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+Low-temperature antiferromagnetic order in orthorhombic CePdAl3
+Vivek Kumar,1, ∗ Andreas Bauer,1, 2 Christian Franz,1, 3 Jan Spallek,1 Rudolf
+Sch¨onmann,1 Michal Stekiel,1 Astrid Schneidewind,3 Marc Wilde,1, 2 and C. Pfleiderer1, 2, 4
+1Physik-Department, Technische Universit¨at M¨unchen, D-85748 Garching, Germany
+2Zentrum f¨ur QuantumEngineering (ZQE), Technische Universit¨at M¨unchen, D-85748 Garching, Germany
+3J¨ulich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), D-85748 Garching, Germany
+4Munich Center for Quantum Science and Technology (MCQST),
+Technische Universit¨at M¨unchen, D-85748 Garching, Germany
+(Dated: January 23, 2023)
+We report the magnetization, ac susceptibility, and specific heat of optically float-zoned single
+crystals of CePdAl3. In comparison to the properties of polycrystalline CePdAl3 reported in the
+literature, which displays a tetragonal crystal structure and no long-range magnetic order, our single
+crystals exhibit an orthorhombic structure (Cmcm) and order antiferromagnetically below a N´eel
+temperature TN = 5.6 K. The specific heat at zero-field shows a clear λ-type anomaly with a broad
+shoulder at TN.
+A conservative estimate of the Sommerfeld coefficient of the electronic specific
+heat, γ = 121 mJ K−2 mol−1, indicates a moderately enhanced heavy-fermion ground state. A twin
+microstructure evolves in the family of planes spanned by the basal plane lattice vectors ao and co,
+with the magnetic hard axis bo common to all twins. The antiferromagnetic state is characterized
+by a strong magnetic anisotropy and a spin-flop transition induced under magnetic field along the
+easy direction, resulting in a complex magnetic phase diagram. Taken together our results reveal a
+high sensitivity of the magnetic and electronic properties of CePdAl3 to its structural modifications.
+I.
+INTRODUCTION
+Cerium-based intermetallic compounds exhibit a vari-
+ety of ground states and various underlying exotic phys-
+ical phenomena, such as unconventional superconductiv-
+ity [1–8], heavy-fermion states [9, 10], non-Fermi liquid
+behavior [11], vibronic hybrid excitations [12–16], and
+complex magnetic order [17–24]. On the phenomenologi-
+cal level, the origin of this remarkable diversity of ground
+states has been attributed to the competition of narrow f-
+electron bands and strong electronic correlations together
+with spin-orbit interaction, crystal electric field (CEF) ef-
+fects, and strong magneto-elastic coupling. An overarch-
+ing theme connecting much of the research in f-electron
+compounds concerns the condition of the emergence of
+magnetic order.
+A class of compounds with the general formula CeTX 3
+(T is a transition metal and X is a p-block element)
+crystallizing in subgroups of the BaAl4-type (I4/mmm)
+tetragonal structure has received special attention [3, 6,
+22–42]. In these compounds, a large number of struc-
+tural variants and diverse magnetic and electrical proper-
+ties can be obtained by changing the transition metal T.
+Many members of this class such as CeRhGe3, CeAuAl3,
+CeCuAl3, and CeCoGe3 adopt a non-centrosymmetric
+tetragonal structure (BaNiSn3-type I4mm) and exhibit
+antiferromagnetic behavior [25–33]. Other members such
+as CeAgAl3 display ferromagnetism with a centrosym-
+metric orthorhombic crystal structure [34, 35]. A spin-
+glass state was reported in non-centrosymmetric tetrago-
+nal CePtAl3 below 0.8 K [32]. Complex magnetic phases
+∗ [email protected]
+have been observed in antiferromagnetic CeNiGe3[22,
+24], CeCoGe3 [23, 36–38] and CePtSi3 [39]. The discov-
+ery of pressure-induced unconventional superconductiv-
+ity in the non-centrosymmetric tetragonal heavy-fermion
+antiferromagnets CeRhSi3, CeIrSi3, CeCoGe3, CeIrGe3,
+and CeRhGe3 even suggests a new direction in condensed
+matter physics [3, 6, 40–42].
+An important aspect is the structural stability of these
+systems and the emergence of different electronic ground
+states. As one of the first examples, CePd2Al2 [13, 15],
+which is closely related to the class of CeTAl3 of ma-
+terials, was found to undergo a structural phase trans-
+formation from a tetragonal to an orthorhombic lat-
+tice at 13.5 K. An inelastic neutron scattering study re-
+vealed three magnetic excitations in the paramagnetic
+phase.
+However, according to Kramer’s theorem, only
+two CEF excitations are expected due to the splitting of
+ground state J = 5/2 of the Ce3+ ion into three doublets
+in tetragonal/orthorhombic point symmetry suggesting
+strong coupling between the crystal fields and the crystal
+structure. Later, Adroja et al. found a similar anomaly
+in CeCuAl3 [14], where a structural instability manifests
+itself in terms of a drastic change in lattice parameters
+of the tetragonal structure around 300 ◦C [29].
+These
+anomalous excitations have been interpreted by means
+of Thalmeier and Fulde’s model of bound states be-
+tween phonons and CEF excitations as generalized to the
+tetragonal point symmetry. Recently, ˇCerm´ak et al. con-
+firmed related hybrid CEF-phonon excitations even for
+weak magnetoelastic coupling in isostructural CeAuAl3
+[43]. Moreover, CePd2Al2, CeCuAl3 and CeAuAl3 order
+antiferromagnetically at low temperatures and exhibit
+incommensurate amplitude-modulated magnetic struc-
+tures [15, 44–46]. The presence of multi-step magnetism
+and complex magnetic phase diagrams suggests the pos-
+arXiv:2301.08617v1  [cond-mat.str-el]  20 Jan 2023
+
+2
+sible existence of topologically non-trivial multi-k struc-
+tures akin to skyrmion lattices [47]. This raises the ques-
+tion, if and how the formation of magnetic order depends
+on the stabilization of specific crystal structure.
+In this paper we focus on CePdAl3. A study of as-
+cast polycrystalline CePdAl3 by Schank et al. in 1994
+revealed a tetragonal I4mm structure with lattice con-
+stants a = 4.343 ˚A and c = 10.578 ˚A [48], where the
+heat treatment at high temperature results in a struc-
+tural phase transformation with an antiferromagnetic or-
+der below TN ≃ 6 K. In contrast, no magnetic order was
+found down to 0.1 K in a recent investigation by Franz
+et al. on single crystalline tetragonal CePdAl3 grown by
+optical float zoning with a growth rate of 6 mm/h [49].
+For the work reported in the following, a single crystal
+was prepared by optical float zoning using a much lower
+growth rate of 1 mm/h. Under these conditions we found
+that CePdAl3 crystallizes in an orthorhombic as opposed
+to a tetragonal structure [50]. In this paper, we report
+comprehensive magnetization, ac susceptibility, and spe-
+cific heat measurements on single crystalline orthorhom-
+bic CePdAl3.
+As our main result we find the charac-
+teristics of antiferromagnetic order below TN = 5.6 K. We
+determine the magnetic phase diagram upto 14 T, where
+we find the emergence of complex magnetic phases un-
+der magnetic fields applied along the easy direction. The
+presence of different structural and magnetic configura-
+tions of CePdAl3 identifies a new example of a material in
+which to search for hybrid excitations and new magnetic
+phases in the future.
+Our paper is organized as follows. After a brief account
+of the experimental methods in Sec. II, we present our ex-
+perimental results in Sec. III. We start with the structural
+properties and notation in Sec. III A, followed by the spe-
+cific heat results in Sec. III B and magnetic susceptibility
+data in Sec. III C. The temperature- and field-dependence
+of the magnetization is presented in Sec. III D. We find
+that the magnetic field-driven transitions for fields ap-
+plied along the easy direction are consistent with the
+specific heat as a function of temperature as presented
+in Sec. III E. In Sec. III F, we examine the magnetic tran-
+sitions in more detail by analyzing the hysteresis of the
+field-dependent magnetic susceptibility. Comprehensive
+datasets allow to infer the magnetic phase diagram pre-
+sented in Sec. III G. The conclusions are summarized in
+Sec. IV.
+II.
+EXPERIMENTAL METHODS
+A single-crystal of CePdAl3 was grown using the op-
+tical floating-zone technique following a process similar
+that described in Ref. [49, 51, 52]. As the main difference,
+the growth rate was reduced from 6 mm/h [49] to 1 mm/h
+which resulted in the formation of an orthorhombic crys-
+tal.
+The crystal structure of CePdAl3 was determined by
+means of single-crystal x-ray diffraction (SCXRD). A
+platelet-shaped crystal with dimensions 50 µm × 40 µm ×
+10 µm was cleaved of the CePdAl3 crystal as grown. The
+platelet was investigated at a Rigaku XtaLAB Synergy-S
+diffractometer, using a Mo x-ray source with λ = 0.71 ˚A
+and a two-dimensional HyPix-Arc 150◦ detector. Bragg
+reflections were indexed using CrysAlisP ro [53] as inte-
+grated with the diffractometer.
+The single crystals were oriented by Laue x-ray diffrac-
+tion and a cuboidal sample was cut with orientations a⋆
+o,
+c⋆
+o and bo as introduced below for the measurement of
+the bulk properties.
+The ac susceptibility, magnetiza-
+tion, and specific heat were measured in a Quantum De-
+sign physical property measurement system (PPMS) at
+temperatures down to 2 K under magnetic fields up to
+14 T. In order to determine the temperature dependence
+of the bulk properties, the sample was first cooled from a
+high temperature, well above TN, to the lowest attainable
+temperature in the absence of a magnetic field. Subse-
+quently, the field was set to the desired value and data
+were collected for increasing temperature. This protocol
+was repeated for different target magnetic fields. The ac
+susceptibility was measured at an excitation amplitude of
+1 mT and an excitation frequency of 911 Hz. The specific
+heat was measured down to 2 K using a large heat-pulse
+method [54]. For temperatures between 0.08 K and 4 K
+the specific heat was measured in a Dryogenic adiabatic
+demagnetization refrigerator using a conventional heat-
+pulse method.
+The field dependence of the magnetization and the ac
+susceptibility was measured using the following temper-
+ature versus field protocol. First, the sample was cooled
+from a high temperature well above TN to the target tem-
+perature in the absence of a magnetic field. Second, data
+as a function of magnetic field were recorded in a se-
+quence of field sweeps from zero-field to 14 T, 14 T to
+-14 T, and -14 T to 14 T.
