. 2015
+ieee/rsj international conference on intelligent robots and systems (iros)
+(p. 6000-6006). 10.1109/IROS.2015.7354231
+Makris, S., Kampourakis, E., Andronas, D.
+(2022).
+On deformable
+object
+handling:
+Model-based
+motion
+planning
+for
+human-robot
+co-manipulation.
+CIRP Annals, 71(1), 29-32.
+Retrieved from
+https://www.sciencedirect.com/science/article/pii/S0007850622000956
+https://doi.org/10.1016/j.cirp.2022.04.048
+
+Springer Nature 2021 LATEX template
+20
+Co-manipulation of soft-materials estimating deformation from depth images
+Mcconachie, D., Dobson, A., Ruan, M., Berenson, D. (2020). Manipulating
+deformable objects by interleaving prediction, planning, and control. The
+International Journal of Robotics Research, 39, 957 - 982.
+MERGING Consortium (2019). Manipulation enhancement through robotic
+guidance and intelligent novel grippers (merging).
+Retrieved from
+http://www.merging-project.eu/
+Miller, S., van den Berg, J.P., Fritz, M., Darrell, T., Goldberg, K., Abbeel,
+P.
+(2012).
+A geometric approach to robotic laundry folding.
+The
+International Journal of Robotics Research, 31, 249 - 267.
+Nicola, G. (2022). Co-manipulation of soft-materials estimating deformation
+from depth images. https://doi.org/10.5281/zenodo.7300247. ([Online;
+accessed 10-November-2022])
+Nicola, G., Villagrossi, E., Pedrocchi, N. (2022). Human-robot co-manipulation
+of soft materials: enable a robot manual guidance using a depth
+map feedback.
+2022 31st ieee international conference on robot and
+human interactive communication (ro-man) (p. 498-504).
+10.1109/
+RO-MAN53752.2022.9900710
+Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., . . .
+Chintala, S. (2019). Pytorch: An imperative style, high-performance
+deep learning library. Advances in neural information processing sys-
+tems 32 (pp. 8024–8035).
+Curran Associates, Inc.
+Retrieved from
+https://proceedings.neurips.cc/paper/2019/file/bdbca288fee7f92f2bfa9f7012727740-
+Paper.pdf
+Sanchez, J., Corrales, J.-A., Bouzgarrou, B.-C., Mezouar, Y.
+(2018).
+Robotic manipulation and sensing of deformable objects in domes-
+tic
+and
+industrial
+applications:
+a
+survey.
+The
+International
+Journal
+of
+Robotics
+Research,
+37(7),
+688-716.
+Retrieved
+from
+htps://doi.org/10.1177/0278364918779698
+https://arxiv.org/abs/
+https://doi.org/10.1177/0278364918779698
+10.1177/0278364918779698
+She, Y., Wang, S., Dong, S., Sunil, N., Rodriguez, A., Adelson, E.H. (2020).
+Cable manipulation with a tactile-reactive gripper. The International
+Journal of Robotics Research, 40, 1385 - 1401.
+Simonyan, K., & Zisserman, A. (2015). Very deep convolutional networks for
+large-scale image recognition. CoRR, abs/1409.1556.
+
+Springer Nature 2021 LATEX template
+Co-manipulation of soft-materials estimating deformation from depth images
+21
+Sirintuna, D., Giammarino, A., Ajoudani, A. (2022). Human-robot collab-
+orative carrying of objects with unknown deformation characteristics.
+ArXiv, abs/2201.10392.
+Tanaka,
+D.,
+Arnold,
+S.,
+Yamazaki,
+K.
+(2018).
+Emd
+net:
+An
+encode–manipulate–decode network for cloth manipulation.
+IEEE
+Robotics and Automation Letters, 3(3), 1771-1778.
+10.1109/LRA.2018.2800122
+Tang, T., & Tomizuka, M.
+(2022).
+Track deformable objects from point
+clouds with structure preserved registration. The International Journal
+of Robotics Research, 41, 599 - 614.
+T¨olgyessy, M., Dekan, M., Chovanec, L. (2021). Skeleton tracking accuracy
+and precision evaluation of kinect v1, kinect v2, and the azure kinect.
+Applied Sciences, 11(12). Retrieved from https://www.mdpi.com/2076-
+3417/11/12/5756
+10.3390/app11125756
+Tsurumine, Y., & Matsubara, T. (2022). Variationally autoencoded dynamic
+policy programming for robotic cloth manipulation planning based
+on raw images.
+2022 ieee/sice international symposium on system
+integration (sii) (p. 416-421). 10.1109/SII52469.2022.9708850
+Vasiliev,
+V.V.,
+&
+Morozov,
+E.V.
+(2018).
+Chapter
+8
+-
+equations
+of
+the
+applied
+theory
+of
+thin-walled
+composite
+struc-
+tures
+(Fourth
+Edition
+ed.).
+Elsevier.
+Retrieved
+from
+https://www.sciencedirect.com/science/article/pii/B9780081022092000086
+https://doi.org/10.1016/B978-0-08-102209-2.00008-6
+Verleysen, A., Biondina, M., Wyffels, F.
+(2020).
+Video dataset of human
+demonstrations of folding clothing for robotic folding. The International
+Journal of Robotics Research, 39, 1031 - 1036.
+Wang, J., & Olson, E. (2016). Apriltag 2: Efficient and robust fiducial detec-
+tion.
+2016 ieee/rsj international conference on intelligent robots and
+systems (iros) (p. 4193-4198). 10.1109/IROS.2016.7759617
+Wang, W., & Balkcom, D.J. (2018). Knot grasping, folding, and re-grasping.
+The International Journal of Robotics Research, 37, 378 - 399.
+
+Springer Nature 2021 LATEX template
+22
+Co-manipulation of soft-materials estimating deformation from depth images
+Yang, P.-C., Sasaki, K., Suzuki, K., Kase, K., Sugano, S., Ogata, T. (2017).
+Repeatable folding task by humanoid robot worker using deep learning.
+IEEE Robotics and Automation Letters, 2(2), 397-403.
+10.1109/LRA.2016.2633383
+

Y9AyT4oBgHgl3EQfvvk3/content/tmp_files/2301.00635v1.pdf.txt

@@ -0,0 +1,1424 @@
+Coexistence of phononic Weyl, nodal line, and threefold excitations
+in chalcopyrite CdGeAs2 and associated thermoelectric properties
+Vikas Saini,1, ∗ Bikash Patra,1, † Bahadur Singh,1 and A. Thamizhavel1, ‡
+1Department of Condensed Matter Physics and Materials Science,
+Tata Institute of Fundamental Research, Mumbai 400005, India
+Realization of topologically protected quantum states leads to unprecedented opportunities for
+fundamental science and device applications.
+Here, we demonstrate the coexistence of multiple
+topological phononic states and calculate the associated thermoelectric properties of a chalcopyrite
+material CdGeAs2 using first-principles theoretical modeling. CdGeAs2 is a direct bandgap semi-
+conductor with a bandgap of 0.65 eV. By analysing the phonon spectrum and associated symmetries,
+we show the presence of nearly isolated Weyl, nodal line, and threefold band crossings in CdGeAs2.
+Specifically, the two triply degenerate points (TDPs) identified on the kz axis are formed by the
+optical phonons bands 7, 8, and 9 with type-II energy dispersion. These TDPs form a time reversal
+pair and are connected by a straight nodal line with zero Berry phase. The TDPs formed between
+bands 14, 15, and 16 exhibit type-I crossings and are connected through the open straight nodal
+line. Our transport calculations show a large thermopower exceeding ∼500 and 200 µV/K for the
+hole and electron carriers, respectively, above 500 K with a carrier doping of 1018 cm−3. The large
+thermopower in p-type CdGeAs2 is a consequence of the sharp density of states appear from the
+presence of a heavy hole band at the Γ point. We argue that the presence of topological states in
+the phonon bands could lead to low lattice thermal conductivity and drive a high figure-of-merit in
+CdGeAs2.
+I.
+INTRODUCTION
+The existence of topologically protected nontrivial
+states has been demonstrated in wide classes of crys-
+talline materials [1–3]. For example, Dirac semimetals [4–
+7] are formed by the crossings of spin degenerate valence
+and conduction bands in the momentum space.
+The
+low-energy excitations around the crossing points mimic
+the linear energy-momentum relation of Dirac fermions.
+These Dirac points are invariant under perturbations pre-
+serving parity and time-reversal symmetries. Breaking
+either of these symmetries splits a Dirac point into two
+Weyl points with opposite chirality [8–11].
+The band
+crossings in presence of non-symmorphic crystallographic
+symmetries can lead to higher fold fermionic excitations
+and have been realized in numerous materials [12–15].
+The triply degenerate point (TDP) semimetal states have
+been established in MoP, WC, ZrSe, etc. materials where
+the presence of three-fold crossings are shown to ex-
+ist between a double degenerate and single degenerate
+bands [16–21]. Importantly, the TDP semimetal phase
+conceptually lies between the Dirac and Weyl phases.
+Topological band crossings can also be classified based on
+the dimensionality of the band touching points. They can
+form nodal points with zero-dimensional point-like cross-
+ings or nodal lines with one-dimensional band crossings.
+Such one-dimensional nodal lines can constitute nodal
+lines, nodal links, nodal knots, nodal rings, and nodal
+chains, etc. [22–26]. Besides the degeneracies and dimen-
+sionalities of the crossing points, chiral charges of the
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+nodal points, energy dispersion and slope of the bands
+are essential to distinguish among a variety of topologi-
+cal states. In particular, the type-II band crossings are
+identified at the touching point of the hole and electron
+pockets and break the Lorentz symmetry [27–31]. The
+topology in the electronic structure has been explored
+over the last several years, and many novel phenomena
+have been proposed and verified in experiments. Com-
+mon to the topology of electronic structure is that they
+are constrained by the Pauli exclusion principle.
+Topology of the bosonic states especially topological
+states in the phonon spectrum is an emerging research
+field. Phonons obey the Bose-Einstein statistics and are
+not constrained by the Fermi energy, giving access to the
+whole spectrum of phonon energy range from the THz
+and IR to probe the topological states [32–34]. Recent
+studies on graphene, FeSi, and other materials uncover
+the presence of topological excitations such as Weyl, dou-
+ble Weyl, and multifold Weyl in the phonon spectrum.
+The higher fold degenerate phonons have been proposed
+for several materials with specific space groups which
+show the topological phonon states protected by various
+crystalline symmetries [35–41]. Motivated by the stud-
+ies of topological phonons in materials and their possible
+effects on thermal transport, we explore the topological
+phonons and transport properties of chalcopyrite mate-
+rial CdGeAs2. Our phonon calculations reveal topolog-
+ical Weyl phonons, triply-degenerate nodal points, and
+nodal lines, among other phases in various phonon bands.
+Specifically, we find Weyl points with chiral charge ±1
+in multiple phonons branches. Moreover, multiple TDPs
+protected by C3v symmetry are found on the kz axis with
+both the type I and type II energy dispersions.
+We also investigate the electronic and thermoelec-
+tric properties of CdGeAs2.
+Electronic structure and
+arXiv:2301.00635v1  [cond-mat.str-el]  2 Jan 2023
+
+2
+transport calculations are carried with both the gener-
+alized gradient approximation (GGA) [42] and mBJ [43]
+exchange-correlation (XC) functionals and with the in-
+clusion of spin-orbit coupling (SOC). The results ob-
+tained with mBJ functional shows a band gap of 0.65
+eV, close to experimentally observed values of 0.57 eV at
+room temperature [44–46]. The calculated thermopower
+(S) of p-type carriers is found to be more than the n-type
+carriers in our considered carrier density range of 1018 -
+1021 cm−3. This large thermopower for p-type carriers is
+attributed to the steepness in the density of states (DOS)
+below the Fermi energy level due to the presence of heavy
+hole-type bands. We also discuss the effect of carrier ef-
+fective mass in generating the large thermopower and fig-
+ure of merit ZT e = S2σ/τ
+ke/τ . Specifically, the upper limit
+of ZT e without considering the lattice thermal conduc-
+tivity, is remarkably large with a value of ∼ 11.5 at 600 K
+and carrier density ∼ 1.1× 1018 cm−3 for p-type carriers.
+In this way, our work identifies CdGeAs2 as a potential
+chalcopyrite material for exploring topological phonons
+and thermoelectric properties.
+(a)
+           (b)
+k x
+k y
+k z
+Z
+N
+Γ
+X
+P
+(c)
+     (d)
+Cd
+Ge
+As
+a
+b
+c
+a
+b
+a
+b
+c
+FIG. 1.
+Crystal structure and Brillouin zone of CdGeAs2.
+(a), (b) Conventional tetragonal unit cell of CdGeAs2. (c)
+Primitive unit cell and (d) associated Brillouin zone structure
+of CdGeAs2. High-symmetry points are marked in red.
+II.
+METHODS
+Electronic
+structure
+calculations
+were
+performed
+within the framework of density functional theory (DFT)
+using the full-potential linearized augmented plane wave
+(FP-LAPW) method as implemented in WIEN2k pack-
+age [47–49]. Self-consistent calculations were performed
+on a dense k-mesh with 10000 k points. Plane-wave cut-
+off for RMTKMAX was set to be 9 (RMT is the muffin tin
+radius and KMAX is the maximum value of reciprocal lat-
+tice vector). Thermoelectric properties were calculated
+by solving the Boltzmann transport equations under the
+constant scattering time approximation (CSTA) as im-
+plemented in the BoltzTraP2 [50, 51]. We used 149784
+irreducible k points to accurately model transport prop-
+erties. The phonon spectrum was obtained using the fi-
+nite displacement method by considering a 2×2×2 super-
+cell employing the VASP [52–54] and Phonopy codes [55].
+The lattice parameters for the first-principles and ther-
+moelectric calculations were obtained from the pow-
+der x-ray diffraction experiments using our synthesized
+CdGeAs2 polycrystalline sample.
+The refined parame-
+ters were obtained as a = 5.94(4) ˚A, c = 11.21(6) ˚A and
+the Wyckoff positions (0.22009, 0.25, 0.125), (0.0, 0.0,
+0.5), and (0.0, 0.0, 0.0) for As, Cd, and Ge atoms, re-
+spectively.
+III.
+RESULTS AND DISCUSSION
+The CdGeAs2 crystallizes in a non-centrosymmetric
+tetragonal crystal lattice with space group I¯42d (No.
+122). The conventional unit cell of CdGeAs2 is shown
+in Fig. 1(a). The As atoms in the top layer are shifted
+with respect to the As atoms in the bottom layer, as
+shown in Fig. 1(b).
+This suggests the absence of in-
+version symmetry in the crystal. Figure 1(c) shows the
+primitive unit cell and Fig. 1(d) illustrates the associated
+Brillouin zone (BZ) with high-symmetry points marked.
+The phonon dispersion of CdGeAs2 along various high
+symmetry paths is shown in Fig. 2(a). The absence of
+negative phonon frequencies suggests the dynamical sta-
+bility of the crystal. From the phonon calculations, we
+found that the acoustic phonons are mainly contributed
+by the heavy Cd atoms, as depicted from the atom pro-
+jected density of states in Fig. 2(b).
+The topological phonon analysis of CdGeAs2 is shown
+in Figs. 2, 3, and 4. The acoustic phonon modes along the
+in-plane Γ-X direction are non-degenerate whereas, the
+two lowest energy acoustic modes are degenerate along
+the Γ - Z direction. Top transverse acoustic (TTA) mode
+possesses linear phonon dispersion along the in- and out-
+of-plane directions.
+For the topological analysis, a few of the phonon bands
+are selected, which are highlighted in Fig. 2(a) from their
+bulk band crossings. Bands 8 and 9 show band crossings
+leading to the symmetry-protected nodal rings, nodal
+line, Weyl, and triply degenerate points inside the BZ.
+All the nodal phases formed by the crossings of these
+bands are summarized in Fig. 3(a).
+The Weyl points
+are located parallel to Γ - X directions but not at the
+kz = 0 plane. The nodal line formed from these bands
+is represented by the green color in Fig. 2(a), at the
+endpoint of the nodal ring band 7 appears to form the
+triply degenerate point along the Γ - Z direction. The
+triply degenerate points of the 7, 8, and 9 bands cross-
+
+3
+TDP
+TDP
+WP
+NL
+NL
+7
+8
+9
+14
+15
+16
+TTA
+FIG. 2. (a) Calculated phonon dispersion of CdGeAs2 along high symmetry Brillouin zone directions. The topological crossings
+are highlighted. (b) Partial and total phonon density of states (DOS) of CdGeAs2.
+ings are protected by the C3z rotational symmetry. The
+dispersion along the kz direction for these three bands
+are shown separately in Fig. 3(b). The triply degenerate
+points are obtained from the crossings of doubly degen-
+erate and non-degenerate bands. The 3D visualization of
+these three bands in kx − ky and kx − kz planes can be
+observed from the Fig. 3(c,d). Fig. 3(c) shows that the
+TDP point is formed by the crossings of two nearly flat
+and one dispersive band. Since the slope of the crossing
+bands appears to be the same, therefore, type-II TDP is
+resolved in the kx −ky plane. The two TDP points along
+the kz axis are connected through a straight doubly de-
+generated Weyl nodal line. The calculated Berry phase
+for this nodal line is 0 suggesting a topologically trivial
+character and that is followed by the in-plane quadratic
+dispersion as shown in Fig. 3(e). The nodal line is formed
+by the bands 8 and 9 as depicted in Fig. 3(a). Below the
+TDP along kz axis two nodal rings intersect which are
+lying on the diagonal mirror planes and the Berry phase
+of both rings is 0, which again reveals the topologically
+trivial character of these rings and the band dispersion
+along the diagonal axis is depicted in Fig. 3(f). The cross-
+ing point is formed by one nearly flat and one dispersive
+band. For the Berry phase calculation, a circle contour
+is defined perpendicular to the nodal plane with a small
+finite radius to avoid inclosing other band crossings.
+Moreover, eight Weyl points are observed parallel to kx
+and ky axes at kz = ± 0.21 ˚A−1. These eight Weyl points
+formed by these bands possess a magnitude of charge 1
+and all the Weyl points are at the same frequency 2.41
+THz as a result of mirror inversion in the mxy diagonal
+planes. For same kx and ky coordinates opposite kz WPs
+have the same chiral charge owing to the absence of the
+mz symmetry thus the projected WPs on the (001) sur-
+face possess a chiral charge of ± 2 and are at the kx and
+ky axes as illustrated in the Fig. 3(i). The Berry curva-
+ture around a positive chiral WP is non-zero as depicted
+in Fig. 3(h). Opposite chiral charge Weyl nodes connect
+by the topologically nontrivial arcs on (001) surface as
+depicted in Fig. 3(g) for phonon energy 2.44 THz.
+Similar to the preceding analysis, we also analyzed
+topological features of the crossings of 14 and 15 phonon
+bands as identified in Fig. 4(a). The nodal line is guided
+in green color and a TDP is present inside the shown box.
+Unlike bands 8 and 9, these phonon bands have two pairs
+of Weyl points along the principle momentum axes at kz
+= 0 plane. Notably, the magnitude of the chiral charges
+remains the same as before. The nodal points of these
+bands are summarized in Fig. 4(a). Along kz axis a pair
+of TDP is observed at kz= ±0.37 ˚A−1. These points are
+connected with the open straight line that possesses zero
+Berry phase and obeys the non-linear band characters
+along the in-plane direction. Whereas along the cross-
+ing line it shows the linear band crossings as represented
+in Fig. 4(e,f). The TDP is formed by the crossings of
+14, 15, and 16 bands, whose 3D dispersion along kxky is
+shown in Fig. 4(c). Bands 15 and 16 are doubly degener-
+ate along kz axis before the TDP crossings and after the
+
+4
+NL
+TDP
+7
+8
+9
+7
+9
+8
+7
+8
+M
+Г
++2
++2
+-2
+-Г
+-2
+NLP
+NRP
+TDP
+TDP
+9
+NL
+TDP
+NR
+NR
+(a)
+(g)
+(f)
+(e)
+(d)
+(c)
+(b)
+-2
+Г-
+X
+Y
+M
+-
+-2
+-2
++2
++2
+-
+-
+(i)
+(h)
+X
+Г
+7
+8
+9
+f = 2.44 THz
+FIG. 3. Topological phonon structure for bands 8 and 9 (see Fig. 2 for band index). (a) Nodal crossing points formed by the
+bands 8 and 9 in 3D momentum space. (b) Triple degenerate point and nodal line along Γ-Z direction. (c-d) 3D dispersion
+around the TDP in kx − ky and kx − kz planes. (e) Phonon dispersion for a nodal line point along in-plane axis. (f) Phonon
+dispersion for a point present at nodal ring along the diagonal axis. (g) Topological fermi arc surface states connecting the
+projected phonon Weyl points on the (001) surface. (h) Non-zero Berry curvature around the positive Weyl point. (i) Schematic
+diagram of the Weyl points on the (001) surface. Blue and yellow circles denote the positive and negative charges.
+crossing bands, 14 and 15 become degenerate for a range
+of k-values as shown in Fig. 4(b,d).
+We also discuss the Weyl phonons which are present
+on the kz = 0 plane along the kx and ky axes at |kx| =
+|ky| = 0.31 ˚A−1. The energy-momentum relations and
+locations of WPs in the 3D are shown in Fig. 4(g). The
+Berry phases around the opposite chiral WPs are calcu-
+lated using the Wilson loop method from the Wannier
+charges. Positive and negative chiral WPs possess π and
+−π Berry phases that can be observed from Fig. 4(h-i).
+WPs on kx axis and ky axis have opposite chiral charges.
+The opposite chiral Weyl points are connected through
+the arcs as depicted in Fig. 4(j) and the corresponding
+surface states of the opposite WPs on to the (001) surface
+are shown in Fig. 4(k,l).
+The topology in the phonon bands leads to the symme-
+try protected states which stays robust against symmetry
+respected perturbations. There are many other bands in
+the phonon spectrum featuring topological states, for ex-
+ample acoustic and optical bands possess several straight
+nodal lines and TDP along kz direction protected by the
+crystalline symmetries. Topological protection of phonon
+states lead to many novel phenomena and increase in
+the phonon scatterings result in the reduction of the lat-
+tice thermal conductivity that leads to the thermoelec-
+tric properties. It has been established for many of the
+materials such as TaSb, TaBi, NbSb etc.
+that having
+the topological phonons lead to the lower lattice thermal
+
+0.2
+0.20
+0.0
+0.25
+kx (A-l)
+-0.2
+0.30Frequency (THz)
+2.94
+2.52
+2.10
+0.2
+0.2
+0.3
+0.0
+0.4
+k
+(A-1)
+-0.23.2
+(zHL)
+2.8
+Frequency
+2.4
+0.2
+-0.2
+0.0
+0.0
+k
+-0.2
+(A-1)
+0.20.4
+(r-y)
+0.0
+K
+-0.4
+-0.4
+0.0
+0.4
+kx(A-l)3.0
+Frequency (THz)
+2.5
+0
+0.52.6
+(THz)
+Frequency
+2.4
+0.5
+1.03.0
+(ZHL) 
+2.5
+Frequency
+2.0
+1.5
+1.0
+T
+Z0.25
+(A-1)
+0.00
+K
+-0.25
+0.25
+-0.25
+0.00
+0.00
+-0.25
+kx (A-l)
+0.255
+14
+15
+TSS
+TSS
++
+-
+WP
+WP
+SS
+-1
++1
+-1
++1
+Г-
+f = 5.99 THz
+16
+15
+15
+14
+14
+16
+NLP
+TDP
+TDP
+NLP
+14
+14
+15
+16
+15
+16
+NL
+TDP
+WP
+NL
+(a)
+(i)
+(h)
+(g)
+(f)
+(e)
+(d)
+(c)
+(b)
+(l)
+(k)
+(j)
+Г
+X
+Г
+Y
+k // ΓX
+k // ΓZ
+1.0
+FIG. 4. Topological phonon structure for bands 14 and 15 (see Fig. 2 for band index). (a) Nodal crossings of bands 14 and 15
+inside the BZ. (b) Isolated part for the TDP and NL along Γ-Z direction. (c-d) 3D phonon dispersion in kx − ky and kx − kz
+planes around the TDP. NL is marked along the kz axis in (d). (e-f) Dispersion along the ky and kz directions around a point
+present at the nodal line. (g) Weyl points distribution on the (001) surface. The WPs are drawn as circles. (h-i) Wannier
+center evolution on a sphere enclosing the positive and negative chiral charge WPs. (d) Topological fermi arc surface states
+connecting opposite chiral WPs on the (001) surface. (k-l) Chiral topological surface states connecting the projected bulk WPs.
+conductivity than the materials which do not have the
+topological characters in the phonons [56–58].
+In par-
+allel to topological phonons, many electronic properties
+such as high band gap value between valence and conduc-
+tion bands that suppress the bipolar effect prevents the
+reduction of thermopower below a certain temperature.
+Heavy hole band for p-type doping gives rise to the large
+thermopower, and the anisotropic nature of hole band
+supports to amplify the thermoelectric power factor and
+ZT by increasing the mobility of charge carriers along
+highly dispersive band directions.
+We present the calculated bulk band structure ob-
+tained with both the GGA and mBJ XC functionals in
+Figs. 5(a-c) in the presence of spin-orbit coupling (SOC).
+A negative bandgap of ∼ 0.08 eV is obtained with GGA
+XC functional with a clear band inversion at Γ point
+[Figs. 5(a) and Fig. 5(b)].
+The orbital resolved band
+structure is shown in Fig. 5(b) where As p (blue color)
+and Ge s (green color) states contribute dominantly near
+the Fermi level. A band inversion between the As p and
+Ge s states is evident at the Γ-point. We also examine
+the electronic structure in the absence of SOC.
+Weyl points of positive chiral charges are located at
+kx axis and negative charges are found at ky axis. For
+surface states, four Weyl points in the bulk are projected
+TABLE I. Location of Weyl nodes on the kz = 0 plane without
+SOC obtained with GGA XC functional.
+Strain
+b
+′=b
+1.01b
+1.02b
+1.03b
+Weyl position
+kx = ky (˚A−1)
+0.028
+0.022
+0.016
+0.008
+onto (001) surface, and two opposite chiral Weyl points
+are connected with the arcs as represented in Fig. 5(g).
+To check the effect of strain on the location of Weyl points
+in momentum space we summarized a few of the strains
+in Table I. Keeping the unit cell volume constant, com-
+pression on the small a-axis of the tetragonal structure
+is applied in the form of the tensile strain on the longest
+b-axis by 1 to 3%. In absence of SOC, the Weyl phase is
+robust with four Weyl nodes on the kz = 0 plane. The
+separation between the Weyl nodes points continuously
+decreases as the strain is increased from 1% to 3% as
+tabulated in Table I.
+GGA XC functional underestimates band gap in the
+materials therefore we have carried out thermoelectric
+calculations in presence of the mBJ XC functional.
+
+0.5
+(A-1
+0.0
+-0.5
+-0.5
+0.0
+0.5
+kx(A-l)1.0
+0.5
+0
+0.5
+k (元/a)6.10
+(ZHL)
+6.05
+Frequency
+6.00
+5.95
+5.90
+0.1
+0.2
+0.3
+0.4
+0.56.10
+(THZ)
+6.05
+Frequency (
+6.00
+5.95
+5.90
+0.0
+0.1
+0.2
+0.3
+0.41.0
+? 0.5
+0
+0.5
+1.0
+k (元/a)6.2
+6.0
+kx (
+0.0
+-0.4
+-0.5
+-0.3
+k, (A-)6.2
+Frequency (THz)
+6.0
+5.8
+-0.2
+0.0
+0.2
+0.0.
+0.2
+-0.2
+k, (A-)kx (A-l)
+0
+k, (A-l)6.10
+ (THz)
+Frequency
+6.05
+6.00
+0
+5.006.04
+(ZHL) 1
+Frequency
+6.02
+6.00
+0
+0.1
+0.26.4
+(ZHL)
+6.2
+Frequency
+6.0
+5.8
+5.6
+Z0.5
+(A-I)
+0.0
+-0.5
+0.25
+-0.25
+0.00
+0.00
+-0.25
+kx (A-)
+0.256
+E - Ef = 0.07 eV
+FIG. 5. (a) Electronic band structure of CdGeAs2 obtained with GGA XC functional. (b) Orbital resolved band structure
+obtained with GGA XC functional. (c) Same as (a) but obtained with the mBJ XC functional. (d) Calculated total DOS
+for the GGA and mBJ XC functionals. (e) Band dispersion of valence and conduction bands for Γ-X and Γ-Z directions.
+(f) Calculated carrier effective masses for in-plane and out-of-plane directions for p-type carriers at 300 K. (g) Topologically
+nontrivial surface states on crystallographic (001) surface obtained for GGA XC functional without SOC. (h) Anisotropic nature
+of topmost hole band below 20 meV energy from the Fermi level for mBJ XC functional calculation. (i) Electron pocket of the
+lowest conduction band at an energy around 1 eV upward from the Fermi level for mBJ XC functional.
+Figure 5(c) shows the band structure obtained with
+mBJ XC functional.
+The band inversion is now dis-
+appeared and the system becomes trivial with a direct
+bandgap of 0.65 eV. This is close to the experimentally
+reported value of 0.57 eV at room temperature [44–46].
+Fig. 5(d) represents the density of states (DOS) for GGA
+and mBJ XC functionals in red and blue colors, respec-
+tively. The DOS below the Fermi energy shows a rapid in-
+crease with energy which leads to the large thermopower
+for p-type doping as discussed below.
+Our first-principles results show heavy hole bands
+along Γ-X direction whereas the bands are highly dis-
+persive along Γ-Z direction [Fig. 5(e)].
+This suggests
+that the charge carriers of in-plane direction (Γ-X) pos-
+sesses high effective mass compared to the out-of-plane
+direction (Γ-Z) which possesses light effective mass and
+high mobility. To see the anisotropic effects, the weighted
+mobility can be defined as µw = µ( m∗
+me )
+3
+2 , where µ is the
+mobility of charge carriers, m∗ = N
+2
+3
+v (mxmymz)
+1
+3 ; Nv
+valley degeneracy, and me is the rest mass of the electron.
+The highly dispersive band along out-of-plane (Γ-Z) di-
+rection gives rise to the higher weighted mobility µw be-
+cause of high mobility as compared to the in-plane Γ-X
+
+0.04
+0.02
+k(A-l)
+0.00
+-0.02
+-0.04
+-0.04
+-0.02
+0.00
+0.02
+0.04
+k(A-l)7
+ !"
+"
+" 
+#$%&'" 
+"()
+*"+
+*",
+*"-
+" 
+"(
+" 
+".
+" 
+/ 
+" 
+/"
+ !"#"'0+
+*1-
+&/  &2
+&1  
+&3  
+&4  
+&5  
+&6  
+&(  
+&
+&7*897:
+&;*897:
+'<-
+5  
+4  
+3  
+1  
+/  
+"  
+ 
+=
+/#$%'" 
+.>2
+*/+
+*",
+*"-
+" 
+"(
+" 
+".
+" 
+/ 
+" 
+/"
+ !"#"'0+
+*1-
+&/  &2
+&1  
+&3  
+&4  
+&5  
+&6  
+&(  
+&
+&7*897:
+&;*897:
+'0-
+ !  "
+ ! "
+ !"
+"
+" 
+2:$%'" 
+"3>+
+*"2
+*",
+*"-
+" 
+"(
+" 
+".
+" 
+/ 
+" 
+/"
+ !"#"'0+
+*1-
+&/  &2
+&1  
+&3  
+&4  
+&5  
+&6  
+&(  
+&7*897:
+&;*897:
+'?-
+5  
+3  
+/  
+ 
+*/  
+='@A$2-
+" 
+"(
+" 
+".
+" 
+/ 
+" 
+/"
+ !"#"'0+
+*1-
+&/  &2
+&1  
+&3  
+&4  
+&5  
+&&7*897:
+&;*897:
+'B-
+&6  
+&(  
+FIG. 6. Calculated thermoelectric properties of CdGeAs2 for p-type and n-type doping. (a) Thermopower plot with varying
+carrier density at constant temperatures. (b) Electrical conductivity divided by relaxation time as a function of carrier density
+in the regime of 1018-1021 cm−3. (c) The power factor divided by relaxation time with carrier density at constant temperatures.
+(d) Electrical thermal conductivity over relaxation time plotted against carrier density for various temperatures.
+direction. The high value of the µw lead to the higher
+power factor and ZT parameters. The anisotropy in ef-
+fective masses is calculated from transport parameters at
+room temperature
+1
+m∗
+αβ = σαβ/τ
+e2n ; where σαβ/τ is electri-
+cal conductivity tensor divided by relaxation time, n is
+the carrier density. Further, the anisotropic band struc-
+ture is consistent with the calculated effective masses for
+ab-plane (mxx) and c-direction (mzz) for p-type CdGeAs2
+as shown in Fig. 5(f).
+In the low carrier density p ∼
+1.27 × 1018 cm−3 the anisotropy in the masses is maxi-
+mum mxx ∼ 1.9 mzz, and as the hole carrier density in-
+creases the anisotropy decreases and the effective masses
+for in- and out-of-plane bands are around mxx = 0.73 me
+and mzz = 0.63 me for p ∼ 1021 cm−3.
+Figures 6, 7, and 8 show the calculated thermoelectric
+properties of CdGeAs2 within constant relaxation time
+approximation [59–63]. The thermopower (S) of p-type
+carriers (solid curves) is greater than n-type carriers (dot-
+ted curves) in the entire carrier density range 1018 - 1021
+cm−3 at a constant temperature as shown in Fig. 6(a).
+The thermopower increases with increasing temperature
+and reaches a value of 567 µV/K at 700 K for p-type
+doping and 270 µV/K at 800 K for n-type at carrier con-
+centration 1018 cm−3.
+To understand the large thermopower of p-type carri-
+ers, we have used the Mott relation in constant relaxation
+time approximation which is given by [64, 65].
+S(n, T) = π2k2
+BT
+3q
+� 1
+n
+dn(E)
+dE
++ 1
+µ
+dµ(E)
+dE
+�
+EF
+(1)
+Here, q denotes the charge, n(E) represents the density
+of states, µ is the mobility, and T denotes the absolute
+temperature. The equation is composed of the addition
+of two derivative terms. It can be simply related from the
+first term of Eqn. 1 that steepness in the DOS enhances
+thermopower. The mBJ XC calculated DOS (Fig 5(d))
+
+8
+ 
+!
+"
+#$$%#&&'(()$$%)&&
+!"
+!*
+!"
+!+
+!"
+ "
+!"
+ !
+ (,-.
+/01
+(#$$%#&&
+()$$%)&&
+!(2(0""(3
+,41
+!5
+! 
+!"
+*
+6
+5
+ 
+"
+"!#
+!"
+!*
+!"
+!+
+!"
+ "
+!"
+ !
+ $%&(,-.
+/01
+( ""(3
+(0""
+(7""
+(6""
+(8""
+(*""
+(9/:;9<
+(=/:;9<
+,>1
+FIG. 7.
+(a) Anisotropic parameters σzz/σxx and Szz/Sxx
+plotted as a function of hole carrier density at T= 300 K. (b)
+The upper limit of the figure-of-merit (ZTe) as a function of
+carrier density at constant temperatures.
+for p-type doping shows a steep rise than the n-type dop-
+ing in CdGeAs2 resulting large thermopower for the p
+doped system. The second term of the Mott relation ex-
+hibits that if the mobility of charge carriers increases with
+energy as a consequence of the critical scatterings near
+the Fermi level then the thermopower can be boosted
+further.
+Thermopower of both types of carriers gradually re-
+duce with increasing carrier density except for the p-type
+800 K plot. For p-type carriers, owing to electronic ther-
+mal excitations, the thermopower reduces with decreas-
+ing carrier density at high temperatures as observed in
+Fig. 6(a) for 800 K.
+The bipolar conduction effect in the low carrier density
+regime for the narrow gap semiconductors is kind of nor-
+mal, as studies suggest, to suppress this effect the band
+gap energy should be more than around 8 kBT [66].
+The band gap of CdGeAs2 (0.65 eV) is reasonably large
+which ensures that the bipolar effect is not seen at tem-
+peratures below 700 K. The band gap of CdGeAs2 is
+higher than the calculated values for the other TE ma-
+terials such as (PbSe (Eg = 0.28) and PbTe (Eg = 0.36)
+from ref. [67]). Also, thermopower at 300 K for the p-
+type CdGeAs2 is more than the PbSe, SnTe and PbTe
+compounds [59, 60, 68].
+The electrical conductivity divided by the relaxation
+time σ
+τ is represented in Fig. 6(b) on a logarithmic scale
+for both types of charge carriers for different tempera-
+tures from 200 to 800 K. The
+σ
+τ of the n-type charge
+carriers is higher than the p-type carriers that is consis-
+tent from the highly dispersive nature of the n-type band
+compared to the p-type band that results to have rela-
+tively small effective mass of n-type band which essen-
+tially increases the electrical conductivity. The increase
+in temperature reduces the electrical conductivity in the
+entire range of carrier density 1018 - 1021 cm−3. How-
+ever, the magnitude of change with the temperature at
+a given carrier concentration is not much for both types
+of carriers. For p-type charge carriers σ
+τ is (0.1 and 0.06)
+× 1018 Ω−1 m−1 s−1 at carrier concentration 1018 cm−3
+and for temperatures 300 and 800 K.
+The power factor divided by the relaxation time S2σ
+τ
+is shown in Fig. 6(c) for the both types of charge carriers
+in the range of carrier density 1018 - 1021 cm−3. It is
+obvious from the figure that the p-type power factor is
+dominated over the n-type power factor in intermediate
+carrier density regime before crossover appears around
+1.07×1020 cm−3 and after the crossover n-type power fac-
+tor is dominated over p-type until carrier density reaches
+up to 1021 cm−3 for each temperature ranging from 200
+to 800 K.