+The bulk properties recorded on different pieces cut
+from the large single crystal ingot were consistent. The
+temperature and field dependent features along a⋆
+o and
+c⋆
+o were qualitatively identical.
+Therefore, comprehen-
+sive data focused on one of these directions, c⋆
+o, were
+recorded. Summarizing the key result of our study, the
+magnetic phase diagrams of CePdAl3 were inferred. Sig-
+natures detected in measurements as a function of tem-
+perature and magnetic field are labelled as Tj and Hj,
+respectively. For clarity, the same subscript j is assigned
+to the transitions corresponding to the same line in the
+phase diagram.
+III.
+EXPERIMENTAL RESULTS
+A.
+Crystal structure and twinning
+Different crystal growth conditions favor a tetrago-
+nal (I4mm) [49] or orthorhombic crystal structures of
+CePdAl3. By means of single crystal x-ray diffraction,
+we determined that the orthorhombic lattice stabilizes
+
+3
+in the Cmcm space group.
+The lattice parameters at
+room temperature are ao = 6.379 ˚A, bo = 10.407 ˚A and
+co = 5.975 ˚A. The orthorhombic phase exhibits a pseudo-
+tetragonal twinning in the basal plane, evident, for in-
+stance, by the splitting of the Bragg reflections shown in
+Fig. 1(a). The twinning law was determined by index-
+ing all measured reflections with components of the four
+twins presented in Fig. 1(b). An illustration of the twin
+orientation is shown in Figs. 1(c) and (d). The three per-
+pendicular cartesian directions of twins for i = 1, 2, 3, 4
+are denoted by ai
+o, bi
+o and ci
+o, where ai
+o and ci
+o construct
+an effective basal plane and bi
+o mutually represents the
+long axis. The volume fraction of the four twins labelled
+i = 1, 2, 3, and 4 are 0.38, 0.26, 0.23, and 0.13, respec-
+tively. The mismatch angle between the twins numbered
+1 and 2, as well as 3 and 4, are around 3◦.
+Measurements on different pieces cleaved of the sin-
+gle crystalline ingot demonstrate the same twinning
+scheme with minor differences in twin fractions of differ-
+ent twins. An attempt to detwin the crystals by means
+of high-temperature treatment, etching, or cleaving of
+micrometer-sized crystals neither affected the twinning
+as such nor the twinning fractions.
+In turn, measurements in any direction in the effective
+basal plane reflect effectively an admixture of ai
+o and ci
+o
+directions due to the four twins. We define, therefore,
+two mutually perpendicular effective sample directions
+a⋆
+o and c⋆
+o, explicitly taking into account the volume frac-
+tions of the four twins. This definition is schematically
+depicted in Figs. 1(c) and (d) where a⋆
+o is nearly aligned
+along a1,2
+o
+and c3,4
+o , while c⋆
+o is aligned to that of c1,2
+o
+and a3,4
+o . The third crystal direction, corresponding to
+the long axis bo, remains unaffected by the twin defor-
+mations.
+B.
+Temperature-dependence of the specific heat
+The temperature dependence of the specific heat C(T)
+of single-crystalline tetragonal (I4mm) and orthorhom-
+bic (Cmcm) CePdAl3, as well as nonmagnetic polycrys-
+talline tetragonal (I4mm) LaPdAl3 measured in the ab-
+sence of a magnetic field are shown in Fig. 2.
+No evi-
+dence suggesting magnetic order was observed in tetrag-
+onal CePdAl3 [49]. In orthorhombic CePdAl3, a λ-type
+anomaly comprising a peak at 5.4 K followed by a shoul-
+der closely above the transition temperature TN = 5.6 K
+is observed, where the magnetization is characteristic of
+antiferromagnetism as reported below. The behavior ob-
+served is consistent with a previous study of polycrys-
+talline CePdAl3 [48]. Moreover, the properties are rem-
+iniscent of the commensurate to incommensurate mag-
+netic transition reported of other strongly correlated sys-
+tems [55, 56].
+A pronounced shoulder in the specific heat has also
+been seen in other systems, notably, the chiral cubic mag-
+net MnSi [54, 57], where it reflects a change of char-
+acter of the critical spin-fluctuations when approach-
+FIG. 1.
+Twin scheme in the basal plane of orthorhombic
+CePdAl3 as derived from single crystal x-ray diffraction. (a)
+X-ray scattering intensity reconstructed in the H0L plane.
+The splitting of the reflections is characteristic of twin for-
+mation. (b) Indexed reflections of panel (a) with the colors
+corresponding to different twin domains. Schematics of the
+lattice vectors ai
+o and ci
+o of twin i in the basal plane H0L of
+the orthorhombic crystal are depicted in the lower panels (c)
+and (d). Four twins labelled i = 1, 2, 3, and 4 were identi-
+fied. a⋆
+o and c⋆
+o are defined as mutual perpendicular sample
+directions comprising the admixtures of twin lattice vectors.
+ing long-range helimagnetic order and a concomitant
+fluctuation-induced first-order transition. Details of the
+low-temperature specific heat of orthorhombic CePdAl3
+at zero-field are presented in Sec. III E below, which also
+includes data collected at different magnetic fields.
+Above TN, the expression C/T = γ + βT 2, where
+γ and β are the electronic and phononic contributions
+to the specific heat, respectively, has been fitted to the
+specific heat data in the range ∼18 to ∼23 K of or-
+thorhombic CePdAl3. The values obtained for γ and β
+are 234 mJ mol−1 K−2 and 3.437 × 10−4 J mol−1 K−4,
+respectively.
+The Debye temperature, ΘD = 305 K,
+associated with β may be derived using the relation
+β = (12/5)π4nR/Θ3
+D, where n is the number of atoms
+per formula unit and R is the gas constant. The phonon
+contribution to the specific heat in the Debye model [or-
+ange line in Fig. 2] is given by
+Cph,Debye = 9nR
+� T
+ΘD
+�3 � xD
+0
+x4ex
+(ex − 1)2 dx
+(1)
+where xD = ΘD/T. At high temperatures the experimen-
+tal data of tetragonal LaPdAl3 and CePdAl3, as well as
+orthorhombic CePdAl3 approach the Dulong-Petit limit,
+3nR = 15R = 124.7 J mol−1 K−1, where n = 5.
+
+(a)
+(b)
+-2
+2
+0
+1it
+0
+2
+-2
+0
+2
+-2
+0
+2
+H
+H
+(c)
+(d)
+Twin volumes
+*
+*
+1: 38 %
+a.ttas
+ch +t c?
+2: 26 %
+3: 23 %
+4: 13 %
+3
+*
+a
+.4
+The large value of γ = 234 mJ mol−1 K−2 obtained
+from the low-temperature specific heat above TN is typ-
+ical for a heavy-fermion system. It has to be borne in
+mind, however, that evaluating γ at the relatively high-
+temperature range above TN is associated with substan-
+tial uncertainties. A lower bound of γ, fitting the exper-
+imental data in the antiferromagnetic state at tempera-
+tures between ∼0.9 K and ∼3.7 K, yields a value of γ =
+121 mJ mol−1 K−2 still characteristic of heavy-fermion
+behaviour.
+At high temperatures (T > 100 K), the specific heat of
+all three compounds exhibits essentially the same tem-
+perature dependence. However, the specific heat of or-
+thorhombic CePdAl3 is slightly smaller than for tetrago-
+nal CePdAl3, suggesting reduced electronic and phononic
+contributions associated with the reduced crystal sym-
+metry. Compared to nonmagnetic LaPdAl3, the specific
+heat of orthorhombic CePdAl3 is also slightly smaller,
+yet within the experimental error of experiment. Indeed,
+a multiplication with a fraction of 0.99 to the total signal
+of LaPdAl3 fully superimposes the data of CePdAl3 as
+shown in Fig. 3(a) of C/T vs T. The corresponding dif-
+ference in specific heats may be attributed to the mag-
+netic contribution of the specific heat of orthorhombic
+CePdAl3.
+Shown in Fig. 3(b) is a sharp peak at T = 5.4 K in
+the magnetic contribution to the specific heat following
+subtraction of the phonon contribution signaling an an-
+tiferromagnetic transition.
+In addition, a broad maxi-
+mum around 30 K may be discerned as characteristic of
+a Schottky anomaly due to crystal electric field contribu-
+tions.
+In the tetragonal as well as the orthorhombic symme-
+try of the lattice, the degeneracy of the sixfold ground
+state multiplet of the Ce3+ ion splits into three doublet
+states. These lift the first and second excited state with
+respect to the ground state resulting in a contribution to
+the specific heat which can be expressed as [58]
+CCEF =R
+Z
+2
+�
+l=0
+gl
+� El
+kT
+�2
+exp
+�
+− El
+kT
+�
+− R
+Z2
+� 2
+�
+l=0
+El
+kT glexp
+�
+− El
+kT
+��2
+(2)
+where
+Z =
+2
+�
+l=0
+glexp
+�
+− El
+kT
+�
+(3)
+is the partition function, and l = 0, 1 and 2 denote the
+ground, first and second excited states, respectively. The
+degeneracy of the three doublet states is g0 = g1 = g2 =
+2.
+The energy difference E1−E0 = ∆1 and E2−E0 = ∆2
+represent the levels of the first and the second excited
+states, respectively. A fit of the data to Eqn. (2) between
+20 K and 100 K yields ∆1 = 25.4 K and ∆2 = 76.0 K,
+0
+50
+100
+150
+200
+0
+50
+100
+150
+0
+5
+10
+15
+20
+25
+0
+8
+16
+Specific heat C (J mol-1 K-1)
+CePdAl3
+Cmcm
+CePdAl3
+I4mm
+ΘD = 305 K
+Dulong-Petit limit
+LaPdAl3
+I4mm
+Temperature T (K)
+(a)
+Temperature T (K)
+(b)
+TN
+FIG. 2.
+(a) Zero-field specific heat of single-crystalline or-
+thorhombic (black) and tetragonal (blue) [49] CePdAl3 as
+a function of temperature. Data of orthorhombic CePdAl3
+were measured in a Dryogenic system between 0.08 K and 4 K,
+and in a PPMS between 2 K and 200 K. Also shown are the
+specific heat of nonmagnetic polycrystalline LaPdAl3 (Gray
+line) and the Debye fit (orange line) calculated from the low-
+temperature specific heat of the Cmcm structure. The Debye
+temperature is ΘD = 305 K. The Dulong-Petit limit for all
+three compounds, 15R = 124.7 J mol−1 K−1 is depicted by
+a dashed line. (b) The low-temperature part of the specific
+heat of orthorhombic CePdAl3 shows a pronounced λ-type
+anomaly with a broad shoulder at the magnetic transition at
+TN.