+The power factor is one of the considerable parame-
+ters to increase the thermoelectric performance of mate-
+rials. However, for the rough estimation of power factor
+if empirically relaxation time is considered in the order
+of 10−14 s then it governs reasonably good values of the
+power factor in the range of mWK−2m−1 to µWK−2m−1
+for the p-type carriers from high to low density regime,
+attributed to the optimized band structure of CdGeAs2.
+The anisotropic ratio of out-of- and in-plane ther-
+mopower Szz/Sxx and electrical conductivity σzz/σxx
+are plotted against hole carrier density at T= 300 K
+in the Fig. 7(a).
+The thermopower along the out-of-
+plane direction is lower than the in-plane direction as
+a consequence of light band along Γ-Z direction which
+leads to the small effective mass. The Szz/Sxx is weakly
+changing in the low carrier density regime. Conversely,
+the electrical conductivity for out-of-plane direction is
+σzz ∼ 1.9 σxx at carrier density p ∼ 1.27×1018 cm−3.
+Beyond this hole density, the anisotropy in conductivity
+decreases up to 1.9×1020 cm−3 and later this increases
+very gradually which attains σzz ∼ 1.16 σxx at carrier
+density p ∼ 1021 cm−3. The anisotropy in the calculated
+transport parameters is consistent with the anisotropic
+effective mass as discussed in Fig. 5(f).
+Finally, the anisotropic nature of the hole band at Γ
+point is shown in Fig. 5 (h) where the Fermi level is
+shifted below 0.2 eV from its pristine value which drives
+the anisotropy in the electrical conductivity that helps to
+
+9
+ !"
+ !#
+ !$
+ 
+ !
+$ 
+$%
+$ 
+$&
+$ 
+$'
+$ 
+# 
+$ 
+#$
+$ 
+##
+"()*+
+,"-
+.(/($ 
+,$01
+)*-
+("  (2
+(0  
+(3  
+(4  
+$!3
+$! 
+ !3
+ 
+ !
+$ 
+$%
+$ 
+$&
+$ 
+$'
+$ 
+# 
+$ 
+#$
+$ 
+##
+"()*+
+,"-
+.(/($ 
+,$"1
+)5-
+("  (2
+(0  
+(3  
+(4  
+ !"
+ !#
+ !$
+ 
+ !
+$ 
+$%
+$ 
+$&
+$ 
+$'
+$ 
+# 
+$ 
+#$
+#()*+
+,"-
+.(/($ 
+,$01
+("  (2
+(0  (
+(3  
+(4  
+(
+)6-
+ !0
+ !"
+ !#
+ !$
+ 
+ !
+$ 
+$%
+$ 
+$&
+$ 
+$'
+$ 
+# 
+$ 
+#$
+#()*+
+,"-
+.(/($ 
+,$"1
+)7-
+FIG. 8. Calculated figure-of-merit ZT for CdGeAs2. (a-b) ZT values as the functions of hole and electron carrier densities at
+a constant scattering time τ = 10−13. (c-d) Estimation of ZT values at τ = 10−14 s, for hole and electron type of carriers,
+respectively.
+increase the weighted mobility µw for out-of-plane (Γ-Z)
+direction thus resulting in the high power factor and ZT
+for out-of-plane direction compared to in-plane direction.
+For comparison, we also have the lower conduction band
+visualization while the Fermi level is around 1 eV up from
+the maxima of the valence band that shows the isotropic
+nature of electron band. Therefore, anisotropic advan-
+tage of the hole bands may enhance the thermoelectric
+properties of p-type carriers compared to the n-type car-
+riers.
+The electronic thermal conductivity (ke) can be de-
+fined as ke= LσT. The calculated ke is shown in Fig. 6(d)
+for both carriers in the temperature range from 200 to
+800 K. The electronic thermal conductivity divided by
+relaxation time
+ke
+τ
+for p-type doping is smaller than
+the n-type doping in the entire carrier density regime
+1018 − 1021 cm−3 at the constant temperatures which
+supports the higher ZT e values compared to n-type dop-
+ing as shown in Fig. 7(b).
+For n-type doping,
+ke
+τ
+increases gradually with in-
+creasing temperature in the entire carrier density range
+1018−1021 cm−3 and for p-type doping ke
+τ increases grad-
+ually with increasing temperature in the entire carrier
+density regime below 700 K. At 700 K and above in low
+density regime 1018 - 1019 cm−3 electronic thermal con-
+ductivity increases much more rapidly with lowering the
+carrier density as shown in Fig. 6(d) which results in the
+reduction of ZTe values for T ≳ 700 K (Fig. 7(b)).
+The upper limit of figure-of-merit ZT e = S2σ/τ
+ke/τ
+in the
+low-carrier carrier density regime are exceptionally high
+for p-type carriers (11.5 at T = 600 K and n ∼ 1.1×
+1018 cm−3) than the n-type carriers (3.1 at T = 600 K
+and n ∼ 1.1× 1018 cm−3) as a consequence of optimized
+band structure as depicted in Fig. 7(b).
+As the phonon dispersion of CdGeAs2 shown in
+Fig. 2(a) uncover that the small frequencies of acoustic
+phonons attribute to the low group velocity which can
+lead to low lattice thermal conductivity (kl = 1
+3CV vgl).
+Moreover, as we discussed above the mixing of acoustic
+and optical modes along with the topological protection
+give rise the enhanced phonon scatterings resulting into
+small mean free path and thus indicate a small lattice
+thermal conductivity in CdGeAs2.
+However, the experimentally observed lattice thermal
+conductivity of CdGeAs2 at 300 K is 4 Wm−1K−1 [69].
+As we discussed the electronic part of the figure-of-merit
+ZTe is remarkably high for p-type CdGeAs2 and notably
+this does not depend on the relaxation time τ. The ther-
+moelectric figure-of-merit can be written in terms of elec-
+tronic part of figure-of-merit as ZT =
+S2σ
+ke+kl =
+ZTe
+(1+kl/ke).
+To estimate ZT we need to have the electronic thermal
+conductivity ke. Since from the calculations, we obtain
+
+10
+ke
+τ therefore to get ke the knowledge of relaxation time
+τ is required. For the materials, τ can be a function of
+temperature and carrier density, at the simplest electron-
+phonon scatterings show the inverse temperature behav-
+ior τ ∝ T −1. To get the idea of ZT values in p- and n-type
+CdGeAs2, we approximate the relaxation time as 10−13
+and 10−14 s independent of temperature and carrier den-
+sity and later we will add the effect of these parameters
+on τ for ZT values.
+Figure 8 depicts the calculated ZT for τ = 10−13 and
+10−14 s. For τ = 10−13 s ZT reaches from 0.25 at 300 K
+to 1.16 at 600 K for p-type dopings as shown in Fig. 8(a).
+Whereas it goes up to 0.37 at 600 K from 0.12 at 300 K,
+there is one thing to note that we have used the tem-
+perature independent lattice thermal conductivity kl as
+4 W/m-K but conventionally kl decreases with increasing
+temperature, therefore, we may expect further increase
+in the ZT values.
+For the τ = 10−14 s maximum values of ZT are around
+0.047 and 0.049 obtained at 300 K, respectively for p- and
+n-type dopings, and with increasing temperature ZT in-
+creases and attains 0.26 and 0.22 at the 600 K for p- and
+n-type dopings, respectively. Similar to the previous dis-
+cussion, drops in the kl with increasing temperature may
+be expected to enhance the ZT values further. Of course
+with increasing temperature τ reduces and without ex-
+perimental data the scaling of carrier density is challeng-
+ing but as the previous studies suggest this makes τ to
+be suppressed [62]. Thus incorporating both of the these
+effects into the τ would lead to the decrease in the esti-
+mated values of ZT.
+These analytical results indicate towards a reasonably
+good ZT values at high temperatures T ≳ 300 K. Exper-
+imental engineering of the reinforcing phonon scatterings
+from defects and grain boundaries will further reduce the
+kl and enhancement of the ke would increase the ZT val-
+ues for the better performance of thermoelectric devices.
+In this work, we have covered the complete comprehen-
+sive study of the CdGeAs2 from the calculated results
+these will be surely helpful for the further leads to ex-
+plore the experimental aspects.
+SUMMARY AND CONCLUSION
+We have studied the topological states in the phonon
+spectrum and resolved the triply degenerate phonons on
+the kz axis. Multiple pairs of Weyl points in the bulk
+structure, and their topological surface states are ob-
+served on the (001) surface. The topological features in
+the phonon spectra could suppress lattice thermal con-
+ductivity enhancing the thermoelectric figure of merit.
+We discuss electronic properties of CdGeAs2 that ex-
+hibit many thermoelectric supportive features and result
+in high thermopower for p-type dopings. The anisotropic
+hole bands lead to the high-weighted mobility that sup-
+ports the high power factor and ZT values. The calcu-
+lated value of electronic figure-of-merit ZTe is high (more
+than 2 for temperatures 300 K and above) in carrier den-
+sity regime 1018 - 1019 cm−3 for p-type CdGeAs2 that
+result in the noble response to the thermoelectric per-
+formance. Our study unfold that CdGeAs2 has topolog-
+ically nontrivial phonon states and exhibits very good
+theromelectric properties that are surely useful for the
+potential applications.
+ACKNOWLEDGEMENT
+We thank Prof.
+Kalobaran Maiti for providing the
+computational resources and acknowledge the TIFR com-
+puting resources. This work was supported by the De-
+partment of Atomic Energy of the government of India
+under Project No. 12-R&D-TFR-5.10-0100.
+DATA AVAILABILITY
+The data supporting the findings of this study are
+available within the article. More data can be provided
+on a reasonable request to the corresponding author.
+[1] B. Singh, H. Lin, and A. Bansil, Adv. Mater. , 2201058.
+[2] A. Bansil, H. Lin, and T. Das, Rev. Mod. Phys. 88,
+021004 (2016).
+[3] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045
+(2010).
+[4] Z. Liu, B. Zhou, Y. Zhang, Z. Wang, H. Weng, D. Prab-
+hakaran, S.-K. Mo, Z. Shen, Z. Fang, X. Dai, et al., Sci-
+ence 343, 864 (2014).
+[5] G. Jenkins, C. Lane, B. Barbiellini, A. Sushkov, R. Carey,
+F. Liu, J. Krizan, S. Kushwaha, Q. Gibson, T.-R. Chang,
+et al., Phys. Rev. B 94, 085121 (2016).
+[6] T. Nie, L. Meng, Y. Li, Y. Luan, and J. Yu, J. Condens.
+Matter Phys. 30, 125502 (2018).
+[7] Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu,
+H. Weng, X. Dai, and Z. Fang, Phys. Rev. B 85, 195320
+(2012).
+[8] Y. Sun, S.-C. Wu, and B. Yan, Phys. Rev. B 92, 115428
+(2015).
+[9] B. Yan and C. Felser, Annu. Rev. Condens. 8, 337 (2017).
+[10] S.-Y. Xu, N. Alidoust, I. Belopolski, C. Zhang, G. Bian,
+T.-R. Chang, H. Zheng, V. Strokov, D. S. Sanchez,
+G.
+Chang,
+et al.,
+arXiv
+preprint
+arXiv:1504.01350
+(2015).
+[11] B. Lv, S. Muff, T. Qian, Z. Song, S. Nie, N. Xu,
+P. Richard, C. E. Matt, N. C. Plumb, L. Zhao, et al.,
+Phys. Rev. Lett. 115, 217601 (2015).
+
+11
+[12] X. Y´ang, T. A. Cochran, R. Chapai, D. Tristant, J.-X.
+Yin, I. Belopolski, Z. b. u. b. a. Ch´eng, D. Multer, S. S.
+Zhang, N. Shumiya, M. Litskevich, Y. Jiang, G. Chang,
+Q. Zhang, I. Vekhter, W. A. Shelton, R. Jin, S.-Y. Xu,
+and M. Z. Hasan, Phys. Rev. B 101, 201105 (2020).
+[13] S. Thirupathaiah, Y. Kushnirenk, K. Koepernik, B. R.
+Piening, B. Buechner, S. Aswartham, J. van den Brink,
+S. Borisenko, and I. C. Fulga, SciPost Phys. 10, 004
+(2021).
+[14] Y. Yang, H.-x. Sun, J.-p. Xia, H. Xue, Z. Gao, Y. Ge,
+D. Jia, S.-q. Yuan, Y. Chong, and B. Zhang, Nat. Phys.
+15, 645 (2019).
+[15] M. Z. Hasan, G. Chang, I. Belopolski, G. Bian, S.-Y. Xu,
+and J.-X. Yin, Nat. Rev. Mater. 6, 784 (2021).
+[16] B. Lv, Z.-L. Feng, Q.-N. Xu, X. Gao, J.-Z. Ma, L.-Y.
+Kong, P. Richard, Y.-B. Huang, V. Strocov, C. Fang,
+et al., Nature 546, 627 (2017).
+[17] N. Kumar, Y. Sun, M. Nicklas, S. J. Watzman, O. Young,
+I. Leermakers, J. Hornung, J. Klotz, J. Gooth, K. Manna,
+et al., Nat. Commun. 10, 1 (2019).
+[18] Z. Zhu, G. W. Winkler, Q. Wu, J. Li, and A. A.
+Soluyanov, Phys. Rev. X 6, 031003 (2016).
+[19] J.-Z. Ma, J.-B. He, Y.-F. Xu, B. Lv, D. Chen, W.-L. Zhu,
+S. Zhang, L.-Y. Kong, X. Gao, L.-Y. Rong, et al., Nat.
+Phys. 14, 349 (2018).
+[20] H. Weng, C. Fang, Z. Fang, and X. Dai, Phys. Rev. B
+93, 241202 (2016).
+[21] S. Mardanya, B. Singh, S.-M. Huang, T.-R. Chang,
+C. Su, H. Lin, A. Agarwal, and A. Bansil, Phys. Rev.
+Mater. 3, 071201 (2019).
+[22] Z. Yan, R. Bi, H. Shen, L. Lu, S.-C. Zhang, and Z. Wang,
+Phys. Rev. B 96, 041103 (2017).
+[23] R. Bi, Z. Yan, L. Lu, and Z. Wang, Phys. Rev. B 96,
+201305 (2017).
+[24] C. Fang, H. Weng, X. Dai, and Z. Fang, Chin. Phys. B
+25, 117106 (2016).
+[25] W. Rui, Y. Zhao, and A. P. Schnyder, Phys. Rev. B 97,
+161113 (2018).
+[26] J. Zhu, W. Wu, J. Zhao, H. Chen, L. Zhang, and S. A.
+Yang, npj Quantum Mater. 7, 1 (2022).
+[27] T.-T. Zhang, Z.-M. Yu, W. Guo, D. Shi, G. Zhang, and
+Y. Yao, J. Phys. Chem. Lett. 8, 5792 (2017).
+[28] A. A. Soluyanov, Phys. 10, 74 (2017).
+[29] Y. Sun, S.-C. Wu, M. N. Ali, C. Felser, and B. Yan, Phys.
+Rev. B 92, 161107 (2015).
+[30] B. Xia, R. Wang, Z. Chen, Y. Zhao, and H. Xu, Phys.
+Rev. Lett. 123, 065501 (2019).
+[31] S.-Y. Xu, N. Alidoust, G. Chang, H. Lu, B. Singh, I. Be-
+lopolski, D. S. Sanchez, X. Zhang, G. Bian, H. Zheng,
+M.-A. Husanu, Y. Bian, S.-M. Huang, C.-H. Hsu, T.-R.
+Chang, H.-T. Jeng, A. Bansil, T. Neupert, V. N. Strocov,
+H. Lin, S. Jia, and M. Z. Hasan, Sci. Adv. 3, e1603266
+(2017).
+[32] H. Zhu, J. Yi, M.-Y. Li, J. Xiao, L. Zhang, C.-W. Yang,
+R. A. Kaindl, L.-J. Li, Y. Wang, and X. Zhang, Science
+359, 579 (2018).
+[33] J. Wang, H. Yuan, M. Kuang, T. Yang, Z.-M. Yu,
+Z. Zhang, and X. Wang, Phys. Rev. B 104, L041107
+(2021).
+[34] A. P. Litvinchuk and M. Y. Valakh, J. Condens. Matter
+Phys. 32, 445401 (2020).
+[35] J. Li, L. Wang, J. Liu, R. Li, Z. Zhang, and X.-Q. Chen,
+arXiv preprint arXiv:1907.08547 (2019).
+[36] H. Miao, T. Zhang, L. Wang, D. Meyers, A. Said,
+Y. Wang, Y. Shi, H. Weng, Z. Fang, and M. Dean, Phys.
+Rev. Lett. 121, 035302 (2018).
+[37] Q.-B. Liu, Z. Wang, and H.-H. Fu, Phys. Rev. B 103,
+L161303 (2021).
+[38] X. Wang, F. Zhou, T. Yang, M. Kuang, Z.-M. Yu, and
+G. Zhang, Phys. Rev. B 104, L041104 (2021).
+[39] C. Xie, Y. Liu, Z. Zhang, F. Zhou, T. Yang, M. Kuang,
+X. Wang, and G. Zhang, Phys. Rev. B 104, 045148
+(2021).
+[40] Y. Liu, X. Chen, and Y. Xu, Adv. Funct. Mater. 30,
+1904784 (2020).
+[41] M. Zhong, Y. Liu, F. Zhou, M. Kuang, T. Yang,
+X. Wang, and G. Zhang, Phys. Rev. B 104, 085118
+(2021).
+[42] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.
+Lett. 77, 3865 (1996).
+[43] F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401
+(2009).
+[44] J. McCrae, R. Hengehold, Y. Yeo, M. Ohmer, and
+P. Schunemann, Appl. Phys. Lett. 70, 455 (1997).
+[45] I. Akimchenko, V. Ivanov, and A. Borshchevsky, Sov.
+Phys. Semiconduct. 7, 309 (1973).
+[46] L. Bai, C. Xu, P. Schunemann, K. Nagashio, R. Feigelson,
+and N. Giles, J. Phys. Condens. Matter 17, 549 (2005).
+[47] P. Blaha, K. Schwarz, G. K. Madsen, D. Kvasnicka,
+J. Luitz, et al., An augmented plane wave+ local orbitals
+program for calculating crystal properties 60 (2001).
+[48] K. Schwarz and P. Blaha, Comput. Mater. Sci. 28, 259
+(2003).
+[49] K. Schwarz, J. Solid State Chem. 176, 319 (2003).
+[50] G. K. Madsen and D. J. Singh, Comput. Phys. Commun.
+175, 67 (2006).
+[51] G. K. Madsen, J. Carrete, and M. J. Verstraete, Comput.
+Phys. Commun. 231, 140 (2018).
+[52] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).
+[53] G. Kresse and J. Furthm¨uller, Phys. Rev. B 54, 11169
+(1996).
+[54] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
+[55] A. Togo and I. Tanaka, Scr. Mater. 108, 1 (2015).
+[56] S. Singh, Q. Wu, C. Yue, A. H. Romero, and A. A.
+Soluyanov, Phys. Rev. Mater. 2, 114204 (2018).
+[57] W. Li, J. Carrete, G. K. Madsen, and N. Mingo, Phys.
+Rev. B 93, 205203 (2016).
+[58] X. Yu and J. Hong, J. Mater. Chem. 9, 12420 (2021).
+[59] D. J. Singh, Phys. Rev. B 81, 195217 (2010).
+[60] D. Parker and D. J. Singh, Phys. Rev. B 82, 035204
+(2010).
+[61] H. Wang, Y. Pei, A. D. LaLonde, and G. J. Snyder, Adv.
+Mater. 23, 1366 (2011).
+[62] J. Sun and D. J. Singh, Phys. Rev. Appl. 5, 024006
+(2016).
+[63] J. Sun and D. J. Singh, APL Mater. 4, 104803 (2016).
+[64] J. He and T. M. Tritt, Science 357, eaak9997 (2017).
+[65] V. Y. Irkhin and Y. P. Irkhin, (Cambridge Int Science
+Publishing, 2007).
+[66] A. M. Dehkordi, M. Zebarjadi, J. He, and T. M. Tritt,
+Mater. Sci. Eng. R Rep. 97, 1 (2015).
+[67] C. E. Ekuma, D. J. Singh, J. Moreno, and M. Jarrell,
+Phys. Rev. B 85, 085205 (2012).
+[68] D. J. Singh, Funct. Mater. Lett. 3, 223 (2010).
+[69] D. Spitzer, J. Phys. Chem. Solids 31, 19 (1970).
+

d9E1T4oBgHgl3EQfeAT4/content/tmp_files/2301.03203v1.pdf.txt

@@ -0,0 +1,902 @@
+Splitting of doubly quantized vortices in holographic
+superfluid of finite temperature
+Shanquan Lan,1, 2, ∗ Xin Li,3, † Jiexiong Mo,1, ‡ Yu Tian,4, 5, §
+Yu-Kun Yan,4, ¶ Peng Yang,4, ∗∗ and Hongbao Zhang6, ††
+1Department of Physics, Lingnan Normal University, Zhanjiang 524048, China
+2Department of Physics, Peking University, Beijing 100871, China
+3Department of Physics, University of Helsinki,
+P.O. Box 64, FI-00014 Helsinki, Finland
+4School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
+5Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
+6Department of Physics, Beijing Normal University, Beijing 100875, China
+(Dated: January 10, 2023)
+The temperature effect on the linear instability and the splitting process of a dou-
+bly quantized vortex is studied. Using the linear perturbation theory to calculate out
+the quasi-normal modes of the doubly quantized vortex, we find that the imaginary
+part of the most unstable mode increases with the temperature till some turning tem-
+perature, after which the imaginary part of the most unstable mode decreases with
+the temperature. On the other hand, by the fully nonlinear numerical simulations,
+we also examine the splitting process of the doubly quantized vortex, where not only
+do the split singly quantized vortex pair depart from each other, but also revolve
+around each other. In particular, the characteristic time scale for the splitting pro-
+cess is identified and its temperature dependence is found to be in good agreement
+with the linear instability analysis in the sense that the larger the imaginary part of
+the most unstable is, the longer the splitting time is. Such a temperature effect is
+expected to be verified in the cold atom experiments in the near future.
+arXiv:2301.03203v1  [hep-th]  9 Jan 2023
+
+2
+I.
+INTRODUCTION
+In gaseous Bose–Einstein condensates (BEC), quantized vortex is a topological defect
+in the order parameter with a quantum number n multiple of 2π phase winding about the
+vortex center. A vortex with n = 1 is called a singly quantized vortex, a vortex with n ≥ 2
+is called a multiply quantized vortex. There are several experimental methods to create
+such quantized vortices, including potential rotation [1, 2], laser beam stirring[3–5], phase
+imprinting[6–8], angular momentum transformation[9]. In addition, the vortex dynamics has
+also been widely studied both theoretically and experimentally[10–12]. In particular, over
+the last two decades, splitting of multiply quantized vortices into many singly quantized
+vortices has been extensively investigated in gaseous BEC [13–30].
+However, studies on
+the effect of finite temperature on vortex splitting are still lacking.
+The partial reason
+for this arises from the fact that the usually prepared temperature for the gaseous BEC
+is extraordinarily low such that the finite temperature dissipation effect can be neglected.
+Consequently, the Gross-Piaevskii (GP) equation for zero temperature BEC is well suited
+to account for various phenomena for the gaseous BEC. But nevertheless, with the very
+recent experimental advance, the temperature for the gaseous BEC can be engineered close
+to the critical temperature[31]. Thus it becomes urgent for us to take into account the finite
+temperature effect. Such a finite temperature effect is usually modelled phenomenologically
+by adding the dissipation term to GP equation by hand.
+Compared to this traditional
+approach, holographic duality provides us a first principle description of finite temperature
+superfluid dynamics by the bulk hairy black hole[32, 33]. Actually, much effort has been
+devoted to the characteristic features of vortex and its non-equilibrium dynamics through the
+lenses of holography[34–47]. The purpose of the current paper is to initiate our holographic
+investigation of vortex splitting by focusing on the splitting of doubly quantized vortex in
+the finite temperature holographic superfluid.
+∗Electronic address: [email protected]
+†Electronic address: xin.z.li@helsinki.fi
+‡Electronic address: [email protected]
+§Electronic address: [email protected]
+¶Electronic address: [email protected]
+∗∗Electronic address: [email protected]
+††Electronic address: [email protected]
+
+3
+This paper is organised as follows. Holographic superfluid model is reviewed in section
+II. For cases of different temperatures, doubly quantized vortices solutions are obtained and
+their stability is analysed by quasi-normal modes in section III. The splitting process of
+multiply quantized vortices are studied by tracing the positions of the split singly vortices
+in section IV. Section V devotes to conclusions and discussions.
+II.
+A BRIEF REVIEW OF HOLOGRAPHIC SUPERFLUID MODEL
+The holographic model, describing the two dimensional superfluid, is a gravitational
+system in asymptotically AdS4 spacetime coupled with a U(1) gauge field Aµ and a complex
+scalar field Ψ with the mass m and the charge q. The corresponding bulk action is given
+by[32, 33]
+S =
+�
+M
+√−gd4x[
+1
+16πG(R + 6
+L2) − 1
+q2(1
+4F 2 + |DΨ|2 + m2|Ψ|2)],
+(1)
+where DµΨ = (∇µ − iAµ)Ψ, L is the AdS radius, and G Newton’s gravitational constant.
+We consider the probe limit, where q is taken to be large such that the backreaction of the
+matter fields is neglected. For our purpose, the bulk geometry is fixed as the Schwarzschild-
+AdS black hole
+ds2 = L2
+z2 (−f(z)dt2 +
+1
+f(z)dz2 + dx2 + dy2),
+(2)
+where f(z) = 1 − ( z
+zh)3. In this coordinate system, z = 0 is the AdS boundary and z = zh
+is the black hole horizon. The Hawking temperature of the black hole is
+T =
+3
+4πzh
+,
+(3)
+which is also identified as the temperature of the dual boundary system. The equations of
+motion for the bulk matter sector are
+DµDµΨ − m2Ψ = 0,
+∇µF µν = Jν,
+(4)
+with Jν = i(Ψ∗DνΨ − ΨDν∗Ψ∗).
+Without loss of generality, we set L = 1, zh = 1. Below we also choose m2 = −2 and
+adopt the axial gauge Az = 0. Then the asymptotic behaviors of the matter fields near the
+
+4
+ψ+
+2
+μ
+(4.064, 0.0)
+4
+6
+8
+10
+12
+14
+0
+500
+1000
+1500
+μ
+ρ
+(4.064, 4.064)
+4
+6
+8
+10
+12
+14
+0
+10
+20
+30
+40
+50
+60
+FIG. 1: The variation of the condensate density (Left panel) and the charge density (Right panel)
+with the chemical potential for the superfluid phase.
+AdS boundary are
+Ψ = z(ψ− + ψ+z + · · · ),
+Aµ = aµ + bµz + · · · .
+(5)
+According to the holographic dictionary for the standard quantization case, the source ψ−
+is required to be turned off, and ψ+ corresponds to the condensate in the superfluid phase.
+In addition, at = µ and −bt = ρ are interpreted as the chemical potential and the charge
+density, respectively. Furthermore, ax,y are related to the superfluid velocities and bx,y are
+related to the conjugate currents.
+For an isotropic uniform static superfluid system, the equations of motion reduce to
+f(z)d2Φ
+dz2 − 3z2dΦ
+dz + ( A2
+t
+f(z) − z)Φ = 0,
+f(z)d2At
+dz2 − 2AtΦ2 = 0,
+(6)
+where Ψ ≡ zΦ. As shown in Fig.1, there exists a critical chemical potential µc = ρc =
+4.064, above which the scalar field can have a non-trivial solution besides the trivial one
+Φ = 0, signaling a phase transition from the normal fluid phase to the superfluid phase.
+Furthermore, as we can see, both the resulting condensate density ψ2
++ and the charged
+density increase with the increase of the chemical potential µ.
+In order to investigate the temperature effect on the condensate density, we like to take
+
+5
+Ψ+
+ 2 =
+Ψ+
+2
+ρs
+2
+T
+Tc
+ =
+ρc
+ρs
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+FIG. 2: The variation of the condensate density with the temperature, where the charge density
+is fixed to be one.
+advantage of the scaling symmetry of the bulk dynamics
+t → ˜t = σt, x → ˜x = σx, z → ˜z = σz, T → ˜T = T
+σ ,
+µ → ˜µ = µ
+σ, ρ → ˜ρ = ρ
+σ2 = 1, ψ+ → ˜ψ+ = ψ+
+σ2 ,
+(7)
+and set σ = √ρs with ρs the charge density for an isotropic uniform static superfluid at
+the fixed temperature given by Eq. (3). As a result, the charge density is scaled to be one
+and the variation of the condensate density with respect to the temperature is plotted in
+Fig.2, whereby one can see that the condensate density decreases with the temperature and
+vanishes at the critical temperature, in accordance with our intuition.
+III.
+LINEAR STABILITY OF THE DOUBLY QUANTIZED VORTEX
+A.
+Vortex configuration
+To obtain the static vortex configuration, we would like to work with the polar coordi-
+nates, in which the background metric reads
+ds2 = 1
+z2(−f(z)dt2 +
+1
+f(z)dz2 + dr2 + r2dθ2).
+(8)
+The corresponding ansatz for the non-vanishing matter fields is given by
+Ψ ≡ zΦ = zψ(z, r)einθ,
+At = At(z, r),
+Aθ = Aθ(z, r),
+(9)
+
+6
+where n is the winding number of the quantized vortex. Then the equations of motion for
+a vortex can be written as
+∂z(f∂zψ) + ∂2
+rψ + 1
+r∂rψ + (A2
+t
+f − (Aθ − n)2
+r2
+− z)ψ = 0,
+(10)
+f∂2
+zAt + ∂2
+rAt + 1
+r∂rAt − 2Atψ2 = 0,
+(11)
+∂z(f∂zAθ) + ∂2
+rAθ − 1
+r∂rAθ − 2(Aθ − n)ψ2 = 0.
+(12)
+The boundary conditions at the AdS boundary z = 0 are prescribed as
+ψ|z=0 = 0, At|z=0 = µ, Aθ|z=0 = 0.
+(13)
+At the horizon z = 1, the regular boundary conditions are imposed. In the r direction, the
+system is cut off at a sufficient large radius R, where the Neumann boundary conditions are
+imposed as
+∂rψ|r=R = 0, ∂rAt|r=R = 0, ∂rAθ|r=R = 0.
+(14)
+At the vortex center r = 0, we impose the boundary conditions as follows
+ψ|r=0 = 0, ∂rAt|r=0 = 0, ∂rAθ|r=0 = 0.
+(15)
+By setting n = 2 and resorting to the pseudo-spetral method with 28 Chebyshev modes
+in the z direction and 40 Chebyshev modes in the r direction, we obtain the desired doubly
+quantized vortex configuration.
+As a demonstration, we plot the configuration of doubly quantized vortex at temperature
+˜T/ ˜Tc = 0.373 in the left panel of Fig.3, where the fitted vortex radius ξ = 6.26 with the
+vortex radius defined as | ˜ψ+|2(˜r = ξ) = 0.9| ˜ψ+max|2. In addition, we also show the variation
+of the vortex radius with the temperature in the right panel of Fig.3. At low temperatures,
+the vortex radius increases slowly with the temperature. But when the temperature gets
+close to the critical one, the vortex radius increases dramatically with the temperature and
+displays a divergent behavior near the critical temperature, in accordance with the standard
+lore that the healing length, order of the vortex radius, become infinite at the critical point.
+
+7
+ψ+
+
+2
+r = σ r
+T
+Tc
+ = 0.373, n = 2
+ξ = 6.26
+0.9
+ψ+
+
+max
+2
+0
+10
+20
+30
+40
+0.1
+0.2
+0.3
+0.4
+0.5
+ξ
+T
+Tc
+
+n = 2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+10
+20
+30
+40
+50
+FIG. 3: Left panel: The configuration for the doubly quantized vortex at ˜T/ ˜Tc = 0.373 with
+the fitted vortex radius ξ = 6.26. Right panel: The variation of the vortex radius of the doubly
+quantized vortex with the temperature.
+B.
+Quasi-normal modes
+Now we are going to investigate the linear stability of the doubly quantized vortex by
+calculating its quasi-normal modes. To this end, we prefer to go from the Schwarzschild
+coordinates to the Eddington-Finkelstein coordinates, in which the background metric takes
+the following form
+ds2 = 1
+z2(−f(z)dt2 − 2dtdz + dr2 + r2dθ2).
+(16)
+In order to guarantee the axial gauge Az = 0 in the above new coordinates, we are required
+to perform the following gauge transformation
+ψ(z, r) → eiλ(z,r)ψ(z, r),
+λ(z, r) = −
+� At(z, r)
+f(z) dz.
+(17)
+As a result, Ar(z, r) = ∂rλ(z, r) no longer vanishes.
+Since the background configurations of our vortex possess the time translation symmetry
+and rotation symmetry, the linear perturbations of the matter fields can be constructed as
+Ψ = zeinθ(ψ(z, r) + δψ1(z, r)e−iωt+ipθ + δψ∗
+2(z, r)eiω∗t−ipθ),
+(18)
+Ψ∗ = ze−inθ(ψ∗(z, r) + δψ∗
+1(z, r)eiω∗t−ipθ + δψ2(z, r)e−iωt+ipθ),
+(19)
+At = At(z, r) + δAt(z, r)e−iωt+ipθ + δA∗
+t(z, r)eiω∗t−ipθ,
+(20)
+
+8
+Ar = Ar(z, r) + δAr(z, r)e−iωt+ipθ + δA∗
+r(z, r)eiω∗t−ipθ,
+(21)
+Aθ = Aθ(z, r) + δAθ(z, r)e−iωt+ipθ + δA∗
+θ(z, r)eiω∗t−ipθ.
+(22)
+Substituting the above expressions into Eq. (4), we obtain the linear perturbation equations
+for quasi-normal modes ω as follows
+(−2i(ω + At)∂z − i∂zAt − ∂z(f∂z) − (∂r − iAr)2 − ∂r − iAr
+r
++ (Aθ − n − p)2
+r2
++ z)δψ1
+−(iψ∂z + 2i∂zψ)δAt + (iψ∂r + 2i∂rψ + 2ψAr + iψ
+r )δAr + ψ
+r2(2Aθ − n − p)δAθ = 0,(23)
+(−2i(ω − At)∂z + i∂At − ∂z(f∂z) − (∂r + iAr)2 − ∂r + iAr
+r
++ (Aθ − n − p)2
+r2
++ z)δψ2
++(iψ∗∂z + 2i∂zψ∗)δAt − (iψ∗∂r + 2i∂rψ∗ − 2ψ∗Ar + iψ∗
+r )δAr
++ψ∗
+r2 (2Aθ − n + p)δAθ = 0,
+(24)
+(−iψ∗∂z + i∂zψ∗)δψ1 + (iψ∂z − i∂zψ)δψ2 + ∂2
+zδAt − (∂z
+r + ∂z∂r)δAr − ip
+r2∂zδAθ = 0, (25)
+(ωψ∗ + 2Atψ∗)δψ1 + (−ωψ + 2Atψ)δψ2 + (p2
+r2 + 2ψψ∗ − iω∂z − f∂2
+z − ∂r
+r − ∂2
+r)δAt
+−(iω
+r + iω∂r)δAr + pω
+r2 δAθ = 0,
+(26)
+(iψ∗∂r − i∂rψ∗ + 2ψ∗Ar)δψ1 + (−iψ∂r + i∂rψ + 2ψAr)δψ2 − ∂z∂rδAt
++(p2
+r2 + 2ψψ∗ − 2iω∂z − f∂2
+z − f ′∂z)δAr + ip
+r2∂rδAθ = 0,
+(27)
+(2Aθ − 2n − p)ψ∗δψ1 + (2Aθ − 2n + p)ψδψ2 − ip∂zδAt + ip(∂r − 1
+r)δAr
++(−2iω∂z − ∂z(f∂z) − ∂2
+r + ∂r
+r + 2ψψ∗)δAθ = 0,
+(28)
+where the fourth equation as a constraint equation will be used only at the AdS boundary
+z = 0 together with the following boundary conditions
+δψ1|z=0 = 0, δψ2|z=0 = 0, δAt|z=0 = 0, δAr|z=0 = 0, δAθ|z=0 = 0.
+(29)
+
+9
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+T˜/Tc
+˜ = 0.745, p = 2
+-0.4
+-0.2
+0.0
+0.2
+0.4
+-0.04
+-0.02
+0.00
+0.02
+0.04
+Re(ω˜)
+Im(ω˜)
+FIG. 4:
+The quasi-normal modes (˜ω = ω/σ) of a doubly quantized vortex at temperature ˜T/ ˜Tc =
+0.745 for p = 2.
+At the horizon z = 1, the regular boundary conditions are imposed as usual for δψ1, δψ2,
+δAr and δAθ. We further impose the following boundary conditions
+∂rδψ1|r=R = 0, ∂rδψ2|r=R = 0, ∂rδAt|r=R = 0, δAr|r=R = 0, ∂rδAθ|r=R = 0,
+(30)
+δψ1|r=0 = 0, δψ2|r=0 = 0, δAt|r=0 = 0, (pδAr + i∂rδAθ)|r=0 = 0, δAθ|r=0 = 0,
+(31)
+at r = R and r = 0, respectively. Then the quasi-normal modes ω can be obtained by solving
+the generalized eigenvalue problem, where we keep employing the pseudo-spectral method
+with 28 Chebyshev modes in the z direction and 40 Chebyshev modes in the r direction.
+Fig.4 shows the quasi-normal modes (˜ω = ω/σ) of a doubly quantized vortex at tem-
+perature ˜T/ ˜
+Tc = 0.745 for p = 2. As we can see, there exists a quasi-normal mode with a
+positive imaginary part, which is indicative of the instability of the doubly quantized vortex.