+respectively. Note that, the normalized subtraction of the
+LaPdAl3 signal may introduce systematic errors in the
+determination of the precise values of the excited states.
+For instance, subtraction of the signal of LaPdAl3 after
+multiplication with a fraction of 0.98 yields ∆1 = 28.6 K
+and ∆2 = 95.5 K.
+Furthermore, we have calculated the magnetic entropy
+S =
+�
+(C/T)dT presented in Fig. 3(c). At the magnetic
+transition temperature, the entropy reaches the theoret-
+ical value of Rln 2 for a doublet ground state expected
+of Ce3+ ions. When increasing the temperature, the en-
+tropy increases and reaches Rln4 around 30 K, approach-
+ing saturation above 100 K consistent with the scheme of
+crystal electric field levels.
+
+5
+0
+8
+16
+0
+1
+2
+3
+0
+50
+100
+150
+200
+0
+6
+12
+0
+4
+8
+0
+6
+∆C (J mol-1 K-1)
+(b)
+∆1 = 25.4 K
+∆2 = 76.0 K
+ Cmag
+ Fit
+C/T (J mol-1 K-2)
+(a)
+ CePdAl3 (Cmcm)
+ LaPdAl3 (99 %)
+ Difference
+Entropy S (J mol-1 K-1)
+Temperature T (K)
+RLn2
+RLn4
+(c)
+S (J mol-1 K-1)
+T (K)
+RLn2
+FIG. 3. Magnetic contribution to the specific heat and crys-
+tal electric field levels. (a) Specific heat per unit tempera-
+ture, C/T, of orthorhombic CePdAl3 and tetragonal LaPdAl3
+(with a multiplication of a fraction of 0.99) as well as their
+difference. (b) Magnetic specific heat, Cmag, and the fit to
+the expression for the crystal electric field contribution to the
+specific heat yields ∆1 = 25.4 K and ∆2 = 76.0 K (c) Magnetic
+contribution to the entropy. The inset shows the entropy at
+low temperatures.
+C.
+Temperature-dependence of the magnetic
+susceptibility
+The real part of the ac susceptibility, Re χac of or-
+thorhombic CePdAl3 as a function of temperature is
+shown in Fig. 4(a) for a⋆
+o, c⋆
+o and bo. A clear magnetic
+transition is observed at TN = 5.6 K in the low temper-
+ature range, characteristic of the onset of antiferromag-
+netic order as indicated by arrows in the inset. Namely,
+0
+0.5
+1
+1.5
+0
+4
+8
+0
+0.5
+1
+1.5
+0
+100
+200
+300
+0
+1
+2
+0
+20
+40
+0
+4
+Re� ac (10-2) 
+ Hac || a
+�
+o 
+ Hac || c
+�
+o 
+ Hac || bo
+(a)
+Re� ac (10-2) 
+T (K)
+TN
+H/M (102) 
+H/M (103) 
+Temperature T (K)
+ H || a
+�
+o 
+ H || c
+�
+o 
+ H || bo
+� 0H = 0.1 T
+(b)
+T (K)
+FIG. 4. (a) Temperature dependence of the real part of the
+ac susceptibility, Re χac of orthorhombic CePdAl3 measured
+along a⋆
+o, c⋆
+o and bo at an excitation amplitude of 1 mT and
+a frequency of 911 Hz. The inset shows the low-temperature
+part of Re χ ac, reflecting the characteristics of an antiferro-
+magnetic transition at TN = 5.6 K. (b) Susceptibility, H /M,
+as a function of temperature for H ∥ a⋆
+o, H ∥ c⋆
+o and H ∥ bo
+measured in a field of 0.1 T. Gray lines are Curie-Weiss fits.
+The inset shows the data for temperatures below 50 K.
+below TN, Re χac monotonically decreases along a⋆
+o and
+c⋆
+o with decreasing temperature, while slightly increas-
+ing along bo. The magnitude of R eχac along different
+axes differs significantly for T <100 K, indicating size-
+able magnetic anisotropy.
+Figure 4(b) shows the normalized susceptibility, H /M,
+as a function of temperature in a field of 0.1 T for H
+∥ a⋆
+o, H ∥ c⋆
+o and H ∥ bo. In the paramagnetic state
+well above TN, a Curie-Weiss dependence is observed.
+A linear fit to the data above 100 K yields Weiss tem-
+peratures Θa⋆
+W = -0.8 K, Θc⋆
+W = -13.5 K and Θb
+W = -33.0 K
+for H ∥ a⋆
+o, H ∥ c⋆
+o and H ∥ bo, respectively, char-
+acteristic of an antiferromagnetic coupling.
+Moreover,
+the effective moments of 2.39, 2.49 and 2.44 µB per ion
+obtained under magnetic field along a⋆
+o, c⋆
+o and bo, re-
+spectively, are close to the value of 2.54 µB expected for
+a free Ce3+ ion. This might suggest a localized nature
+of the Ce moments in CePdAl3. The deviation of H /
+M from the Curie-Weiss dependence for TN < T < 100 K
+
+6
+shown in the inset of Fig. 4(b) may be related to CEF
+effects and electronic correlations. Furthermore, despite
+the twin deformations, a significant difference between
+the susceptibilities along a⋆
+o and c⋆
+o in the paramagnetic
+state indicates a large anisotropy in the basal plane, char-
+acteristic of an easy-axis system.
+D.
+Magnetization
+The magnetic field dependence of the isothermal mag-
+netization of orthorhombic CePdAl3 at 2 K for H ∥ a⋆
+o, H
+∥ c⋆
+o and H ∥ bo is shown in Fig. 5(a). No hysteresis is ob-
+served. The magnetization varies linearly in the low-field
+region up to 1 T as shown in the inset of Fig. 5(a) con-
+sistent with antiferromagnetic order. For fields along a⋆
+o
+and c⋆
+o, an S-shaped rise is observed in the magnetization
+when further increasing field. A kink around 5.5 T sug-
+gests a field-driven transition. The magnetization values
+at 5.5 T are 0.85 µB for H ∥ a⋆
+o, 0.44 µB for H ∥ c⋆
+o, and
+0.18 µB for H ∥ bo. The magnetization increases mono-
+tonically above this transition where the moments along
+a⋆
+o and c⋆
+o at 14 T, the highest field strength of studied,
+are 1.3 and 0.7 µB per Ce atom, respectively. In compar-
+ison, the magnetization increases linearly with field for
+H ∥ bo. The moment at 14 T is 0.4 µB per Ce atom.
+Keeping in mind the twinned crystal structure, the
+magnetization along a⋆
+o and c⋆
+o represent a mixture of
+the crystallographic ao and co axes. A large quantita-
+tive difference in the magnetization at 5.5 T along a⋆
+o and
+c⋆
+o makes it unlikely, that a metamagnetic transition oc-
+curs at the same field value in the ao and co directions
+in a single twin domain. Instead, it appears most likely
+that the increase in the magnetization corresponds to a
+spin-flop in the ai
+o easy direction of each twin only.
+Shown in Fig. 5(b), (c) and (d) are the isothermal mag-
+netization at various temperatures for H ∥ a⋆
+o, H ∥ c⋆
+o
+and H ∥ bo, respectively.
+The spin-flop transition in
+M (H ) for a⋆
+o and c⋆
+o shifts to lower fields under increas-
+ing temperature and vanishes above TN. In contrast, the
+variation in M (H ) along bo is essentially temperature
+independent at and above TN.
+In order to trace the field-driven magnetic transition,
+we have calculated the differential susceptibility, dM /dH
+from the isothermal magnetization at various tempera-
+tures presented in Fig. 6(a), (b), and (c) for H ∥ a⋆
+o, H
+∥ c⋆
+o and H ∥ bo, respectively. For fields along a⋆
+o and
+c⋆
+o, the transition is characterized by a broad peak at
+∼5.2 T at 2 K, which resolves into two peaks at elevated
+temperatures.
+These peaks exist below TN as marked
+by arrows at the transition fields H3 and H4, following
+the labelling scheme described in Sec. II. With increasing
+temperature the field range between the peaks increases
+and both peaks shift to lower field values. No indication
+exists of a field-induced transition in dM /dH for field
+along bo.
+The evolution of the field-induced transitions may be
+traced in further detail by the temperature dependence
+-12
+-6
+0
+6
+12
+-1.2
+-0.6
+0
+0.6
+1.2
+0
+6
+12
+0
+0.6
+1.2
+0
+6
+12 0
+6
+12
+-1
+0
+1
+-0.05
+0
+0.05
+H || bo
+Magnetization M (� B f.u.-1)
+Magnetic field�� 0H (T)
+T = 2 K
+Magnetic field�� 0H (T)
+H || a
+�
+o 
+H || c
+�
+o 
+(a)
+(b)
+H || a
+�
+o 
+(c)
+M (� B f.u.-1)
+H || c
+�
+o 
+  2 
+  3 
+  4 
+  5 
+  6 
+ 10 
+(d)
+T (K)
+H || bo
+M (� B f.u.-1)
+� 0H (T)
+FIG. 5.
+(a) Isothermal magnetization of orthorhombic
+CePdAl3 at 2 K measured in a field along a⋆
+o, c⋆
+o and bo up
+to 14 T. The arrow indicates the direction of increasing mag-
+netic field. The inset shows the linear variation of the mag-
+netization below 1 T. Typical field dependence of isothermal
+magnetization at various temperatures for (b) H ∥ a⋆
+o, (c) H
+∥ c⋆
+o and (d) H ∥ bo. A field-driven spin-flop transition at
+∼5.5 T is observed below TN for H ∥ a⋆
+o (blue arrow) and H
+∥ c⋆
+o (red arrow).
+0
+2
+4
+6
+0
+2
+4
+6
+0
+0.1
+0.2
+0
+2
+4
+6
++0.025
+H || a
+�
+o
+H || c
+�
+o
+H || bo
+(c)
+10 K
+6 K
+5.5 K
+5 K
+4.5 K
+4 K
+3.5 K
+3 K
+2.5 K
+2 K
+Magnetic field�� 0H (T)
++0.025
+H4
+H3
+(a)
+10 K
+6 K
+5.5 K
+5 K
+4.5 K
+4 K
+3.5 K
+3 K
+2.5 K
+2 K
++0.025
+H4 H3
+(b)
+2 K
+2.5 K
+10 K
+3 K
+3.5 K
+4 K
+4.5 K
+5 K
+5.5 K
+6 K
+Susceptibility dM/dH
+FIG. 6. (a) Susceptibility, dM /dH, calculated from the mea-
+sured magnetization of orthorhombic CePdAl3 for (a) H ∥ a⋆
+o,
+(b) H ∥ c⋆
+o and (c) H ∥ bo. Data are shifted by 0.025 for clar-
+ity. Peaks correspond to field-induced transitions marked by
+arrows at H3 (pink) and H4 (sky blue). The peaks disappear
+above TN.
+
+7
+0
+0.4
+0.8
+0
+0.4
+0.8
+1.2
+0
+0.2
+0.4
+2
+4
+6
+8 10
+0
+2
+4
+2
+4
+6
+8 10
+0
+3
+6
+2
+4
+6
+8 10
+0.4
+0.5
+0.6
+0.7
+T3
+T1
+(b)
+M (� B f.u.-1)
+T4
+T1
+(a)
+� 0H (T)
+T3
+T1
+(c)
+T (K)
+ 1 
+ 2
+ 3
+ 4 
+ 5 
+ 6 
+ 7 
+ 8 
+ 9 
+ 10 
+ 11 
+ 12 
+ 13 
+ 14
+H || c
+�
+o
+T1
+T3
++0.3
+T4
+(e)
+Re� ac (10-2)
+T (K)
+H || a
+�
+o
+T1
+T4
++0.4
+T3
+(d)
+T (K)
+H || bo
+T1
++0.3
+(f)
+FIG. 7.
+Temperature dependence of magnetization, M (T)
+and real part of ac susceptibility, Reχac(T) of orthorhombic
+CePdAl3 in magnetic fields up to 14 T. M (T) is shown in
+panels (a), (b) and (c), and Reχac(T) in (d), (e) and (f) for
+H ∥ a⋆
+o, H ∥ c⋆
+o and H ∥ bo, respectively. Reχac(T) is shifted
+for clarity. Magnetic transitions are marked by vertical lines
+at temperatures T1 (red), T3 (blue) and T4 (green). A complex
+behavior with multiple transitions is present for field along a⋆
+o
+and c⋆
+o between 2 T and 6 T.
+of the magnetization M (T) and the ac susceptibility
+Reχac(T). Shown in Fig. 7 is M (T) and Reχac(T) at
+various fields up to 14 T. By decreasing the temperature,
+orthorhombic CePdAl3 undergoes a phase transforma-
+tion from paramagnetism to antiferromagnetic order at
+a transition temperature T1 (marked by red lines). This
+transition is visible in Reχac(T) in all crystallographic
+directions. The transition at T1 shifts to lower temper-
+atures under increasing field but does not vanish upto
+the highest field of 14 T studied.
+In the intermediate
+field range from 2 T to 6 T, clear changes in M (T) and
+Reχac(T) for field along a⋆
+o [Fig. 7(a) and (d)]) and
+c⋆
+o [Fig. 7(b) and (e)] point to two additional phase
+transitions at temperatures denoted T3 (blue line) and
+T4 (green line).
+These transitions disappear at fields
+above 6 T. For H ∥ bo, only the first transition at T1 is
+observed in M (T) and Reχac(T) [Fig. 7(c) and (f)].
+E.
+Field-dependence of the specific heat
+The specific heat of orthorhombic CePdAl3 as a
+function of temperature at different magnetic fields for
+H ∥ c⋆
+o is presented in Fig. 8.
+At zero magnetic field
+0
+8
+16
+0
+8
+16
+0
+2
+4
+6
+0
+8
+16
+0
+2
+4
+6 0
+2
+4
+6
+0
+1
+2
+3
+4
+5
+6
+0
+8
+16
+� 0H = 0 
+T1
+T2
+(b)
+1 T
+T1
+T2
+T4
+(c)
+2 T
+T1
+T2
+T4
+(d)
+Specific heat C (J K-1 mol-1)
+3 T
+T2
+T1
+T3
+T4
+(e)
+4 T
+T1
+T2
+T3
+T4
+(f)
+5 T
+T1
+T2
+T4
+T3
+(g)
+6 T
+T1
+T2
+T3
+(h)
+Temperature T (K)
+9 T
+T1
+T2
+(i)
+14 T
+T1
+T2
+(j)
+Specific heat C (J K-1 mol-1)
+Temperature T (K)
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+� 0H (T)
+H || c
+�
+o
+(a)
+FIG. 8.
+Specific heat of orthorhombic CePdAl3 as a func-
+tion of temperature under selected magnetic fields up to 14 T
+applied along the c⋆
+o axis. Data measured in the Dryogenic
+system between 0.08 K and 4 K are combined with data mea-
+sured in the PPMS above 2 K. At H = 0 the magnetic transi-
+tion displays a peak at T2 preceded by a broad shoulder with
+a point of inflection at T1. Additional peaks emerge at T3 and
+T4 for magnetic fields between 2 T and 6 T.
+[Fig. 8(b)], a broad shoulder with a point of inflection
+is observed at T1 followed by a sharp peak at T2.
+Increasing the applied field results in a broadening of the
+peak at T2 [Fig. 8(c)] and a splitting with an additional
+peak emerging at a lower temperature T4.
+For even
+higher fields up to 6 T [Fig. 8(d) to (h)], the position
+of T4 continues to shift to lower temperatures with
+the emergence of another peak at T3.
+The emergence
+of the peaks at T3 and T4 in the specific heat in the
+intermediate field range from 2 to 6 T is consistent with
+the phase transitions deduced from the magnetization
+and the ac susceptibility (see Figs. 6 and 7). For fields
+above 6 T, a noticeable shift of T1 and T2 to lower
+temperatures is observed.
+F.
+Field-dependence of the magnetic susceptibility
+In order to investigate the qualitative difference be-
+tween the transitions labelled as H3 and H4 in dM /dH
+
+8
+(see Fig. 6), we have measured the magnetic susceptibility
+as a function of magnetic field between 0 and 14 T. Fig-
+ure 9 shows the real part of the ac susceptibility, Reχac,
+and the susceptibility calculated from the magnetization,
+dM /dH, as a function of increasing and decreasing field.
+At 2 K, dM /dH exhibits two peaks under increasing field,
+first, a pronounced peak at 5.15 T, followed by a second
+broad peak at 5.3 T for both H ∥ c⋆
+o [Fig. 9(a) and (d)]
+and H ∥ a⋆
+o [Fig. 9(c) and (f)].
+The first peak shifts
+to 5 T resulting in a hysteresis, while the second peak
+remains at the same field value under decreasing field.
+Similar effects exists in Reχac where the first peak be-
+comes less pronounced with a smaller hysteresis and a
+slightly lower field of 5.05 T. Also, the value of Reχac
+is slightly lower around the transition. At higher tem-
+peratures, both peaks are shifted to lower field values.
+The hysteresis in Reχac decreases significantly and drops
+below the noise level at 5 K [Fig. 9(b) and (e)].
+Here,
+the magnitude of Reχac matches well with dM /dH ex-
+cept around the first peak. The difference in character
+of the transitions labelled as H3 and H4 suggest their
+intrinsic origin rather than being related to the twinned
+microstructure.
+On the one hand, the hysteresis observed in dM /dH
+and Reχac is reminiscent of changes of population of mul-
+tidomain states. On the other hand, the smaller ampli-
+tude of Reχac as compared to dM /dH indicates the pres-
+ence of slow relaxation processes around the phase tran-
+sition. Similar features are known to trace spin textures
+like helimagnetic disclination or skyrmions in magnetic
+materials [59–61].
+Further experimental investigations
+are needed to explore such a possibility in orthorhombic
+CePdAl3.
+G.
+Magnetic phase diagram
+Combining the features detected in the magnetization
+and the specific heat, we infer the magnetic phase dia-
+grams for field parallel to c⋆
+o and bo shown in Fig. 10(a)
+and (b), respectively. Due to the twinned microstructure,
+the response of the magnetization, specific heat, and ac
+susceptibility are qualitatively alike for H ∥ a⋆
+o and H ∥
+c⋆
+o. In addition, the enhanced signal observed along a⋆
+o
+as compared to c⋆
+o indicates that a⋆
+o reflects a larger frac-
+tion of the easy axis, ao. Therefore, the transitions along
+both a⋆
+o and c⋆
+o reflect equally the phenomenon belonging
+to the easy ao axis of the untwinned single domain.
+Four magnetic regions may be distinguished for field
+along c⋆
+o, denoted AF-I, AF-II, AF-III and AF-IV. At
+low temperature and zero-field, the ground state is de-
+noted as AF-I. With increase temperature, AF-II appears
+at 5.4 K before entering in the paramagnetic (PM) state
+above 5.6 K. Signatures of the AF-II region are detected
+only in the specific heat. The application of a magnetic
+field at low temperature drives a spin-flop transition from
+AF-I to AF-IV with an intermediate region AF-III in a
+narrow field range only. For finite field applied along the
+0
+6
+12
+0
+2
+4
+0
+6
+12
+0
+0.6
+1.2
+2.0
+2.5
+3.0
+0.9
+1.0
+0
+6
+12
+0
+4
+8
+4.8
+5.1
+5.4
+2.0
+2.4
+2.8
+4.8
+5.1
+5.4
+4
+6
+H || c
+�
+o
+T = 2 K
+Magnetic field�� 0H (T)
+(a)
+Re� ac, dM/dH (10-2)
+H || c
+�
+o
+T = 5 K
+Magnetic field�� 0H (T)
+Re� ac, dM/dH (10-2)
+(b)
+1
+Magnetic field�� 0H (T)
+(e)
+Magnetic field�� 0H (T)
+H || a
+�
+o
+T = 2 K
+(c)
+dM/dH
+Re� ac
+Magnetic field�� 0H (T)
+(d)
+H3
+H4
+Magnetic field�� 0H (T)
+(f)
+FIG. 9.
+Details of the magnetic transitions labelled as
+H3 and H4.
+Shown are the real part of ac susceptibility,
+Reχac, and the susceptibility calculated from the magneti-
+zation, dM /dH of orthorhombic CePdAl3 as a function of
+increasing and decreasing field for (a) H ∥ c⋆
+o at 2 K, (b) H
+∥ c⋆
+o at 5 K, and (c) H ∥ a⋆
+o at 2 K. (d), (e) and (f) show the
+magnetic transition regions corresponding to the blue rectan-
+gles in (a), (b) and (c), respectively. Colors denote dM /dH
+for increasing (orange) and decreasing (green) magnetic field.