+We find that there is no unstable mode for p > 2 at all temperatures. For the case of p < 2,
+the imaginary part of the unstable mode is smaller than that of p = 2. So the most unstable
+mode with the largest positive imaginary part appears at p = 2. We plot the variation of
+the imaginary part of the most unstable mode with the temperature in Fig.5. As one can
+see, the imaginary part of the most unstable mode rises with the temperature at first, peaks
+at ˜T/ ˜Tc = 0.745, and then drops when the temperature is approaching the critical one.
+
+10
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+0.012
+T/TC
+
+Im (ω)
+FIG. 5: The variation of the imaginary part of the most unstable modes of a doubly quantized
+vortice with the temperature, where the maximum occurs at ˜T/ ˜Tc = 0.745.
+IV.
+SPLITTING PROCESS OF THE DOUBLY QUANTIZED VORTICES
+In the previous section, we find that the doubly quantized vortex at finite temperature is
+unstable under the linear perturbation. A natural question is where the doubly quantized
+vortex is driven by such a linear instability. In order to answer this question, below we
+shall perform fully non-linear real time numerical simulations. To improve the numerical
+accuracy for such simulations, we prefer to work with the rectangular coordinates, in which
+the background metric reads
+ds2 = 1
+z2(−f(z)dt2 − 2dtdz + dx2 + dy2).
+(32)
+The equations of motion for the matter fields can be written explicitly as follows
+∂t∂zΦ = iAt∂zΦ + 1
+2[i∂zAtΦ + f∂2
+zΦ + f ′∂zΦ + (∂ − iA)2Φ − zΦ],
+(33)
+∂z(∂zAt − ∂ · A) = i(Φ∗∂zΦ − Φ∂zΦ∗),
+(34)
+∂t∂zA = 1
+2[∂z(∂At + f∂zA) + (∂2A − ∂∂ · A)
+−i(Φ∗∂Φ − Φ∂Φ∗)] − AΦ∗Φ,
+(35)
+∂t∂zAt = ∂2At + f∂z∂ · A − ∂t∂ · A − 2AtΦ∗Φ
++if(Φ∗∂zΦ − Φ∂zΦ∗) − i(Φ∗∂tΦ − Φ∂tΦ∗).
+(36)
+
+11
+To proceed, we are required to prescribe the initial data for our numerical simulations.
+First note that the previous vortex configuration is obtained in the polar coordinates, and
+the corresponding matter field functions are
+Φ(z, r, θ) = einθψ(z, r), At(z, r), Ar(z, r), Aθ(z, r).
+(37)
+So in the rectangular coordinates, the corresponding matter field functions become
+Φ(z, x, y), At(z, x, y), Ax(z, x, y), Ay(z, x, y),
+(38)
+where
+Ax(z, x, y) = Ar(z, r) cos θ − Aθ(z, r)
+r
+sin θ,
+Ay(z, x, y) = Ar(z, r) sin θ + Aθ(z, r)
+r
+cos θ.
+(39)
+The initial data for Φ, Ax, and Ay are prescribed as follows
+Φ(t = 0) = Φ
+2
+�
+1 − tanh c(r2 − r2
+m)
+�
+,
+Ax(t = 0) = Ax
+2
+�
+1 − tanh c(r2 − r2
+m)
+�
+,
+Ay(t = 0) = Ay
+2
+�
+1 − tanh c(r2 − r2
+m)
+�
+,
+(40)
+where rm is chosen to be much larger than the doubly quantized vortex radius. Whence the
+initial data for At can be obtained by solving the constraint equation (36) with the following
+boundary conditions
+At(z = 0) = µ
+2
+�
+1 − tanh c(r2 − r2
+m)
+�
+,
+∂zAt(z = 0) = −ρ
+2
+�
+1 − tanh c(r2 − r2
+m)
+�
+,
+(41)
+where µ and ρ are the corresponding chemical potential and charge density of the doubly
+quantized vortex, respectively.
+As demonstrated in Fig.6 for ˜T/ ˜Tc = 0.373, not only do the above initial data reproduce
+the doubly quantized vortex configuration within the region r < rm, but also make the
+data at the square boundary vanish such that the periodic boundary conditions can be
+employed in our later numerical simulations.
+In particular, the pseudo-spectral method
+with 28 Chebyshev modes in the z direction, 101 Fourier modes in the x, y directions, and
+
+12
+the fourth order Runge-Kutta method in the time direction are employed in our numerical
+simulations. As our numerical simulation shows, the initial doubly quantized vortex does
+split into two singly quantized votices. We also plot the trajectories of the two split singly
+quantized vortices in Fig.6 starting from ˜t = 540, because the split singly quantized vortices
+are too close to each other to be identified before ˜t = 540. Although the doubly quantized
+vortex is initially placed in the coordinate origin (0, 0), the center of the vortices is seen to
+deviate a little bit from the center of the coordinate during the evolution process. This is
+reasonable because the square boundary we are using breaks the rotation symmetry of the
+whole system. Furthermore, one can see that not only do the split singly vortices depart from
+each other, but also revolve around the center anti-clockwise. So below we shall quantify the
+whole splitting process by examining the temporal evolution of the two quantities, namely
+the separation distance ˜d(˜t) between the two vortices, and the angle θ(˜t) between the straight
+line connecting the two vortices and the x-axis.
+Fig.7 displays the temporal evolution of the separation distance ˜d(˜t) as well as the sep-
+aration velocity ˜v(˜t) from ˜t = 540 to 3240 at ˜T/ ˜Tc = 0.373. When ˜t > 540, the separation
+velocity is slow at first, becomes faster gradually, reaches its maximum at the critical time
+˜t = 1204 with ˜d = 8.29, and then decreases slowly. It is noteworthy that the overlap of
+two vortices is smaller than 10% at the critical time, which motivates us to identify this
+critical time as the splitting time τ of the doubly quantized vortex into two singly quantized
+vortices and half of the corresponding separation distance as the previously defined radius ξ
+of the singly quantized vortex. Fig.8 further shows the temporal evolution of the angle θ(˜t)
+between the straight line connecting the two vortices and the x-axis, where the left panel
+and the right one are equivalent as the latter is the jointed line of the former. According
+to the slope, we find that the angular velocity is big before the splitting time and becomes
+small after the splitting time.
+We further plot the temperature dependence of the time scale τ required for a doubly
+quantized vortex to split into two separated singly quantized vortices in Fig.9. As we see,
+the result for the temperature dependence of the splitting time from the fully non-linear
+numerical simulations is amazingly consistent with that for the temperature dependence of
+the imaginary part of the most unstable modes by linear perturbation analysis in Fig.5.
+The larger the imaginary part of the most unstable mode is, the more unstable the system
+becomes, leading to the shorter time for a doubly quantized vortex to split into two singly
+
+13
+x
+y
+-15
+-10
+-5
+5
+10
+15
+-15
+-10
+-5
+5
+10
+15
+FIG. 6:
+Splitting process of a doubly quantized vortex at temperature ˜T/ ˜Tc = 0.373.
+The
+condensates at time ˜t = 0, 1204, 3240 and the trajectories of the two singly quantized vortices from
+˜t = 540 to 3240.
+quantized vortices.
+V.
+CONCLUSION AND DISCUSSION
+By virtue of holography, we have investigated the linear instability and splitting process
+of doubly quantized vortices in superfluids at different temperatures. To this end, we first
+obtain the numerical solutions of doubly quantized vortices. In particular, we find that
+at low temperatures, the vortex radius increases slowly with the temperature, while near
+the critical temperature, the dramatically increased vortex radius gets divergent. Then we
+analyze the linear instability of the doubly quantized vortices by calculating the quasi-normal
+modes. It is found that when the temperature is ˜T/ ˜Tc = 0.745, the imaginary part of the
+most unstable quasi-normal mode of the vortex is the largest, indicating that the vortex
+
+0.4
+50
+0.2
+0.0
+0
+-50
+y
+0
+X
+-50
+500.4
+50
+0.2
+0.0
+0
+-50
+y
+0
+X
+-50
+500.4
+50
+0.2
+0.0
+0
+-50
+y
+0
+X
+-50
+5014
+T
+Tc
+ = 0.373
+(1204, 8.29)
+t
+d
+0
+500
+1000
+1500
+2000
+2500
+3000
+0
+5
+10
+15
+20
+25
+30
+35
+(1204, 0.0175)
+v
+t
+0
+500
+1000
+1500
+2000
+2500
+3000
+0.000
+0.005
+0.010
+0.015
+0.020
+FIG. 7: The temporal evolution of the separation distance ˜d(˜t) (on the left panel) and the separation
+˜v(˜t) (on the right panel) between the two split vortices from ˜t = 540 to 3240 at ˜T/ ˜Tc = 0.373,
+where the point in red marks the moment at which the maximal separation velocity is reached. The
+corresponding splitting time of the doubly quantized vortex and the radius of a singly quantized
+vortex can be inferred as (τ = 1204, ξ = 8.29/2).
+t
+θ
+(1204, 0.30)
+T
+Tc
+ = 0.373
+500
+1000
+1500
+2000
+2500
+3000
+-3
+-2
+-1
+0
+1
+2
+3
+θ
+t
+T
+Tc
+ = 0.373
+(1204, 25.43)
+500
+1000
+1500
+2000
+2500
+3000
+0
+5
+10
+15
+20
+25
+30
+FIG. 8: The angle θ(˜t) between the straight line connecting the two vortices and the x-axis from
+˜t = 540 to 3240 at temperature ˜T/ ˜Tc = 0.373. The left panel and the right panel are equivalent as
+the latter is the jointed line of the former.
+is the most unstable. When the temperature is lower or higher, the vortex will becomes
+less unstable. We further resort to the fully non-linear numerical simulations to explore
+the splitting process of the doubly quantized vortices, where the characteristic time scale
+for the splitting process is identified and and its temperature dependence is found to be in
+good agreement with the previous linear instability analysis in the sense that the larger the
+imaginary part of the most unstable mode is, the longer the splitting time is. We expect
+
+15
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0
+500
+1000
+1500
+2000
+2500
+T/TC
+
+τ
+FIG. 9:
+The temperature dependence of the splitting time τ required for a doubly quantized
+vortex to split into two separated singly quantized vortices.
+that our findings, especially such a characteristic feature of the temperature dependence of
+the splitting time, can be verified by cold atom experiments in the near future.
+Acknowledgments
+This work is partly supported by the National Key Research and Development Program of
+China Grant No. 2021YFC2203001, National Natural Science Foundation of China (Grant
+Nos. 12005088, 11975235, 12035016 and 12075026), and Guangdong Basic and Applied Basic
+Research Foundation of China (Grant Nos. 2022A1515011938, 2022A1515012425). Shan-
+quan Lan acknowledges the support from Lingnan Normal University Project (Grants No.
+YL20200203 and No. ZL1930). Xin Li acknowledges the support from China Scholarship
+Council (CSC No. 202008610238)
+[1] K.W. Madison, F. Chevy, V. Bretin, J. Dalibard, Stationary States of a Rotating Bose-Einstein
+Condensate: Routes to Vortex Nucleation, Phys. Rev. Lett. 86 (2001) 4443.
+[2] J.R. Abo-Shaeer, C. Raman, J.M. Vogels, W. Ketterle, Observation of Vortex Lattices in
+Bose-Einstein Condensates, Science 292 (2001) 476.
+[3] K.W. Madison, F. Chevy, W. Wohlleben, J. Dalibard, Vortex Formation in a Stirred Bose-
+Einstein Condensate, Phys. Rev. Lett. 84 (2000) 806.
+
+16
+[4] C. Raman, J.R. Abo-Shaeer, J.M. Vogels, K. Xu, W. Ketterle, Vortex Nucleation in a Stirred
+Bose-Einstein Condensate, Phys. Rev. Lett. 87 (2001) 210402.
+[5] T.W. Neely, E.C. Samson, A.S. Bradley, M.J. Davis, B.P. Anderson, Observation of Vortex
+Dipoles in an Oblate Bose-Einstein Condensate, Phys. Rev. Lett. 104 (2010) 160401.
+[6] J.E. Williams, M. J. Holland, Preparing topological states of a Bose–Einstein condensate,
+Nature 401 (1999) 568.
+[7] M.R. Matthews, B.P. Anderson, P.C. Haljan, D.S. Hall, C.E. Wieman, E.A. Cornell, Vortices
+in a Bose-Einstein Condensate, Phys. Rev. Lett. 83 (1999) 2498.
+[8] A.E. Leanhardt, A. Gorlitz, A.P. Chikkatur, D. Kielpinski, Y. Shin, D.E. Pritchard, W.
+Ketterle, Imprinting Vortices in a Bose-Einstein Condensate using Topological Phases, Phys.
+Rev.Lett. 89 (2002) 190403.
+[9] M. F. Andersen, C. Ryu, Pierre Clad´e, Vasant Natarajan, A. Vaziri, K. Helmerson, and W. D.
+Phillips, Quantized Rotation of Atoms from Photons with Orbital Angular Momentum, Phys.
+Rev.Lett. 97 (2006) 170406.
+[10] C. F. Barenghi, R. J. Donnelly, W. F. Vinen, Quantized Vortex Dynamics and Superfluid
+Turbulence, (Springer, Berlin, Heidelberg, 2001).
+[11] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, (Cambridge Uni-
+versity Press, Cambridge, UK, 2002).
+[12] L. Madeira, A. Cidrim, M. Hemmerling, M. A. Caracanhas, F. E. A. dos Santos, and V. S. Bag-
+nato, Quantum turbulence in Bose-Einstein condensates: Present status and new challenges
+ahead, AVS Quantum Sci. 2 (2020) 035901.
+[13] Y. Shin, M. Saba, M. Vengalattore, T. A. Pasquini, C. Sanner, A. E. Leanhardt, M. Prentiss,
+D. E. Pritchard, and W. Ketterle, Dynamical Instability of a Doubly Quantized Vortex in a
+Bose-Einstein Condensate, Phys. Rev. Lett. 93 (2004) 160406.
+[14] Y. Kawaguchi and T. Ohmi, Splitting instability of a multiply charged vortex in a Bose-Einstein
+condensate, Phys. Rev. A 70 (2004) 043610.
+[15] T. Karpiuk, M. Brewczyk, M. Gajda and K. Rza˙zewski, Decay of multiply charged vortices at
+nonzero temperatures, J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 095301.
+[16] P. Kuopanportti, M. M¨ott¨onen, Splitting dynamics of giant vortices in dilute Bose-Einstein
+condensates, Phys. Rev. A 81 (2010) 033627.
+[17] Q. Zhu, L. Pan, Splitting of a Multiply Quantized Vortex for a Bose-Einstein Condensate in
+
+17
+an Optical Lattice, J. Low. Temp. Phys. 203 (2021) 392–400.
+[18] T. P. Simula, S. M. M. Virtanen, and M. M. Salomaa, Stability of multiquantum vortices in
+dilute Bose-Einstein condensates, Phys. Rev. A 65 (2002) 033614.
+[19] J. A. M. Huhtam¨aki, M. M¨ott¨onen, and S. M. M. Virtanen, Dynamically stable multiply
+quantized vortices in dilute Bose-Einstein condensates, Phys. Rev. A 74 (2006) 063619.
+[20] A. Mu˜noz Mateo, V. Delgado, Dynamical Evolution of a Doubly Quantized Vortex Imprinted
+in a Bose-Einstein Condensate, Phys. Rev. Lett. 97 (2006) 180409.
+[21] T. Isoshima, M. Okano, H. Yasuda, K. Kasa, J. A. M. Huhtam¨aki, M. Kumakura, and Y.
+Takahashi, Spontaneous Splitting of a Quadruply Charged Vortex, Phys. Rev. Lett. 99 (2007)
+200403.
+[22] M. Okano, H. Yasuda, K. Kasa, M. Kumakura, Y. Takahashi, Splitting of a Quadruply Quan-
+tized Vortex in the Rb Bose-Einstein Condensate, J. Low. Temp. Phys. 148 (2007) 447–451.
+[23] T. Kuwamoto, H. Usuda, S. Tojo, and T. Hirano, Dynamics of Quadruply Quantized Vortices
+in 87Rb Bose-Einstein Condensates Confined in Magnetic and Optical Traps, J. Phys. Soc.
+Jpn. 79 (2010) 034004.
+[24] B. Xiong, T. Yang, Y. Lin and D. Wang, Controllable splitting dynamics of a doubly quantized
+vortex in a ring-shaped condensate, J. Phys. B: At. Mol. Opt. Phys. 53 (2020) 075301.
+[25] J. A. M. Huhtam¨aki, M. M¨ott¨onen, T. Isoshima, V. Pietil¨a, and S. M. M. Virtanen, Splitting
+Times of Doubly Quantized Vortices in Dilute Bose-Einstein Condensates, Phys. Rev. Lett.
+97 (2006) 110406.
+[26] K. Gawryluk, T. Karpiuk, M. Brewczyk, and K. Rza˙zewski, Splitting of doubly quantized
+vortices in dilute Bose-Einstein condensates, Phys. Rev. A 78 (2008) 025603.
+[27] H. Takeuchi, M. Kobayashi, and K. Kasamatsu, Is a Doubly Quantized Vortex Dynamically
+Unstable in Uniform Superfluids?, J. Phys. Soc. Jpn. 87 (2018) 023601.
+[28] J. R¨abin¨a, P. Kuopanportti, M. I. Kivioja, M. M¨ott¨onen, and T. Rossi, Three-dimensional
+splitting dynamics of giant vortices in Bose-Einstein condensates, Phys. Rev. A 98 (2018)
+023624.
+[29] S. Ishino, M. Tsubota, and H. Takeuchi, Counter-rotating vortices in miscible two-component
+Bose-Einstein condensates, Phys. Rev. A 88 (2013) 063617.
+[30] Pekko Kuopanportti, Soumik Bandyopadhyay, Arko Roy, and D. Angom, Splitting of singly
+and doubly quantized composite vortices in two-component Bose-Einstein condensates, Phys.
+
+18
+Rev. A 100 (2019) 033615.
+[31] W. J. Kwon, G. Del Pace, K. Xhani, L. Galantucci, A. Muzi Falconi, M. Inguscio, F. Scazza, G.
+Roati, Sound emission and annihilations in a programmable quantum vortex collider, Nature
+600 (2021) 64-69.
+[32] S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a Holographic Superconductor, Phys.
+Rev. Lett. 101 (2008) 031601.
+[33] S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic Superconductors, J. High. Energ.
+Phys. 12 (2008) 015.
+[34] Y. Yan, S. Lan, Y. Tian, P. Yang, S. Yao, H. Zhang Towards an effective description of
+holographic vortex dynamics, arXiv:2207.02814.
+[35] M.Y. Guo, E. Keski-Vakkuri, H. Liu, et al, Dynamical Phase Transition from Nonequilibrium
+Dynamics of Dark Solitons, Phys. Rev. Lett. 124 (2020) 031601.
+[36] V. Keranen, E. Keski-Vakkuri, S. Nowling and K.P. Yogendran, Inhomogeneous Structures in
+Holographic Superfluids: II. Vortices, Phys. Rev. D 81 (2010) 126012.
+[37] S. Lan, W. Liu and Y. Tian, Static structures of the BCS-like holographic superfluid in AdS4
+spacetime, Phys. Rev. D 95 (2017) 066013.
+[38] C. Xia, H. Zeng, H. Zhang, Z. Nie, Y. Tian, and X. Li, Vortex lattice in a rotating holographic
+superfluid, Phys. Rev. D 100 (2019) 061901.
+[39] X. Li, Y. Tian, H.B. Zhang, Generation of vortices and stabilization of vortex lattices in
+holographic superfluids, J. High Energ. Phys. 02 (2020) 104.
+[40] W. Yang, C. Xia, H. Zeng, and H. Zhang, Phase separation and exotic vortex phases in a
+two-species holographic superfluid, Eur.Phys.J.C 81 (2021) 1.
+[41] P. M. Chesler, H. Liu, A. Adams, Holographic vortex liquids and superfluid turbulence, Science,
+341 (2013) 6144.
+[42] C. Ewerz, T. Gasenzer, M. Karl and A. Samberg, Non-Thermal Fixed Point in a Holographic
+Superfluid, J. High Energ. Phys. 05 (2015) 070.
+[43] S. Lan, Y. Tian and H. Zhang, Towards Quantum Turbulence in Finite Temperature Bose-
+Einstein Condensates, J. High Energ. Phys. 07 (2016) 092.
+[44] Y. Du, C. Niu, Y. Tian and H. Zhang, Holographic thermal relaxation in superfluid, J. High
+Energ. Phys. 12 (2015) 018.
+[45] S. Lan, G. Li, J. Mo, and X. Xu, Attractive interaction between vortex and anti-vortex in
+
+19
+holographic superfluid, J. High Energ. Phys. 02 (2019) 122.
+[46] P. Wittmer, C. Schmied, T. Gasenzer, and C. Ewerz, Vortex Motion Quantifies Strong Dissi-
+pation in a Holographic Superfluid, Phys. Rev. Lett. 127 (2021) 101601.
+[47] C. Ewerz, A. Samberg, P. Wittmer, Dynamics of a vortex dipole in a holographic superfluid,
+J. High Energ. Phys. 2021 (2021) 199.
+

fNFMT4oBgHgl3EQf1TG9/content/tmp_files/2301.12440v1.pdf.txt

@@ -0,0 +1,1165 @@
+arXiv:2301.12440v1  [physics.atom-ph]  29 Jan 2023
+On the status of channeling radiation and laser based radiation sources
+Khokonov M.Kh.∗
+(Dated: January 31, 2023)
+A unified description of channeling radiation (CR) in oriented crystals (OC) and radiation in
+the field of an intense plane wave (laser radiation sources – LRS) is proposed in terms of two
+Lorentz invariants, which can be determined using the target (crystal or laser) and the incident
+beam parameters. The case of planar positron channeling (quasi-channeling) and linear polarized
+laser wave is considered in detail up to TeV energies. The crucial difference between LRS and CR is
+that in the former case both invariants are independent, while in channeling they are linearly related
+to each other. This leads to a strong limitation on the range of possible values of these invariants
+in OC. On the other hand, OC make it possible to study QED processes in strong non-uniform
+external fields.
+I.
+INTRODUCTION
+Channeling radiation (CR) in oriented crystals (OC)
+[1, 2] provides a promising source of intense gamma
+radiation, as well as makes it possible to experimentally
+study such phenomena as the radiation reaction [3–
+7], strong field effects [2, 8, 9] and trident production
+[10, 11].
+Over the past two decades,
+interest in
+interaction of relativistic electrons with intense laser
+fields (referred to below as a laser radiation sources, LRS)
+has grown significantly [12, 13] as an alternative method
+of producing X-ray and gamma radiation [14, 15] and
+as a tool for studying the QED strong field effects [16–
+22].
+Therefore, it is of interest to establish the status
+of channeling and lasers in the context of the problems
+outlined.
+Close similarity between CR radiation in crystals and
+lasers has been studied in Refs.
+[14, 23].
+During
+channeling, a relativistic electron interacts with a static
+electric field of atomic chains or planes, while for lasers,
+the interaction occurs with a field close to the field of
+a plane wave. The appearance of higher harmonics in
+the radiation spectra arising in the field of an intense
+laser wave is due to the process of absorption of several
+photons from the laser field with subsequent emission
+a photon whose energy is Doppler shifted toward the
+harder frequency range. The same process happens in
+OC, where an electron absorbs the virtual photons of the
+electrostatic atomic string (plane) potential of a crystal.
+The
+external
+field
+is
+considered
+strong
+if
+the
+Schwinger’s field parameter, χ, exceeds unity, χ
+=
+eℏ|Fµνpν|/(m3c4),
+where
+Fµν
+is
+an
+external
+field
+electromagnetic tensor, pν is an electron 4-momentum,
+e and m are a charge and rest mass of an electron, c is
+the velocity of light in vacuum. In oriented crystals, χ =
+ℏFγ/(m2c3), where γ = E/mc2, is a Lorenz-factor of an
+electron with energy E, F = |∇U| is a force acting on an
+electron from the continuum potential, U, of atomic axes
+or planes of the crystal. In the case of axial channeling,
+∗ Kabardino-Balkarian
+State
+University,
+Nalchik,
+Russian
+Federation.; Electronic address: [email protected]
+F ≈ 2Ze2/(daF ), where Z is an atomic number of the
+crystal, d is a distance between atoms in the atomic
+string, aF is a Thomas-Fermi screening parameter. For
+example, in silicon ⟨110⟩ crystal (Z = 14), F ≈ 520
+eV/˚A, and χ ≈ 1 for electron energies E ≈150 GeV. The
+strong field effects in radiation, like quantum recoil due
+to hard photon emission, is significant already at χ >
+0.1.
+In what follows the Lorentz factor corresponding
+to the initial energy of an electron (positron) before the
+interaction with an external field will be denoted by γ,
+and the current time-dependent Lorentz-factor will be
+γ(t) (or, γ(δ), δ is an invariant phase of a plane wave).
+In contrast with LRS channeling radiation occurs only
+at relativistic energies.
+Let an electron undergoing
+arbitrary extreme relativistic motion with γ = (1 −
+β2)−1/2 ≫ 1 encounter an external field with a potential
+|U| ≪ E, β = cv, v is an electron’s velocity.
+The
+angle of deviation of an electron by the external field,
+θe, is then small enough such that the component of
+an electron’s velocity transverse to the velocity of the
+average rest frame (ARF) is nonrelativistic, i.e. β⊥ =
+v⊥/c ≈ θe ≪ 1. As is well known the peculiarities of
+the radiation spectra of relativistic electrons depend on
+the so-called “non-dipole parameter” D ≈ θeγ, which is
+Lorentz-invariant, since the product, γβ⊥, is invariant,
+[24] (chapter 11.4).
+If the deflection angle is larger
+than the characteristic radiation angle ∼ 1/γ, then the
+condition D ≫ 1 is fulfilled, which means that only a
+small part of the electron trajectory defines the emission
+spectrum which is described by simple synchrotron like
+formulas (constant field approximation – CFA). In the
+opposite limit, D ≪ 1, the dipole approximation is valid,
+and the formulas for the emission spectrum are also
+substantially simplified.
+Note, that the “longitudinal”
+Lorentz-factor, corresponding to the velocity of the ARF,
+v0, γ0 = (1 − β2
+0)−1/2, is in connection with the total
+Lorenz-factor as γ = γ0(1 + D2)1/2 (in the absence of
+the longitudinal oscillations, such that, β2 = β2
+0 + β2
+⊥).
+We will assume that γ ≫ 1 and γ0 ≫ 1.
+In what
+follows, the angle brackets will denote averaging over the
+period, T , of the electron transverse oscillations, ⟨(...)⟩ =
+T −1 � T
+0 (...)dt. The precise definition of the invariant, D,
+in the present paper is, D = ⟨β⊥(t)2γ(t)2⟩1/2, both for
+
+2
+CR and LRS.
+The intensity of a plane electromagnetic wave is
+characterized by the Lorentz-invariant parameter, ν0 =
+eE0/(
+√
+2mcω0), where E0 is the amplitude of the electric
+field of a plane wave with a frequency ω0. As we will
+see below, the exact relation holds for LRS, ν0 = D.
+This equality underlies the comparison of CR with LRS
+(see also Refs. [23], [25]). We also assume that γ ≫ D,
+which provides a small-angle electron scattering by an
+external field, θe ≪ 1.
+In channeling this condition
+is always satisfied, since charged particles incident on
+a single crystal with small angles to crystallographic
+directions and, θe ∼ θL = (2Um/E)1/2, θL is a Lindhard
+critical channeling angle, [1, 2], [26], Um is a depth of
+a continuum potential well of atomic planes or strings.
+For LRS, θe ∼ ν0/γ, and this condition is violated for
+petawatt lasers at energies below several hundred MeV.
+We will need one more invariant, which depends on the
+electron energy, a = 2ℏγΩ0/(mc2), where Ω0 = 2π/T , is
+a period of the transverse oscillations. For LRS it can be
+written in the form, a = 2ℏ(k0p)/(mc)2, where k0 and p
+are the 4-wave vector of the incoming plane wave, and p
+is a 4-momentum of an electron.
+II.
+EQUATIONS OF MOTION
+The trajectory of ultrarelativistic particle interacting
+with a relatively weak external field, |U| ≪ E (and
+θe ≪ 1), can be conveniently divided into transverse and
+longitudinal parts. We direct the longitudinal component
+along the z axis, which in the case of channeling is
+parallel to the atomic chains (or planes) of the crystal,
+and for LRS it coincides with the direction of the initial
+electron (positron) velocity. The longitudinal velocity of
+an electron is then expressed in terms of its transverse
+velocity
+βz(t) ≈ 1 −
+1
+2γ2 − β2
+⊥(t)
+2
+,
+(1)
+where cβ⊥ = dr⊥/dt, r⊥ is the transverse coordinate.
+The time dependence of βz is due to the transverse
+component only. The basic correction to Eq.(1) is equal
+to, −(U/E)γ−2, which is negligible compared to the last
+two terms in Eq.(1). The subsequent terms are of the
+order of, ∼ β4
+⊥, (β⊥/γ)2, γ−4.
+The trajectory, r(t) = (r⊥(t), z(t)), is expressed in
+terms of its transverse part with
+z(t)≈ct
+�
+1− 1
+2γ2
+�
+− c
+2
+� t
+0
+β2
+⊥(τ)dτ.
+(2)
+The
+transverse
+motion
+is
+non-relativistic
+and
+satisfies Newton’s equations with a relativistic mass,
+mγd2r⊥/dt2 = F, where in channeling, F = −∇U.
+Conventionally, for channeling equations (1) and (2) are
+considered as exact equations [1]. For LRS F represents
+the electromagnetic Lorentz force.
+In the absence
+of secondary effects such as multiple scattering and
+radiation damping, the transverse energy in channeling,
+E⊥ = p2
+⊥/(2γm) + U(r⊥), is an integral of motion, as is
+the longitudinal momentum, pz = γmcβz [1]. We do not
+consider the secondary factors.
+The limits of applicability of equations (1) and (2) are
+γ ≫ 1,
+γ ≫ D.
+(3)
+These relations summarize inequalities given in the
+Introduction.
+We will compare the planar channeling of positrons
+with electrons moving head-on the plane-polarized laser
+field. Positrons move in the continuum potential of the
+atomic planes, which potential is close to the parabolic,
+U(x) = 4Umx2/d2
+p, [1, 26], where x is a transverse
+coordinate, dp is a distance between atomic planes. The
+transverse motion of positrons is one-dimensional and
+occurs along the x axis.
+The origin of the coordinate
+system lies between the atomic planes along which the
+z axis is directed. Radiation for planar channeling has
+been studied in detail in Refs. [27–29], and in the recent
+papers [30, 31].
+For LRS electrons are assumed to move along z axis
+and interact head on with a plane wave which electric
+field, E = −E0 sin δ, is directed parallel to x-axis, the
+invariant phase here is, δ = ω0(t + z/c).
+Equations of motion, both for positron channeling and
+for LRS, in the external fields described above, within
+the limits of applicability of Eqs. (3) are
+x(t) = xm sin(Ω0t),
+z(t) = ctβ0 − x2
+mΩ0
+8c
+sin(2Ω0t),
+(4)
+β0 = ⟨βz⟩ = 1 −
+1
+2γ2 − ⟨β2
+⊥⟩
+2
+,
+where ⟨β2
+⊥⟩ = (xmΩ0/2c)2, β0 is the velocity of the ARF.
+Different quantities in Eqs. (4) for channeling and LRS
+are given in the Table 1.
+It
+is
+reasonable to
+compare Eqs.(4)
+with
+exact
+solutions.
+In the case considered the exact classical
+equations of motion for LRS have the form
+x = xm sin δ ,
+z = β0ct − x2
+mΩ0
+8c
+sin 2δ,
+(5)
+Ω0t = δ +
+x2
+mΩ2
+0
+8c2(1 + β0) sin 2δ ,
+γ(δ) = γ0
+�
+1 + ν2
+0
+�
+1 +
+x2
+mΩ2
+0
+4c2(1 + β0) cos 2δ
+�
+,
+(6)
+where the quantities Ω0 and xm are given in the Table 1.
+
+3
+Table 1. Correspondence between different quantities
+in the field of lasers and planar positron channeling.
+Here, γ is the initial Lorentz-factor, for LRS
+γ = γ0(1 + ν2
+0)1/2, Ω0 is the frequency of transverse
+oscillations, θL = (2Um/E)1/2, E = γmc2
+Laser
+Channeling
+xm
+√
+2 ν0c
+Ω0γ
+dp
+2
+�
+E⊥
+Um
+Ω0
+(1 + β0)ω0
+2c
+dp θL
+χ
+eE0ℏγ
+m2c3 (1 + β0)
+| ∇U |
+m2c3 ℏγ
+D
+ν0
+√E⊥E
+mc2
+a
+2ℏω0
+mc2 (1 + β0)γ
+4ℏc
+mc2dp
+�
+2Umγ
+mc2
+Eqs.(5) and (6) satisfy the Lorentz equations of
+motion, mcdvµ/dτ = eF µνvν, where vµ is a 4-velocity, τ
+is a proper time. Space components of this equations give
+Eqs.(5), while the time component leads to the Eq.(6).
+It follows from (5) and (6) that, γ2(δ)β2
+x(δ) = 2ν2
+0 cos2 δ.
+Therefore, the average over the phase gives for LRS,
+D ≡ ⟨β⊥(δ)2γ(δ)2⟩1/2 = ν2
+0.
+The average Lorentz-
+factor, according to Eq.(6), is ⟨γ(δ)⟩ = γ0
+�
+1 + ν2
+0. This
+value should be taken as the initial Lorentz-factor of an
+electron before the interaction with an external field, i.e.
+in infinity.
+At high energies, if conditions (3) take place, the term
+in the third line of Eq.(5), (xmΩ0/c)2 ∼ (D/γ)2 ≪ 1, is
+negligibly small, such that we can replace the arguments
+of sine in the first two lines by, δ ≈ Ω0t. We arrive then
+to Eqs.(4). Note, that in this limit the oscillating term
+in Eq.(6) is also negligible, and the Lorentz-factor is time
+independent, equal to its initial value, γ = γ0
+√
+1 + D2.
+III.
+RADIATION SPECTRUM
+Within the frame of the quasiclassical method, which
+applicability is consistent with conditions (3),
+the
+spectral and angular distribution of energy emitted by a
+relativistic electron moving along an arbitrary trajectory
+is [32]
+Iωn ≡
+d2I
+dωdΩ = e2ω2
+8π2c
+E2 + E′2
+E′2
+×
+(7)
+×
+�
+| n × [n × jω] |2 + ℏ2ω2γ−2
+E2 + E′2 J
+�
+,
+where E′
+=
+E − ℏω is the final electron energy,
+n = (sin θ cos ϕ, sin θ sin ϕ, cos θ) is a unit vector in the
+direction of radiation,
+jω =
+� +∞
+−∞
+β(t) exp
+�
+iω′(t − rn/c)
+�
+dt,
+(8)
+J =
+����
+� +∞
+−∞
+exp
+�
+iω′(t − rn/c)
+�
+dt
+����
+2
+,
+where ω′ = ωE/(E − ℏω).
+For a quasiperiodic motion, r⊥(t) = r⊥(t + T ), given
+by Eqs. (4), Ω0 = 2π/T , general expressions (7), (8) give
+the following result for photon number spectrum emitted
+per unit phase of the transverse motion, ψ = Ω0t, both
+for CR and LRS, as a function of invariants D and a only
+d2Nγ(u, a, D)
+N0 dudψ
+= 3
+πa
+∞
+�
+k=km
+� 2π
+0
+ddϕ
+��
+1 + uu′
+2
+�
+gk + uu′
+4D2 j2
+zk
+�
+,
+(9)
+gk = j2
+xk + η2
+k
+2D2 j2
+zk −
+√
+2 ηk
+D jxk jzk cos ϕ ,
+(10)
+where u = ℏω/E is the photon energy measured in the
+units of electron’s initial energy, u′ = u/(1 − u), η2
+k =
+ak/u′ − ν2
+0 − 1, ηk = γθk, θk is a polar angle of radiation
+for k-th harmonic, ϕ is the azimuth angle of radiation,
+km = 1 + E(u′(1 + D2)/a), E(x) is the integer part of
+x. N0 = dN class
+γ
+/dψ = (2/3)αD2 is a number of emitted
+quanta in the classical limit of Thomson scattering, α =
+1/137. For fixed harmonic number, k, the emitted photon
+energy changes within the interval
+0 < u < umk = ak/(1 + D2 + ak).
+(11)
+Fourier components of the electron current jxk and jzk
+are expressed through the Bessel functions Jm(x):
+jxk = B−1
+∞
+�
+m=−∞
+(k + 2m) Jm(A) Jk+2m(B) ,
+(12)
+jzk =
+∞
+�
+m=−∞
+Jm(A) Jk+2m(B) ,
+where A
+=
+(2a)−1D2u′, B
+=
+2
+√
+2 a−1Du′ηk cos ϕ.
+Expressions like (12) are well known in the theory of
+LRS [33], planar positron channeling [1], [34, 35] and
+undulator radiation [36].
+Formulas (9) – (12) are greatly simplified in two
+limiting cases: D ≪ 1 (dipole approximation) and D ≫
+1 (CFA – constant field approximation, also called as
+synchrotron approximation).
+In the limit D ≪ 1 Eqs. (9) – (12) give a quantum
+dipole spectrum
+d2Nγ
+N0 dudψ = 3
+2a
+�
+1 − 2u′
+a + 2u′2
+a2 + uu′
+2
+�
+,
+(13)
+
+4
+where 0 < u < a/(1 + a).
+The shape of the dipole
+spectrum depends only on one invariant a.