+Black circles correspond to Reχac for increasing (filled sym-
+bols) and decreasing (open symbols) field, respectively. dM /
+dH was calculated after smoothing the data.
+hard axis, i.e., H ∥ bo [Fig. 10(b)] only the AF-I and PM
+phases were observed, possibly due to the lack of specific
+heat data for finite fields along the hard axis. However,
+the AF-II transition was observed in zero field and the
+AF-II regime is shown in the phase diagram in Fig. 10(b)
+for consistency.
+While the magnetization suggests a collinear antifer-
+romagnetic structure along ao in the AF-I phase, and
+AF-IV shows the characterisics of a spin-flop phase,
+the nature of AF-II and AF-III remain completely un-
+known. Neutron scattering studies under magnetic field
+are needed to determine the nature of the four antifer-
+romagnetic phases we observed in orthorhombic single
+crystal CePdAl3.
+
+9
+0
+2
+4
+6
+0
+4
+8
+12
+0
+4
+8
+12
+Magnetic field � 0H (T)
+Temperature T (K)
+PM
+H || bo
+AF-I
+AF-II
+(b)
+1
+2
+Magnetic field � 0H (T)
+H || c
+�
+o 
+PM
+AF-II
+AF-IV
+AF-I
+(a)
+1
+2
+4
+3
+ C(T)
+ M(T)
+M(H) 
+ T1    
+ T1
+ T2
+ T3    
+ T3    
+ H3
+ T4    
+ T4    
+ H4
+AF-IIl
+FIG. 10.
+Magnetic phase diagram of orthorhombic CePdAl3
+for (a) H ∥ c⋆
+o and (b) H ∥ bo as inferred from the magneti-
+zation and specific heat. Due to crystal twinning, the phase
+diagram for H ∥ a⋆
+o qualitatively resembles the phase diagram
+for H ∥ c⋆
+o shown in (a). Phase transitions are guided by the
+lines which are denoted by numerals j = 1, 2, 3, and 4. The
+associated temperature and field values are labelled as Tj and
+Hj, respectively. Four magnetically ordered phases may be
+distinguished as discussed in the text.
+IV.
+CONCLUSIONS
+In summary, we measured the magnetization, ac
+susceptibility, and specific heat of a single crystal of
+CePdAl3 grown by optical float-zoning.
+A highly
+anisotropic behavior with a twinned orthorhombic crystal
+symmetry was observed. Antiferromagnetic order with
+TN = 5.6 K was observed in terms of transitions in the
+ac susceptibility and specific heat.
+The magnetization
+is characteristic of antiferromagnetic order with an easy
+ao direction in the basal plane. Field-driven transitions
+were detected in the magnetization along the easy di-
+rection, consistent with the ac susceptibility and specific
+heat. Taken together, our study reveals a strong inter-
+play of electronic correlations, complex magnetic order
+and structural modifications in CePdAl3.
+ACKNOWLEDGMENTS
+We wish to thank A. Engelhardt, S. Mayr, and W.
+Simeth for fruitful discussions and assistance with the
+experiments. We thank T. E. Schrader on measurements
+with the Rigaku single-crystal diffractometer in the x-ray
+labs of the J¨ulich Centre for Neutron Science (JCNS).
+This study has been funded by the Deutsche Forschungs-
+gemeinschaft (DFG, German Research Foundation) un-
+der TRR80 (From Electronic Correlations to Function-
+ality, Project No. 107745057, Project E1), SPP2137
+(Skyrmionics, Project No. 403191981, Grant PF393/19),
+and the excellence cluster MCQST under Germany’s Ex-
+cellence Strategy EXC-2111 (Project No. 390814868).
+Financial support by the European Research Council
+(ERC) through Advanced Grants No. 291079 (TOPFIT)
+and No. 788031 (ExQuiSid) is gratefully acknowledged.
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@@ -0,0 +1,2706 @@
+arXiv:2301.01698v1  [hep-th]  4 Jan 2023
+Quantum Energy Inequalities along stationary worldlines
+Christopher J. Fewster
+1,∗ and Jacob Thompson
+1,2,†
+1 Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom.
+2 School of Mathematics and Statistics, The University of Sheffield, Hicks Building, Hounsfield Road,
+Sheffield S3 7RH, United Kingdom.
+January 5, 2023
+Abstract
+Quantum energy inequalities (QEIs) are lower bounds on the averaged energy density of a
+quantum field. They have been proved for various field theories in general curved spacetimes but
+the explicit lower bound is not easily calculated in closed form. In this paper we study QEIs for the
+massless minimally coupled scalar field in four-dimensional Minkowski spacetime along stationary
+worldlines – curves whose velocity evolves under a 1-parameter Lorentz subgroup – and find closed
+expressions for the QEI bound, in terms of curvature invariants. Our general results are illustrated
+by specific computations for the six protoypical stationary worldlines. When the averaging period
+is taken to infinity, the QEI bound is consistent with a constant energy density along the worldline.
+For inertial and uniformly linearly accelerated worldlines, this constant value is attained by the
+Minkowski and Rindler vacuums respectively. It is an open question as to whether the bounds for
+other stationary worldlines are attained by other states of interest.
+1
+Introduction
+Even if a classical field theory obeys local energy conditions, such as positivity of energy density, the
+corresponding quantum field theory (QFT) will fail to do so, as a result of a general theorem [7].
+In fact, it is typical that the expectation value of energy density at any given point can be made
+arbitrarily negative by a suitable choice of the quantum state [9]. Nonetheless, in many QFT models,
+local averages of the expected energy density are bounded below by Quantum Energy Inequalities
+(QEIs), independent of the state.
+Starting with results of Ford and Roman [17, 18, 19] QEIs have been derived for a variety of
+quantum fields in flat and curved spacetimes. References and discussion may be found in the recent
+reviews [11, 32]. For example, consider the real scalar field of mass m ≥ 0 in any globally hyperbolic
+spacetime (M, g), recalling that global hyperbolicity demands only that the spacetime possesses a
+global Cauchy surface. Let γ(s) be any smooth timelike curve, parameterised by proper time. It was
+shown in [8] that the energy density of the quantum field along γ obeys the QEI
+� ∞
+−∞
+ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ −
+� ∞
+0
+dα
+π
+�
+g ⊗ gT(−α, α) > −∞,
+(1.1)
+which holds for all real-valued compactly supported smooth test functions g, and all Hadamard states
+ω of the field. Here, the hat denotes a Fourier transform, defined according to the convention ˆg(α) =
+� ∞
+−∞ ds eiαsg(s), and we employ units where ℏ = c = 1, which will be in force throughout this paper. On
+the left-hand side, the normal ordering is conducted with respect to an arbitrary Hadamard reference
+state ω0, whose two-point function is used to construct the distribution T(s, s′) that appears on the
+right-hand side. Recall also that the Hadamard states form a large class of physically reasonable states,
+determined by their short-distance structure [29, 37]. The two most important features of the QEI (1.1)
+∗[email protected]
+†[email protected]
+1
+
+are that the right-hand side is completely independent of the state ω, and that the bound is finite –
+which is proved using the microlocal properties of Hadamard states uncovered by Radzikowski [39].
+Discussion of QEIs for other QFTs, including non-free models, may be found in [11, 32]; see [21] for
+a very recent development.
+Although the lower bound in (1.1) is explicit and rigorous, it is not easy to compute in closed
+form except in special cases.
+To the best of our knowledge this has only been achieved when T
+exhibits translational invariance T(s + r, s′ + r) = T(s, s′) which occurs, for instance, when (M, g)
+is a stationary spacetime, γ is a timelike Killing orbit and ω0 is stationary. Translational invariance
+allows us to write, with an abuse of notation, T(s, s′) = T(s − s′), from which one easily finds that
+the QEI (1.1) simplifies to
+� ∞
+−∞
+ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ −
+� ∞
+−∞
+dα|ˆg(α)|2Q(α),
+(1.2)
+where
+Q(α) =
+1
+2π2
+� α
+−∞
+du ˆT(u);
+(1.3)
+the QEI (1.2) is also valid for complex-valued g. Taking the massless free field as an example, averaging
+along an inertial worldline in Minkowski space and using the Minkowski vacuum as the reference state
+ω0, this results in Q(α) = α4Θ(α)/(16π3). Using the evenness of |ˆg|2 together with Parseval’s theorem
+then yields
+� ∞
+−∞
+ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ −
+1
+16π2
+� ∞
+−∞
+ds|g′′(s)|2.
+(1.4)
+Similar expressions are known for massive fields and in Minkowski spacetime of general dimension [12];
+for some curved spacetime examples see [15]. Another explicit example arises where γ is a uniformly
+linearly accelerated worldline in four-dimensional Minkowski spacetime with proper acceleration a, in
+which case the QEI (1.2) becomes [13]
+� ∞
+−∞
+ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ −
+1
+16π2
+� ∞
+−∞
+ds
+�
+|g′′(s)|2 + 2a2|g′(s)|2 + 11
+30a4|g(s)|2
+�
+,
+(1.5)
+and is again valid for all Hadamard states ω and complex-valued test functions g.
+Using such expressions the scaling behaviour of the bound is easily understood and phenomena
+such as ‘quantum interest’ may be explored [20, 16, 11]. For example, let gλ(s) = λ−1/2g(s/λ), where
+g is normalised so that
+� ∞
+−∞ ds|g(s)|2 = 1. Then (1.5) implies
+lim inf
+λ−→∞
+� ∞
+−∞
+ds|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − 11a4
+480π2 ,
+(1.6)
+reducing to the Averaged Weak Energy Condition (AWEC)
+lim inf
+λ−→∞
+� ∞
+−∞
+dτ|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ 0
+(1.7)
+in the limit a → 0, which can also be obtained directly from (1.4). An interesting observation is that
+the lower bound in (1.6) is exactly the constant energy density of the Rindler vacuum state along the
+accelerated worldline, while the lower bound in (1.7) is the energy density of the Minkowski vacuum.
+As closed form expressions for QEI bounds are relatively few in number, it is of interest to find
+others. The purpose of this paper is to present a calculation of the QEI bound for a massless scalar
+field along any stationary worldline in 4-dimensional Minkowski spacetime. By a stationary worldline,
+we mean any timelike curve γ(s), parameterised by proper time s, whose velocity vector evolves under
+a 1-parameter subgroup of the Lorentz group: ˙γ(s) = exp(sM) ˙γ(0) for some fixed M ∈ so(1, 3) and
+future-pointing unit timelike ˙γ(0).
+Stationary worldlines have a long history. Kottler [33], Synge [41] and Letaw [34] (see also [36]) all
+obtained them as the solutions to four-dimensional Frenet-Serret equations subject to constancy of the
+curvature invariants; the name ‘stationary worldlines’ is due to Letaw. The three curvature invariants
+2
+
+are the curvature, which measures the proper acceleration, and the torsion and hypertorsion, which
+specify its proper angular velocity. More details are given in Section 2. Stationary worldlines are
+equivalently described as the orbits of timelike Killing vector fields in Minkowski spacetime. There
+are also overlaps with the theory of rigid motions in special relativity that goes back to Born [1] and
+Herglotz [25]; in particular, any rotational rigid motion is the flow of a timelike Killing vector by the
+Herglotz–Noether theorem, although the same theorem allows any C2 timelike curve to be a flow line
+of an irrotational rigid motion. See [22] for discussion and references.
+By a Poincar´e transformation, any stationary worldline can be reduced to one of six prototypes:
+the inertial, uniformly linearly accelerated, and uniformly rotating worldlines are all familiar, while
+the three remaining ones have spatial projections corresponding to a semicubical parabola, a catenary
+or a helix. We will give more detail as we discuss each case separately.
+The main result of this paper is that the QEI (1.2) along any stationary worldline in Minkowski
+spacetime may be given explicitly as
+�
+ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ −
+1
+16π2
+� ∞
+−∞
+ds
+�
+|g′′(s)|2 + 2A|g′(s)|2 + B|g(s)|2�
+,
+(1.8)
+where A and B are expressed in terms of the curvature κ, torsion τ, and hypertorsion υ as
+A = κ2 + τ 2 + υ2
+(1.9)
+and
+B = 1
+90
+�
+3κ4 + 62κ2τ 2 + 30(κ2 + τ 2 + υ2)2�
+,
+(1.10)
+and the inequality (1.8) holds for all Hadamard states ω and all smooth compactly supported test
+functions g.
+To interpret the QEI (1.8), it is useful to consider its scaling behaviour. As before, we take a test
+function gλ which is just a scaled version of the test function g, namely gλ(s) = λ−1/2g(s/λ), so the
+support width of gλ is proportional to λ. Observing that g(k)
+λ (s) = λ−k−1/2g(k)(s/λ), we find
+�
+ds|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ − ∥g′′∥2
+16π2λ4 − A∥g′∥2
+8π2λ2 − Treg(0)∥g∥2,
+(1.11)
+where the norms are those of L2(R), i.e., ∥g∥2 =
+� ∞
+−∞ ds|g(s)|2.
+Here we have written Treg(0) =
+B/(16π2) for reasons that will become clear later – see, for example, equation (1.13) and the ar-
+guments presented in Section 3.
+For sampling times shorter than the curvature scales, i.e., λ ≪
+min{κ−1, τ −1, υ−1}, the leading term dominates, reflecting the fact that any worldline is approxi-
+mately inertial on short enough timescales. At intermediate and long timescales relative to curvature
+scales, the bound will receive corrections from, and eventually be dominated by the last two terms
+in (1.11), showing that the QEI is sensitive to the curvature invariants of the worldline γ. In the limit
+λ → +∞, and with g normalised so that ∥g∥ = 1, we obtain the remarkably simple formula
+lim inf
+λ−→∞
+� ∞
+−∞
+ds|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ −Treg(0),
+(1.12)
+which bounds the average energy density along the entire trajectory. In particular, the QEI is con-
+sistent with the existence of a constant renormalised energy density −Treg(0) along γ, and this is the
+most negative value that any constant energy density could take. An intriguing question is whether
+or not this value is attained by some Hadamard state, or a sequence of Hadamard states in a limiting
+sense, which we will address in Section 6.
+The derivation of (1.8) requires a number of innovations. Although the point-split energy density
+can be obtained easily enough for any given stationary worldline, its Fourier transform does not have
+a closed form – as far as we know – for three of the six prototypes. In Section 3, we develop a new
+method for computing the QEI bound for massless fields in four-dimensions that avoids the use of
+Fourier transforms. The result is that the QEI will take the form (1.8) provided that the point-split
+energy density takes the form
+T(s, s′) = lim
+ǫ→0+
+�
+3
+2π2(s − s′ − iǫ)4 −
+A
+4π2(s − s′ − iǫ)2
+�
++ Treg(s − s′),
+(1.13)
+3
+
+where the regular part Treg must satisfy various conditions, whereupon the coefficient B is given by B =
+16π2Treg(0) as before. In Section 4, we apply these ideas to stationary worldlines, resulting in formulae
+for the point-split energy density in terms of functions easily computed from the Lorentzian distance
+between two points on the curve and a tetrad that is adapted to it, in a manner we describe. Most of the
+required conditions on Treg follow directly from this analysis, and the values A and B are identified
+in terms of Taylor coefficients of these functions. Appendix A gives more detail on our methods,
+while in Appendix B the relevant Taylor coefficients are evaluated in terms of curvature invariants
+thus establishing (1.9) and (1.10). In Section 5, we work through each prototype in turn, providing
+explicit formulae for the point-split energy density that allow the remaining technical condition to
+be verified, and also as a check on our Taylor series calculations. In three cases, (inertial worldlines,
+linearly accelerated worldlines and the semicubical parabola), a closed form may be found for ˆT,
+and we can also check our calculations by using (1.2) and (1.3). Finally, in Section 6, we discuss
+the physical significance of our results and some open problems. Two further appendices contain
+additional computations: Appendix C computes a quantum inequality for the Wick square along
+stationary worldlines following the same general method of the main text, while Appendix D records
+the calculation of the minimally coupled stress-energy tensor in the Rindler vacuum and Rindler
+thermal states, which is needed for our discussion.
+2
+Stationary worldlines
+Throughout this paper we work on 4-dimensional Minkowski spacetime, with metric η = dt2 − dx2 −
+dy2 − dz2, and we employ the inertial coordinates (t, x, y, z) except where otherwise specified.
+A
+stationary worldline is any smooth curve γ : R → R4, whose velocity vector ˙γ is a future-pointing unit
+timelike vector evolving under a 1-parameter subgroup of the Lorentz group SO(1, 3), i.e.,
+˙γµ(s) = exp(sM)µ
+ν ˙γν(0),
+(2.1)
+where M is any fixed element of so(1, 3) (which requires precisely that Mµν is antisymmetric). As
+every component of exp(sM) is analytic in s, it follows that the Cartesian components of ˙γ(s) and, in-
+tegrating, the Cartesian coordinates of γ(s) are also s-analytic. An equivalent definition of a stationary
+worldline is that γ is an orbit of a future-pointing timelike Killing vector field
+ξµ(x) = Mµ
+ν(xν − γ(0)ν) + ˙γµ(0),
+(2.2)
+which is necessarily timelike in a neighbourhood of γ and future-pointing unit vector on γ.
+Finally, stationary worldlines can also be described as the solutions to the Frenet-Serret equations
+with constant curvatures [33, 41, 34]. Here, the curvature invariants of a general timelike curve γ(s),
+parameterised by proper time, are defined as follows. Suppose a right-handed tetrad eµ
+a has been
+chosen along γ so that
+γ(k+1)(s) ∈ span{e0(s), . . . , ek(s)}
+(0 ≤ k ≤ 3),
+and
+˙γ(s) = e0(s),
+(2.3)
+in which case we say that eµ
+a is adapted to γ. If the tetrad also satisfies
+e1(s)µ¨γ(s)µ ≤ 0,
+e2(s)µ...γ (s)µ ≤ 0,
+(2.4)
+then it will be called a Frenet–Serret tetrad. If the tetrad is defined by ea(s) = exp(sM)ea(0), then it
+is adapted (respectively, Frenet–Serret) if and only if (2.3) holds at s = 0 (resp., (2.3) and (2.4) hold
+at s = 0). Explicit formulae resulting from a Gram–Schmidt procedure are given in [34]. Expanding
+the derivatives of the tetrad vectors in terms of the tetrad, one obtains the generalized Frenet–Serret
+equations
+˙eµ
+a = K b
+a eµ
+b ,
+(2.5)
+where Kab is antisymmetric and tridiagonal (due to (2.3)). Thus it takes the form
+K••(s) =
+
+
+
+
+0
+−κ(s)
+0
+0
+κ(s)
+0
+−τ(s)
+0
+0
+τ(s)
+0
+−υ(s)
+0
+0
+υ(s)
+0
+
+
+
+ ,
+(2.6)
+4
+
+which defines the curvature κ, torsion τ and hypertorsion υ. Here, and elsewhere in this paper, bullets
+are used to indicate tensorial type, when displaying tensorial components in vector or matrix form.
+Explicitly, one has
+κ = e0µ ˙eµ
+1 = −e1µ ˙eµ
+0,
+τ = e1µ ˙eµ
+2 = −e2µ ˙eµ
+1,
+υ = e2µ ˙eµ
+3 = −e3µ ˙eµ
+2.
+(2.7)
+The choices made when specifying the Frenet–Serret tetrad ensure that κ and τ are nonnegative, while
+υ can take any real value.
+As the curvature invariants are constant along stationary worldlines, it is easy to compute higher
+derivatives of the tetrad,
+dk
+dsk eµ
+a = (Kk) b
+a eµ
+b ,
+(Kk) b
+a = K c1
+a
+K
+c2
+c1
+· · · Kck−1
+b.
+(2.8)
+For example, the first three derivatives of the velocity u = ˙γ may be computed as
+˙uµ = ˙eµ
+0 = κeµ
+1,
+¨uµ = κ2eµ
+0 + κτeµ
+2,
+...u µ = κ(κ2 − τ 2)eµ
+1 + κτυeµ
+3.
+(2.9)
+It is also possible to give a general formula for γ(s) in terms of M, γ(0) and ˙γ(0). As M•
+• is
+antisymmetric with respect to η, there is a unique decomposition
+˙γ(0)µ = Mµ
+νvν + kµ,
+(2.10)
+where Mµ
+νkν = 0. One then has
+γ(s)µ = exp(sM)µ
+νvν + skµ + γ(0)µ − vµ.