+In the opposite limit of CFA, D ≫ 1, harmonics with
+k ≫ 1 play a decisive role. In this case the spectrum has
+the form
+d2Nγ
+dudψ =
+� 2π
+0
+d2N (CF A)(α0)
+dudψ
+dα0
+2π ,
+(14)
+where
+the
+integrand
+is
+the
+well-known
+quantum
+synchrotron formula [37]
+d2N (CF A)
+γ
+N0dudψ
+=
+√
+3
+πaD2
+(15)
+×
+�
+(2 + uu′)K2/3(ξ) −
+� ∞
+ξ
+K1/3(η)dη
+�
+,
+where ξ
+=
+2u′/(3χ).
+Here,
+the field parameter
+�� is expressed through invariants a and D:
+χ
+=
+aD sin α0/
+√
+2.
+Integration over α0 in Eq.(14) means
+averaging over the period of an electron transverse
+motion. Convenient representation of Eq. (15) without
+special functions is given in [38, 39].
+The spin
+contribution in Eqs.
+(9), (13) and (15) is due to the
+terms containing the product uu′.
+IV.
+THE D ÷ a DIAGRAM
+Despite the similarity of the CR and LRS, outlined
+in the previous sections, there is a significant difference
+between them, associated with the different behavior of
+invariants a and D as a function of the electron (positron)
+energy. For LRS D and a are independent parameters,
+D does not depend on energy and a ∼ γ. For channeling,
+both D and a, demonstrate the same type of energy
+dependence, proportional to γ1/2, because in this case,
+Ω0 ∼ γ−1/2. It means that in channeling D and a are
+not independent and represent a line in the (D÷a) space.
+Further, we will also be interested in the axial channeling
+of electrons, which we consider approximately. In this
+case we will take Um ≃ 2Ze2/d, θL = (4Ze2/dE)1/2,
+Ω0 ≈ cθL/aF . By means of these relations and formulas
+of the Section 2 we obtain the connection between a and
+D for axial and planar channeling
+a = (4λc/dp)(2Um/E⊥)1/2D ,
+planar positrons,(16)
+a ≈ (2λc/aF)D ,
+axial electrons,
+(17)
+where λc = ℏ/mc is a Compton wave length.
+The
+planar Eq.(16) is exact, whereas the axial Eq.(17) is
+approximate. For channeling, only those values of a and
+D are possible that satisfy Eqs.(16) and (17), while there
+are no restrictions for LRS. The choice of a target crystal
+does not affect much the positions of the D−a lines (16),
+(17).
+The above results can be conveniently illustrated using
+a diagram in the a versus D plane on the logarithmic
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+-8
+-7
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+2
+CS
+CD
+QD
+Q
+P
+O
+C
+D
+B
+A
+N
+QS
+M
+ln D
+ln a
+K
+1
+FIG. 1.
+The D ÷ a diagram. Different regions are explained
+in the text.
+Lines 1 and 2 correspond to planar positron,
+Eq.(16), and axial electron, Eq.(17) channeling, respectively.
+scale similar to that done in Ref.[40] for LRS. A diagram
+presented in the Fig.1 classifies CR and LRS in terms of
+invariants X ≡ ln D (abscissa) and Y ≡ ln a (ordinate),
+so that the origin of the coordinate system corresponds
+to the D and a values of unity.
+In this coordinate
+system, the region of strong fields is to the right of
+the Y - axis, while the region where quantum effects in
+the radiation spectrum occur lies in the vicinity of the
+X- axis and above this axis. The region to the left of
+the dashed line MN in Fig.1 represents weak external
+fields:
+D < 0.1 (X = –2.3).
+The dashed line AKB
+defines the boundaries of the area of relatively small
+energies for which a < 0.1 (Y = −2.3).
+Accordingly,
+the AKN region CD corresponds to the classical dipole
+approximation (Thomson backscattering, Eq.(13) for
+u ≪ 1).
+The quantum dipole spectrum (Compton
+backscattering, Eq.(13)) takes place in the AKM region
+QD. In the last two regions, the radiation spectrum is
+characterized by high monochromaticity and displays a
+single peak terminating at the frequencies, umk, given by
+the formula (11) for k = 1.
+The condition of the quantum CFA (D2 > a and
+D
+>
+1) appears in Fig.(1) as 2X
+> Y , and the
+corresponding region extends to the right of the OP line.
+The radiation spectrum is determined by the quantum
+synchrotron formula (15) and has a shape determined
+by a single parameter χ ≈ aD.
+The region of the
+classical synchrotron approximation (χ < 0.1; D ≥ 1)
+
+5
+0,00
+0,05
+0,10
+0,15
+0,0
+0,1
+0,2
+0,3
+0,4
+0,5
+0,6
+0,7
+0,00
+0,01
+0,02
+0,03
+0,04
+0,05
+0,0
+0,2
+0,4
+0,6
+0,8
+1,0
+1,2
+1,4
+1,6
+Intensity
+Photon energy, u
+A
+B
+1
+2
+FIG. 2.
+Radiation intensity spectra for a = 0.0183 (Y = −4,
+centered triangles in Fig.1). Solid lines are exact calculations
+according to (9). A: D=1 (X=0, curve 1); D=2.718 (X=1,
+curve 2); dashed lines, spectra in the CFA (14), (15).
+B:
+D=0.5 (X = −0.69); dashed line, dipole approximation (13).
+Beam energy for LRS is 1.2 GeV.
+is determined by the inequality X + Y
+< –2.3 and
+corresponds to the CS sector (DBQ). In this case, the
+radiation spectrum appears as the classical synchrotron
+spectrum. By the same token, the QS sector corresponds
+to the quantum synchrotron spectrum (QBOP). The
+validity of the CFA (14), (15) depends also on the emitted
+photon energy and has the form [14]
+D2a−1u(1 − u)−1 ≫ 1.
+(18)
+The other regions shown in Fig.1 require exact
+calculations taking into account the generation of higher
+harmonics. The formulas of classical electrodynamics are
+valid in the NKBD region, whereas the exact quantum
+expressions (7)-(12) have to be used in the MKBOP
+region.
+In the case of LRS, all values of a and D in Fig 1 are
+available.
+The energy of an electron for LRS in Fig.1
+is E = a(mc2)2/4ℏω0.
+For a laser with ℏω0 = 1 eV
+(this energy of the laser photon will be used in numerical
+calculations below) the origin of the coordinate system
+in Fig. 1 corresponds to E=65.3 GeV.
+For a crystal, only those values of D and a are
+possible that lie on the lines satisfying Eqs.(16), (17).
+Calculations are done for Si crystal (Z = 14). The line
+1 in Fig 1 represents the planar channeling of positrons
+(16) in Si (110) crystal, Um = 21 eV, dp = 1.92 ˚A and
+E⊥ = Um. The line 1 has an equation, Y = X − 4.48. It
+starts at positron energy E ≈ 31 MeV and ends at E ≈ 5
+TeV. Axial channeling refers to the line 2 in Fig. 1. The
+calculation is for Si ⟨110⟩, aF = 0.2 ˚A, d = 3.84 ˚A. The
+axial channeling line (17) has the form, Y = X − 3.25. It
+0,0
+0,2
+0,4
+0,6
+0,8
+1,0
+0,0
+0,1
+0,2
+0,3
+0,4
+0,5
+0,6
+0,0
+0,2
+0,4
+0,6
+0,8
+0,0
+0,2
+0,4
+0,6
+0,8
+1,0
+Photon energy, u
+Intensity
+2
+1
+A
+B
+FIG. 3.
+Radiation intensity spectra for a = 1 (Y = 0,
+centered circles in Fig.1). Solid lines represent calculations
+according to Eq.(9).
+A: D=1 (X=0, curve 1); D=2.718
+(X=1, curve 2, dots show the spin contribution); dashed lines
+are spectra in the CFA (14), (15). B: D=0.5 (X = −0.69);
+dashed line, dipole approximation (13). Beam energy for LRS
+is 65.3 GeV.
+starts at E ≈23 MeV end ends at E ≈3.7 TeV. Crosses
+on the lines 1 and 2 correspond to the beam energy 300
+GeV. Electron energies of LRS at these points are 3.6
+GeV for planar line 1 in Fig. 1 and 39 GeV for axial line
+2. Beam energies for channeling lines in Fig. 1 can be
+calculated as
+E =
+a2
+2Um
+� dp
+4λc
+mc2
+�2
+,
+planar positrons,
+(19)
+E = a2d
+4Ze2
+� aF
+2λc
+mc2
+�2
+,
+axial electrons.
+(20)
+Calculations of the radiation intensity spectra for
+points shown by the centered symbols in Fig.1 are
+presented in Figs. 2 – 4. Intensity is given in the units
+of N0, i.e. the right hand sides of Eqs.(9), (13), (15),
+multiplied by u = ℏω/E are shown as functions of u.
+Fig.2 represents classical radiation spectra, a ≪ 1, for the
+points shown by triangles in Fig. 1. The LRS electron
+beam energy is 1.2 GeV. CFA adequately describes the
+spectrum already for D > 2.
+Quantum effects of emitted photon recoil and spin
+become clearly expressed at a = 1, as is seen in Fig.3
+(circles in Fig.1). The LRS energy in this case is 65.3
+GeV. For D = 1 CFA is applicable only for hard photons
+in accordance with condition (18), but for D = 2.78 CFA
+describes the spectrum well, except for the relatively low-
+frequency region, where individual harmonics are clearly
+pronounced.
+Intensity spectra for very high energies (483 GeV for
+LRS) are shown in Fig.
+4 (a = 7.4, Y = 2, square
+
+6
+0,0
+0,2
+0,4
+0,6
+0,8
+1,0
+0,00
+0,02
+0,04
+0,06
+0,08
+0,10
+0,0
+0,2
+0,4
+0,6
+0,8
+1,0
+0,0
+0,2
+0,4
+0,6
+0,8
+Intensity
+Photon energy
+1
+2
+Intensity
+1
+2
+A
+B
+FIG. 4.
+Radiation intensity spectra for a = 7.4 (Y = 2,
+centered squares in Fig.1). Solid lines are exact calculations
+according to (9). A: D=1 (X=0); dashed curve 1 is a dipole
+approximation (13); dashed curve 2 is CFA (14), (15).
+B:
+D=2.718 (X=1, curve 1); D=7.4 (X=2, curve 2); dashed
+lines, spectra in the CFA (14), (15).
+Dots show the spin
+contribution. Beam energy for LRS is 483 GeV.
+characters in the Fig.1). The spin contribution dominates
+at high energy photon region, u > 0.6. CFA describes the
+radiation spectrum at relatively large values of D > 7.
+As it follows from Fig. 4, even for D = 7.4 there is a
+pronounced peak in the soft part of the spectrum.
+It is well known, that the validity of the CFA in
+channeling becomes better while the electron energy
+increases, since D ∼ γ1/2. There is a reverse trend in the
+case of LRS. For fixed laser field intensity, D, the justice
+of CFA worsens with an increase in electron energy. This
+is clear from Figs. 2 – 4 for D = 2.718 (symbols in Fig.
+1 for X = 1). At low energies (curve 2 in Fig. 2) CFA
+describes the whole radiation spectrum. At higher energy
+corresponding to curve 2 in Fig. 3 individual harmonics
+are clearly visible in the soft part of the spectrum, while
+at very high energies (curve 1 in Fig. 4, B), CFA is valid
+only in the hard photon region, u > 0.9.
+V.
+QUASI-CHANNELING
+The
+transverse
+trajectories
+of
+the
+above-barrier
+particles are infinite, E⊥ > Um. For LRS this kind of
+trajectories does not exist with except of the scattering
+of the incoming particle beam by the intense laser beam
+with ν0 > γ.
+In planar quasi-channeling, the above-
+barrier particles cross atomic planes with a period, Tq,
+equal to the interplanar travel time. The definition of
+the invariant D in this case should take into account
+that the average transverse velocity is not zero: D =
+⟨∆β2
+⊥⟩1/2γ, where ⟨∆β2
+⊥⟩ = ⟨β2
+⊥⟩ − ⟨β⊥⟩2.
+We define
+also the invariant a for quasi-channeled particles as:
+a = 2ℏγΩq/(mc2), where Ωq = 2π/Tq.
+The mean square variation of the transverse velocity
+can be calculated analytically for some realistic model
+potentials. We shall restrict ourselves with the case of
+large transverse energies, E⊥ ≫ Um. In this limit the
+result is: ⟨∆β2
+⊥⟩ ≈ ξU 2
+m/(E⊥E), where for planar quasi-
+channeling in the parabolic potential ξ = 2/45.
+This
+formula is valid up to the terms ∼ (Um/E⊥)2.
+For
+invariant parameters a and D we obtain
+a = 4π λc
+dp
+�2E⊥γ
+mc2
+�1/2
+,
+(21)
+D = Um
+�
+ξγ
+mc2E⊥
+�1/2
+.
+(22)
+The corresponding line on the D ÷ a diagram is
+a = 4π λc
+dp
+E⊥
+Um
+�2
+ξ
+�1/2
+D.
+(23)
+The positron energy on the line (23) is
+E =
+a2
+2E⊥
+� dp
+4πλc
+mc2
+�2
+.
+(24)
+Formulas
+(21)–(24)
+demonstrate
+a
+significant
+difference between the dependence of parameters D
+and a on the transverse energy for channeling (presented
+in the Table 1) and quasi-channeling. The parameter D
+for quasi-channeling decreases with increasing transverse
+energy as ∼
+E−1/2
+⊥
+, which causes the radiation to
+become more dipole and the transverse trajectories to
+become more rectilinear, so that the transverse velocity
+tends to β⊥ → (2E⊥/E)1/2.
+The applicability of the
+CFA becomes less acceptable as the transverse energy
+increases. This agrees with the results of Refs. [41–43],
+in which the field non-uniformity parameter, ν, coincides
+with the present parameter D up to a factor of the order
+of unity (see Eq.(12) in [41]).
+For 300 GeV positrons in Si (110) and E⊥/Um = 10
+(it corresponds to the angles ∼ 3θL with respect to the
+atomic plane):
+D =0.33 and a =0.55, which values
+belong to the region of quantum dipole spectrum in the
+
+7
+diagram on Fig.1. The D − a line for quasi-channeling
+(23) lies above the line for channeled particles, and the
+higher, the greater the transverse energy (not shown
+in Fig.1).
+For example, for E⊥/Um = 10 the quasi-
+channeling line (23) is parallel to lines 1 and 2 on the
+diagram in Fig.1 and goes through the point Y = 0.53
+and X = 0 (D = 1). The dipole approximation for such
+positrons is violated at TeV energies, (D = 1 for E = 2.8
+TeV when a = 1.7).
+VI.
+SUMMARY AND CONCLUDING
+REMARKS
+Radiation
+spectrum
+of
+ultrarelativistic
+electrons
+(positrons) in the fields of lasers and oriented crystals
+can be expressed in terms of two invariants, one of
+which, a, characterizes the significance of quantum effects
+in radiation and depends on the electron (positron)
+energy, while the non-dipole parameter D determines
+the influence of the external field.
+In the particular
+case of the linear polarized laser wave and planar
+positron channeling the radiation spectra are described
+by the same formula (9), where parameters a and D are
+determined by expressions shown in the Table 1. The
+crucial difference between LRS and CR is that in the
+former case parameters a and D are independent, while
+in channeling they are linearly related to each other. This
+leads to a strong limitation on the range of possible values
+of the parameters a and D for channeling, since they lie
+on a line in the space of X = ln D and Y = ln a, as
+shown in the diagram on Fig.1 (the D − a line).
+For
+example, the quantum dipole radiation spectrum with a
+single peak in the hard photon region, ℏω ∼ E, is not
+possible in channeling, while for LRS it corresponds to
+the Compton back scattering in the weak field, D ≪ 1,
+a ≥ 1.
+The D parameter for LRS does not depend on
+particle’s energy, while for CR it increases as ∼ γ1/2. The
+consequence of this is that the applicability of the CFA
+for channeled particles becomes better as their energy
+increases. For LRS, conversely, the condition for CFA
+applicability is violated as the energy grows for fixed D.
+In contrast to channeled particles, the non-dipole
+parameter D for quasi-channeled particles decreases
+with increasing the transverse energy as ∼ E−1/2
+⊥
+(22).
+Accordingly, the D − a line of quasi-channeled particles
+(23) lies the higher, the greater their transverse energy,
+and the applicability of the CFA deteriorates. We can say
+that CFA is valid if the angles of entry of particles into
+an oriented crystal do not exceed the critical Lindhard
+angle, θL. Since in real crystals the transverse energy
+tends to increase due to multiple scattering, the particles
+continuously move to higher D−a lines as they penetrate
+through the target.
+The influence of the spatial non-
+uniformity of the field of atomic chains (planes) on
+radiation become stronger. For example, a suppression
+of radiation for TeV quasi-channeled electrons in about
+the whole spectrum compared with that for CFA may
+take place [41]. Thus, the OC makes it possible to study
+QED processes in strong non-uniform external fields.
+[1] V.V. Beloshitsky, F.F. Komarov // Phys. Reports. 93
+(1982) 117-197.
+[2] U.I. Uggerhøj. // Rev. Mod. Phys. 77 (2005) 1131-1171.
+[3] A. Di Piazza, T.N. Wistisen, U.I. Uggerhøj // Phys. Lett.
+B. 765 (2017) 1-5.
+[4] T.N. Wistisen,
+A. Di Piazza,
+H.V. Knudsen,
+U.I.
+Uggerhøj // Nature Communications, 9: 795 (2018).
+[5] M.Kh.Khokonov // Physics Letters B, Volume 791,
+Pages 281-286, 2019.
+[6] T. N. Wistisen, A. Di Piazza, C. F. Nielsen, A. H.
+Sorensen, U. I. Uggerhøj // Phys. Rev. Research 1,
+033014 (2019)
+[7] Ch. F. Nielsen, J. B. Justesen, A. H. Sorensen, U. I.
+Uggerhøj, R. Holtzapple // Phys. Rev. D 102, 052004
+(2020)
+[8] A. Di Piazza, T. N. Wistisen, M. Tamburini, U. I.
+Uggerhøj // Phys. Rev. Lett. 124, 044801 (2020)
+[9] F. C. Salgado, N. Cavanagh, M. Tamburini, et. al. //
+New J. Phys. 24, 015002 (2021)
+[10] J. Esberg, K. Kirsebom, H. Knudsen, et. al. // Phys.
+Rev. D 82, 072002 (2010)
+[11] Ch. F. Nielsen, R. Holtzapple, M. M. Lund et. al. //
+arXiv:2211.02390 (2022)
+[12] A. Di Piazza, C. Muller, K.Z. Hatsagortsyan, Ch.H.
+Keitel // Rev. of Mod. Phys. 84 1177-1228 (2012)
+[13] H. Abramowicz et.al. Conceptual design report for the
+LUXE experiment // The European Physical Journal
+Special Topics volume 230, pages 2445–2560 (2021).
+[14] A.Kh.Khokonov, M.Kh.Khokonov, A.A.Kizdermishov //
+Technical Physics, V.47, pp.1413-1419, (2002)
+[15] B. King and S. Tang // Phys. Rev. A 102 (2020), No. 2,
+022809
+[16] A. Gonoskov, T. G. Blackburn, M. Marklund, and S. S.
+Bulanov // Rev. Mod. Phys. 94, 045001 (2022)
+[17] T. Podszus and A. Di Piazza // Phys. Rev. D 99 (2019),
+no. 7, 076004
+[18] A. Ilderton // Phys. Rev. D 99 (2019), no. 8, 085002
+[19] A. Mironov, S. Meuren, and A. Fedotov // Phys. Rev. D
+102 (2020) 053005
+[20] C. Baumann, E. N. Nerush, A. Pukhov, and I. Y.
+Kostyukov // Sci. Rep. 9 (2019) 9407
+[21] H. Hu, C. Muller, and C. H. Keitel // Phys. Rev. Lett.
+105 (2010) 080401.
+[22] A. Ilderton // Phys. Rev. Lett. 106 (2011) 020404.
+[23] M.Kh.Khokonov, R.A. Carrigan Jr. // Nucl.Inst. and
+Meth. B, V.145, P.133-141, (1998).
+[24] Jackson, J D, 1999, Classical Electrodynamics (John
+Wiley & Sons. Inc., New York).
+[25] A.Kh. Khokonov, M.Kh. Khokonov, R.M. Keshev //
+Nucl.Inst. and Meth. B, V.145, P.54-59, (1998).
+
+8
+[26] J. Lindhard // Mat-Fys. Medd. Dan. Vid. Selsk. 34 No.
+14 (1965).
+[27] V. V. Beloshitsky and M. A. Kumakhov // Sov. Phys.
+JETP 47 (4), 652-658 (1978). (Zh. Eksp. Teor. Fi. 74,
+1244-1256, 1978).
+[28] N. K. Zhevago // Sov. Phys. JETP 48(4), 701-707 (1978)
+(Zh. Eksp. Teor. Fiz. 75, 1389-1401, 1978)
+[29] M.A. Kumakhov, Kh. G. Trikalinos // Sov. Phys. JETP
+51(4), 815-821 (1980) (Zh. Eksp. Teor. Fiz. 78, 1623,
+1980).
+[30] T.N. Wistisen and A. Di Piazza // Phys. Rev. A 98,
+022131 (2018)
+[31] T.N. Wistisen and A. Di Piazza // Phys. Rev. D, 99,
+116010 (2019).
+[32] V.N.
+Baier,
+V.M.
+Katkov,
+V.M.
+Strakhovenko.
+Electromagnetic Processes at High Energies in Oriented
+Single Crystals, World Scientific Pub Co Inc, 1998.
+[33] A.I. Nikishov and V.I. Ritus // Soviet Physics JETP V.
+19, No. 2 529-541 (1964) (J. Exp. Theor. Phys. USSR 46,
+776-796, 1964)
+[34] V. N. Baier, V. M. Katkov, and V. M. Strakhovenko //
+Zh. Eksp. Teor. Fiz. 80, 1348-1360 (1981)
+[35] V. A. Bazylev, V. V. Beloshitskil, V. I. Glebov et. al. //
+Sov. Phys. JETP 53 (2), p. 306 - 316 (1981).
+[36] R. Walker. Insertion devices: undulators and wigglers.
+In: CERN accelerator school. Synchrotron radiation and
+free electron lasers. Geneva, 1998, p. 129. ISBN 92-9083-
+118-9
+[37] N. G. Klepikov, Zh. Eksp. Teor. Fiz. 26, 19 (1954)
+[38] M.Kh. Khokonov // Physica Scripta V.55, No.5, P.513-
+519, (1997).
+[39] M.Kh. Khokonov // JETP, V.99, No.4, pp. 690-707
+(2004).
+[40] A.Kh.Khokonov, M.Kh.Khokonov // Tech. Phys. Lett.,
+V.31, No.2, p. 154-156 (2005).
+[41] M.Kh. Khokonov, H. Nitta // Phys. Rev. Lett. 89 (2002)
+094801.
+[42] H. Nitta, M.Kh. Khokonov, Y. Nagata, S. Onuki // Phys.
+Rev. Lett. V. 93, No.18, 180407 (2004)
+[43] Y. Nagata, H. Nitta and M. Kh. Khokonov // Nucl. Inst.
+Meth. in Phys. Res., B 234, p. 159-167 (2005).
+

htE3T4oBgHgl3EQf4Quq/content/tmp_files/2301.04771v1.pdf.txt

@@ -0,0 +1,2356 @@
+Variational Inference: Posterior Threshold Improves Network
+Clustering Accuracy in Sparse Regimes
+Xuezhen Li
+[email protected]
+Department of Statistics
+University of California, Davis
+Davis, CA 95616-5270, USA
+Can M. Le
+[email protected]
+Department of Statistics
+University of California, Davis
+Davis, CA 95616-5270, USA
+Abstract
+Variational inference has been widely used in machine learning literature to fit various Bayesian
+models. In network analysis, this method has been successfully applied to solve the community
+detection problems. Although these results are promising, their theoretical support is only for
+relatively dense networks, an assumption that may not hold for real networks. In addition, it has been
+shown recently that the variational loss surface has many saddle points, which may severely affect
+its performance, especially when applied to sparse networks. This paper proposes a simple way
+to improve the variational inference method by hard thresholding the posterior of the community
+assignment after each iteration. Using a random initialization that correlates with the true community
+assignment, we show that the proposed method converges and can accurately recover the true
+community labels, even when the average node degree of the network is bounded. Extensive
+numerical study further confirms the advantage of the proposed method over the classical variational
+inference and another state-of-the-art algorithm.
+Keywords: Variational Inference, Community Detection, Posterior Threshold, Non-convex Opti-
+mization, Sparse Network
+1. Introduction
+Variational inference (VI) is arguably one of the most important methods in Bayesian statistics (Jordan
+et al., 1999). It is often used to approximate posterior distributions in large-scale Bayesian inference
+problems when the exact computation of posterior distributions is not feasible. For example, this is
+the case for many high-dimensional and complex models which involve complicated latent structures.
+Mean-field variational method (Beal, 2003; Jordan et al., 1999) is the simplest VI algorithm, which
+approximates the posterior distribution of latent variables by a product measure. This method has
+been applied in a wide range of fields including network analysis (Airoldi et al., 2008; Celisse
+et al., 2012), computer science (Wang and Blei, 2013) and neuroscience (Grabska-Barwi´nska et al.,
+2017). The mean-field variational approximation is especially attractive in large-scale data analysis
+compared to alternatives such as Markov chain Monte Carlo (MCMC) (Gelfand and Smith, 1990)
+due to its computational efficiency and scalability (Blei et al., 2017).
+Although mean-field VI has been successfully applied to various Bayesian models, the theoretical
+behavior of these algorithms has not been fully understood. Most of the existing theoretical work
+focused on the global minimum of the variational method (Blei et al., 2003; Bickel et al., 2013;
+©2022 Xuezhen Li and Can M. Le.
+License: CC-BY 4.0, see https://creativecommons.org/licenses/by/4.0/.
+arXiv:2301.04771v1  [stat.ML]  12 Jan 2023
+
+LI AND LE
+Zhang and Gao, 2020; Wang and Blei, 2019). For example, Bickel et al. (2013) showed that the
+global minimum of the variational method is consistent under the stochastic block model and also
+obtained the asymptotic normality under the assumption of relatively dense networks. However, it is
+often intractable to compute the exact global minimizers of high-dimensional and complex models
+in practice. Instead, iterative algorithms such as Batch Coordinate Ascent Variational Inference
+(BCAVI) are often applied to estimate them (Blei et al., 2017).
+This paper focuses on statistical properties of BCAVI for estimating node labels of a network
+generated from stochastic block models (SBM) (Holland et al., 1983). Let n be the number of nodes,
+indexed by integers i ∈ [n] = {1, 2, ..., n}, and A ∈ {0, 1}n×n be the adjacency matrix with Aij = 1
+if i and j are connected. We only consider undirected networks without self-loops, so A is symmetric
+and Aii = 0 for all i. Under SBM, nodes are partitioned into K communities with community labels
+zi ∈ [K] drawn independently from a multinomial distribution with parameter vector π ∈ RK. The
+label vector z = (z1, ..., zn)T ∈ [K]n can also be encoded by a membership matrix Z ∈ Rn×K such
+that the i-th row of Z represents the membership of node i with Zizi = 1 and Zik = 0 if k ̸= zi.
+Conditioned on Z, {Aij, i < j} are independent Bernoulli random variables with corresponding
+probabilities {Pij}. For SBM, the edge probability Pij is determined by the block memberships of
+node i and j:
+Pij = Bzizj,
+where B ∈ RK×K is the block probability matrix that may depend on n. The goal of community
+detection is to recover z and estimate B.
+Many algorithms have been proposed for solving the community detection problem (Abbe,
+2018), and BCAVI is one of the most popular approaches. Given the adjacency matrix A and
+an initial estimate Z(0) ∈ [K]n of the true membership matrix Z, BCAVI aims to calculate the
+posterior P(Z|A) and use it to recover the true label vector Z, for example, by setting the estimated
+label of node i to argmaxk∈[K]P(Zik|A). Since the exact calculation of P(Z|A) is infeasible
+because it involves summing P(A, Z) over an exponentially large set of label assignments Z, BCAVI
+approximates this posterior by a much simpler and tractable distribution Q(Z(t)) and iteratively
+updates Q(Z(t)), t ∈ N, to improve the accuracy of this approximation.
+1.1 Our Contribution
+This paper considers the application of BCAVI in the sparse regime when the average node degree
+may be bounded. Our contribution is two-fold. First, we propose to improve BCAVI by hard
+thresholding the posterior of the label assignment at each iteration: After Z(t) has been calculated
+by BCAVI in the t-th iteration, the largest entry of each row Z(t)
+i
+is set to one, and all other entries
+are set to zero. In view of the uninformative saddle points (Sarkar et al., 2021) of BCAVI, this step
+appears to be a naive way to project the posterior back to the set of “reasonable” label assignments.
+However, as the hard threshold discards part of the label information in Z(t), it is not a priori clear
+why such a step is beneficial. Surprisingly, exhaustive simulations show that this adjustment often
+leads to significant improvement of BCAVI, especially when the network is sparse or the accuracy of
+the label initialization is poor. The resulting algorithm also outperforms a state-of-the-art method of
+Gao et al. (2017), which is rate-optimal only when the average degree is unbounded.
+Second, we prove that BCAVI, with the threshold step, accurately estimates the community labels
+and model parameters even when the average node degree is bounded. This nontrivial result extends
+the theoretical guarantees for BCAVI in literature to the sparse regime. It is in parallel with the
+2
+
+VARIATIONAL INFERENCE WITH POSTERIOR THRESHOLD
+extension of spectral clustering to the sparse regime by removing nodes of unusually large degrees
+Chin et al. (2015); Le et al. (2017). However, in contrast to that approach, our algorithm does not
+require any pre-processing step for the observed network.
+Similar to existing work on BCAVI (Zhang and Zhou, 2020; Zhang and Gao, 2020; Sarkar
+et al., 2021; Yin et al., 2020), we emphasize that the goal of this paper is not to propose the most
+competitive algorithm for community detection, which may not exist for all model settings. Instead,
+we aim to refine VI, a popular but still not very well-understood machine learning method, and
+provide a deeper understanding of its properties, with the hope that the new insights developed in this
+paper can help improve the performance of VI for other problems (Blei et al., 2017). As a proof of
+concept, we also apply the threshold strategy to cluster data points generated from Gaussian mixtures
+and numerically show that it consistently improves the classical variational inference in this setting;
+theoretical analysis for the Gaussian mixtures is very different from that for SBM and is left for
+future research.
+1.2 Related Work
+Regarding the consistency of BCAVI, Zhang and Zhou (2020) shows that if Z(0) is sufficiently
+close to the true membership matrix Z then Z(t) converges to Z at an optimal rate. In Sarkar et al.
+(2021), the authors give a complete characterization of all the critical points for optimizing Q(Z)
+and establish the behavior of BCAVI when Z(0) is randomly initialized in such a way so that it is
+correlated with Z. However, their theoretical guarantees only hold for relatively dense networks,
+an assumption that may not be satisfied for many real networks (Chin et al., 2015; Le et al., 2017;
+Gao et al., 2017). Moreover, BCAVI often converges to uninformative saddle points that contain
+no information about community labels when networks are sparse. Yin et al. (2020) addresses this
+problem by introducing dependent posterior approximation, but they still require strong assumptions
+on the network density.
+2. BCAVI with Posterior Threshold
+2.1 Classical BCAVI
+To simplify the presentation, we first describe the classical BCAVI for SBM without the threshold
+step. In particular, consider SBM with K communities and the likelihood given by
+P(A, Z) =
+�
+1≤i<j≤n
+�
+1≤a,b≤K
+�
+BAij
+ab (1 − Bab)1−Aij
+�ZiaZjb ·
+n
+�
+i=1
+K
+�
+a=1
+πZia
+a
+.
+The goal of mean-field VI is to compute the posterior distribution of Z given A:
+P(Z|A) = P(Z, A)
+P(A) ,
+P(A) =
+�
+Z∈Z
+P(Z, A),
+where Z = {0, 1}n×K is the set of membership matrices. Since P(A) involves a sum over the expo-
+nentially large set Z, computing P(Z|A) exactly is not infeasible. The mean-field VI approximates
+this posterior distribution by a family of product probability measures Q(z) = �n
+i=1 qi(zi). The
+optimal Q(z) is chosen to minimize the Kullback–Leibler divergence:
+Q∗(z) = argmin
+Q
+KL(Q(Z), P(Z|A)).
+3
+
+LI AND LE
+An important property of this divergence is that
+KL(Q(Z), P(Z|A)) = EQ[log Q(Z)] ��� EQ[log P(Z, A)] + log P(A),
+where EQ denotes the expectation with respect to Q(Z). This property enables us to approximate the
+likelihood of the observed data P(A) by its evidence lower bound (ELBO), defined as
+ELBO(Q) = EQ[log P(Z, A)] − EQ[log Q(Z)].
+Optimizing ELBO(Q) also yields the optimal Q∗(Z) and ELBO(Q∗) is the best approximation of
+P(A) we can obtain.
+We consider Q(Z) of the form Q(Z) = �n
+i=1
+�K
+a=1 ΨZia
+ia , where Ψ = EQ[Z]. Then
+ELBO(Q)
+=
+�
+1≤i<j≤n
+�
+1≤a,b≤K
+ΨiaΨjb
+�
+Aij log Bab + (1 − Aij) log(1 − Bab)
+�
++
+n
+�
+i=1
+K
+�
+a=1
+Ψia log (πa/Ψia) .
+(1)
+To optimize ELBO(Q), we alternatively update Ψ and {B, π} at every iteration. In particular, we
+first fix Ψ (the first iteration requires an initial value for Ψ) and set new values for {B, π} to be the
+solution of the following equations:
+∂ELBO(Q)
+∂Bab
+= 0,
+∂ELBO(Q)
+∂πa
+= 0,
+a, b ∈ [K].
+Depending on whether a = b, the first equation yields the following update for Bab:
+Baa =
+�
+i<j AijΨiaΨja
+�
+i<j ΨiaΨja
+,
+Bab =
+�
+i<j Aij (ΨiaΨjb + ΨibΨja)
+�
+i<j (ΨiaΨjb + ΨibΨja)
+,
+a ̸= b.
+(2)
+To update π, we treat πa, a ∈ [K − 1], as independent parameters and πK = 1 − �K
+a=1 πa, For each
+a ∈ [K − 1], taking the derivative with respect to πa and setting it to zero, we get
+∂ELBO(Q)
+∂πa
+=
+�n
+i=1 Ψia
+πa
+−
+�n
+i=1 ΨiK
+πK
+= 0,
+which yields
+πa =
+�n
+i=1 Ψia
+�n
+i=1
+�K
+b=1 Ψib
+,
+a ∈ [K].
+(3)
+Holding {B, π} fixed, we then update Ψ based on the equation
+∂ELBO(Q)
+∂Ψia
+= 0,
+i ∈ [n],
+a ∈ [K − 1],
+which gives for each i ∈ [n] and a ∈ [K]:
+Ψia =
+πa exp
+��
+j̸=i
+�K
+b=1 Ψjb [Aij log Bab + (1 − Aij) log(1 − Bab)]
+�
+�
+a′ πa′ exp
+��
+j̸=i
+�K
+b=1 Ψjb [Aij log Ba′b + (1 − Aij) log(1 − Ba′b)]
+�.
+(4)
+4
+
+VARIATIONAL INFERENCE WITH POSTERIOR THRESHOLD
+2.2 Posterior Threshold
+The main difference between the proposed method and BCAVI is that we add a threshold step after
+each update of Ψ, setting the largest value of each row Ψi to one and all other values of the same row
+to zero. That is, for each i ∈ [n], we further update:
+Ψiki = 1,
+Ψia = 0,
+a ̸= ki,
+(5)
+where ki = arg maxa{Ψia} and if there are ties, we arbitrarily choose one of the values of ki. The
+summary of this algorithm, which we will refer to as Threshold BCAVI or T-BCAVI, is provided in
+Algorithm 1.
+Intuitively, thresholding the label posterior forces variational inference to avoid saddle points by
+performing a majority vote at each iteration. However, unlike the naive majority vote that does not
+take into account model parameters or the majority vote with penalization Gao et al. (2017), which
+requires a careful design of the penalty term, T-BCAVI performs this step effortlessly by completely
+relying on its posterior approximation. That is, once a strategy for posterior approximation is chosen,
+T-BCAVI does not require an extra model-specific (and often nontrivial) step to determine efficient
+majority vote updates. This property allows us to readily extend this algorithm to other applications
+of variational inference, such as clustering mixtures of exponential families (see Appendix B).
+2.3 Initialization
+Similar to other variants of BCAVI (Zhang and Zhou, 2020; Sarkar et al., 2021; Yin et al., 2020), our
+algorithm requires an initial value Ψ = Z(0) that is correlated with the true membership matrix Z.
+A natural way to obtain such an initialization is through network data splitting (Chin et al., 2015;
+Li et al., 2020), a popular approach in machine learning and statistics. For this purpose, we fix
+τ ∈ (0, 1/2) and create A(init) ∈ {0, 1}n×n by setting all entries of the adjacency matrix A to zero
+independently with probability 1 − τ. A standard spectral clustering algorithm (Abbe, 2018) is
+applied to A(init) to find the membership matrix Z(0). We then apply our algorithm on A − A(init)
+using the initialization Ψ = Z(0). It is known that under certain conditions on SBM, Z(0) is provably
+correlated with the true membership matrix, even when the average node degree is bounded (Chin
+et al., 2015; Le et al., 2017). Numerical study of this approach is given in Section 4.
+Algorithm 1: Threshold BCAVI
+Input: Adjacency matrix A, initialization Ψ(0) and number of iterations s.