+(2.11)
+Any stationary worldline γ may be related to one of six basic types by a proper orthochronous
+Poincar´e transformation. Note that γ(s) is determined by the initial position, γ(0) ∈ R4, the initial
+four-velocity ˙γ(0) and the element M ∈ so(1, 3) that fixes the evolution ˙γ(s) = exp(sM) ˙γ(0). Under
+a Poincar´e transformation x �→ Λx + w, γ is mapped to ˜γ(s) = Λγ(s) + w, whose velocity evolves
+according to the 1-parameter Lorentz subgroup exp
+�
+sΛMΛ−1�
+and which is therefore also a stationary
+worldline. As the Lorentz transformation maps a Frenet–Serret tetrad for γ to a Frenet–Serret tetrad
+for ˜γ, it follows from (2.7) that the curvature invariants of ˜γ are identical to those of γ. Using the
+classification of conjugacy classes in so(1, 3) [40], we may choose Λ in such a way that ˜
+M = ΛMΛ−1
+is one of five possible types: (a) the zero element, generating the trivial subgroup of SO(1, 3), (b) a
+generator of boosts in the tx-plane, corresponding to a hyperbolic subgroup of SO(1, 3), (c) a generator
+of rotations in the yz-plane, corresponding to an elliptic subgroup of SO(1, 3), (d) a generator of a null
+rotation that fixes the null vector ∂t + ∂x but acts nontrivially on all other null vectors, corresponding
+to a parabolic subgroup of SO(1, 3); (e) the sum of a generator of boosts in the tx-plane and a
+generator of rotations in the yz plane, corresponding to a loxodromic subgroup of SO(1, 3). In each
+case, Lorentz transformations that commute with the 1-parameter subgroup in question can be used
+to arrange that ˙˜γ(0) takes a convenient form.
+Taking these possibilities in turn: in case (a), all Lorentz transformations commute with the trivial
+subgroup, so we may without loss assume that ˜γ(s) = (s, 0, 0, 0). In case (b), the subgroup of boosts
+parallel to the x-axis commutes with itself and the subgroup of rotations in the yz-plane. Thus, we
+may arrange that ˙˜γ(0) = cosh χ∂t + sinh χ∂y for some χ ∈ R,1 leading to two subcases: χ = 0, in
+which case (after possible translation)
+˜γ(s) = (a−1 sinh as, a−1 cosh as, 0, 0)
+(2.12)
+is a uniformly linearly accelerated worldline with a ̸= 0, or χ ̸= 0, in which case (up to translations)
+˜γ(s) = (a−1 cosh χ sinh as, a−1 cosh χ cosh as, −s sinh χ, 0)
+(2.13)
+is a catenary. The curvature invariants (in either subcase) are κ = |a| cosh χ and τ = |a sinh χ|, while
+the hypertorsion is υ = 0. For convenience, the curvature invariants for all six prototypes are tabulated
+in Table 1, in agreement with [36].
+1We could even arrange that χ ≥ 0, but it is convenient not to insist on this.
+5
+
+Inertial
+Linear Acc.
+Catenary
+Parabolic
+Elliptic
+Loxodromic
+κ = τ = υ = 0
+κ > 0
+κ > τ > 0
+κ = τ > 0
+τ > κ > 0
+κ, τ > 0
+τ = υ = 0
+υ = 0
+υ = 0
+υ = 0
+υ ̸= 0
+κ
+0
+|a|
+|a| cosh χ
+|a|
+rω2
+√
+C2a2 + V 2ω2
+τ
+0
+0
+|a sinh χ|
+|a|
+|ω|
+�
+1 + (rω)2
+(a2 + ω2)C|V |/κ
+υ
+0
+0
+0
+0
+0
+aω/κ
+Table 1: Curvature invariants for the stationary worldlines.
+In case (c), the 1-parameter parabolic subgroup takes the form
+P •
+•(s) =
+
+
+
+
+1 + (as)2/2
+−(as)2/2
+0
+as
+(as)2/2
+1 − (as)2/2
+0
+as
+0
+0
+1
+0
+as
+−as
+0
+1
+
+
+
+ = exp
+
+
+
+
+0
+0
+0
+as
+0
+0
+0
+as
+0
+0
+0
+0
+as
+−as
+0
+0
+
+
+
+
+(2.14)
+for some constant nonzero a ∈ R, and commutes with Lorentz transformations of the form
+Λ•
+• =
+
+
+
+
+1 + r2/2
+−r2/2
+r cos θ
+r sin θ
+r2/2
+1 − r2/2
+r cos θ
+r sin θ
+r cos θ
+−r cos θ
+1
+0
+r sin θ
+−r sin θ
+0
+1
+
+
+
+
+(2.15)
+which can be used to bring the initial velocity into the form ˙˜γ(0) = cosh χ∂t+sinh χ∂x for some χ ∈ R.
+Conjugating P •
+•(s) with a boost in the tx-plane results in P •
+•(λs) for some λ > 0; in other words
+effectively rescaling a. Therefore there is no loss of generality in assuming that the initial 4-velocity
+is ˙˜γ(0) = ∂t, in which case the worldline (up to translation) is the semicubical parabola,
+˜γ(s) =
+�
+s + 1
+6a2s3, 1
+6a2s3, 0, 1
+2as2
+�
+.
+(2.16)
+Next, the elliptic subgroup in case (d) commutes with boosts in the tx-plane and rotations in
+the yz-plane. Accordingly, we may arrange the initial velocity to be ˙˜γ(0) = cosh χ∂t + sinh χ∂z for
+some χ ∈ R; the special case χ = 0 corresponds to inertial motion and may be discarded. Up to a
+translation, this results in the uniformly rotating worldline
+˜γ•(s) = (s cosh χ, 0, r cos ωs, r sin ωs) ,
+(2.17)
+where the radius r > 0 and proper angular velocity ω ̸= 0 are related to the initial rapidity by
+rω = sinh χ. The proper acceleration is κ = rω2, while the torsion is τ = |ω|
+�
+1 + (rω)2 and the
+hypertorsion vanishes.
+Lastly, in case (e), the loxodromic subgroup is generated by a linear combination of a tx-boost
+generator and a yz-rotation generator.
+As it commutes with tx-boosts and yz-rotations, we may
+assume without loss that the initial velocity is ˙˜γ(0) = cosh χ∂t + sinh χ∂z for χ ∈ R \ {0}; the
+possibility χ = 0 corresponds to a hyperbolic worldline and is rejected. Up to a translation, this
+results in the worldline
+γ•(s) = (Ca−1 sinh(as), Ca−1 cosh(as), V ω−1 cos(ωs), V ω−1 sin(ωs)),
+(2.18)
+where C = cosh χ and V = sinh χ, which undergoes both rotation in the yz-plane at constant proper
+angular velocity ω ̸= 0 and constant distance |V/ω| from the x-axis, while undergoing uniform acceler-
+ation in the x-direction controlled by a ̸= 0 (the cases where one or both of a or ω vanish are already
+covered under (a), (b) and (d)). The curvature invariants for this worldline are
+κ =
+�
+C2a2 + V 2ω2,
+τ = (a2 + ω2)C|V |/κ,
+υ = aω/κ.
+(2.19)
+6
+
+3
+Reformulation of the QEI bound
+We study the massless minimally coupled scalar field in 4-dimensional Minkowski spacetime, with field
+equation □φ = ηµν∇µ∇νφ = 0 and classical stress-energy tensor
+Tµν = (∇µφ)∇νφ − 1
+2ηµνηαβ(∇αφ)∇βφ.
+(3.1)
+Consider an observer following a timelike curve γ, parameterised by proper time, with 4-velocity
+uµ = ˙γµ. This observer sees energy density
+Tµνuµuν = 1
+2
+3
+�
+a=0
+(eµ
+a∇µφ)2,
+(3.2)
+where eµ
+a (0 ≤ a ≤ 3) is a tetrad defined around γ with eµ
+0|γ = uµ.
+In quantum field theory, the stress-energy tensor requires renormalisation. Let
+G(x, x′) = ⟨φ(x)φ(x′)⟩ω
+(3.3)
+be the Wightman function of the field in a state ω. The Wick square has expectation value
+⟨:φ2(x):⟩ω = (G − G0)(x, x),
+(3.4)
+where
+G0(x, x′) = lim
+ǫ→0+
+−1
+4π2((t − t′ − iǫ)2 − ∥x − x′∥2)
+(3.5)
+is the Wightman function of the Poincar´e invariant vacuum ω0. This expression makes sense if (like
+ω0) ω is a Hadamard state [29, 37], because the difference G−G0 is then a smooth function. Similarly,
+the renormalised stress-energy tensor has expectation value
+⟨:Tµν(x):⟩ω = Dµν(x) − 1
+2ηµνηαβDαβ(x),
+(3.6)
+where
+Dµν(x) = [[(∇ ⊗ ∇)(G − G0)]]µν (x)
+(3.7)
+and the double square brackets denote a coincidence limit.
+Although the classical energy density (3.2) is everywhere nonnegative, the quantised energy density
+may assume negative expectation values. The QEIs provide lower bounds on averaged expectation
+values, for which a prototype is a lower bound on the following expression
+�
+ds|g(s)|2⟨:(Qφ)2:⟩ω(γ(s)),
+(3.8)
+where Q is a partial differential operator with smooth real coefficients and g ∈ C∞
+0 (R) is a smooth
+real-valued test function. In the case where Q is the identity, (3.8) is an averaged Wick square, while
+by considering a sum of similar terms for Qa = 2−1/2eµ
+a∇µ for 0 ≤ a ≤ 3, we can bound averages of
+the energy density along γ.
+A lower bound on (3.8) was established in [8] – in fact the bound applies to general timelike curves
+in arbitrary globally hyperbolic spacetimes for massive as well as massless fields. In our case it asserts
+that
+� ∞
+−∞
+ds|g(s)|2⟨:(Qφ)2:⟩ω(γ(s)) ≥ −
+� ∞
+0
+dα
+π
+÷
+g ⊗ gT (−α, α) > −∞
+(3.9)
+holds for all real-valued compactly supported smooth test functions g, and all Hadamard states ω,
+where
+T(s, s′) = ⟨Qφ(γ(s))Qφ(γ(s′))⟩ω0 = ((Q ⊗ Q)G0)(γ(s), γ(s′)).
+(3.10)
+Here, the vacuum two-point function enters because normal ordering is performed relative to the
+vacuum; the general results of [8] also allow for any Hadamard state to be used as the reference state
+for this purpose. At a more formal level, T is the pull-back of the distribution (Q ⊗ Q)G0 by the map
+7
+
+(s, s′) �→ (γ(s), γ(s′)), and its existence is owed to the special properties of the Hadamard condition
+and the fact that γ is timelike – see [8] for full details and rigorous proofs.
+As already mentioned, a QEI for the energy density involves a sum of such bounds, leading to (1.1)
+with
+T(s, s′) = 1
+2
+3
+�
+a=0
+((∇ea ⊗ ∇ea)G0)(γ(s), γ(s′)).