+1 for k ← 1 to s do
+2
+Compute B(k) according to Equation (2) ;
+3
+Compute π(k) according to Equation (3);
+4
+Compute Ψ(k) according to Equation (4);
+5
+Compute Ψ(k) by hard threshold according to Equation (5);
+Output: Estimation of label matrix Ψ(s), estimation of block probability matrix B(s) and
+estimation of parameters π(s).
+3. Theoretical Results
+This section provides the theoretical guarantees for the Threshold BCAVI algorithm described in
+Section 2. Although this algorithm works for any stochastic block model, for the theoretical analysis,
+5
+
+LI AND LE
+we will focus on a simplified setting studied by Sarkar et al. (2021) and Yin et al. (2020). Specifically,
+we consider SBM with K = 2 communities of equal sizes and the block probability matrix given by
+B =
+�p
+q
+q
+p
+�
+.
+We assume that p > q > 0 can vary with n so that p ≍ q ≍ p − q ≍ ρn, where for two sequences
+(an)∞
+n=1 and (bn)∞
+n=1 of positive numbers, we write an ≍ bn if there exists a constant C > 0 such
+that an/C ≤ bn ≤ Can. This model provides a benchmark for studying the behavior of various
+community detection algorithms (Mossel et al., 2012; Amini et al., 2013; Gao et al., 2017; Abbe,
+2018). In the context of VI, it significantly simplifies the update rules of BCAVI and makes the
+theoretical analysis tractable (Sarkar et al., 2021; Yin et al., 2020).
+Let C1, C2 ⊂ [n] be the sets of nodes in the two communities. When K = 2, we can use a vector
+Z ∈ {0, 1}n to encode the community membership; for example, Z = 1C1 indicates that Zi = 1 for
+i ∈ C1 and Zi = 0 otherwise (Z = 1C2 is an equivalent alternative). In particular, the variational
+posterior Ψ = EZ is also an n-dimensional vector. It admits the following simple update rule:
+Ψ(s) = h
+�
+ξ(s)�
+,
+ξ(s) = 4t(s) �
+A − λ(s)(1n1T
+n − In)
+� �
+Ψ(s−1) − 1/2 · 1n
+�
+,
+s ≥ 1,
+where h(x) = 1 if x > 0, h(x) = 0 if x ≤ 0, h
+�
+ξ(s)�
+is the component-wise evaluation, 1n = 1[n]
+is the all-one vector, In ∈ Rn×n is the identity matrix,
+t(s) = 1
+2 log p(s) �
+1 − q(s)�
+q(s) �
+1 − p(s)�,
+λ(s) =
+1
+2t(s) log 1 − q(s)
+1 − p(s) ,
+and p(s), q(s) are the estimates of p, q at the s-th iteration, given as follows:
+p(s) =
+(Ψ(s−1))
+T AΨ(s−1) + (1n − Ψ(s−1))T A(1n − Ψ(s−1))
+(Ψ(s−1))T (1n1Tn − In)Ψ(s−1) + (1n − Ψ(s−1))T (1n1Tn − In)(1n − Ψ(s−1))
+,
+q(s) =
+(Ψ(s−1))T A(1n − Ψ(s−1))
+(Ψ(s−1))T (1n1Tn − In)(1n − Ψ(s−1)).
+These formulas follow from a direct calculation; for details, see for example (Sarkar et al., 2021).
+For the theoretical analysis, we assume that the initialization Z(0) can be obtained from the true
+label vector Z by independently perturbing its entries. This assumption is slightly stronger than
+what we can obtain from the practical initialization involving data splitting and spectral clustering
+described in Section 2.3. It simplifies our analysis and helps us highlight the essential difference
+between T-BCAVI and BCAVI; a similar assumption is also used in Sarkar et al. (2021).
+Assumption 3.1 (Random initialization) Let ε ∈ (0, 1/2) be a fixed error rate and Z = 1C1 be
+the true label vector. Assume that the initialization Z(0) is a vector of independent Bernoulli random
+variables with P(Z(0)
+i
+= Zi) = 1 − ε. Moreover, Z(0) and the adjacency matrix A are independent.
+The following proposition shows that if the initialization is better than random guessing in the
+sense that ε < 1/2, then p(s) and q(s) are sufficiently accurate after only a few iterations.
+6
+
+VARIATIONAL INFERENCE WITH POSTERIOR THRESHOLD
+Proposition 1 (Parameter estimation) Fix ε ∈ (0, 1/2) and consider an initialization for the
+Threshold BCAVI that satisfies Assumption 3.1. In addition, assume that p > q > 0 and p ≍
+q ≍ p − q ≍ ρn. Then there exist constants C, C1, C2, c > 0 only depending on ε such that if
+d = n(p + q)/2 > C then with high probability 1 − n−r for some constant r > 0,
+t(1) ≥ C1,
+���λ(1) − p + q
+2
+��� ≤ C2ρn.
+Moreover, for s ≥ 2,
+��p(s) − p
+�� ≤ p exp(−cd),
+��q(s) − q
+�� ≤ q exp(−cd),
+��t(s) − p
+�� ≤ t exp(−cd),
+��λ(s) − λ
+�� ≤ λ exp(−cd).
+Proposition 1 shows that when d grows with n then the estimation biases of p(s), q(s), t(s), and
+λ(s) vanish exponentially fast. These biases remain relatively small when d is bounded, which proves
+sufficient for showing the accuracy of the Threshold BCAVI in estimating community labels.
+Theorem 2 (Clustering accuracy of Threshold BCAVI) Fix ε ∈ (0, 1/2) and consider an initial-
+ization for the Threshold BCAVI that satisfies Assumption 3.1. In addition, assume that p > q > 0
+and p ≍ q ≍ p − q ≍ ρn. Then there exist constants C, c > 0 only depending on ε such that if
+d = n(p + q)/2 ≥ C then with high probability 1 − n−r for some constant r > 0,
+∥Ψ(s) − 1C1∥1 ≤ n exp(−cd),
+for every s ≥ 1, where ∥.∥1 denotes the ℓ1 norm.
+Since Ψ(s) is a binary vector, the bound in Theorem 2 implies that the fraction of incorrectly
+labeled nodes is exponentially small in the average degree d. This result is comparable with the
+bound obtained by Sarkar et al. (2021) for BCAVI in the regime when d grows at least as log n.
+Besides estimating community labels, Proposition 1 shows that the Threshold BCAVI also
+provides good estimates of the unknown parameters. This property is essential not only for proving
+Theorem 2 but also for statistical inference purposes. In this direction, Bickel et al. (2013) shows
+that if the exact optimizer of the variational approximation can be calculated, then the parameter
+estimates of VI for SBM converge to normal random variables. Our following result shows that the
+same property also holds for the Threshold BCAVI in the regime that d grows at least as log n. It
+will be interesting to see if the condition d ≫ log n can be removed.
+Theorem 3 (Limiting distribution of parameter estimates) Suppose that conditions of Theorem 2
+hold and d ≥ C′
+ε log n for some large constant C′
+ε only depending on ε. Then for every s ≥ 2, as n
+tends to infinity, (p(s), q(s)) converges in distribution to a bi-variate Gaussian vector:
+n
+��p(s)
+q(s)
+�
+−
+�p
+q
+��
+→ N
+��0
+0
+�
+,
+�4p
+0
+0
+4q
+��
+.
+Given the asymptotic distribution of (p(s), q(s)), we can construct joint confidence intervals for
+the unknown parameters p and q.
+7
+
+LI AND LE
+4. Numerical Studies
+This section compares the performance of Theshold BCAVI (T-BCAVI), the classical version without
+the threshold step (BCAVI), majority vote (MV) (which iteratively updates node labels by assigning
+nodes to the communities they have the most connections), and the version of majority vote with
+penalization (P-MV) of Gao et al. (2017). For a thorough numerical analysis, we will consider both
+balanced (equal community sizes) and unbalanced (different community sizes) network models with
+K = 2 or K = 3 communities, although our theoretical results in Section 3 are only proved for
+balanced models with two communities.
+As mentioned in Section 1, the threshold step also improves the performance of variational
+inference in clustering data points drawn from Gaussian mixtures. Numerical results for this setting
+are provided in Appendix B.
+4.1 Simulated Networks
+(a) Two communities with idealized initialization. We first provide numerical support for the
+theoretical results in Section 3 and compare the methods listed above. To this end, we consider SBM
+with n = 600 nodes and K = 2 communities of sizes n1 and n2; both balanced (n1 = n2) and
+unbalanced (n1/n2 = 2/3) settings are included. The true model parameters p and q are chosen so
+that the ratio p/q is fixed to be 10/3 while the expected average degree d = n(p + q)/2 can vary. We
+generate initializations Z(0) for all methods from true label vectors Z according to Assumption 3.1
+with various values of the error rate ε. The accuracy of the estimated label vector Ψ ∈ {0, 1}n is
+measured by the fraction of correctly labeled nodes:
+max {1 − ||Ψ − 1C1||1/n, 1 − ||ψ − 1C2||1/n} .
+Note that according to this measure, choosing node labels independently and uniformly at random
+results in the baseline accuracy of approximately 1/2. For each setting, we report this clustering
+accuracy averaged over 100 replications with one standard deviation band. For reference, the actual
+accuracy of initializations Z(0) (RI), which is similar to 1 − ε, is also included.
+Figure 1 shows that T-BCAVI performs uniformly better than BCAVI, MV, and P-MV, for both
+balanced and unbalanced networks. In particular, the improvement is significant when d is small or ε
+is large. Since real-world networks are often sparse and the accuracy of initialization is usually poor,
+this improvement highlights the practical importance of our threshold strategy.
+In addition, we supply experimental results for the accuracy of parameter estimation. We report
+the relative errors of p(s), q(s) and p(s)/q(s), defined by p(s)−p
+p
+, q(s)−q
+q
+, and p(s)/q(s)−p/q
+p/q
+, respectively.
+Figure 2a shows that, overall, T-BCAVI is much more accurate than BCAVI in parameter estimation
+for sparse networks. Both algorithms overestimate q and underestimate p, although the bias is smaller
+for T-BCAVI. When the network is dense, two algorithms have a negligible relative error. Figure 2b
+shows that when the initialization contains a large amount of wrong labels (ε = 0.4), T-BCAVI still
+provides meaningful estimates of p and q while the estimation errors of BCAVI do not reduce as the
+average degree increases. This is because BCAVI always produces p(s) = q(s) when s is sufficiently
+large. Figure 2c further confirms that T-BCAVI is much more robust with respect to initialization
+than BCAVI is.
+(b) Two communities with initialization by spectral clustering. Since the initialization in Assump-
+tion 3.1 is not available in practice, we now evaluate the performance of all methods using ini-
+8
+
+VARIATIONAL INFERENCE WITH POSTERIOR THRESHOLD
+5
+10
+15
+20
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Average Degree
+Clustering Accuracy
+RI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(a) ε = 0.2, n1 = n2 = 300
+5
+10
+15
+20
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Average Degree
+Clustering Accuracy
+RI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(b) ε = 0.4, n1 = n2 = 300
+0.10
+0.15
+0.20
+0.25
+0.30
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Error Rate
+Clustering Accuracy
+RI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(c) d = 8, n1 = n2 = 300
+5
+10
+15
+20
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Average Degree
+Clustering Accuracy
+RI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(d) ε = 0.2, n1 = 240, n2 = 360
+5
+10
+15
+20
+0.5
+0.6
+0.7
+0.8
+0.9
+Average Degree
+Clustering Accuracy
+RI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(e) ε = 0.4, n1 = 240, n2 = 360
+0.10
+0.15
+0.20
+0.25
+0.30
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Error Rate
+Clustering Accuracy
+RI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(f) d = 8, n1 = 240, n2 = 360
+Figure 1: Performance of Threshold BCAVI (T-BCAVI), the classical BCAVI, majority vote (MV),
+and majority vote with penalization (P-MV) in different settings. Networks are generated from
+SBM with n = 600 nodes, K = 2 communities of sizes n1 and n2, p/q = 10/3, and average
+degree d = ((n2
+1 + n2
+2)p + 2n1n2q)/n. Initializations are generated from true node labels according
+to Assumption 3.1 with error rate ε, resulting in actual clustering initialization accuracy (RI) of
+approximately 1 − ε.
+5
+10
+15
+20
+−1.0
+0.0
+0.5
+1.0
+1.5
+Average Degree
+Relative Error
+p 
+q
+p/q
+p (T)
+q (T)
+p/q (T)
+(a) ε = 0.2
+5
+10
+15
+20
+−1.0
+0.0
+0.5
+1.0
+1.5
+Average Degree
+Relative Error
+p 
+q
+p/q
+p (T)
+q (T)
+p/q (T)
+(b) ε = 0.4
+0.10
+0.15
+0.20
+0.25
+0.30
+−1.0
+0.0
+0.5
+1.0
+1.5
+Error Rate
+Relative Error
+p 
+q
+p/q
+p (T)
+q (T)
+p/q (T)
+(c) d = 8
+Figure 2: Relative errors of parameter estimation by Threshold BCAVI (T) and the classical BCAVI
+in different settings. Networks are generated from SBM with n = 600 nodes, K = 2 communities
+of equal sizes, p/q = 10/3, and average degree d = n(p + q)/2. Initializations are generated from
+true node labels according to Assumption 3.1 with error rate ε.
+tialization generated from network data splitting (Chin et al., 2015; Li et al., 2020). According
+to the discussion in Section 2, we fix a sampling probability τ ∈ (0, 1/2) and sample edges in A
+independently with probability τ; denote by A(init) the adjacency matrix of the resulting sampled
+network. To get a warm initialization Z(0), we apply spectral clustering algorithm (Von Luxburg,
+9
+
+LI AND LE
+2007) on A(init). All methods are then performed on the remaining sub-network A − A(init) using the
+initialization Z(0). For reference, we also report the accuracy of Z(0) (SCI).
+Similar to the previous setting when Z(0) are generated according to Assumption 3.1, Figure 3
+shows that T-BCAVI performs much better than other methods for both balanced and unbalanced
+networks, especially when the spectral clustering initialization is almost uninformative (accuracy
+close to random guess of 1/2 when the network is balanced). This observation again highlights that
+T-BCAVI often requires a much weaker initialization than what BCAVI and variants of majority
+vote need. Network data splitting also provides a practical way to implement our algorithm in the
+real-world data analysis.
+10
+15
+20
+25
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Average Degree
+Clustering Accuracy
+SCI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(a) τ = 0.5, n1 = n2 = 300
+10
+15
+20
+25
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Average Degree
+Clustering Accuracy
+SCI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(b) τ = 0.2, n1 = n2 = 300
+0.1
+0.2
+0.3
+0.4
+0.5
+0.5
+0.6
+0.7
+0.8
+0.9
+Sampling Probability
+Clustering Accuracy
+SCI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(c) d = 12, n1 = n2 = 300
+10
+15
+20
+25
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Average Degree
+Clustering Accuracy
+SCI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(d) τ = 0.5, n1 = 240, n2 = 360
+10
+15
+20
+25
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Average Degree
+Clustering Accuracy
+SCI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(e) τ = 0.2, n1 = 240, n2 = 360
+0.1
+0.2
+0.3
+0.4
+0.5
+0.5
+0.6
+0.7
+0.8
+0.9
+Sampling Probability
+Clustering Accuracy
+SCI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(f) d = 12, n1 = 240, n2 = 360
+Figure 3: Performance of Threshold BCAVI (T-BCAVI), the classical BCAVI, majority vote (MV),
+and majority vote with penalization (P-MV) in different settings. Networks are generated from SBM
+with n = 600 nodes, K = 2 communities of sizes n1 and n2, p/q = 10/3, and average degree
+d = ((n2
+1 + n2
+2)p + 2n1n2q)/n. Initializations are computed by spectral clustering (SCI) applied to
+sampled sub-networks A(init) with sampling probability τ while T-BCAVI and BCAVI are performed
+on remaining sub-networks A − A(init).
+Figure 4a further shows that T-BCAVI outperforms BCAVI in estimating the model parameters
+of sparse networks. When the average degree is large, both algorithms have the similar and negligible
+errors. This is consistent with the results in Figure 2a and 2b when initialization are generated
+according to Assumption 3.1. Figure 4b shows a much better improvement of T-BCAVI over BCAVI;
+this is because τ = 0.2 (compared to τ = 0.5 in Figure 4a), indicating that the initialization is poor
+due to the sparsity of the sub-network A(init) to which spectral clustering is applied. Figure 4c further
+confirms that T-BCAVI is more robust than BCAVI with respect to the sampling probability τ.
+(c) Three communities with idealized and spectral clustering initializations. Although our theoretical
+guarantees are only for SBM with two communities, we emphasize that T-BCAVI can be carried out
+10
+
+VARIATIONAL INFERENCE WITH POSTERIOR THRESHOLD
+10
+15
+20
+25
+−1.0
+0.0
+0.5
+1.0
+1.5
+Average Degree
+Relative Error
+p 
+q
+p/q
+p (T)
+q (T)
+p/q (T)
+(a) τ = 0.5
+10
+15
+20
+25
+−1.0
+0.0
+0.5
+1.0
+1.5
+Average Degree
+Relative Error
+p 
+q
+p/q
+p (T)
+q (T)
+p/q (T)
+(b) τ = 0.2
+0.1
+0.2
+0.3
+0.4
+0.5
+−1.0
+0.0
+0.5
+1.0
+1.5
+Sampling Probability
+Relative Error
+p 
+q
+p/q
+p (T)
+q (T)
+p/q (T)
+(c) d = 12
+Figure 4: Relative errors of parameter estimation by Threshold BCAVI (T) and the classical BCAVI
+in different settings. Networks are generated from SBM with n = 600 nodes, K = 2 communities
+of equal sizes, p/q = 10/3, and average degree d = n(p + q)/2. Initializations are computed by
+spectral clustering (SCI) applied to sampled sub-networks A(init) with sampling probability τ while
+T-BCAVI and BCAVI are performed on remaining sub-networks A − A(init).
+for any SBM, as described in Section 2. To that end, we now consider SBM with n = 600 nodes,
+K = 3 communities of sizes n1, n2, n3, and the block probability matrix given by
+B =
+�
+�
+p
+q
+q
+q
+p
+q
+q
+q
+p
+�
+� ,
+where the ratio p/q is fixed to be 10/3. The measure of accuracy for estimated labels in the setting
+K = 2 can be extended to the case K = 3 in a natural way. Note, however, that choosing node labels
+independently and uniformly at random now results in the baseline accuracy of approximately 1/3 in
+the balanced network. Figures 5a and 5c show the performance of all methods using initializations
+generated according to Assumption 3.1 with error rate ε = 0.4; Figures 5b and 5d show the case when
+these methods use initializations computed by spectral clustering with sampling probability τ = 0.25.
+Similar to the case of the two communities, T-BCAVI does not require accurate initializations and
+completely outperforms other methods for both balanced and unbalanced networks.
+4.2 Real Data Examples
+We first consider the political blogosphere network data set (Adamic and Glance, 2005). It is related
+to the 2004 U.S. presidential election, where the nodes are blogs focused on American politics and
+the edges are hyperlinks connecting the blogs. These blogs have been labeled manually as either
+“liberal” or “conservative” (Adamic and Glance, 2005). We ignore the direction of the hyperlinks
+and conduct our analysis on the largest connected component of the network (Karrer and Newman,
+2011), which contains 1490 nodes, and the average node degree is 22.44. This network includes 732
+conservative blogs and 758 liberal blogs.
+Another political data set, which is also related to the 2004 presidential election, is a network
+of books about US politics published around the 2004 presidential election and sold by the online
+bookseller Amazon.com. Edges between books represent frequent co-purchasing of books by the
+same buyers. The network was compiled by Krebs (2022) and was posted by Newman (2013). The
+books are labeled as ”liberal,” ”neutral,” or ”conservative” based on their descriptions and reviews
+11
+
+LI AND LE
+5
+10
+15
+20
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Average Degree
+Clustering Accuracy
+RI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(a) ε = 0.4, n1 = n2 = n3 = 200
+10
+15
+20
+25
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Average Degree
+Clustering Accuracy
+SCI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(b) τ = 0.25, n1 = n2 = n3 = 200
+5
+10
+15
+20
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Average Degree
+Clustering Accuracy
+RI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(c) ε = 0.4, n1 = 150, n2 = 210,
+n3 = 240
+10
+15
+20
+25
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Average Degree
+Clustering Accuracy
+SCI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(d) τ = 0.25, n1 = 150, n2 = 210,
+n3 = 240
+Figure 5: Performance of Threshold BCAVI (T-BCAVI), the classical BCAVI, majority vote (MV),
+and majority vote with penalization (P-MV) in different settings. Networks are generated from
+SBM with n = 600 nodes, K = 3 communities of sizes n1, n2 and n3, p/q = 10/3, and average
+degree d = ((n2
+1 + n2
+2 + n2
+3)p + 2(n1n2 + n2n3 + n3n1)q)/n. Plots (a) and (c): Initializations
+are generated from true node labels according to Assumption 3.1 with error rate ε = 0.4. Plots (b)
+and (d): Initializations are computed by spectral clustering (SCI) applied to sampled sub-networks
+A(init) with sampling probability τ = 0.25 while T-BCAVI and BCAVI are performed on remaining
+sub-networks A − A(init).
+of the books (Newman, 2013). This network has an average degree of 8.40, and it contains 49
+conservative books, 43 liberal books, and 13 neutral books.
+The last data set we analyze in this section is the network of common adjectives and nouns in the
+novel “David Copperfield” by Charles Dickens, as described by Newman (2006). Nodes represent
+the most commonly occurring adjectives and nouns in the book. Node labels are 0 for adjectives
+and 1 for nouns. Edges connect any pair of words that occur in an adjacent position in the text of the
+book. The network has 58 adjectives and 54 nouns with an average degree of 7.59.
+Similar to Section 4.1, to implement all methods, we first randomly sample a sub-network and
+apply spectral clustering algorithm to get a warm initialization. We then run these algorithms on
+the remaining sub-network for comparison. Figure 6a for political blogosphere shows that they all
+improve the accuracy of the initialization, but T-BCAVI outperforms other methods for almost all
+sampling probability τ. Figure 6b for the book network shows that BCAVI, MV, and P-MV do
+not improve the initialization much while T-BCAVI still improves the accuracy of the initialization
+considerably. Finally, Figure 6c shows that T-BCAVI is still consistently better than other methods,
+although the initializations are close to the random guess, resulting in a small improvement of all
+algorithms.
+12
+
+VARIATIONAL INFERENCE WITH POSTERIOR THRESHOLD
+0.1
+0.2
+0.3
+0.4
+0.5
+0.5
+0.6
+0.7
+0.8
+0.9
+Sampling Probability
+Clustering Accuracy
+SCI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(a) Politics blogosphere (K=2)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Sampling Probability
+Clustering Accuracy
+SCI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(b) Politics book (K=3)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.50
+0.54
+0.58
+0.62
+Sampling Probability
+Clustering Accuracy
+SCI
+T-BCAVI
+BCAVI
+P-MV
+MV
+(c) Adjectives and nouns (K=2)
+Figure 6: Performance of regularized BCAVI (T-BCAVI) and the classical BCAVI in real data
+examples. Initializations are computed by spectral clustering (SCI) applied to sampled sub-networks
+A(init) with sampling probability τ while T-BCAVI and BCAVI are performed on remaining sub-
+networks A − A(init). Plot (a): Two communities of sizes 732 and 758. Plot (b): Three communities
+of sizes 49, 43, and 13. Plot (c): Two communities of sizes 58 and 54.
+5. Discussion
+This paper studies the batch coordinate update variational inference algorithm for community
+detection under the stochastic block model. Existing work in this direction only establishes the
+theoretical support for this iterative algorithm in the relatively dense regime. The contribution of
+this paper is two-fold. First, we extend the validity of the variational approach to the sparse setting
+when node degrees may be bounded. Second, we propose a simple but novel threshold strategy that
+significantly improves the accuracy of the classical variational inference method, especially in the
+sparse regime or when the accuracy of the initialization is poor.
+While we only have theoretical results for the stochastic block model with two communities, we
+believe that similar results also hold for more general settings and leave their analysis for future work.
+In addition, Assumption 3.1 requires that each entry of the initialization is perturbed independently
+with the same error rate. Simulations in Section 4 suggest that our algorithm works for much weaker
+assumptions on the initialization.
+Finally, extending our results beyond stochastic block models and network problems is a promis-
+ing research direction. For example, the numerical results for Gaussian mixtures in Appendix B
+show that the improvement of the threshold step is not network-specific. Obtaining theoretical results
+for this setting is the first step in the direction we plan to pursue.
+13
+
+LI AND LE
+A. Proofs of Results in Section 3
+Let us first recall the updates of Ψ(s) and ξ(s) described in Section 3:
+Ψ(s) = h
+�
+ξ(s)�
+,
+ξ(s) = 4t(s) �
+A − λ(s)(1n1T
+n − In)
+� �
+Ψ(s−1) − 1/2 · 1n
+�
+,
+s ≥ 1.
+(A6)
+To analyze the threshold BCAVI, we consider the population version of ξ(s), following Sarkar et al.
+(2021):
+¯ξ(s) := 4t(s)M(s) �
+Ψ(s−1) − 1/2 · 1n
+�
+,
+(A7)
+where M(s) := P − λ(s)(1n1T
+n − In) and P = E[A|Z] is a 2 × 2 block matrix with constant values
+p or q within each block. A direct calculation shows that eigenvalues of M(s) are
+ν(s)
+1
+= n
+�p + q
+2
+− λ(s)
+�
+−(p−λ(s)),
+ν2 = n(p − q)
+2
+−(p−λ(s)),
+ν(s)
+i
+= −(p−λ(s)),
+i ≥ 3.
+The corresponding eigenvectors for ν(s)
+1
+and ν(s)
+2
+are u1 = 1n and u2 = 1C1 − 1C2, respectively. We
+decompose Ψ(s) according to these eigenvectors as
+Ψ(s) = ζ(s)
+1 u1 + ζ(s)
+2 u2 + v(s),
+(A8)
+where
+ζ(s)
+i
+= ⟨Ψ(s), ui⟩/n,
+i = 1, 2.
+(A9)
+Formula (A8) yields the following coordinate-wise version of (A7):
+¯ξ(s)
+i
+=
+4t(s)n
+��
+ζ(s−1)
+1
+− 1
+2
+� �p + q
+2
+− λ(s)
+�
++ σiζ(s−1)
+2
+p − q
+2
+�
++4t(s) �
+λ(s) − p
+� ��
+ζ(s−1)
+1
+− 1
+2
+�
++ σiζ(s−1)
+2
++ v(s−1)
+i
+�
+=:
+na(s−1)
+σi
++ b(s−1)
+i
+.
+(A10)
+Unlike Sarkar et al. (2021), we need to control the irregularity of sparse networks carefully. To
+that end, denote by A′ the adjacency matrix of the network obtained from the observed network by
+removing some edges (in an arbitrary way) of nodes with degrees greater than C0d so that all node
+degrees of the new network are at most C0d, where C0 > 0 is a fixed constant. Note that A′ is only
+used for the proofs, and our algorithm does not need to specify it.
+In the following lemmas, we assume that the assumptions of Proposition 1 and Theorem 2 are
+satis��ed.
+Lemma 4 (Number of removed edges) Assume that κ ≤ d ≤ η log n for a large constant κ > 0
+and a constant η ∈ (0, 1). Then with high probability, we have
+�
+i,j
+(Aij − A′
+ij) ≤ 2n exp(−c0d),
+where c0 is an absolute constant.
+14
+
+VARIATIONAL INFERENCE WITH POSTERIOR THRESHOLD
+Proof [of Lemma 4] It follows from Proposition 1.12 of Benaych-Georges et al. (2019) that with
+high probability,
+�
+i,j
+(Aij − A′
+ij) ≤ 2
+∞
+�
+k=C0d+1
+kn exp(−f(k)),
+where
+f(x) = x log
+�x
+d
+�
+− (x − d) − log
+√
+2πx.
+Notice that for k > C0d,
+f(k) − log k ≥ k(log C0 − 1) − log
+√
+2πk − log k ≥ k(log C0 − 1)/2.
+This implies the following inequality
+�
+i,j
+(Aij − A′
+ij) ≤ 2n
+∞
+�
+k=C0d+1
+exp
+�
+−log C0 − 1
+2
+k
+�
+≤ 2n exp(−c0d)
+occurs with high probability for some constant c0.
+Lemma 5 (Concentration of regularized adjacency matrices) Let P = E[A|Z]. With high prob-
+ability,
+||A′ − EA|| = O(
+√
+d),
+where ∥.∥ denotes the spectral norm.
+Proof [of Lemma 5] This lemma is a special case of Theorem 1.1 in Le et al. (2017).
+The next lemma provides crude bounds for the parameters of the threshold BCAVI after the first
+iteration.
+Lemma 6 (First step: Parameter estimation) Fix ε ∈ (0, 1/2) and consider an initialization for
+the threshold BCAVI that satisfies Assumption 3.1. Then there exist positive constants C, C1, C2 only
+depending on ε such that if d = n(p + q)/2 > C then with high probability,
+t(1) ≥ C1,
+���λ(1) − p + q
+2
+��� ≤ C2ρn.
+Proof [of Lemma 6] Recall the first update p(1) described in Section 3:
+p(1) =
+(Z(0))T AZ(0) + (1n − Z(0))T A(1n − Z(0))
+(Z(0))T (Jn − In)Z(0) + (1n − Z(0))T (Jn − In)(1n − Z(0)),
+where Jn = 1n1T
+n. We first analyze the denominator of p(1).
+15
+
+LI AND LE
+Since Z(0) satisfies Assumption 3.1, it follows that �
+i Z(0)
+i
+= n/2 + OP (√n) by the standard
+Chernoff bound. Therefore, the denominator of p(1) can be approximated as follows. First,
+Z(0)T (Jn − In)Z(0) =
+�
+i̸=j
+Z(0)
+i
+Z(0)
+j
+=
+� n
+�
+i=1
+Z(0)
+i
+�2
+−
+n
+�
+i=1
+(Z(0)
+i
+)2
+=
+� n
+�
+i=1
+Z(0)
+i
+�2
+−
+n
+�
+i=1
+Z(0)
+i
+= n2/4 + OP (n3/2).
+Similarly,
+(1n − Z(0))T (Jn − In)(1n − Z(0)) = n2/4 + OP (n3/2).
+These estimates give
+(Z(0))T (Jn − In)Z(0) + (1n − Z(0))T (Jn − In)(1n − Z(0)) = n2/2 + Op(n3/2).
+(A11)
+By replacing A with P + (A − P), we decompose the numerator of p(1) as the sum of signal
+and noise terms:
+signal
+:=
+(Z(0))T PZ(0) + (1n − Z(0))T P(1n − Z(0)),
+noise
+:=
+(Z(0))T (A − P)Z(0) + (1n − Z(0))T (A − P)(1n − Z(0)).
+Again, by Assumption 3.1 and the Chernoff bound,
+�
+i∈C1
+Z(0)
+i
+= n(1 − ε)/2 + OP (√n),
+�
+i∈C2
+Z(0)
+i
+= nε/2 + OP (√n).
+Therefore,
+(Z(0))T PZ(0)
+=
+�
+�
+�
+��
+i∈C1
+Z(0)
+i
+�
+�
+2
+−
+�
+i∈C1
+(Z(0)
+i
+)2 +
+�
+��
+i∈C2
+Z(0)
+i
+�
+�
+2
+−
+�
+i∈C2
+(Z(0)
+i
+)2
+�
+� p
++2
+�
+�
+�
+��
+i∈C1
+Z(0)
+i
+�
+�
+�
+��
+i∈C2
+Z(0)
+i
+�
+�
+�
+� q
+=
+�
+n2(1 − ε)2/4 + n2ε2/4
+�
+p +
+�
+n2ε(1 − ε)/2
+�
+q + OP (n3/2ρn).
+Using a similar estimate for (1n − Z(0))T P(1n − Z(0)), we get
+signal =
+�n2[ε2 + (1 − ϵ)2]
+2
+�
+p + [n2ε(1 − ε)]q + OP (n3/2ρn).
+(A12)
+We now analyze the noise term. Since
+E[(Z(0))T (A − P)Z(0)|Z(0)] = 0,
+16
+
+VARIATIONAL INFERENCE WITH POSTERIOR THRESHOLD
+we have
+Var[(Z(0))T (A − P)Z(0)] = E
+�
+Var[(Z(0))T (A − P)Z(0)|Z(0)]
+�
+= 4E
+�
+��
+i<j
+Z(0)
+i
+Z(0)
+j Var[Aij]
+�
+� ≤ 2n2p.
+By Chebyshev’s inequality,
+(Z(0))T (A − P)Z(0) = OP (n√ρn).
+Similarly,
+(1n − Z(0))T (A − P)(1n − Z(0)) = OP (n√ρn).
+These bounds give
+noise = OP (n√ρn).
+(A13)
+From (A11), (A12), and (A13) we get
+p(1) = p − 2ε(1 − ε)(p − q) + OP (ρn/√n).
+(A14)
+A similar analysis gives
+q(1) = q + 2ε(1 − ε)(p − q) + OP (ρn/√n).
+(A15)
+Since
+t(1) = 1
+2 log p(1)(1 − q(1))
+q(1)(1 − p(1)),
+λ(1) =
+1
+2t(1) log 1 − q(1)
+1 − p(1) ,
+it follows from (A14) and (A15) that
+t(1) ≥ C1,
+���λ(1) − p + q
+2
+��� ≤ C2ρn
+with high probability for some constants C1, C2 only depending on ε. The proof is complete.
+The next lemma provides the accuracy of the label estimates after the first iteration.
+Lemma 7 (First step: Label estimation) With high probability,
+||Ψ(1) − 1C1||1 ≤ n exp(−cd),
+where c is a constant only depending on ε.
+Proof [of Lemma 7] According to the population update in (A10) with s = 1, we have
+¯ξ(1)
+i
+=
+4t(1)n
+��
+ζ(0)
+1
+− 1
+2
+� �p + q
+2
+− λ(1)
+�
++ σiζ(0)
+2
+p − q
+2
+�
++4t(1) �
+λ(1) − p
+� ��
+ζ(0)
+1
+− 1
+2
+�
++ σiζ(0)
+2
++ v(0)
+i
+�
+=:
+na(0)
+σi + b(0)
+i ,
+(A16)
+17
+
+LI AND LE
+where σi = 1 if i ∈ C1 and σi = −1 otherwise, and according to (A9),
+ζ(0)
+1
+=
+(Z(0))T 1n/n = 1/2 + Op(1/√n),
+ζ(0)
+2
+=
+(Z(0))T (1C1 − 1C2)/n = (1 �� 2ε)/2 + Op(1/√n).
+(A17)
+It follows from (A16), (A17) and Lemma 6 that na(0)
+σi = Ω(nρn) and b(0)
+i
+= O(ρn) with high
+probability. Moreover, the dominated term of na(0)
+σi is nt(1)σi(1 − 2ε)(p − q).
+Returning to the sample updates in (A6) with s = 1, we have
+ξ(1)
+i
+= na(0)
+σi + b(0)
+i
++ 4t(1)r(0)
+i
+,
+where the error term r(0)
+i
+is
+r(0)
+i
+=
+�
+j̸=i
+(Aij − Pij)(Z(0)
+j
+− 1/2).
+To show the accuracy of Ψ(1)
+i , we will prove that the noise level |4t(1)r(0)
+i
+| is small compared to
+the signal strength nt(1)(1 − 2ε)(p − q) in the population update. To this end, we claim that node i
+is correctly labeled if
+|r(0)
+i
+| < (1 − 2ε)(p − q)
+2(p + q)
+d.
+Let Yij = (Aij − Pij)(Z(0)
+j
+− 1/2), δ be a constant such that 0 < δ < (1−2ε)(p−q)
+2(p+q)
+, and {Y ∗
+ij} be an
+independent copy of {Yij}. For an event E, denote by 1(E) the indicator of E. Then by the triangle
+inequality,
+1
+�
+|r(0)
+i
+| > δd
+�
+= 1
+�
+|
+�
+j̸=i
+Yij| > δd
+�
+≤ 1
+����
+i−1
+�
+j=1
+Yij
+��� +
+���
+n
+�
+j=i+1
+Yij
+��� > δd
+�
+≤ 1
+����
+i−1
+�
+j=1
+Yij
+��� +
+���
+n
+�
+j=i+1
+Yij
+��� +
+���
+i−1
+�
+j=1
+Y ∗
+ij
+��� +
+���
+n
+�
+j=i+1
+Y ∗
+ij
+��� > δd
+�
+≤ 1
+����
+i−1
+�
+j=1
+Yij
+��� +
+���
+n
+�
+j=i+1
+Y ∗
+ij
+��� > δd
+2
+�
++ 1
+����
+i−1
+�
+j=1
+Y ∗
+ij
+��� +
+���
+n
+�
+j=i+1
+Yij
+��� > δd
+2
+�
+.
+Applying Bernstein’s inequality for {Yij} with E[Yij] = 0, |Yij| < 1/2 and E[Y 2
+ij] ≤ p/4, we get
+P
+����
+i−1
+�
+j=1
+Yij
+��� +
+���
+n
+�
+j=i+1
+Y ∗
+ij
+��� > δd
+2
+�
+≤ P
+����
+i−1
+�
+j=1
+Yij
+��� > δd
+4
+�
++ P
+����
+n
+�
+j=i+1
+Y ∗
+ij
+��� > δd
+4
+�
+≤ 4 exp
+�
+−(δd/4)2/2
+d/2 + (δd/4)/6
+�
+= 4 exp
+�
+−δ2d
+16 + 4δ/3
+�
+.
+Since (�i−1
+j=1 Yij, �n
+j=i+1 Y ∗
+ij), 1 ≤ i ≤ n, are independent, by Bernstein’s inequality,
+n
+�
+i=1
+1
+����
+i−1
+�
+j=1
+Yij
+��� +
+���
+n
+�
+j=i+1
+Y ∗
+ij
+��� > δd
+2
+�
+≤ 4(1 + o(1))n exp
+�
+−δ2d
+16 + 4δ/3
+�
+18
+
+VARIATIONAL INFERENCE WITH POSTERIOR THRESHOLD
+with probability at least 1 − n−r for some constant r > 0. Therefore, the following event
+A1 =
+� n
+�
+i=1
+1(|r(0)
+i
+| > δd) ≤ 8(1 + o(1))n exp
+�
+−δ2d
+16 + 4δ/3
+��
+occurs with high probability. Clearly, A1 implies ||Ψ(1) − 1C1||1 ≤ n exp(−cd) for some constant c
+only depending on ε.
+Using the bound for Ψ(1) in Lemma 7, we now provide estimates for parameters of the threshold
+BCAVI in the second iteration.
+Lemma 8 (Second step: Parameter estimation) With high probability,
+|p(2) − p| ≤ p exp(−cd),
+|q(2) − q| ≤ q exp(−cd)
+|t(2) − t| ≤ t exp(−cd),
+|λ(2) − λ| ≤ λ exp(−cd)
+where c > 0 is a constant only depending on ε.
+Proof [of Lemma 8] According to Section 3, the estimate of p in the second step is
+p(2) =
+(Ψ(1))T AΨ(1) + (1n − Ψ(1))T A(1n − Ψ(1))
+(Ψ(1))T (Jn − In)Ψ(1) + (1n − Ψ(1))T (Jn − In)(1n − Ψ(1)).
+(A18)
+Using the decomposition A = A′ + (A − A′), we have
+(Ψ(1))T AΨ(1) = 1T
+C1A1C1 + 2(Ψ(1) − 1C1)T A1C1 + (Ψ(1) − 1C1)T A(Ψ(1) − 1C1)
+= 1T
+C1A1C1 + 2(Ψ(1) − 1C1)T A′1C1 + (Ψ(1) − 1C1)T A′(Ψ(1) − 1C1)
++ 2(Ψ(1) − 1C1)T (A − A′)1C1 + (Ψ(1) − 1C1)T (A − A′)(Ψ(1) − 1C1).
+Note that by definition, all node degrees of the network with adjacency matrix A′ are at most C0d.
+From Lemma 4 and Lemma 7, we get
+|(Ψ(1))T AΨ(1) − 1T
+C1A1C1| ≤ 3||Ψ(1) − 1C1||1 max
+i
+d′
+i + 3
+�
+i,j
+(Aij − A′
+ij)
+≤ 3n exp(−cd)C0d + 6n exp(−c0d).
+Similarly,
+|(1n − Ψ(1))T A(1n − Ψ(1)) − 1T
+C2A1C2| ≤ 3n exp(−cd)C0d + 6n exp(−c0d).
+Therefore, the numerator of p(2) satisfies
+|(Ψ(1))T AΨ(1) + (1n − Ψ(1))T A(1n − Ψ(1)) − 1T
+C1A1C1 − 1T
+C2A1C2|
+≤ 6n exp(−cd)C0d + 12n exp(−c0d).
+(A19)
+Moreover, notice that the denominator of p(2) in (A18) is:
+1T
+n(Jn − In)1n − 2(1n − Ψ(1))T (Jn − In)Ψ(1) = n2 − n − 2||1n − Ψ(1)||1||Ψ(1)||1,
+19
+
+LI AND LE
+and by Lemma 7,
+n2
+4 (1 − 2 exp(−2cd)) ≤ ||1n − Ψ(1)||1||Ψ(1)||1 ≤ (n/2)2,
+which implies
+|1T
+n(Jn − In)1n − 2(1n − Ψ(1))T (Jn − In)Ψ(1) − n2/2| ≤ 2n2 exp(−2cd).
+(A20)
+Therefore, by (A19), (A20), and the concentration property of 1T
+C1A1C1 + 1T
+C2A1C2 we obtain that
+|p(2) − p| ≤ p exp(−cd)
+holds with high probability, where c only depends on ε. The same argument can be used to show that
+|q(2) − q| ≤ q exp(−cd)
+with high probability.
+Since
+t(2) = 1
+2 log p(2)(1 − q(2))
+q(2)(1 − p(2)),
+λ(2) =
+1
+2t(2) log 1 − q(2)
+1 − p(2) ,
+it follows directly that
+|t(2) − t| ≤ t exp(−cd),
+|λ(2) − λ| ≤ λ exp(−cd)
+with high probability.
+Next, we bound the error of the label estimation in the second iteration.
+Lemma 9 (Second step: Label estimation) With high probability,
+||Ψ(2) − 1C1||1 ≤ n exp(−cd),
+where c > 0 is a constant only depending on ε.
+Proof [of Lemma 9] According to (A9) with s = 1, by Lemma 7 we have
+���ζ(1)
+1
+− 1
+2
+��� =
+���(Ψ(1))T 1n/n − 1
+2
+��� ≤ exp(−cd),
+ζ(1)
+2
+= (Ψ(1))T (1C1 − 1C2)/n ≥ (1 − 2 exp(−cd))/2.
+(A21)
+Therefore, according to (A10), (A21) and Lemma 8 the dominant part in na(1)
+σi is at least t(2)nσi(1 −
+4 exp(−cd))(p − q). Define
+r(1)
+i
+=
+�
+j̸=i
+(Aij − A′
+ij)(Ψ(1)
+j
+− Zj) +
+�
+j̸=i
+(A′
+ij − ˜Pij)(Ψ(1)
+j
+− Zj) +
+�
+j̸=i
+(Aij − Pij)(Zj − 1
+2)
+=: r(1)
+1i + r(1)
+2i + r(1)
+3i .
+20
+
+VARIATIONAL INFERENCE WITH POSTERIOR THRESHOLD
+Denote the events A2, A3, A4 and A5 by
+A2 =
+� n
+�
+i=1
+1(|r(1)
+i
+| > (1 − 4 exp(−cd))(p − q)
+2(p + q)
+d) ≤ n exp(−cd)
+�
+,
+A3 =
+� n
+�
+i=1
+1(|r(1)
+1i | > δ1d) ≤ n
+3 exp(−cd)
+�
+,
+A4 =
+� n
+�
+i=1
+1(|r(1)
+2i | > δ2d) ≤ n
+3 exp(−cd)
+�
+,
+A5 =
+� n
+�
+i=1
+1(|r(1)
+3i | > δ3d) ≤ n
+3 exp(−cd)
+�
+,
+where δ1 + δ2 + δ3 = (1−4 exp(−cd))(p−q)
+2(p+q)
+. Since A3 ∩ A4 ∩ A5 implies A2, we have
+P(A2) ≥ P(A3 ∩ A4 ∩ A5) ≥ 1 − P(Ac
+3) − P(Ac
+4) − P(Ac
+5).
+To show A2 occurs with high probability, we only need show A3, A4 and A5 occurs with high
+probability. By using the same argument as in the first iteration in Lemma 7, it can be shown that A5
+occurs with high probability as long as d ≥ C for some constant C only dependent of ε.
+To show A4 occurs with high probability, first notice the following inequality holds with high
+probability by Lemma 5:
+n
+�
+i=1
+|r(1)
+2i |2 ≤
+�
+||A′ − P|| · ||Ψ(1) − Z||2
+�2
+≤ O(d)n exp(−cd).
+This implies
+n
+�
+i=1
+1(|r(1)
+2i | > δ2d) ≤ O(d)n exp(−cd)
+δ2
+2d2
+≤ n
+3 exp(−cd)
+as long as d ≥ C.
+To show A3 occurs with high probability, first notice the following inequality holds with high
+probability by Lemma 4:
+n
+�
+i=1
+|r(1)
+1i | ≤
+�
+ij
+(Aij − A′
+ij) ≤ 2n exp(−c0d).
+This implies
+n
+�
+i=1
+1(|r(1)
+1i | > δ3d) ≤ 2n exp(−c0d)
+δ3d
+≤ n
+3 exp(−cd)
+as long as d ≥ C and c < c0.
+We have shown that A2 occurs with high probability. Since A2 implies ||Ψ(2) − 1C1||1 ≤
+n exp(−cd), the following inequality
+||Ψ(2) − 1C1||1 ≤ n exp(−cd)
+21
+
+LI AND LE
+holds with high probability.
+Using the above lemmas, we are now ready to prove Proposition 1 and Theorem 2 by induction.
+Proof [of Proposition 1 and Theorem 2] When s = 1 and s = 2, the claim follows directly from
+Lemma 7, Lemma 8 and Lemma 9. For s ≥ 3, assume that the claim in Lemma 8 holds for
+p(s−1), q(s−1), t(s−1) and λ(s−1). Assume that the claim in Lemma 9 holds for Ψ(s−1). To repeat
+the arguments in Lemma 8, we can use the same decomposition for (Ψ(s−1))T AΨ(s−1). Since
+||Ψ(s−1) − 1C1||1 ≤ n exp(−cd) by assumption, we have
+|(Ψ(s−1))T AΨ(s−1) − 1T
+C1A1C1| ≤ 3 exp(−cd)C0d + 6n exp(−c0d)
+by Lemma 4. A similar bound holds for (1n − Ψ(s−1))T A(1n − Ψ(s−1)) and the denominator of
+p(s) still has the same bound as in (A20). Therefore, we obtain that
+|p(s) − p| ≤ p exp(−cd),
+where c is only dependent on ε. The remaining arguments for q(s), t(s) and λ(s) are analogous.
+To repeat the arguments in Lemma 9, notice ||Ψ(s−1)−1C1||1 ≤ n exp(−cd) holds by assumption
+and the claim in Lemma 8 holds for p(s), q(s), t(s) and λ(s). Then by using the same arguments as in
+Lemma 9 we obtain that
+||Ψ(s) − 1C1||1 ≤ n exp(−cd),
+where c is only dependent on ε. Therefore, all claims in Proposition 1 and Theorem 2 hold for every
+s ≥ 3 by induction.
+Proof [of Theorem 3] For notation simplicity, we use Ψ to denote Ψ(s−1) in this proof. The update
+equation for p(s) (s > 1) is :
+p(s) =
+ΨT AΨ + (1n − Ψ)T A(1n − Ψ)
+ΨT (Jn − In)Ψ + (1n − Ψ)T (Jn − In)(1n − Ψ).
+Similarly as in equation (A19), the numerator of p(s) satisfies
+|(Ψ)T AΨ + (1n − Ψ)T A(1n − Ψ) − 1T
+C1A1C1 − 1T
+C2A1C2|
+≤ Θ(nd) exp(−cd)
+with high probability. And similarly as in equation (A20), the denominator satisfies
+|1T
+n(Jn − In)1n − 2(1n − Ψ)T (Jn − In)Ψ − n2/2| ≤ 2n2 exp(−2cd).
+with high probability. By Berry-Esseen theorem the asymptotic normality holds for 1T
+C1A1C1 +
+1T
+C2A1C2 since n2ρn → ∞. That is,
+n
+√4p
+�
+1T
+C1A1C1 + 1T
+C2A1C2
+n2/2
+− p
+�
+→ N(0, 1).
+22
+
+VARIATIONAL INFERENCE WITH POSTERIOR THRESHOLD
+To obtain the asymptotic normality of p(s), by Slutsky’s theorem we need nd exp(−cd) = o(
+√
+nd),
+which holds when d ≥ C′
+ε log n for some large constant C′
+ε only depending on ε. The analysis for
+q(s) is analogical and by the multidimensional version of Slutsky’s theorem, we showed that p(s) and
+q(s) are jointly asymptotically normally distributed:
+n
+��p(s)
+p(s)
+�
+−
+�p
+q
+��
+→ N
+��0
+0
+�
+,
+�4p
+0
+0
+4q
+��
+.
+The proof is complete.
+B. Mixture of Gaussians
+In this section, we provide empirical evidence that the proposed threshold strategy, studied in detail
+for networks in the main text, also improves the accuracy of clustering data points generated from
+a mixture of Gaussians (Blei et al., 2017). This observation suggests that the threshold step is not
+network-specific and may be applicable to other problems.
+Consider a set of n data points x1, ..., xn ∈ Rr drawn from a relatively balanced Gaussian
+mixture of K clusters as follows. First, the cluster means µ1, ..., µK ∈ Rr are independently drawn
+from N(0, σ2
+r Ir), and the labels z1, ..., zn are drawn independently and uniformly from {1, ..., K};
+conditioned on µ1, ..., µK and z1, ..., zn, xi are then independently generated from the corresponding
+Gaussian clusters N(µzi, 1
+rIr). Similar to the network setting, the variational inference approach can
+be used to alternatively estimate the label posteriors and the model parameters. Thresholding the
+label posteriors is performed after each round of label posterior updates.
+Figure 7 compares the threshold CAVI (T-CAVI) with the classical CAVI for various values of
+σ2 in three different settings when the accuracy of random initialization is relatively good (ε = 0.3),
+moderate (ε = 0.5) and bad (ε = 0.8). For reference, we also include the actual accuracy of random
+initialization (RI), which is approximately 1 − ε. It can be seen from Figure 7 that T-CAVI performs
+consistently better than CAVI. The largest improvement is achieved when the variance of cluster
+means µ1, ..., µK is moderate. This is because small variances render the clustering problem hard
+and large variances make it simple.
+A more complete picture of the dependence of T-CAVI and CAVI on the error rate ε is given by
+Figure 8, which also shows the dominance of T-CAVI over CAVI and their performance drop as ε
+increases.
+Figure 9 compares the two methods under various values of data dimension r. Overall, T-BCAVI
+performs better than BCAVI, and both methods suffer when r is large. This is likely due to the fact
+that they need to estimate a large number of parameters in that setting.
+Figure 10 shows the behaviors of the two methods when the sample size n varies. Again, T-
+BCAVI performs uniformly better than BCAVI, and not surprisingly, both methods become more
+accurate as the sample size increases.
+References
+Emmanuel Abbe. Community detection and stochastic block models: Recent developments. Journal
+of Machine Learning Research, 18(177):1–86, 2018.
+23
+
+LI AND LE
+0
+1
+2
+3
+4
+0.3
+0.5
+0.7
+0.9
+Variance of Cluster Means
+Clustering Accuracy
+RI
+CAVI
+T-CAVI
+(a) ε = 0.3
+0
+1
+2
+3
+4
+0.3
+0.5
+0.7
+0.9
+Variance of Cluster Means
+Clustering Accuracy
+RI
+CAVI
+T-CAVI
+(b) ε = 0.5
+0
+1
+2
+3
+4
+0.3
+0.5
+0.7
+0.9
+Variance of Cluster Means
+Clustering Accuracy
+RI
+CAVI
+T-CAVI
+(c) ε = 0.8
+Figure 7: Performance of threshold CAVI (T-CAVI) and the classical CAVI in the Bayesian mixture
+of Gaussians with n = 50, K = 5 and p = 6. Initializations (RI) are randomly generated from true
+labels with error rate ε.
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.3
+0.5
+0.7
+0.9
+Error Rate
+Clustering Accuracy
+RI
+CAVI
+T-CAVI
+(a) n = 50
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.3
+0.5
+0.7
+0.9
+Error Rate
+Clustering Accuracy
+RI
+CAVI
+T-CAVI
+(b) n = 150
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.3
+0.5
+0.7
+0.9
+Error Rate
+Clustering Accuracy
+RI
+CAVI
+T-CAVI
+(c) n = 300
+Figure 8: Performance of threshold CAVI (T-CAVI) and the classical CAVI in the Bayesian mixture
+of Gaussians with σ2 = 1.5, K = 5 and p = 10. Initializations (RI) are randomly generated from
+true labels with error rate ε.
+Lada A Adamic and Natalie Glance. The political blogosphere and the 2004 us election: divided they
+blog. In Proceedings of the 3rd international workshop on Link discovery, pages 36–43, 2005.
+Edoardo Maria Airoldi, David M Blei, Stephen E Fienberg, and Eric P Xing. Mixed membership
+stochastic blockmodels. Journal of machine learning research, 2008.
+Arash A Amini, Aiyou Chen, Peter J Bickel, and Elizaveta Levina. Pseudo-likelihood methods for
+community detection in large sparse networks. The Annals of Statistics, 41(4):2097–2122, 2013.
+Matthew James Beal. Variational algorithms for approximate Bayesian inference. University of
+London, University College London (United Kingdom), 2003.
+Florent Benaych-Georges, Charles Bordenave, and Antti Knowles. Largest eigenvalues of sparse
+inhomogeneous erd˝os–r´enyi graphs. The Annals of Probability, 47(3):1653–1676, 2019.
+Peter Bickel, David Choi, Xiangyu Chang, and Hai Zhang. Asymptotic normality of maximum
+likelihood and its variational approximation for stochastic blockmodels. The Annals of Statistics,
+41(4):1922–1943, 2013.
+David M Blei, Andrew Y Ng, and Michael I Jordan. Latent dirichlet allocation. the Journal of
+machine Learning research, 3:993–1022, 2003.
+24
+
+VARIATIONAL INFERENCE WITH POSTERIOR THRESHOLD
+0
+5
+10
+15
+20
+25
+30
+0.2
+0.4
+0.6
+0.8
+1.0
+Dimension
+Clustering Accuracy
+RI
+CAVI
+T-CAVI
+(a) ε = 0.3
+0
+5
+10
+15
+20
+25
+30
+0.2
+0.4
+0.6
+0.8
+Dimension
+Clustering Accuracy
+RI
+CAVI
+T-CAVI
+(b) ε = 0.5
+0
+5
+10
+15
+20
+25
+30
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Dimension
+Clustering Accuracy
+RI
+CAVI
+T-CAVI
+(c) ε = 0.8
+Figure 9: Performance of threshold CAVI (T-CAVI) and the classical CAVI in the Bayesian mixture
+of Gaussians with n = 50, K = 5 and σ2 = 1.5. Initializations (RI) are randomly generated from
+true labels with error rate ε.
+50
+100
+150
+200
+0.3
+0.5
+0.7
+0.9
+Sample Size
+Clustering Accuracy
+RI
+CAVI
+T-CAVI
+(a) ε = 0.3
+50
+100
+150
+200
+0.3
+0.5
+0.7
+0.9
+Sample Size
+Clustering Accuracy
+RI
+CAVI
+T-CAVI
+(b) ε = 0.5
+50
+100
+150
+200
+0.3
+0.5
+0.7
+0.9
+Sample Size
+Clustering Accuracy
+RI
+CAVI
+T-CAVI
+(c) ε = 0.8
+Figure 10: Performance of threshold CAVI (T-CAVI) and the classical CAVI in the Bayesian mixture
+of Gaussians with σ2 = 1.5, K = 5 and p = 10. Initializations (RI) are randomly generated from
+true labels with error rate ε.
+David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians.
+Journal of the American statistical Association, 112(518):859–877, 2017.
+Alain Celisse, Jean-Jacques Daudin, and Laurent Pierre. Consistency of maximum-likelihood and
+variational estimators in the stochastic block model. Electronic Journal of Statistics, 6:1847–1899,
+2012.
+Peter Chin, Anup Rao, and Van Vu. Stochastic block model and community detection in sparse
+graphs: A spectral algorithm with optimal rate of recovery. In Conference on Learning Theory,
+pages 391–423. PMLR, 2015.
+Chao Gao, Zongming Ma, Anderson Y Zhang, and Harrison H Zhou. Achieving optimal misclassifi-
+cation proportion in stochastic block models. The Journal of Machine Learning Research, 18(1):
+1980–2024, 2017.
+Alan E Gelfand and Adrian FM Smith. Sampling-based approaches to calculating marginal densities.
+Journal of the American statistical association, 85(410):398–409, 1990.
+Agnieszka Grabska-Barwi´nska, Simon Barthelm´e, Jeff Beck, Zachary F Mainen, Alexandre Pouget,
+and Peter E Latham. A probabilistic approach to demixing odors. Nature neuroscience, 20(1):
+98–106, 2017.
+25
+
+LI AND LE
+Paul W Holland, Kathryn Blackmond Laskey, and Samuel Leinhardt. Stochastic blockmodels: First
+steps. Social networks, 5(2):109–137, 1983.
+MI Jordan, Zoubin Ghahramani, TS Jaakkola, and Lawrence K Saul. Anintroduction to variational
+methods for graphical models. Learning in graphical models, pages 105–161, 1999.
+Brian Karrer and Mark EJ Newman. Stochastic blockmodels and community structure in networks.
+Physical review E, 83(1):016107, 2011.
+Valdis Krebs. Orgnet. http://http://www.orgnet.com, 2022. [Online; accessed 18-May-
+2022].
+Can M Le, Elizaveta Levina, and Roman Vershynin. Concentration and regularization of random
+graphs. Random Structures & Algorithms, 51(3):538–561, 2017.
+Tianxi Li, Elizaveta Levina, and Ji Zhu. Network cross-validation by edge sampling. Biometrika,
+107(2):257–276, 2020.
+Elchanan Mossel, Joe Neeman, and Allan Sly. Stochastic block models and reconstruction. arXiv
+preprint arXiv:1202.1499, 2012.
+Mark Newman. Network data. http://http://www-personal.umich.edu/%7Emejn/
+netdata//, 2013. [Online; accessed 18-May-2022].
+Mark EJ Newman. Finding community structure in networks using the eigenvectors of matrices.
+Physical review E, 74(3):036104, 2006.
+Purnamrita Sarkar, YX Rachel Wang, and Soumendu Sundar Mukherjee. When random initializations
+help: a study of variational inference for community detection. Journal of Machine Learning
+Research, 22:22–1, 2021.
+Ulrike Von Luxburg. A tutorial on spectral clustering. Statistics and computing, 17(4):395–416,
+2007.
+Chong Wang and David M Blei. Variational inference in nonconjugate models. Journal of Machine
+Learning Research, 14(4), 2013.
+Yixin Wang and David M Blei. Frequentist consistency of variational bayes. Journal of the American
+Statistical Association, 114(527):1147–1161, 2019.
+Mingzhang Yin, YX Rachel Wang, and Purnamrita Sarkar. A theoretical case study of structured vari-
+ational inference for community detection. In International Conference on Artificial Intelligence
+and Statistics, pages 3750–3761. PMLR, 2020.
+Anderson Y Zhang and Harrison H Zhou. Theoretical and computational guarantees of mean field
+variational inference for community detection. The Annals of Statistics, 48(5):2575–2598, 2020.
+Fengshuo Zhang and Chao Gao. Convergence rates of variational posterior distributions. The Annals
+of Statistics, 48(4):2180–2207, 2020.
+26
+

ktFLT4oBgHgl3EQfdC-Z/content/tmp_files/2301.12085v1.pdf.txt

@@ -0,0 +1,1230 @@
+This paper appears in the Proceedings of IEEE International Conference on Communications (ICC) 2023.
+Please feel free to contact us for questions or remarks.
+Resource Allocation of Federated Learning Assisted
+Mobile Augmented Reality System in the Metaverse
+Xinyu Zhou, Yang Li, Jun Zhao
+Nanyang Technological University
+Abstract—Metaverse has become a buzzword recently. Mobile
+augmented reality (MAR) is a promising approach to providing
+users with an immersive experience in the Metaverse. However,
+due to limitations of bandwidth, latency and computational re-
+sources, MAR cannot be applied on a large scale in the Metaverse
+yet. Moreover, federated learning, with its privacy-preserving
+characteristics, has emerged as a prospective distributed learning
+framework in the future Metaverse world. In this paper, we
+propose a federated learning assisted MAR system via non-
+orthogonal multiple access for the Metaverse. Additionally, to
+optimize a weighted sum of energy, latency and model accuracy,
+a resource allocation algorithm is devised by setting appropriate
+transmission power, CPU frequency and video frame resolution
+for each user. Experimental results demonstrate that our pro-
+posed algorithm achieves an overall good performance compared
+to a random algorithm and greedy algorithm.
+Index Terms—Resource allocation, federated learning, aug-
+mented reality, Metaverse, NOMA.
+I. INTRODUCTION
+Metaverse has become a buzzword in recent years. It seeks
+to create a society integrated virtual/augmented reality and
+allows millions of people to communicate online with a virtual
+avatar. Augmented Reality (AR) technology, which has been
+expected to one of the most significant component of the Meta-
+verse, is an enhanced version of the real physical world that
+is achieved through the use of digital visual elements, sound,
+or other sensory stimuli and delivered via technology. Mobile
+Augmented Reality (MAR) is to implement AR technology on
+mobile devices and allow users to experience services through
+AR devices (e.g. smart glasses, headsets, controllers, etc.).
+Motivation. According to Moore’s law, the storage capacity
+and computing power of mobile devices will be further im-
+proved in the future, making it possible to implement machine
+learning models on mobile devices [1]. However, limited by
+the small amount of personal data, it is difficult to train a high-
+performing MAR model on a single device. Federated learning
+(FL) [2], presented in 2017, allows models from diverse
+participants to train a global model with better performance
+while protecting user privacy. With FL, each device only needs
+to do local training with its own data and upload the model
+parameters to the server without sharing any information
+with other devices. The server will aggregate the parameters
+from different participants, form a better-performing global
+model and send it back for further training. Therefore, it is
+worth investigating how to use FL to enhance the MAR-based
+Metaverse experience.
+Base Station
+User 1
+User 2
+User N
+Object Detection
+Rider
+Pedestrain
+Tree
+···
+w1,w2
+wn-1,wn
+w
+Scenario
+w
+Frequency
+Power
+NOMA
+User 1
+User 2
+User N-1
+SIC at the base station 
+obeying NOMA protocol
+Fig. 1. Mobile augmented reality (MAR) with NOMA for the Metaverse.
+Besides, non-orthogonal multiple access (NOMA) was pro-
+posed as an effective solution in 5G system [3]–[5]. In contrast
+to the conventional orthogonal multiple access (OMA), it
+introduces an extra power domain. It allows different users
+to be multiplexed on the same channel to maximize the
+throughput [6]. Specifically, it uses superposition coding at
+the transmitter and applies successive interference cancellation
+(SIC) at the receivers to differentiate signals from multiple
+users in the power domain. Hence, with the help of NOMA,
+we devise an FL-assisted MAR system in the Metaverse.
+Challenges. It still faces several challenges to deploy FL to
+MAR applications in the Metaverse: (1) Each device is often
+given limited bandwidth, so the upper bound of transmission
+rate is affected according to Shannon’s formula, which causes
+long delay during the communication period, further influences
+the convergence of global model. (2) The massive energy
+consumption caused by the training of local model is a
+challenge for the power supply of mobile devices. (3) Video
+frame resolution is also a critical factor in the accuracy of the
+MAR model. Training with higher resolution could improve
+the performance but also means higher computational resource
+consumption for participants. Therefore, how to adjust video
+frame resolutions to balance model performance and total
+consumption is also a problem to ponder.
+Related work. To improve the quality of experience in
+MAR applications, some studies tackled the resource alloca-
+tion problem to utilize limited resources and achieved good
+fairness performance [7]–[12]. [11] designed an MAR-based
+model for the Metaverse and proposed a transmission resource
+allocation algorithm to allocate transmission power and reso-
+lution for each device to achieve the best utility. NOMA had
+been deployed massively in multi-user mobile edge computing
+(MEC) networks to improve the communication resource, and
+quality of service [13]–[15]. For instance, [13] considered
+1
+arXiv:2301.12085v1  [cs.SI]  28 Jan 2023
+
+This paper appears in the Proceedings of IEEE International Conference on Communications (ICC) 2023.
+Please feel free to contact us for questions or remarks.
+dynamic user pairing NOMA-based offloading and established
+an energy consumption minimization framework by joint opti-
+mizing pairing. There is few work incorporating FL into MAR,
+and we only found one paper [16]. An FL-based mobile edge
+computing paradigm was proposed in [16] to solve object
+recognition and classification problem without considering
+optimizing the resource allocation in the framework.
+Novelty. The previously mentioned studies [13]–[15] were
+only about NOMA and MEC systems without integrating FL
+and MAR. [16] was about FL and AR without considering
+resource allocation. Additionally, although [7]–[12] were re-
+lated to resource allocation and MAR, they did not apply FL
+to their systems. In this paper, we consider a basic FL-assisted
+MAR system via NOMA, and design an algorithm to jointly
+optimize time, energy and model accuracy simultaneously.
+Besides, we take into account the different requirements in
+diverse situations. For example, we prefer to optimize energy
+consumption as much as possible when the mobile device is
+low on power. So, the objective function includes a weighted
+combination of total energy consumption, completion time
+and model accuracy and allows the weights to be adjusted
+to achieve different optimization results.
+Contributions. The contributions of our paper are as fol-
+lows:
+• To our best knowledge, we are the first to introduce FL
+assisted NOMA system in MAR to enhance the model
+performance, which could improve the experience of
+users in the Metaverse.
+• An optimization algorithm is proposed to optimize time,
+energy, and accuracy jointly. The weights of them could
+be adjusted freely according to practical situations.
+• Detailed comparative experiments, convergence analysis
+and time complexity, are provided to show the robustness
+and effectiveness of our method.
+II. SYSTEM MODEL
+We consider an MAR network with N cellular-connected
+MAR devices. All MAR devices transmit data to a BS through
+the uplink NOMA transmission protocol. Each device trains its
+own object detection model locally and collaboratively trains
+a global model through FL, as shown in Fig. 1. In this paper,
+bold symbols represent vectors. If a symbol x is a solution to
+a problem, then x∗ means the optimal solution.
+Uplink NOMA. In an uplink-NOMA system, through chan-
+nel and power assignment, signals of devices are multiplexed
+on each channel. Assume there are N users and K channels.
+The higher the number of devices multiplexed on each chan-
+nel, the higher the hardware complexity and latency. Thus,
+following [17], [18] and considering practicality, we suppose
+there are 2 users multiplexed on the k-th subchannel.
+Suppose a user n belongs to the subchannel k, and it is the
+i-th (i = 1, 2) user in the subchannel k. Define gk,i as the
+channel gain between the base station and the i-th device on
+channel k. Without loss of generality, assume gk,i−1 < gk,i on
+channel k and the information of the user with better channel
+gain will be decoded first by the base station.
+User pairing. Similar to [14], we consider using the fol-
+lowing user pairing schemes: (a) random selection: randomly
+selecting two users to form a group, (b) nearest-user-pairing:
+pairing two nearest users, then the next nearest users, and
+so forth, (c) nearest-farthest-pairing: the nearest user and the
+farthest user are formed into a group, then pairing the second
+nearest and the second farthest user, and so forth. In the
+resource allocation algorithm (Algorithm 2) stated in section
+III-E, three user pairing schemes are used, and the best one
+will be selected as the final result.
+Federated learning. Assume there are Dn (i.e., Dk,i)
+samples on each device n (i.e., the i-th device on chan-
+nel k). As shown in Fig. 1, each mobile device runs its
+own model locally and collaboratively solves the problem
+minω F(ω) = �N
+n=1
+Dn
+�N
+n=1 Dn ln(ω), where ln(·) denotes the
+loss function per sample and ω is the global model. After
+each device finishes its local training for a certain number
+of iterations, it will upload the model parameter to the base
+station. Next, the base station will calculate the weighted
+average model parameter
+Dnωn
+�N
+n=1 Dn and send it back to each
+device. Such uploading and broadcasting process is called one
+global communication round.
+A. Energy and Time Consumption
+We only study the process between two global communica-
+tion rounds. We define the total energy consumption as E, and
+it includes wireless transmission energy and local computation
+energy. The total time consumption is defined as T . It consists
+of transmission time and local computation time. Following
+[19], [20], the energy consumed at the base station is not
+considered.
+Transmission energy. In light of the fact that the base
+station’s output power is significantly greater than the uplink
+transmission power of a mobile device, the downlink time
+is ignored in this work. Hence, according to the Shannon
+formula, the data transmission rate of the i-th user on channel
+k is
+rk,i =Bklog2(1+
+pk,igk,i
+BkNk + �i−1
+j=0pk,jgk,j
+),
+(1)
+and we define pk,0 = 0. We denote the total bandwidth is B,
+and Bk =
+B
+K . pk,i refers to the transmission power. gk,i is
+the channel between the base station and the i-th device on
+channel k. Nk is the noise power spectral density of Gaussian
+noise. Suppose the transmission data size of each device is
+dk,i, so the transmission time of the i-th user on channel k is
+T trans
+k,i
+= dk,i/rk,i.
+(2)
+Therefore, the corresponding transmission energy is
+Etrans
+k,i
+= pk,iT trans
+k,i
+.
+(3)
+Local computation energy. We incorporate You Only Look
+Once (YOLO) algorithm [21] to handle the object detection
+tasks on each device. Note that the main structure of YOLO
+is convolutional neural network (CNN). Assume on the i-
+th device on channel k, the video frame resolution used for
+training is sk,i × sk,i pixels. Thus, due to the influence of
+the frame resolution used for training on computing resources
+2
+
+This paper appears in the Proceedings of IEEE International Conference on Communications (ICC) 2023.
+Please feel free to contact us for questions or remarks.
+[22] and motivated by [23], the local computation energy is
+defined as
+Ecmp
+k,i
+= κηξs2
+k,ick,iDk,if 2
+k,i,
+(4)
+where κ represents the effective switched capacitance, η is
+the number of local iterations, s2
+k,i is the pixels of the frame
+resolution, Dk,i is the number of samples, fk,i is the CPU
+frequency and ck,i is the number of CPU cycles per standard
+sample. We define the standard sample as a video frame
+with a resolution of s0 × s0 pixels and ξ =
+1
+s2
+0 . This means
+if a video frame with s2
+k,i pixels and sk,i = s0, the local
+computation energy will be κηck,iDk,if 2
+k,i, which is the same
+as the definition in [23].
+Hence, the total energy consumption E is
+E =
+K
+�
+k=1
+2
+�
+i=1
+(Etrans
+k,i
++ Ecmp
+k,i ).
+(5)
+Transmission time. This is given in Eq. (2).
+Computation time. The local computation time of the i-th
+device on channel k in one global iteration is
+T cmp
+k,i
+= η
+ξs2
+k,ick,iDk,i
+fk,i
+.
+(6)
+Thus, the total completion time is
+T =max{T trans
+k,i
++T cmp
+k,i }, i = 1, 2, k ∈ [1, K].
+(7)
+B. Accuracy Analysis
+Denote A be the training accuracy of the whole federated
+learning process and define it as a function of sk,i, which is
+A(s1,1, s1,2, · · · , sK,1, sK,2) = �K
+k=1
+�2
+i=1 Ak,i.
+The accuracy model from [24] is used in this work. Based
+on YOLO algorithm, [24] constructs the accuracy function
+regarding different video frame resolutions. Thus, the accuracy
+function of the i-th user on channel k is defined as
+Ak,i = 1 − 1.578e−6.5×10−3sk,i.
+(8)
+III. JOINT OPTIMIZATION OF ENERGY, TIME AND
+ACCURACY WITH FIXED USERS
+In this section, problem formulation, problem decompo-
+sition and solutions to the optimization problem will be
+illustrated, respectively.
+A. Problem Formulation
+A joint optimization of energy E, time T and accuracy
+A problem is formulated in this section. The optimization
+problem is as follows:
+min
+sk,i, fk,i, pk,iαE + βT − γA,
+(9)
+subject to, pmin ≤ pk,i ≤ pmax, k ∈ [1, K], i = 1, 2, (9a)
+f min ≤ fk,i ≤ f max, k ∈ [1, K], i = 1, 2,
+(9b)
+sk,i ∈ {s1, s2, s3},
+(9c)
+where sk,i, fk,i and pk,i are three optimization variables.
+α, β and γ are three weight parameters, and α + β = 1,
+α, β ∈ [0, 1], and γ ≥ 0. Constraints (9a) and (9b) limit the
+ranges of the transmission power and CPU frequency of each
+device. Constraint (9c) sets three choices for the video frame
+resolution and s1 < s2 < s3.
+Due to the max function of T , problem (9) is non-convex
+and difficult to be decomposed. To avoid this difficulty, an
+auxiliary variable T is introduced, and the problem becomes:
+min
+sk,i, fk,i, pk,iα(
+K
+�
+k=1
+2
+�
+i=1
+Ecmp
+k,i
++ Etrans
+k,i
+) + βT − γA,
+(9)
+subject to, (9a), (9b), (9c)
+T trans
+k,i
++T cmp
+k,i
+≤T, i ∈ {1, 2}, k = 1, · · ·,K, (10a)
+where constraint (10a) is considered an upper bound of the
+total time consumption.
+B. Problem Decomposition
+Due
+to
+the
+original
+optimization
+problem
+(9)
+being
+non-convex and quite complex, we split it into two subprob-
+lems (SP1 and SP2) to make it easier to solve. Since pk,i only
+appears in Etrans
+k,i
+, two subproblems are constructed, one with
+the optimization variable fk,i and sk,i, and the other with pk,i.
+Subproblems 1 and 2 write as follows:
+SP1:
+min
+sk,i,fk,iα(
+K
+�
+k=1
+2
+�
+i=1
+Ecmp
+k,i )+βT −γA,
+(11)
+subject to, (9b), (9c), (10a).
+SP2:
+min
+pk,i α
+K
+�
+k=1
+2
+�
+i=1
+pk,idk,i
+rk,i
+,
+(12)
+subject to, (9a), (10a).
+C. Solution to SP1
+Since the video frame resolution sk,i is discrete, we relax it
+into a continuous variable ˆsk,i to make SP1 easier to tackle.
+Next, we handle Ak,i(ˆsk,i). Because our research mainly fo-
+cuses on the application of federated learning in NOMA, so we
+introduce a simple but effective linear approach to construct
+the function Ak,i (8). By using two points (s1, Ak,i(s1))
+and (s3, Ak,i(s3)),we approximate Ak,i( ˆ
+sk,i) as the following
+linear function Ak,i(ˆsk,i):
+ˆAk,i(ˆsk,i) = ˆkk,i(ˆsk,i − s1) + Ak,i(s1),
+(13)
+where ˆkk,i = Ak,i(s3)−Ak,i(s1)
+s3−s1
+.
+Then, the SP1 becomes:
+min
+sk,i,fk,iα
+K
+�
+k=1
+2
+�
+i=1
+Ecmp
+k,i +βT −γ
+K
+�
+k=1
+2
+�
+i=1
+ˆAi,k(ˆsk,i)
+(14)
+subject to, (9b), (10a),
+s1 ≤ ˆsk,i ≤ ˆs3, ∀k ∈ [1, K], i = 1, 2.
+(14a)
+It is easy to verify that the function of SP1 is convex, and
+the constraints are also. Karush-Kuhn-Tucker (KKT) approach
+works well to get the optimal solutions for this optimization
+problem.
+After applying KKT conditions, we get
+f ∗
+k,i =
+3�
+λk,i
+2ακ, ˆs∗
+k,i =
+γˆkk,i
+2ηξck,iDk,i(ακf 2
+k,i+
+λk,i
+fk,i ),
+(15)
+β = �K
+k=1
+�2
+i=1 λk,i,
+(16)
+where λk,i is the Lagrange multiplier associated with the
+inequality constraint (10a).
+Through Eq. (15), we can use λk,i to represent fk,i and ˆs2
+k,i.
+3
+
+This paper appears in the Proceedings of IEEE International Conference on Communications (ICC) 2023.
+Please feel free to contact us for questions or remarks.
+Then, we can get the dual problem as follows:
+max
+λk,i
+K
+�
+k=1
+2
+�
+i=1
+−
+γ2ˆk2
+k,i
+4h(2− 2
+3 + 2
+1
+3 )
+λ
+− 2
+3
+k,i + T up
+k,iλk,i + γˆkk,is1
+− γAk,i(s1)
+(17)
+subject to, (16), λk,i ≥ 0,
+where h = ηξck,iDk,i(ακ)
+1
+3 . Obviously, this dual problem is
+a simple convex optimization problem. In this paper, we use
+CVX [25] to solve it and get the optimal λ∗ = [λ∗
+1,1, ..., λ∗
+k,2].
+Then we can leverage the λ∗ to calculate the optimal f ∗ and ˆs∗
+through Eq. (15). With the constraint of f min ≤ fk,i ≤ f max,
+we can get f ∗
+k,i in the below:
+f ∗
+k,i = min(f max, max(f ∗
+k,i, f min)).
+(18)
+Since the frame resolution sk,i is discrete, we adopt the
+following formula to map ˆsk,i to sk,i:
+s∗
+k,i =
+�
+�
+�
+�
+�
+s3, if
+ˆ
+sk,i > s2+s3
+2
+s2, if s1+s2
+2
+≤ ˆ
+sk,i ≤ s2+s3
+2
+s1, if
+ˆ
+sk,i < s1+s2
+2
+(19)
+SP1 is solved. Next, SP2 will be explained.
+D. Solution to SP2
+The objective function (12) of SP2 is non-convex, since
+it is easy to verify that its Hessian matrix is not positive
+semidefinite. On channel k, the forms of rk,1 and rk,2 are
+different. Hence, to continuously simplify SP2, we decompose
+it into another two subproblems—SP2 1 with optimization
+variable pk,1 and SP2 2 with pk,2.
+SP2 1 : min
+pk,1 α(
+K
+�
+k=1
+pk,1dk,1
+Bk log2(1 + pk,1gk,1
+BkNk )),
+(20)
+subject to, (9a), rk,1 ≥ rmin
+k,1 ,
+(20a)
+SP2 2 : min
+pk,2 α(
+K
+�
+k=1
+pk,2dk,2
+Bk log2(1 +
+pk,2gk,2
+BkNk+pk,1gk,1 )),
+(21)
+subject to, (9a), rk,2 ≥ rmin
+k,2 ,
+(21a)
+where rmin
+k,1 =
+dk,1
+T −
+ηck,1Dk,1
+fk,1
+, and rmin
+k,2 =
+dk,2
+T −
+ηck,2Dk,2
+fk,2
+.
+In fact, SP2 1 and SP2 2 are two minimization sum-of-
+ratios problems, which are NP-complete [26] and challenging
+to solve. Therefore, to further make them solvable, we trans-
+form SP2 1 and SP2 2 into the epigraph form. To be concise
+and due to the same property of SP2 1 and SP2 2, we write
+a general form as follows. Besides, for SP2 1, ˆpk = pk,1; for
+SP2 2, ˆpk = pk,2.
+SP2 epi : min
+ˆpk,Γk α
+K
+�
+k=1
+Γk,
+(22)
+subject to, (9a), Constraint1,
+where Γk is the auxiliary variable and the constraint
+Constraint1 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+SP2 1: (20a),
+pk,1dk,1
+ˆrk
+≤ Γk,
+ˆrk = Bk log2(1 + pk,1gk,1
+BkNk ),
+SP2 2: (21a), pk,2dk,2
+ˆrk
+≤Γk,
+ˆrk = Bk log2(1 +
+pk,2gk,2
+BkNk+pk,1gk,1 ).
+(23)
+However, the above form is still non-convex. In SP2 1,
+since the only variable is pk,1, it is obvious that pk,1dk,1
+is convex, and Bk log2(1 + pk,1gk,1
+BkNk ) is concave. This means
+each term of the objective function has the characteristics—
+the numerator is convex and the denominator is concave. In
+addition, SP2 2 has the same characteristics. Because of such
+characteristics, we provide the following lemma to transform
+the subproblems into subtractive-form problems, which are
+equivalent to the original subproblems.
+Lemma 1. If (ˆp∗
+k, Γ∗
+k) is the solution of SP2 epi, the following
+problem has ˆp∗
+k as its solution if there exist νk = ν∗
+k, Γk = Γ∗
+k,
+k = 1, · · · , K.
+SP2 sub : min
+ˆpk
+K
+�
+k=1
+νk[ˆpk ˆdk − Γkˆrk],
+(24)
+subject to :(9a), Constraint2,
+where
+Constraint2 =
+�
+SP2 1:(20a),
+SP2 2:(21a).
+For SP2 1, ˆpk ˆdk = pk,1dk,1; for SP2 2, ˆpk ˆdk = pk,2dk,2.
+Additionally, with νk = ν∗
+k, Γk = Γ∗
+k and ˆpk = ˆp∗
+k, the
+following equations are satisfied:
+ν∗
+k = α
+ˆrk
+, Γ∗
+k = ˆp∗
+k ˆdk
+ˆrk
+, k = 1, · · · , K.
+(25)
+proof. The proof is provided by Lemma 2.1 in [27].
+In brief, Lemma 1 proves that SP2 epi and SP2 sub are
+equivalent and have the same optimal solution. Hence, both
+subproblems SP2 1 and SP2 2 can be transformed into the
+form of SP2 sub.
+Thus, to solve SP2 epi, we can solve SP2 sub to obtain
+ˆpk with given νk and Γk. Next, with the obtained ˆpk, we can
+calculate the new νk and Γk through Eq. (25).
+To solve SP2 sub, applying KKT conditions, we can get
+ˆp∗
+k =
+�
+�
+�
+(νkγ+µk)Bk
+νkdk,1 ln 2 −Λ, µk = 0
+(2
+ˆrmin
+k
+Bk −1)Λ, µk > 0,
+(26)
+where µk = [2
+ˆrmin
+k
+Bk Λ(ln 2)νk ˆdk/Bk − νkΓk]+ and [x]+
+means max(0, x). In addition,
+�
+SP2 1 : ˆpk = pk,1, ˆrmin
+k
+= rmin
+k,1 , Λ = BkNk
+gk,1 ,
+SP2 2 : ˆpk = pk,2, ˆrmin
+k
+= rmin
+k,2 , Λ = BkNk+pk,1gk,1
+gk,2
+.
+(27)
+We have finished converting SP1 and SP2. The algorithm for
+optimizing SP2 is listed in Algorithm 1. The original algorithm
+is given in [27] and is a Newton-like method.
+In Algorithm 1, we define ϕ(Γk, νk) = [ϕT
+1 (Γk), ϕT
+2 (νk)]T ,
+where
+ϕ1(Γk)=[−ˆpk ˆdk + Γkˆrk]T, ϕ2(νk)=[−α+νkˆrk]T, k∈[1, K].
+(28)
+4
+
+This paper appears in the Proceedings of IEEE International Conference on Communications (ICC) 2023.
+Please feel free to contact us for questions or remarks.
+The Jacobian matrices of ϕ1(Γk) and ϕ2(νk) are
+ϕ′
+1(Γk) = diag(ˆrk), ϕ′
+2(νk) = diag(ˆrk), k ∈ [1, K],
+(29)
+where diag() stands for a diagonal matrix.
+Algorithm 1: Optimization of SP2 1/SP2 2
+1 Initialize j = 0, ζ ∈ (0, 1), ϵ ∈ (0, 1) and feasible ˆp0
+k.
+2 repeat
+3
+Calculate (ν(j)
+k , Γ(j)
+k ) according to Eq. (25).
+4
+Get (ˆp(j+1)
+k
+) by calculating Eq. (26).
+5
+Let i be the smallest integer satisfying
+|ϕ(Γk + ζiσ(j)
+k,1, νk + ζiσ(j)
+k,2|
+≤ (1 − ϵζi)|ϕ(Γ(j)
+k , ν(j)
+k )|,
+(30)
+where
+σ(j)
+k,1 = −[ϕ′
+1(Γ(j)
+k )]−1ϕ1(Γ(j)
+k ),
+σ(j)
+k,2 = −[ϕ′
+2(ν(j)
+k )]−1ϕ2(ν(j)
+k ),
+(31)
+6
+Update
+(Γ(j+1)
+k
+, ν(j+1)
+k
+)=(Γ(j)
+k +ζiσ(j)
+k , νk+ζiσ(j)
+k,2). (32)
+7
+j ← j + 1.
+8 until ϕ(Γk, νk) = 0 or reaching the maximum
+iteration number J;
+E. Resource Allocation Algorithm
+Based on Eq. (26), a resource allocation algorithm is pro-
+posed, which is an iterative optimization algorithm, as shown
+in Algorithm 2. It first initially assigns a feasible solution set
+within the range of f, p and s. Next, iteratively solving SP1
+(problem (17)) and SP2 (SP2 1 & SP2 2) to obtain (f, s) and
+p respectively until convergence.
+Algorithm 2: Resource Allocation Algorithm
+1 Initialize S(0) ← (f (0), s(0), p(0)) of problem (9).
+Iteration number i = 1, the maximum number of
+iterations M.
+2 while |S(i) − S(i−1)| > ε and i ≤ M do
+3
+Solve problem (17), which is the dual problem of
+SP1, to obtain (f (i), s(i)) through CVX given
+p(i−1).
+4
+Given (f (i), s(i)), call Algorithm 1 (ˆpk = pk,1) to
+solve SP2 1 and get p(i)
+k,1.
+5
+Given p(i)
+k,1, call Algorithm 1 (ˆpk = pk,2) to solve
+SP2 2 to obtain p(i)
+k,2.
+6
+p(i) ← [p(i)
+1,1, · · · , p(i)
+K,1, p(i)
+1,2, · · · , p(i)
+K,2].
+7
+S(i) ← (f (i), s(i), p(i)).
+8
+i ← i + 1.
+9 end
+F. Time Complexity and Convergence Analysis
+Time complexity. We use floating point operations (flops)
+to analyze the time complexity. One flop is any mathematical
+operation (e.g., addition/subtraction/multiplication/division).
+Since steps 2–8 are the bulk of Algorithm 2, we mainly
+analyze this part. Because Algorithm 1 is called in steps 4
+and 5, we turn the view to Algorithm 1 first.
+In Algorithm 1, step 3, 4 and 6 take O(K) flops. Step 5
+takes O((i + 1)K), where i is the smallest integer satisfying
+the inequality (30). Therefore, Algorithm 1’s time complexity
+is O((i + 4)K). Besides, in Algorithm 2, step 3 solves
+problem (17) through CVX. Due to the use of the interior-point
+algorithm in CVX, the worst time complexity is O(K4.5 log 1
+ϵ )
+[28]. Note that Algorithm 2 solves SP2 by calling Algorithm
+1 iteratively. Thus, the time complexity of Algorithm 2 is
+O(K4.5 log 1
+ϵ + 2(i + 4)K).
+Convergence Analysis. Algorithm 1 is called in Algorithm
+2, so first, we discuss the convergence of Algorithm 1.
+The convergence proof is provided by Theorem 3.2 in [27].
+Additionally, according to Theorem 3.2 in [27], Algorithm 1
+converges with a linear rate at any starting point (ν0, Γ0) and
+a quadratic convergence rate of the solution’s neighborhood.
+Therefore, Algorithm 2, which iteratively solves SP1 to get
+(f, s) and SP2 1 to get p, will converge eventually.
+Algorithm 3: Benchmark Greedy Algorithm
+1 Initialize s = s1, Cmin = ∞, k ∈ [1, K]
+2 P = [p0, ..., p10], pi = pmin + 0.1i(pmax − pmin),
+3 F = [f0, ..., f10], fi = f min + 0.1i(f max − f min)
+4 for fk,1 in F and fk,2 in F do
+5
+for pk,1 in P and pk,2 in P do
+6
+The energy of channel k: E ← Eq. (3)+Eq. (4).
+7
+The time consumption of channel k: T ← Eq.
+(2)+ Eq. (6).
+8
+C ← αE + βT.
+9
+if C < Cmin then
+10
+Update Cmin ← C, Eout ← E, Tout ← T
+11
+end
+12
+end
+13 end
+14 output : Cmin, Eout, Tout
+IV. EXPERIMENTAL RESULTS
+In this section, we evaluate experimental results: 1) the ef-
+fectiveness of our proposed algorithm in the joint optimization
+of energy and time consumption. 2) How the global model
+accuracy varies with the weight parameter γ.
+A. Parameter Settings
+The overall system settings are provided in Table I.
+Recall that the constant ξ equals 1
+s2
+0 , and the accuracy metric
+A(s1,1, s1,2, ..., sk,i) should be �K
+k=1
+�2
+i=1 Ak,i(sk,i).
+The optimization objective is αE + βT − γA from (9). We
+“normalize” the weight parameters such that α + β = 1 by
+dividing each of them by α+β. The intuition for doing this is
+that T and A are considered as ”costs” and A is the ”gain”,
+then α + β = 1 means the coefficient for the cost part is 1.
+5
+
+This paper appears in the Proceedings of IEEE International Conference on Communications (ICC) 2023.
+Please feel free to contact us for questions or remarks.
+B. System Parameters
+We have three weight parameters for the optimization
+problem: α, β and γ, and α + β = 1. If more focus is
+on energy consumption, the parameter α should be greater
+than β. If the delay is the objective to mainly optimize, β
+should be set as a larger value. Besides, the value of γ affects
+object detection in a similar way. To explore the influence of
+three parameters separately, we first implement experiments
+under different (α, β) with fixed γ and then show results under
+different γ with fixed (α, β) = (0.5, 0.5).
+We compare three pairs of weight parameters (α, β) =
+(0.9, 0.1), (0.5, 0.5) and (0.1, 0.9) with random allocation
+strategy and one greedy algorithm (provided in Algorithm
+3). (α, β) = (0.9, 0.1) is carried out when devices are low-
+battery to save energy. (α, β) = (0.5, 0.5) represents the or-
+dinary situation to equally consider time and energy. Besides,
+(α, β) = (0.1, 0.9) stresses the time-sensitive scenario.
+Fig. 2(a), (b) and (c) show the total energy consumption E,
+time consumption T and αE + βT under different maximum
+transmission power limits. It can be observed that as the max-
+imum transmission power increases, T and αE +βT decrease
+while E slightly increases. This is because as the range of pmax
+expands, there will be a more optimal solution to decrease
+the time consumption. Our proposed algorithm is superior to
+the random allocation strategy and greedy algorithm in terms
+of energy optimization and αE + βT . In terms of total time
+consumption, the proposed algorithm performs worse than the
+greedy algorithm. When (α = 0.1, β = 0.9), However, it
+is evident that the gap is much smaller than that of energy
+consumption, and this is why our proposed algorithm still
+performs better in terms of αE + βT .
+Moreover, Fig. 3 demonstrates the performance of different
+algorithms under different maximum CPU frequencies. From
+Fig. 3(a), it can be seen the proposed algorithm performs much
+better than greedy algorithm in terms of energy consumption.
+Although the random strategy is slightly better than the green
+line (Proposed, α = 0.1, β
+= 0.9) in terms of energy
+consumption, there is a huge gap in terms of time consumption
+as shown in Fig. 3(b). Still, when maximum CPU frequency
+TABLE I
+SYSTEM PARAMETER SETTING
+Parameter
+Value
+The path loss model
+128.1 + 37.6 log(d (km))
+The standard deviation of shadow fading
+8 dB
+Noise power spectral density Nk
+−174 dBm/Hz
+The number of users N
+50
+The number of channels K
+25
+Total Bandwidth (B)
+20 MHz
+CPU frequency fmax, fmin
+2 GHz, 0 GHz
+The number of CPU cycles cn
+Randomly assigned in
+[1, 3] × 104
+Transmission power pmax, pmin
+12 dBm, 0 dBm
+The number of local iterations η
+10
+The data size uploaded dn
+28.1 kbits
+The number of samples Dn
+500
+Effective switched capacitance kappa
+1028
+Video frame resolutions (s0, s1, s2, s3)
+(100, 160, 320, 640) pixels
+5
+10
+15
+20
+0
+1
+2
+3
+4
+5
+6
+(a)
+Proposed, =0.1 =0.9
+Proposed, =0.5 =0.5
+Proposed, =0.9 =0.1
+random
+5
+10
+15
+20
+0
+0.5
+1
+1.5
+2
+(b)
+5
+10
+15
+20
+0
+0.5
+1
+1.5
+(c)
+greedy, =0.1 =0.9
+greedy, =0.5 =0.5
+greedy, =0.9 =0.1
+Fig. 2. Consumption under different maximum transmit power. γ=1.
+1
+1.5
+2
+0
+1
+2
+3
+4
+5
+(a)
+1
+1.5
+2
+0
+0.5
+1
+1.5
+2
+2.5
+3
+(b)
+Proposed, =0.1 =0.9
+Proposed, =0.5 =0.5
+Proposed, =0.9 =0.1
+random
+1
+1.5
+2
+0
+0.5
+1
+1.5
+(c)
+greedy, =0.1 =0.9
+greedy, =0.5 =0.5
+greedy, =0.9 =0.1
+Fig. 3. Consumption under different maximum CPU frequency. γ=1.
+increases, the gap between proposed algorithm and greedy
+algorithm in terms of E becomes larger and larger. Although
+the performance of the proposed algorithm in terms of T
+is slightly worse than the greedy algorithm, the proposed
+algorithm is roughly more advantageous than the greedy
+algorithm in terms of αE + βT .
+C. Accuracy analysis
+To analyze the selection of the frame resolution for each
+device under different γ and illustrate the result concisely, the
+number of users is set as 4. We fix the weight parameters
+(α, β) = (0.5, 0.5) and choose different γ, to calculate
+corresponding optimal video resolution for each user as shown
+in Fig. 4(a) and illustrate the relationship between γ and model
+accuracy in Fig. 4(b). YOLOv5m [29], which is one of the
+object detection architectures You Only Look Once (YOLO),
+is implemented for each user under federated learning setting.
+The dataset is COCO [30]. This experiment environment is
+on a workstation with three NVIDIA RTX 2080 Ti GPUs for
+computation acceleration.
+There are 4 lines in Fig. 4(a) which represent the resolution
+choices for 4 users. Obviously, as γ increases, which means the
+system will pay more attention on model accuracy, users are
+more inclined to choose higher video resolution. Because such
+selections of s could bring prominent improvement for model
+accuracy, which could be noticed in Fig. 4(b). When γ is
+lower than around 0.85, 4 users will always choose the lowest
+resolution s1 = 160 and the model accuracy is only about 0.32.
+After γ reach 0.85, users begin to pick resolution s2 = 320
+and the model accuracy increase to about 0.40. Besides, if γ
+gets bigger, users will adopt the highest available resolution
+s3 = 640 and so there is a dramatically improvement in
+accuracy, reaching approximately 0.68.
+V. CONCLUSION
+In this work, an FL-assisted MAR system via NOMA is
+proposed. We have explored the problem of joint optimiza-
+6
+
+This paper appears in the Proceedings of IEEE International Conference on Communications (ICC) 2023.
+Please feel free to contact us for questions or remarks.
+0
+0.5
+1
+1.5
+2
+160
+320
+640
+Frame resolution sn
+(a)
+user 1
+user 2
+user 3
+user 4
+0
+0.5
+1
+1.5
+2
+0.3
+0.4
+0.5
+0.6
+0.7
+Accuracy
+(b)
+Fig. 4.
+Frame resolution and accuracy under different γ. Here (α, β) =
+(0.5, 0.5).
+tion of energy, time and accuracy to allocate appropriate
+transmission power, computational frequency and video frame
+resolution for each device in the system. Through adjusting
+three weight parameters in the optimization problem, our
+proposed algorithm can be adapted to various scenarios. Time
+complexity and convergence analysis are also provided for
+the proposed resource allocation algorithm. In experimental
+results, it can be observed that our proposed algorithm is
+particularly effective in optimizing energy consumption com-
+pared to random allocation strategy and a benchmark greedy
+algorithm. Our paper also provides new insights into how
+federated learning and MAR can be applied to the Metaverse.
+REFERENCES
+[1] J. D. N. Dionisio, W. G. B. III, and R. Gilbert, “3D virtual worlds and
+the metaverse: Current status and future possibilities,” ACM Computing
+Surveys (CSUR), vol. 45, no. 3, pp. 1–38, 2013.
+[2] B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas,
+“Communication-efficient learning of deep networks from decentralized
+data,” in Artificial intelligence and statistics.
+PMLR, 2017, pp. 1273–
+1282.
+[3] Q. C. Li, H. Niu, A. T. Papathanassiou, and G. Wu, “5G network
+capacity: Key elements and technologies,” IEEE Vehicular Technology
+Magazine, vol. 9, no. 1, pp. 71–78, 2014.
+[4] Y. Chen, B. Wang, Y. Han, H.-Q. Lai, Z. Safar, and K. R. Liu, “Why time
+reversal for future 5G wireless?[perspectives],” IEEE Signal Processing
+Magazine, vol. 33, no. 2, pp. 17–26, 2016.
+[5] C. He, H. Wang, Y. Hu, Y. Chen, X. Fan, H. Li, and B. Zeng,
+“Mcast: High-quality linear video transmission with time and frequency
+diversities,” IEEE Transactions on Image Processing, vol. 27, no. 7, pp.
+3599–3610, 2018.
+[6] A. Benjebbour, Y. Saito, Y. Kishiyama, A. Li, A. Harada, and T. Naka-
+mura, “Concept and practical considerations of non-orthogonal multiple
+access (noma) for future radio access,” in 2013 International Symposium
+on Intelligent Signal Processing and Communication Systems.
+IEEE,
+2013, pp. 770–774.
+[7] T. Song, X. Tan, J. Ren, W. Hu, S. Wang, S. Xu, X. Wang, G. Sun, and
+H. Yu, “Dram: A drl-based resource allocation scheme for mar in mec,”
+Digital Communications and Networks, 2022. [Online]. Available:
+https://www.sciencedirect.com/science/article/pii/S2352864822000633
+[8] A. Al-Shuwaili and O. Simeone, “Energy-efficient resource allocation
+for mobile edge computing-based augmented reality applications,” IEEE
+Wireless Communications Letters, vol. 6, no. 3, pp. 398–401, 2017.
+[9] X. Chen and G. Liu, “Energy-efficient task offloading and resource
+allocation via deep reinforcement learning for augmented reality in
+mobile edge networks,” IEEE Internet of Things Journal, vol. 8, no. 13,
+pp. 10 843–10 856, 2021.
+[10] X. Chen and G. Liu, “Joint optimization of task offloading and resource
+allocation via deep reinforcement learning for augmented reality in
+mobile edge network,” in 2020 IEEE International Conference on Edge
+Computing (EDGE).
+IEEE, 2020, pp. 76–82.
+[11] P. Si, J. Zhao, H. Han, K.-Y. Lam, and Y. Liu, “Resource allocation
+and resolution control in the metaverse with mobile augmented reality,”
+arXiv preprint arXiv:2209.13871, 2022.
+[12] M. Makolkina, A. Paramonov, and A. Koucheryavy, “Resource alloca-
+tion for the provision of augmented reality service,” in Internet of Things,
+Smart Spaces, and Next Generation Networks and Systems, O. Galinina,
+S. Andreev, S. Balandin, and Y. Koucheryavy, Eds.
+Cham: Springer
+International Publishing, 2018, pp. 441–455.
+[13] J. Li, F. Wu, K. Zhang, and S. Leng, “Joint dynamic user pairing,
+computation offloading and power control for noma-based mec system,”
+in 2019 11th International Conference on Wireless Communications and
+Signal Processing (WCSP).
+IEEE, 2019, pp. 1–6.
+[14] Y. Ye, R. Q. Hu, G. Lu, and L. Shi, “Enhance latency-constrained
+computation in mec networks using uplink noma,” IEEE Transactions
+on Communications, vol. 68, no. 4, pp. 2409–2425, 2020.
+[15] S. Gupta, D. Rajan, and J. Camp, “Noma-enabled computation and
+communication resource trading for a multi-user mec system,” IEEE
+Transactions on Vehicular Technology, 2022.
+[16] D. Chen, L. J. Xie, B. Kim, L. Wang, C. S. Hong, L.-C. Wang,
+and Z. Han, “Federated learning based mobile edge computing for
+augmented reality applications,” in 2020 international conference on
+computing, networking and communications (ICNC).
+IEEE, 2020, pp.
+767–773.
+[17] Z. Zhang, H. Sun, and R. Q. Hu, “Downlink and uplink non-orthogonal
+multiple access in a dense wireless network,” IEEE Journal on Selected
+Areas in Communications, vol. 35, no. 12, pp. 2771–2784, 2017.
+[18] C. He, Y. Hu, Y. Chen, and B. Zeng, “Joint power allocation and channel
+assignment for noma with deep reinforcement learning,” IEEE Journal
+on Selected Areas in Communications, vol. 37, no. 10, pp. 2200–2210,
+2019.
+[19] C. T. Dinh, N. H. Tran, M. N. Nguyen, C. S. Hong, W. Bao, A. Y.
+Zomaya, and V. Gramoli, “Federated learning over wireless networks:
+Convergence analysis and resource allocation,” IEEE/ACM Trans. on
+Networking, vol. 29, no. 1, pp. 398–409, 2021.
+[20] Z. Yang, M. Chen, W. Saad, C. S. Hong, and M. Shikh-Bahaei, “Energy
+efficient federated learning over wireless communication networks,”
+IEEE Trans. on Wireless Comm., vol. 20, no. 3, pp. 1935–1949, 2021.
+[21] J. Redmon, S. Divvala, R. Girshick, and A. Farhadi, “You Only Look
+Once: Unified, Real-Time Object Detection,” in Proceedings of the IEEE
+Conference on Computer Vision and Pattern Recognition, 2016, pp. 779–
+788.
+[22] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet Classifica-
+tion with Deep Convolutional Neural Networks,” Advances in Neural
+Information Processing Systems, vol. 25, 2012.
+[23] Y. Mao, J. Zhang, and K. B. Letaief, “Dynamic Computation Offloading
+for Mobile-Edge Computing with Energy Harvesting Devices,” IEEE
+Journal on Selected Areas in Communications, vol. 34, no. 12, pp. 3590–
+3605, 2016.
+[24] Q. Liu, S. Huang, J. Opadere, and T. Han, “An Edge Network Orches-
+trator for Mobile Augmented Reality,” in IEEE Conference on Computer
+Communications (INFOCOM).
+IEEE, 2018, pp. 756–764.
+[25] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex
+programming, version 2.1,” 2014.
+[26] R. W. Freund and F. Jarre, “Solving the sum-of-ratios problem by an
+interior-point method,” Journal of Global Optimization, vol. 19, no. 1,
+pp. 83–102, 2001.
+[27] Y. Jong, “An Efficient Global Optimization Algorithm for Nonlinear
+Sum-of-Ratios Problem,” Optimization Online, 2012.
+[28] Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semidefinite
+Relaxation of Quadratic Optimization Problems,” IEEE Signal Process-
+ing Magazine, vol. 27, no. 3, pp. 20–34, 2010.
+[29] G. Jocher, A. Chaurasia, A. Stoken, J. Borovec, NanoCode012, Y. Kwon,
+TaoXie, K. Michael, J. Fang, imyhxy, Lorna, C. Wong, Z. Yifu, A. V,
+D. Montes, Z. Wang, C. Fati, J. Nadar, Laughing, UnglvKitDe, tkianai,
+yxNONG, P. Skalski, A. Hogan, M. Strobel, M. Jain, L. Mammana, and
+xylieong, “ultralytics/yolov5: v6.2 - YOLOv5 Classification Models,
+Apple M1, Reproducibility, ClearML and Deci.ai integrations,” Aug.
+2022. [Online]. Available: https://doi.org/10.5281/zenodo.7002879
+[30] T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan,
+P. Doll´ar, and C. L. Zitnick, “Microsoft coco: Common objects in
+context,” in European conference on computer vision.
+Springer, 2014,
+pp. 740–755.
+7
+

m9AyT4oBgHgl3EQflPjP/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1476e3ac60ce07cd550514653d9e77248cc720f1f67097cfe72c9fbfb4fc8b42
+size 233871

m9E3T4oBgHgl3EQfKgnW/content/tmp_files/2301.04355v1.pdf.txt

@@ -0,0 +1,1494 @@
+SIR–MODEL FOR HOUSEHOLDS
+PHILIPP D¨ONGES∗, THOMAS G¨OTZ†, TYLL KR¨UGER‡, KAROL NIEDZIELEWSKI§,
+VIOLA PRIESEMANN∗, AND MORITZ SCH¨AFER†
+Abstract.
+Households play an important role in disease dynamics. Many infections hap-
+pening there due to the close contact, while mitigation measures mainly target the transmission
+between households. Therefore, one can see households as boosting the transmission depending
+on household size. To study the effect of household size and size distribution, we differentiated
+the within and between household reproduction rate. There are basically no preventive measures,
+and thus the close contacts can boost the spread. We explicitly incorporated that typically only
+a fraction of all household members are infected. Thus, viewing the infection of a household of a
+given size as a splitting process generating a new, small fully infected sub–household and a remain-
+ing still susceptible sub–household we derive a compartmental ODE–model for the dynamics of
+the sub–households. In this setting, the basic reproduction number as well as prevalence and the
+peak of an infection wave in a population with given households size distribution can be computed
+analytically. We compare numerical simulation results of this novel household–ODE model with
+results from an agent–based model using data for realistic household size distributions of different
+countries. We find good agreement of both models showing the catalytic effect of large households
+on the overall disease dynamics.
+Key words. COVID–19, Epidemiology, Disease dynamics, SIR–model, Social Structure
+MSC2020:
+MSC codes. 92D30, 93-10
+1. Introduction. The spread of an infectious diseases strongly depends on
+the interaction of the considered individuals. Traditional SIR–type models assume
+a homogeneous mixing of the population and typically neglect the increased trans-
+mission within closed subcommunities like households or school–classes. However,
+literature indicates, that household transmission plays an important role [3,4,9].
+In case of the COVID–pandemic, several studies have quantified the secondary
+attack rate within households, i.e. the probability household-members get infected,
+given that one household member is infected [3,7,9–12]. The secondary attack rate
+depends on the virus variant, immunity and vaccination status, cultural differences
+and mitigation measures within a household (like e.g. early quarantine). In De-
+cember 2021, when both the Delta and Omicron variants were spreading, it ranged
+between about 19 % (Delta variant in Norway) [9] to 39 % (Omicron in Spain) [3].
+Hence, interestingly, the secondary attack rate within a household is far from 100
+% despite the close contacts.
+There are several models reported that try to include the contribution of in–
+household transmission to the overall disease dynamics.
+The Reed–Frost model
+describing in–household infections as a Bernoulli–process has be used by [4,5]. An
+average model assuming households always get completely infected and ignoring
+the temporal dynamics has been proposed in [2]. Their findings for the effective
+reproduction number agree with our results, see (3.2). Extensions of differential
+equation based SIR–models in case of a uniform household distribution have be
+proposed by [6, 8]. However, in real populations, the household distribution is far
+from uniform, see [14]. Ball and co–authors provided in [1] an overview of challenges
+posed by integrating household effects into epidemiological models, in particular in
+the context of compartmental differential equations models.
+∗Max
+Planck
+Institute
+for
+dynamics
+and
+self–organization,
+G¨ottingen,
+Germany
+([email protected]).
+†Mathematical
+Institute,
+University
+Koblenz,
+Germany
+([email protected];
+[email protected]; [email protected]).
+‡Polytechnic University Wroclaw, Poland ([email protected]).
+§University Warsaw, Poland ([email protected]).
+1
+arXiv:2301.04355v1  [q-bio.PE]  11 Jan 2023
+
+2
+P. D¨ONGES, T. G¨OTZ, T. KR¨UGER, E.A.
+In this paper we develop an extended ODE SIR–model for disease transmission
+within and between households of different sizes. We will treat the infection of a
+given household as a splitting process generating two new sub–households or house-
+hold fragments of smaller size - representing the susceptible (S) or fully infected
+(I) members. Each of the susceptible household fragments can get infected and
+split later in time, while infected household segments recover with a certain rate
+and are then immune (recovered or removed, R). The dynamics of the susceptible,
+infected and recovered household segments is modeled in Section 2. In Section 3,
+we compute the basic reproduction number for our household model combining the
+attack rate inside a single households and the transmission rate between individual
+households. The prevalence, as the limit of the recovered part of the population can
+be computed analytically — at least in case of small maximal household size, see
+Section 4. The peak of a single epidemic wave in a population with given household
+distribution is considered in Section 5. Again, for small maximal household size,
+we will be able to compute analytically the maximal number of infected. Numeri-
+cal simulations based on realistic household size distributions for different countries
+and a comparison with an agent–based model demonstrate the applicability of the
+presented model. As outlook with focus on non–pharmaceutical interventions we
+consider an extension of our model quarantining infected sub–households based on
+a certain detection rate for a single infected.
+2. Household Model. Instead of real households we consider effective sub–
+households or household fragments, still called households throughout this paper,
+being either fully susceptible, infected or recovered. Let Sj, Ij and Rj denote the
+number of the respective households consisting of j persons, where 1 ≤ j ≤ K
+and K denoting the maximal household size. Then Hj = Sj + Ij + Rj equals to
+the total number of current sub–households of size j. Furthermore, we introduce
+H = �K
+j=1 Hj as the total number of households and hj = Hj/H.
+The total
+population is given by N = �K
+j=1 jHj. For further reference we also introduce the
+first two moments of the household size distribution
+µ1 :=
+K
+�
+j=1
+jhj
+and
+µ2 :=
+K
+�
+j=1
+j2hj .
+If an infection is brought into a susceptible sub–household of size j, secondary
+infections will occur inside the household. We assume that each of the remaining
+j−1 household members can get infected with probability a, called the in–household
+attack rate. On average, we expect
+Ej = a(j − 1) + 1
+infections (including the primary one) inside a household of size j. Existing field
+studies, see [7, 12] indicate an in–household attack rate in the range of 16%–30%
+depending on the overall epidemiological situation, household size and vaccinations.
+For the sake of tractability and simplicity, our model assumes a constant attack rate
+a independent of the household size.
+Let bj,k denote the probability, that a primary infection in a household of size j
+generates in total k infections inside this household, where 1 ≤ k ≤ j. The secondary
+infections give rise to a splitting of the initial household of size j into a new, fully
+infected sub–household of size k and another still susceptible sub–household of size
+j − k.
+An infected household of size k recovers with a rate γk and contributes to the
+overall force of infection by an out–household infection rate βk. We assume that
+
+SIR–MODELS FOR HOUSEHOLDS
+3
+the out–household reproduction number is independent of the household size, i.e.
+(2.1)
+R∗ = βk
+γk
+= constant independent of k .
+Now, the dynamical system governing the dynamics of the susceptible, infected
+and recovered households of size k reads as
+S′
+k = Y
+�
+�−kSk +
+K
+�
+j=k+1
+jSj · bj,j−k
+�
+� ,
+(2.2a)
+I′
+k = −γkIk + Y
+K
+�
+j=k
+jSj · bj,k ,
+(2.2b)
+R′
+k = γkIk ,
+(2.2c)
+where
+Y := 1
+N
+K
+�
+k=1
+βk · kIk
+denotes the total force of infection.
+For the recovery rate of an infected household of size k we can consider the two
+extremal cases and an intermediate case.
+1. Simultaneous infections: All members of the infected household get infected
+at the same time and recover at the same time, hence the recovery rate
+γk = γ1 is independent of the household size. Assumption (2.1) leads to a
+constant out–household infection rate βk = β.
+2. Sequential infections: All members of the household get infected one after
+another and the total recovery time for the entire household equals to k
+times the individual recovery time. Hence the recovery rate γk = γ1/k and
+by (2.1) we get βk = β1/k.
+3. Parallel infections: The recovery times Ti, i = 1, . . . , k for each of the k
+infected individuals are modeled as independent exponentially distributed
+random variables. Hence the entire household is fully recovered at time
+max(T1, . . . , Tk). The recovery rate γk equals to the inverse of the expected
+recovery time, i.e.
+γk =
+1
+E[max(T1, . . . , Tk)] =
+1
+E[T1] · θk
+= γ1
+θk
+,
+where θk = 1 + 1
+2 + · · · + 1
+k ∼ log k + g denotes the k–th harmonic number
+and g ≈ 0.5772 . . . denotes the Euler–Mascheroni constant.
+All these cases can be subsumed to
+γk = γ1
+ηk
+,
+βk = β1
+ηk
+,
+where ηk models the details of the temporal dynamics inside an infected household.
+In Figure 2.1 we compare these three cases in the scenario of a population
+with maximal household size K = 6. Each household size represents 1/6 of the
+entire population. Initially, 1� of the population is infected. The out–household
+reproduction number is assumed to be R∗ = 1.33, in–household attack rate equals
+a = 0.2 and the recovery rate γ1 for an infected individual equals 0.1.
+Shown
+are the incidences, i.e. the daily new infections over time. The three cases differ
+
+4
+P. D¨ONGES, T. G¨OTZ, T. KR¨UGER, E.A.
+in the timing and the height of the peak of the infection; for the simultaneous
+infections (γk constant) the disease spreads fastest and for the sequential infections
+(γk = γ1/k) the spread is significantly delayed. The cases of parallel infections,
+i.e. γk ≃ γ1/(log k + g), lies between theses two extremes.
+0
+50
+100
+150
+200
+250
+300
+Time t
+0
+100
+200
+300
+400
+500
+Incidence
+ constant
+ scales with k
+ scales with log k
+Fig. 2.1.
+Simulation of one epidemic wave in case of the three different scalings of the
+recovery rates for households of size k. Shown is the incidence, i.e. daily new infections over
+time. The green curve corresponds equal recovery rates for all households, i.e. γk = γ1. The red
+curve corresponds to γk = γ1/k (sequential infections). The blue curve corresponds to recovery
+rates obtaind from the maximum of exponential distributions, i.e. γk ≃ γ1/(log k).
+Example 2.1. Modeling the infections inside the household by a Bernoulli–
+process with in–household attack rate (infection probability) a ∈ [0, 1], the total
+number of infected persons inside the household follows a binomial distribution
+(2.3)
+bj,k :=
+�j − 1
+k − 1
+�
+ak−1(1 − a)j−k .
+For the binomial in–household infection (2.3) it holds, that Ej = a(j − 1) + 1.
+Theorem 2.2. In model (2.2) the total population N = �K
+k=1 k ·(Sk +Ik +Rk)
+is conserved.
+Proof. We have
+N ′ = Y
+�
+�−
+K
+�
+k=1
+k2Sk +
+K
+�
+k=1
+k
+K
+�
+j=k+1
+jSjbj,j−k +
+K
+�
+k=1
+k
+K
+�
+j=k
+jSjbj,k
+�
+�
+We consider the last two summands separately and reverse the order of summation.
+Hence
+K
+�
+k=1
+k
+K
+�
+j=k
+jSjbj,k =
+K
+�
+j=1
+jSj
+j
+�
+k=1
+kbj,k =
+K
+�
+j=1
+jEjSj
+
+SIR–MODELS FOR HOUSEHOLDS
+5
+and analogously
+K
+�
+k=1
+k
+K
+�
+j=k+1
+jSjbj,j−k =
+K
+�
+j=2
+jSj
+j−1
+�
+k=1
+kbj,j−k
+=
+K
+�
+j=2
+jSj
+j−1
+�
+l=1
+(j − l)bj,l =
+K
+�
+j=2
+jSj
+j
+�
+l=1
+(j − l)bj,l
+=
+K
+�
+j=2
+jSj
+�
+j
+j
+�
+l=1
+bj,l −
+j
+�
+l=1
+lbj,l
+�
+=
+K
+�
+j=2
+j2Sj − jEjSj
+=
+K
+�
+j=1
+j2Sj − jEjSj − (S1 − E1S1) .
+Due to E1 = 1, the last term vanishes. Summing all the contributions, we finally
+get
+N ′ = Y
+�
+�
+K
+�
+j=1
+−(j2Sj) + jEjSj + (j2Sj − jEjSj)
+�
+� = 0 .
+3. Basic Reproduction Number. To compute the basic reproduction num-
+ber for the household–model (2.2), we follow the next generation matrix approach
+by Watmough and van den Driessche, see [15]. We split the state variable x =
+(S1, . . . , RK) ∈ R3K into the infected compartment ξ = (I1, . . . IK) ∈ RK and the
+remaining components χ ∈ R2K. The infected compartment satisfies the differential
+equation
+ξ′
+k = Fk(ξ, χ) − Vk(ξ, χ)
+where Fk = Y �K
+j=k jSj bj,k, Vk = γkξk and Y =
+1
+N
+�K
+j=1 βjjξj. Now, the Jaco-
+bians at the disease free equilibrium (0, χ∗) are given by
+Fkj = ∂ Fk
+∂ ξj
+(0, χ∗) = 1
+N jβj
+K
+�
+m=k
+mHm bm,k
+Vkk = ∂ Vk
+∂ ξk
+(0, χ∗) = γk
+and Vkj = 0 for k ̸= j
+The next generation matrix G ∈ RK×K is given by
+G = FV −1 = 1
+N
+�
+jβj
+γj
+K
+�
+m=k
+mHm bm,k
+�
+k,j
+Using the relation (2.1) we obtain
+G = R∗
+�
+j
+K
+�
+m=k
+mHm
+N
+bm,k
+�
+k,j
+= R∗
+�
+�
+�
+�
+�
+�
+�K
+m=1
+mHm
+N
+bm,1
+�K
+m=2
+mHm
+N
+bm,2
+...
+KHK
+N
+bKK
+�
+�
+�
+�
+�
+�
+·
+�1, 2, . . . , K�
+where the last term represents the next generation matrix as a dyadic product
+G = R∗ · dcT .
+
+6
+P. D¨ONGES, T. G¨OTZ, T. KR¨UGER, E.A.
+The basic reproduction number R = ρ(G) is defined as the spectral radius ρ(G)
+of the next generation matrix G ∈ RK×K. As a dyadic product, G has rank one
+and hence there is only one non–zero eigenvalue.
+Lemma 3.1. Let c, d ∈ Rn be two vectors with cT d ̸= 0. The non–zero eigen-
+value of the dyadic product A = dcT ∈ Rn×n is given by cT d = tr(A).
+Proof. Let v be an eigenvector of A to the eigenvalue λ. Then λv = Av =
+(dcT )v = (cT v)d, where cT v ∈ R. If λ ̸= 0, then v = cT v
+λ d. Set µ = cT v
+λ
+∈ R. Now,
+Av = A(µd) = µ(dcT )d = (cT d)µd = (cT d)v; hence λ = cT d.
+As an immediate consequence we obtain
+Theorem 3.2. The basic reproduction number of system (2.2) is given by
+(3.1)
+R = R∗
+K
+�
+i=1
+i
+K
+�
+m=i
+mHm
+N
+bm,i = R∗
+K
+�
+m=1
+mHm
+N
+Em .
+Corollary 3.3. In the particular situation, when the expected number of in-
+fections inside a household of size m is given by Em = a(m − 1) + 1, the basic
+reproduction number equals
+(3.2)
+R = R∗
+�
+1 + a
+�µ2
+µ1
+− 1
+��
+.
+Assuming, that the infection of any member of a household results in the in-
+fection of the entire household, i.e. in–household attack rate a = 1, our result
+R = µ2
+µ1 R∗ agrees with the result obtained by Becker and Dietz in [2, Sect. 3.2].
+4. Computing the prevalence. Let z = 1
+N
+�K
+k=1 kRk denote the fraction of
+recovered individuals. Then z satisfies the ODE
+z′ = Y
+R∗ .
+In the sequel we will derive an implicit equation for the prevalence limt→∞ z(t)
+in two special cases:
+1. for maximal household size K = 3. The procedure used here allows for im-
+mediate generalization but the resulting expression gets lengthy and provide
+only minor insight into the result.
+2. for in–household attack rate a = 1 and arbitrary household sizes. In this
+setting the equations (2.2a) for the susceptible households decouple and
+allow and complete computation of the equation for the prevalence.
+We will start with the second case. Let us consider a = 1, i.e. inside households
+infections are for sure and Ek = k. Then the ODE for the susceptible households
+reads as S′
+k = −Y kSk and we can insert the recovered z
+S′
+k = −kR∗Skz .
+After integration with respect to t from 0 to ∞, we get
+ln Sk(∞)
+Sk(0) = kR∗ (z(0) − z(∞)) .
+Assuming initially no recovered individuals, i.e. z(0) = 0 and considering the total
+population at the the end of time, i.e. N = Nz(∞) + �
+k kSk(∞), we arrive at the
+system
+N = Nz(∞) +
+K
+�
+k=1
+kSk(0)e−kR∗z(∞) .
+
+SIR–MODELS FOR HOUSEHOLDS
+7
+Scaling with N, i.e. introducing sk,0 = Sk(0)/N, we get
+1 = z +
+K
+�
+k=1
+k sk,0 e−kR∗z .
+(4.1)
+In the limit R∗z ≪ 1 we obtain the approximation
+z ∼
+1 − �
+k ksk,0
+1 − R∗ �
+k k2sk,0
+.
+Figure 4.1 shows the numerical solution of the prevalence equation (4.1) in case
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+R
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Z(
+)
+mixed households
+many singles
+many doubles
+Fig. 4.1.
+Prevalence in case of maximal household size K = 2 vs. reproduction number.
+If double households dominate (green), the prevalence is larger than in the case of mostly single
+households (red), since households speed up the infection dynamics.
+of K = 2 for reproduction numbers R∗ ∈ [0, 3] and initially 2% of the entire
+population being infected. The three different graphs correspond to different initial
+values for the susceptible households: both single and double households contain
+49% of the population (blue), 89% of susceptibles live in single households and only
+9% in double households (red) or just 9% in single households and 89% in double
+households (green).
+In case of arbitrary in–household attack rates a ∈ [0, 1], we consider the situa-
+tion for K = 3. Setting z = Z/N, the relevant equations read as
+S′
+3 = −3R∗z′ S3
+S′
+2 = R∗z′ (−2S2 + 3b3,1S3)
+S′
+1 = R∗z′ (−S1 + 2b2,1S2 + 3b3,2S3) .
+Solving the equations successively starting with S3(t) = S3(0)e−3R∗z(t) and using
+variation of constants, we arrive at
+S2(t) = S2(0)e−2R∗z(t) + 3b3,1S3(0)
+�
+1 − e−R∗z(t)�
+e−2R∗z(t)
+S1(t) = S1(0)e−R∗z(t) + 2b2,1 (S2(0) + 3b3,1S3(0))
+�
+1 − e−R∗z(t)�
+e−R∗z(t)
++ 3
+2 (b3,2 − 2b2,1b3,1) S3(0)
+�
+1 − e−2R∗z(t)�
+e−R∗z(t) .
+
+8
+P. D¨ONGES, T. G¨OTZ, T. KR¨UGER, E.A.
+For the prevalence z = limt→∞ z(t) we obtain the implicit equation
+(4.2)
+1 = z +
+�
+s1,0 + 2b2,1s2,0 + 3(b2,1b3,1 + 1
+2b3,2)s3,0
+�
+e−R∗z
++ 2(1 − b2,1) [s2,0 + 3b3,1s3,0] e−2R∗z
++ 3
+�
+1 + (b2,1 − 2)b3,1 − 1
+2b3,2
+�
+s3,0e−3R∗z
+or in short
+1 = z + c1e−R∗z + c2e−2R∗z + c3e−3R∗z ,
+where the coefficients c1, c2 and c3 are the above, rather lengthy expressions involv-
+ing the initial conditions and the in–household infection probabilities bj,k. In case
+of K = 2, i.e. setting s3,0 = 0, we get
+(4.3)
+1 = z + (s1,0 + 2b2,1s2,0) e−R∗z + 2 (1 − b2,1) s2,0e−2R∗z .
+In case of arbitrary household size K > 3, the resulting equation for the prevalence
+will have the same structure.
+For the binomial infection distribution with a = 1 this reduces to
+1 = z + s1,0e−R∗z + 2s2,0e−2R∗z + 3s3,0e−3R∗z
+and in case of a = 0 we arrive at
+1 = z + (s1,0 + 2s2,0 + 3s3,0) e−R∗z .
+In case of small initial infections, i.e. s1,0 + 2s2,0 + 3s3,0 = c1 + c2 + c3 ≈ 1,
+Eqn. (4.2) allows for the trivial, disease free solution z = 0. However, above the
+threshold Rc =
+1
+c1+2c2+3c3 the non–trivial endemic solution shows up. Expanding
+the exponential for R∗z ≪ 1, we arrive at
+1 ≈ z + (c1 + c2 + c3) − R∗z (c1 + 2c2 + 3c3) + R∗2z2
+2
+(c1 + 4c2 + 9c3)
+and hence
+z (R∗(c1 + 2c2 + 3c3) − 1) ≈ R∗2z2
+2
+(c1 + 4c2 + 9c3) .
+Besides the trivial solution z = 0, this approximation has the second solution
+(4.4)
+z ≃ 2 (R∗(c1 + 2c2 + 3c3) − 1)
+R∗2 (c1 + 4c2 + 9c3)
+.
+If R∗ > Rc = (c1 + 2c2 + 3c3)−1, this second root is positive.
+The following Figure 4.2(left) shows the numerical solution of the prevalence
+equation (4.3) in case of K = 2 for reproduction numbers R∗ ∈ [0, 3] and initially
+2% of the entire population being infected while both single and double households
+contain 49% of the susceptible population. The different curves show the results
+depending on the in–household attack rate a. The graph on the right depict the
+case K = 3 and s1,0 = s2,0 = s3,0 = 1
+6 for three different in–household attack rates
+a. The approximation (4.4) is shown by the dashed curves. For R∗ < Rc, the trivial
+disease free solution z = 0 is the only solution.
+
+SIR–MODELS FOR HOUSEHOLDS
+9
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+R
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Z(
+)
+a=0.2
+a=0.4
+a=0.6
+a=0.8
+a=1.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+R
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Prevalence z(
+)
+a=0.1
+a=0.1 approx
+R_crit
+a=0.2
+a=0.2 approx
+R_crit
+a=0.3
+a=0.3 approx
+R_crit
+Fig. 4.2.
+Visualization of the prevalence vs. out–household reproduction number R∗ and for
+different in–household attack rate a. (Left:) Maximal household size K = 2. (Right:) Maximal
+household size K = 3. Also shown are the asymptotic approximation (4.4) and the critical value
+Rc.
+5. Computing the peak of the infection. Let J = �K
+k=1 kIk denote the
+total number of infected. Then J satisfies
+J′ =
+K
+�
+k=1
+kI′
+k = Y
+�
+�− N
+R∗ +
+K
+�
+k=1
+k
+K
+�
+j=k
+jSjbj,k
+�
+� = Y
+�
+�− N
+R∗ +
+K
+�
+j=1
+jSj
+j
+�
+k=1
+kbj,k
+�
+�
+= Y
+�
+�− N
+R∗ +
+K
+�
+j=1
+jSjEj
+�
+� .
+For K = 2 we have to consider the problem
+S′
+1 = Y [−S1 + 2S2b2,1]
+S′
+2 = Y [−2S2]
+J′ = Y
+�
+− N
+R∗ + S1 + 2E2S2
+�
+.
+Note, that E2 = b2,1 + 2b2,2 and b2,1 + b2,2 = 1, hence b2,1 = 2 − E2. For sake of
+shorter notation and easier interpretation, we introduce the scaled compartments
+s1 = S1/N, s2 = S2/N and j = J/N. Writing s1 as a function of s2, we get
+ds1
+ds2
+= E2 − 2 + s1
+2s2
+with the solution
+s1 = c√s2 − 2(2 − E2)s2 ,
+where c = 2(2 − E2)√s2,0 + s1,0/√s2,0. For j we have the equation
+dj
+ds2
+= −1/R∗ + s1 + 2E2s2
+−2s2
+=
+1
+2R∗s2
+− ds1
+ds2
+− E2
+with the solution given by
+j =
+1
+2R∗ ln s2
+s2,0
+− s1(s2) − E2(s2 − s2,0) .
+
+10
+P. D¨ONGES, T. G¨OTZ, T. KR¨UGER, E.A.
+The maximum of j is either attained in case of an under–critical epidemic at initial
+time at the necessary condition j′ = 0 has to hold. Inserting s1 as a function of s2
+we arrive at the quadratic equation
+4(E2 − 1)x2 + cx − 1
+R∗ = 0
+for x = √s2 with the positive root
+x = √s2 =
+c
+8(E2 − 1)
+��
+1 + 16
+R∗
+E2 − 1
+c2
+− 1
+�
+.
+This solution is only meaningful, if s1 + 2s2 ≤ 1, i.e.
+(5.1)
+1
+R∗ ≤ 1 + 2(E2 − 1)x2
+which is an implicit equation for a threshold value of R∗.
+To obtain the value of j at the maximum, we plug this root into the above
+solution for j and arrive at
+jmax = 1 −
+1
+2R∗
+�
+1 + ln s2,0
+x2
+�
+− cx
+2 .
+Hence, the peak of infection jmax is a function of the out–household reproduction
+number R∗ = βk/γk, the expected infections in double households E2 and the initial
+conditions s2,0, s1,0. In particular, it is independent of the scaling of the recovery
+periods γk, as can be seen also in the more complex setting presented in Figure 2.1.
+The following Figure 5.1 shows the peak of infection jmax versus the out–
+household reproduction number R. Solutions are only plotted, if the condition (5.1)
+is satisfied. The left figure shows the situation for E2 = 1.5 and different initial
+conditions. The right figure shows the variation with respect to E2 = 1 + a keeping
+the initial conditions fixed as s1,0 = 0.5 and s2,0 = 0.25.
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+2.4
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+jmax
+s20 = 0.25
+s20 = 0.05
+s20 = 0.45
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+2.50
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+jmax
+a = 0.2
+a = 0.4
+a = 0.5
+a = 0.6
+a = 0.8
+Fig. 5.1.
+Peak height jmax of a single epidemic wave vs. out–household reproduction
+number R∗. (Left:) Variation with respect to different initial household size distributions. (Right:)
+Variation with respect to different in–household attack rates.
+6. Simulations and comparison with agent–based model. The analy-
+sis of our household model presented in the previous sections, shows that larger
+households have a strong effect on the spread of the epidemic. To illustrate this,
+we simulate a single epidemic wave in populations with close to realistic house-
+hold distribution. For comparison we choose the household size distributions for
+Bangladesh (BGD), Germany (GER) and Poland (POL) as published by the UN
+statistics division in 2011, see [14]. Using a sample of almost 1 000 000 individuals,
+we obtain the distribution shown in Table 6.1.
+
+SIR–MODELS FOR HOUSEHOLDS
+11
+Number Hk of households of size
+Total
+k = 1
+k = 2
+k = 3
+k = 4
+k = 5
+k ≥ 6
+Bangladesh
+999 995
+7 366
+24 351
+44 022
+55 989
+42 037
+53 960 (k = 7)
+Germany
+999 999
+173 640
+154 920
+67 846
+48 585
+15 201
+7 106 (k = 6)
+Poland
+999 996
+85 032
+91 220
+71 221
+57 605
+26 156
+22 523 (k = 7)
+Table 6.1
+Distribution of household sizes in Bangladesh, Germany and Poland for the year 2011,
+see [14]. The sample is based on a scaled population of 1 000 000. Differences are due to round–off
+effects.
+Remark 6.1. Note, that in Table 6.1 we have redistributed for Poland and
+Bangladesh the fraction of population living in households of size 6 or bigger to
+household of size equal 7 to match the total population. How one treats the popu-
+lation represented by the tail in the household distribution can make a substantial
+difference in the simulation outcomes. To visualize that, we show in Figure 6.1
+simulations for out–household reproduction number R∗ = 0.9, in–household attack
+rate a = 0.2 and three different versions of how to treat the tail in the Polish house-
+hold size distribution. The blue curve represents the population distribution given
+in Table 6.1. The green curve uses the extended census data including households
+up to size 10+. All households of size 10+ are treated as households of size equal
+to 10. The red curve show a simplified treatment of the tail, where all households
+of size 6+ are treated as being of size equal to 6.
+The simulation redistributing the households of size 6+ to size 7 (blue) agrees
+within the simulation accuracy with the detailed distribution (green). The simplified
+treatment (red) shows already a significant difference. This difference will be even
+more pronounced for smaller reproduction numbers R∗ since the smaller households
+first get subcritical with decreasing R∗. This result clearly visualizes the need for
+careful treatment of the tail in the household size distribution, since big households
+have an overproportional impact on the dynamics. Unfortunately, most publicly
+available data truncates the household size distribution at size 6. From a modeling
+and simulation point of view, there is a need for more detailed data on the household
+size distribution including information for households of size bigger than 6.
+0
+100
+200
+300
+400
+500
+Time t
+0
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+Total Infected
+POL, R_0=0.90
+redistribution to K=7
+cut-off at K=10
+cut-off at K=6
+Fig. 6.1.
+Simulation of one infection wave using different treatment of the tail in the
+household size distribution. (Blue:) Population distribution given in Table 6.1 where households
+of size 6+ are redistributed to size 7. (Green:) Polish census data including household sizes up
+to 10+. (Red:) Households of size 6+ treated as being of size equal to 6.
+
+12
+P. D¨ONGES, T. G¨OTZ, T. KR¨UGER, E.A.
+As initial condition for our simulation we assume 100 infected single households.
+The recovery rates inside the households are described using the model of parallel
+infections, i.e. γk = γ1/(1 + 1/2 + · · · + 1/k). For the out–household reproduction
+number R∗ we consider the range 0.6 ≤ R∗ ≤ 2.3 and the in–household attack is
+chosen as a = 0.25. The ODE–system (2.2) is solved using a standard RK4(5)–
+method.
+Figures 6.2 and 6.3 show a comparison of the three countries. The prevalance
+and the maximum number of infected shown in Fig. 6.2 clearly visualize that large
+households are drivers of the infection. For moderate out–household reproduction
+number R∗ ≃ 1, Bangladesh, with an average household size µ1 = 4.4, faces a rela-
+tive prevalence being approx. 50% higher than Germany with an average household
+size of µ1 = 2.1 persons. Also the peak number of infected is almost twice as high
+in Bangladesh compared to Germany for R∗ ∈ [1, 1.25]. Fig. 6.3 shows the simu-
+lation of a single infection wave for moderate out–household reproduction number
+R∗ = 1.1 and in-household attack rate a = 0.25. The graph illustrates the faster
+and more severe progression of the epidemics in case of larger households. The
+peak of the wave occurs in Bangladesh 60 days earlier and affects almost double the
+number of individuals compared to Germany.
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+out-household Repro-number R *
+0.0
+0.2
+0.4
+0.6
+0.8
+rel. Prevalence
+GER, a=0.25
+POL, a=0.25
+BGD, a=0.25
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+out-household Repro-number R *
+0
+50
+100
+150
+200
+250
+300
+350
+max. Infected [x 1000]
+GER, a=0.25
+POL, a=0.25
+BGD, a=0.25
+Fig. 6.2.
+Simulation for countries with different household size distributions according
+to Table 6.1. The x–axis shows the out–household reproduction number R∗. The in–household
+attack rate is fixed as a = 0.25. Left: Relative Prevalence z after T = 2 000 days. Right: Absolute
+height of peak number of infected Jmax (in thousands).
+0
+50
+100
+150
+200
+250
+300
+Time [days]
+0
+20
+40
+60
+80
+100
+120
+140
+Infected [x 1000]
+GER, R * =1.1, a=0.25
+POL, R * =1.1, a=0.25
+BGD, R * =1.1, a=0.25
+Fig. 6.3.
+Simulation of one infection wave for countries with different household size
+distributions according to Table 6.1.
+Out–household reproduction number R∗ = 1.1 and in–
+household attack rate a = 0.25. Plotted is the total number of infected J versus time.
+
+SIR–MODELS FOR HOUSEHOLDS
+13
+To validate our model, we compared it to a stochastic, microscopic agent–based
+model developed at the Interdisciplinary Centre for Mathematical and Computa-
+tional Modelling at the University of Warsaw. Complete details of this model are
+given in [13]. In this model agents have certain states (susceptible, infected, re-
+covered, hospitalized, deceased, etc.)
+and infection events occur in certain con-
+text, i.e. in households, on the streets or workplaces and several more.
+Besides
+the household–context, the street–context is used to capture infection events out-
+side households.
+Since the ODE–model (2.2) is a variant of an SIR–model, the
+agent–based model also just uses the SIR–states and ignores all other states. The
+agent–based model uses for each infected individual a recovery time that is sampled
+from an exponential distribution with mean 10 days. Based on the household dis-
+tribution for Poland, Figure 6.4 provides a comparison of the computed prevalence
+vs. the out–household reproduction number for our model 2.2 and the agent–based
+model (blue squares). The solid lines show the results of the ODE–model for dif-
+ferent in–household attack rates. The relative prevalence plotted in the figure is
+defined as the total number of recovered individuals for time t → ∞; here we use
+T = 2 000 days for practical reasons.
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+out-household Repro-number
+0.0
+0.2
+0.4
+0.6
+0.8
+rel. Prevalence
+Polish Household Distribution, Prevalence
+ODE-model, a= 0.16
+ODE-model, a= 0.20
+ODE-model, a= 0.24
+ODE-model, a= 0.28
+Agent-based
+Fig. 6.4.
+Simulation of one infection wave, plotted is the relative prevalence z versus
+the out–household reproduction number R∗. Initially 100 infected single households in a popu-
+lation according to Table 6.1. The solid lines show the results of the ODE model for different
+in–household attack rates. The results of the agent–based model are shown by the blue squares
+(exponentially distributed recovery time).
+Figure 6.5 shows the results for the peak of the infection wave. The graph on
+the left shows the peak of the incidences, i.e. the maximum number of daily new
+infections and the graph on the right shows the time, when this peak occurs.
+For all three presented criteria: prevalence, peak height and peak time, our
+ODE–household model (2.2) matches quite well with the agent–based simulation.
+Best agreement can be seen for in–household attack rate a in the range between
+0.16 and 0.20.
+7. Effect of household quarantine. In this section we will extend our ba-
+sic model (2.2) to households put under quarantine after an infection is detected.
+Therefore, we introduce the additional compartments Qk of quarantined households
+
+14
+P. D¨ONGES, T. G¨OTZ, T. KR¨UGER, E.A.
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+out-household Repro-number
+0
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+40000
+Incidence
+Polish Household Distribution, Peak of Incidence
+ODE-model, a= 0.16
+ODE-model, a= 0.20
+ODE-model, a= 0.24
+ODE-model, a= 0.28
+Agent-based
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+out-household Repro-number
+0
+50
+100
+150
+200
+250
+300
+Day
+Polish Household Distribution, Peak of Incidence
+ODE-Model, a= 0.16
+ODE-Model, a= 0.20
+ODE-Model, a= 0.24
+ODE-Model, a= 0.28
+Agent-based
+Fig. 6.5.
+Simulation of one infection wave, initially 100 infected single households in a
+population according to Table 6.1.
+Left: Peak number of daily new infections versus the out–
+household reproduction number R∗. Right: Time, when the peak occurs vs. R∗. The solid lines
+show the results of the ODE model for different in–household attack rates. The results of the
+agent–based model are shown by the blue squares.
+of size k. The extended household model reads as
+S′
+k = Y
+�
+�−kSk +
+K
+�
+j=k+1
+jSj · bj,j−k
+�
+�
+(7.1a)
+I′
+k = −γkIk − qkIk + Y
+K
+�
+j=k
+jSj · bj,k
+(7.1b)
+Q′
+k = qkIk
+(7.1c)
+R′
+k = γkIk
+(7.1d)
+Here qk > 0 denotes the detection and quarantine rate for a household of size k.
+Let q > 0 be the probability of infected individual to get detected. The probabil-
+ity that a household consisting of k infected gets detected equals dk(q) = 1−(1−q)k.
+Given recovery and detection rates γk and qk for a household of size k, the proba-
+bility, that the household gets detected before recovery equals qk/(γk + qk), which
+equals dk(q). Therefore
+qk
+γk
+=
+dk(q)
+1 − dk(q) ∼ kq
+for q ≪ 1.
+To compute the basic reproduction number, we repeat the computations for the
+next generation matrix. The vector ξ of infected compartments remains unaltered
+and the new quarantine compartments Q1, . . . , QK are included in the vector χ.
+The gain term Fk for the infected compartments remains unaltered as well, but
+the loss term reads as Vk = (γk + qk)ξk.
+Accordingly, its Jacobian modifies to
+Vkk = γk + qk and finally the next generation matrix G reads as
+G = β
+N
+�
+j
+γj + qj
+K
+�
+m=i
+mHm bm,i
+�
+i,j
+= R∗
+�
+�
+�
+�
+�
+�
+�K
+m=1
+mHm
+N
+bm,1
+�K
+m=2
+mHm
+N
+bm,2
+...
+KHK
+N
+bKK
+�
+�
+�
+�
+�
+�
+·
+�
+1
+1+qk/γk ,
+2
+1+q2/γ2 , . . . ,
+K
+1+qK/γK
+�
+
+SIR–MODELS FOR HOUSEHOLDS
+15
+Its non–zero eigenvalue equals
+Rq := R∗
+K
+�
+k=1
+k
+1 + qk/γk
+K
+�
+m=k
+mHm
+N
+bm,k = R∗
+K
+�
+m=1
+mHm
+N
+m
+�
+k=1
+kbm,k
+1 + qk/γk
+.
+Due to denominator 1 + qk/γk we cannot interpret the last sum as the expected
+infected cases in a household of size m. In the asymptotic case of small quarantine
+rates qk ≪ 1, we can use the series expansion (1 + qk/γk)−1 ∼ 1 − qk/γk ∼ 1 − kq
+and get
+Rq ∼ R∗
+K
+�
+m=1
+mHm
+N
+�
+Em − q
+m
+�
+k=1
+k2bm,k
+�
+.
+So, the second moment of the in–household infection distribution comes into play.
+Remark 7.1. If the expectation Em and the the second moment �m
+k=1 k2bm,k of
+the in–household infections scale linear with the household size m, then the above
+reproduction number depends on also on the third moment of the household size
+distribution. This leads to a straight–forward extension of the result (3.2).
+In case of a binomial distribution of the in–household infections, it holds that
+Em = a(m − 1) + 1 and �m
+k=1 k2bm,k = a(m − 1) (3 + a(m − 2)) + 1.
+For the
+reproduction number we arrive at the expression
+Rq ∼ R∗
+�
+1 − q
+γ + a
+�
+1 − 3q
+γ (1 − a)
+� �µ2
+µ1
+− 1
+�
+− q
+γ a2
+�µ3
+µ1
+− 1
+��
+,
+where µ3 := �K
+k=1 k3hk denotes the third moment of the household size distribution.
+The effectiveness of quarantine measures can be assessed by relating the reduc-
+tion in prevalence to the quarantined individuals. For a given detection probability
+q > 0, let Z(q) denote the prevalence observed under these quarantine measures
+and let Q(q) denote the total number of individuals quarantined. We define the
+effectiveness of the quarantine measures as
+ηq := |Z(q) − Z(0)|
+Q(q)
+.
+The numerator equals to the reduction in prevalence and hence η can be interpreted
+as infected cases saved per quarantined case.
+In Figure 7.1 we compare the effectiveness in the setting of household distribu-
+tions resembling Germany (predominantly small households) and Bangladesh (large
+households are dominant) for out–household reproduction numbers in the range be-
+tween 0.9 and 1.2. The results indicate, that quarantine measures are less effective
+in societies with larger households since infection chains inside households are not
+affected by the quarantine. On the other hand, quarantine measures seem most
+effective in presence of small households and for low reproduction numbers.
+8. Conclusion and Outlook. The findings of the compartmental household
+model (2.2) are in rather good agreement with microscopic agent–based simulations.
+Despite its several simplifying assumptions, the prevalence and the peak of the
+infection wave are reproduced quite well.
+However, field data suggests that the in–household attack rate a is significantly
+higher for 2–person households than for larger households, see [7,12]. Analyzing the
+data given in [7] the attack rate in two–person households is by a factor of 1.25–1.5
+larger than in households with three or more members. According to [12] this may
+be caused by spouse relationship to the index case reflecting intimacy, sleeping in
+
+16
+P. D¨ONGES, T. G¨OTZ, T. KR¨UGER, E.A.
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+Quarantine Probability q
+0
+2
+4
+6
+8
+10
+Effectiveness 
+GER R * =0.9
+GER R * =1.0
+GER R * =1.1
+GER R * =1.2
+BGD R * =0.9
+BGD R * =1.0
+BGD R * =1.1
+BGD R * =1.2
+Fig. 7.1.
+Effectiveness η of quarantine measures for the different household distributions in
+Germany (GER) and Bangladesh (BGD), see Table 6.1.
+the same room and hence longer or more direct exposure to the index case. An
+extension of the model (2.3) to attack rates ak depending on the household size k
+is straightforward. Carrying over the analytical computation of the reproduction
+number (3.2) would lead to a variant of (3.2) including weighted moments of the
+household size distribution.
+When considering quarantine for entire households, the model shows some in-
+teresting outcomes. Defining the ratio of reduction in prevalence and number of
+quarantined persons as the effectivity of a quarantine, the model suggests that quar-
+antine is more effective in populations where small households dominate. Whether
+this finding is supported by field data is a topic for future literature research.
+Vaccinations are not yet included in the model.
+From the point of view of
+public health, the question arises whether the limited resources of vaccines should
+be distributed with preference to larger households. Given the catalytic effect of
+large households on the disease dynamics, one could suggest to vaccinate large
+households first. However, a detailed analysis of this question will be subject of
+follow–up research.
+When extending our ODE SIR–model to an SIRS–model including possible
+reinfection a recombination of the recovered sub–households is required.
+Again,
+this extension will be left for future work. A relaxation of the distribution of the
+recovered sub–household distribution to the overall household distribution seems a
+tractable approach to tackle this issue.
+REFERENCES
+[1] F. Ball, T. Britton, T. House, V. Isham, D. Mollison, L. Pellis, and G. S. Tomba,
+Seven challenges for metapopulation models of epidemics, including households models,
+Epidemics, 10 (2015), pp. 63–67.
+[2] N. G. Becker and K. Dietz, The effect of household distribution on transmission and
+control of highly infectious diseases, Mathematical biosciences, 127 (1995), pp. 207–219.
+[3] J. Del ´Aguila-Mej´ıa, R. Wallmann, J. Calvo-Montes, J. Rodr´ıguez-Lozano, T. Valle-
+Madrazo, and A. Aginagalde-Llorente, Secondary attack rate, transmission and
+incubation periods, and serial interval of sars-cov-2 omicron variant, spain, Emerging
+Infectious Diseases, 28 (2022), p. 1224.
+[4] C. Fraser, D. A. Cummings, D. Klinkenberg, D. S. Burke, and N. M. Ferguson,
+Influenza transmission in households during the 1918 pandemic, American journal of
+epidemiology, 174 (2011), pp. 505–514.
+[5] K. Glass, J. McCaw, and J. McVernon, Incorporating population dynamics into household
+
+SIR–MODELS FOR HOUSEHOLDS
+17
+models of infectious disease transmission, Epidemics, 3 (2011), pp. 152–158.
+[6] T. House and M. J. Keeling, Deterministic epidemic models with explicit household struc-
+ture, Mathematical biosciences, 213 (2008), pp. 29–39.
+[7] T. House, H. Riley, L. Pellis, K. B. Pouwels, S. Bacon, A. Eidukas, K. Jahanshahi,
+R. M. Eggo, and A. Sarah Walker, Inferring risks of coronavirus transmission
+from community household data, Statistical Methods in Medical Research, 31 (2022),
+pp. 1738–1756.
+[8] G. Huber, M. Kamb, K. Kawagoe, L. M. Li, B. Veytsman, D. Yllanes, and D. Zig-
+mond, A minimal model for household effects in epidemics, Physical Biology, 17 (2020),
+p. 065010.
+[9] S. B. Jørgensen, K. Nyg˚ard, O. Kacelnik, and K. Telle, Secondary attack rates for
+omicron and delta variants of sars-cov-2 in norwegian households, JAMA, 327 (2022),
+pp. 1610–1611.
+[10] W. Li, B. Zhang, J. Lu, S. Liu, Z. Chang, C. Peng, X. Liu, P. Zhang, Y. Ling, K. Tao,
+et al., Characteristics of household transmission of covid-19, Clinical Infectious Dis-
+eases, 71 (2020), pp. 1943–1946.
+[11] F. P. Lyngse, L. H. Mortensen, M. J. Denwood, L. E. Christiansen, C. H. Møller,
+R. L. Skov, K. Spiess, A. Fomsgaard, R. Lassauni`ere, M. Rasmussen, et al., House-
+hold transmission of the sars-cov-2 omicron variant in denmark, Nature communica-
+tions, 13 (2022), pp. 1–7.
+[12] Z. J. Madewell, Y. Yang, I. M. Longini, M. E. Halloran, and N. E. Dean, House-
+hold transmission of sars-cov-2: a systematic review and meta-analysis, JAMA network
+open, 3 (2020), pp. e2031756–e2031756.
+[13] K. Niedzielewski,
+J. M. Nowosielski,
+R. P. Bartczuk,
+F. Dreger,
+�L. G´orski,
+M. Gruziel-S�lomka, A. Kaczorek, J. Kisielewski, B. Krupa, A. Moszy´nski, et al.,
+The overview, design concepts and details protocol of icm epidemiological model (pdyn
+1.5), (2022).
+[14] UN statistics division, Demographic and social statistics. https://unstats.un.org/unsd/
+demographic-social/products/dyb/documents/household/4.pdf, 2015. Accessed: 2023-
+01-09.
+[15] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold en-
+demic equilibria for compartmental models of disease transmission, Mathematical bio-
+sciences, 180 (2002), pp. 29–48.
+