+(3.11)
+While it is usually not hard to obtain the distribution T for a given timelike curve in Minkowski
+spacetime, assuming that G0 is given, it is not usually possible to find the Fourier transform required
+to compute the QEI bound (3.9) in closed form.
+The situation is somewhat simplified if T(s, s′) is translationally invariant, in which case one has
+the bound given by (1.2) and (1.3). This can be taken a little further, on observing that |ˆg(α)|2 is
+even, so only the even part Qeven(α) = 1
+2(Q(α) + Q(−α)) of Q contributes to (1.2), resulting in the
+bound
+�
+ds|g(s)|2⟨:(Qφ)2:⟩ω(γ(s)) ≥ −
+� ∞
+−∞
+dα|ˆg(α)|2Qeven(α),
+(3.12)
+which is the final form of our prototypical quantum inequality.
+A convenient expression for Qeven may be found by manipulating equation (1.3) in the following
+way:
+Qeven(α) =
+1
+4π2
+�� α
+−∞
+ˆT(u) du +
+� −α
+−∞
+ˆT(u) du
+�
+=
+1
+4π2
+�
+2
+� 0
+−∞
+ˆT(u) du +
+� α
+0
+ˆT(u) du −
+� α
+0
+ˆT(−u) du
+�
+=
+1
+2π2
+�� 0
+−∞
+ˆT(u) du +
+� α
+0
+ˆTodd(u) du
+�
+,
+(3.13)
+where ˆTodd(u) = 1
+2( ˆT(u) − ˆT(−u)). In the above calculation, ˆT is assumed to be continuous, as is the
+case for the examples we will study.
+Evaluating Qeven from (3.13) requires several steps. Computing T is a tedious but straightforward
+calculation best handled using computer algebra. In the simplest cases, the transform may be eval-
+uated in closed form, which (as will be seen later) is the case for the inertial, uniformly accelerated
+and semicubical parabola worldlines, but is not possible (to our knowledge) in the case of the other
+stationary worldlines. However, this obstacle can be circumvented, as we now describe.
+Using the Minkowski vacuum as the reference state, we will show in Section 4 that the point-split
+energy density along a stationary worldline may be written in the form
+T(s, s′) = Tsing(s − s′) + Treg(s − s′),
+(3.14)
+where Tsing is given by the distributional limit
+Tsing(s) = lim
+ǫ→0+
+�
+3
+2π2(s − iǫ)4 −
+A
+4π2(s − iǫ)2
+�
+(3.15)
+for some constant A (the sign is chosen for later convenience) and Treg is smooth, real and even, and
+decaying as O(s−2) as |s| → ∞. In particular, Treg is absolutely integrable and has a well-defined
+Fourier transform that is continuous, real and even. Therefore it does not contribute to ˆTodd. Turning
+to Tsing, its leading singularity is universal, essentially because all stationary worldlines resemble
+inertial worldlines on sufficiently short timescales. The specific coefficient is fixed by the Hadamard
+form and the definition of the energy density along the curve. Meanwhile the coefficient A carries
+information about the specific curve at hand. The Fourier transform of Tsing, in our convention, is
+ˆTsing(u) = 1
+2π(u3 + Au)Θ(u),
+(3.16)
+8
+
+where Θ is the Heaviside distribution. Evidently Tsing does not contribute to the first term in (3.13),
+while the odd part of ˆT is
+ˆTodd(u) = 1
+4π (u3 + Au),
+(3.17)
+recalling that ˆTreg is even. We now have Qeven in the form
+Qeven(α) =
+1
+2π2
+�� 0
+−∞
+du ˆTreg(u) + 1
+4π
+� α
+0
+du(u3 + Au)
+�
+=
+1
+32π3 (α4 + 2Aα2) + Treg(0)
+2π
+,
+(3.18)
+where we have again used the evenness of ˆTreg and the Fourier inversion formula. Inserting (3.18)
+into (3.12) and using Parseval’s theorem gives the QEI bound
+�
+ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ −
+1
+16π2
+� ∞
+−∞
+ds
+�
+|g′′(s)|2 + 2A|g′(s)|2 + B|g(s)|2�
+,
+(3.19)
+where B = 16π2Treg(0).
+The upshot of this analysis is a direct route to the QEI once the point-split expression T is obtained;
+all that is needed is to isolate the appropriate values of A and Treg(0), avoiding the need to compute ˆT
+explicitly. This apparent royal road is made possible because of the special structure of the Minkowski
+vacuum two-point function for the massless scalar field in four dimensions – closely related to Huygens’
+principle. A similar analysis for a QI on the Wick square can be found in Appendix C.
+4
+Computation of the point-split energy density
+In this section we establish that the point-split energy density along stationary worldlines obeys
+equations (3.14) and (3.15), and also that Tsing and Treg have the properties mentioned above, with
+one exception that will be treated by examining the six prototypical cases in Section 5.
+Let γ be any stationary worldline with ˙γ(s) = exp(sM)˙γ(0) and ˙γ(0) a future-pointing unit timelike
+vector. Suppose that
+ea(s) = exp(sM)ea(0)
+(0 ≤ a ≤ 3)
+(4.1)
+is an adapted frame on γ satisfying (2.3). In general there may be many possible adapted tetrads of
+this type. However, if ˜ea(s) is any other then it is related to ea(s) by a rigid rotation, i.e., ˜e0(s) = e0(0)
+and ˜ei(0) = R j
+i ej(0) (summing j over 1, 2, 3), where δimR j
+i R
+n
+m = δmn, det R = 1. This must be true
+for some R at s = 0, and extends to all s as both tetrads evolve under exp(sM).
+Next, recall that the vacuum 2-point function may be given as a distributional limit
+G0(x, x′) = lim
+ǫ→0+ F(σǫ(x, x′))
+(4.2)
+where F(z) = 1/(4π2z) and
+σǫ(x, x′) = −ηµν(x − x′ − iǫ∂t)µ(x − x′ − iǫ∂t)ν
+(4.3)
+is the regulated signed squared geodesic separation of x and x′. As usual, we have identified Minkowski
+spacetime with its tangent spaces at all points.
+Distributional derivatives may be taken under the limit in (4.2), giving
+1
+2(∇µ ⊗ 1)G0(x, x′) = − lim
+ǫ→0+ F ′(σǫ(x, x′))(x − x′ − iǫ∂t)µ
+(4.4)
+and
+1
+2(∇µ ⊗ ∇ν)G0(x, x′) = lim
+ǫ→0+
+�
+F ′(σǫ(x, x′))ηµν − 2F ′′(σǫ(x, x′))(x − x′ − iǫ∂t)µ(x − x′ − iǫ∂t)ν
+�
+.
+(4.5)
+9
+
+Contracting with ea(x)µea(x′)ν (without summing on a) and pulling back to the worldline, we find
+1
+2((∇ea ⊗ ∇ea)G0)(γ(s), γ(s′)) = lim
+ǫ→0+ F ′(σǫ(γ(s), γ(s′)))Ca(s, s′)
++ lim
+ǫ→0+ 2F ′′(σǫ(γ(s), γ(s′)))Da(s, s′)Da(s′, s).
+(4.6)
+(note the order of variables in the last two factors in the second term) where
+Ca(s, s′) = ηµνeµ
+a(s)eν
+a(s′),
+Da(s, s′) = (γ(s) − γ(s′))µeµ
+a(s).
+(4.7)
+Under a change of frame from ea to ˜ea as described above, one has ˜C0 = C0, ˜D0 = D0, while
+˜Di = R j
+i Dj and ˜Ci(s, s′) = R j
+i R k
+i ηµνeµ
+j (s)eν
+k(s′). By orthogonality, this implies that �3
+i=1 ˜Ci(s, s′) =
+�3
+i=1 Ci(s, s′) and �3
+i=1 ˜Da(s, s′) ˜Da(s′, s) = �3
+i=1 Da(s, s′)Da(s′, s).
+In Appendix A, we give some further details to justify the above distributional manipulations and
+prove the following result, where κ, τ and υ are the curvature invariants of γ as described in Section 2.
+Lemma. (a) With the choice of tetrad just described, Ca(s, s′) and Da(s, s′) are translationally in-
+variant, depending only on s − s′. There are entire analytic functions Ga and Ha such that
+Ca(s, s′) = Ga(κ2(s − s′)2),
+Da(s, s′)Da(s′, s) = −(s − s′)2Ha(κ2(s − s′)2),
+(4.8)
+where, in the limit z → 0,
+3
+�
+a=0
+Ga(z) = −2 + τ 2 + υ2
+κ2
+z + (κτ)2 − (τ 2 + υ2)2
+κ4
+z2 + O(z3),
+(4.9)
+and
+3
+�
+a=0
+Ha(z) = 1 + z
+12 + κ2 + 19τ 2
+360κ2
+z2 + O(z3).
+(4.10)
+(b) The signed square geodesic separation of points along γ obeys
+σ0(γ(s), γ(s′)) = −(s − s′)2Υ(κ2(s − s′)2),
+(4.11)
+where Υ is entire analytic with
+Υ(z) = 1 + 1
+12z + κ2 − τ 2
+360κ2 z2 + O(z3)
+(4.12)
+as z → 0. Furthermore, for z ∈ [0, ∞), Υ(z) is real with Υ(z) ≥ 1.
+The Lemma now allows us to compute the point-split energy density by evaluating the right-hand
+side of (4.6) and summing over a. We use the fact (explained in Appendix A) that
+lim
+ǫ→0+
+(s − s′)2j
+σǫ(γ(s), γ(s′))k =
+(−1)k
+Υ(κ2(s − s′)2)k lim
+ǫ→0+
+1
+(s − s′ − iǫ)2(k−j) ,
+(4.13)
+where the limits are taken in the sense of distributions, as is the multiplication by a smooth prefactor
+on the right-hand side. If j = k, the distributional limit on the right-hand side may be replaced
+by unity. In particular, when calculating T(s, s′) from (4.6), the factor (s − s′)2 in Da(s, s′)Da(s′, s)
+cancels a factor of (s − s′ − iǫ)2 in the denominator, as ǫ → 0+. The upshot is that
+T(s, s′) = − 1
+4π2 lim
+ǫ→0+
+K(κ2(s − s′)2)
+(s − s′ − iǫ)4 ,
+where
+K(z) =
+3
+�
+a=0
+�Ga(z)
+Υ(z)2 − 4Ha(z)
+Υ(z)3
+�
+(4.14)
+is a meromorphic function that is analytic in a neighbourhood of the positive real axis (on which Υ is
+bounded away from zero).
+10
+
+The singular part is easily isolated by splitting off the first two terms of the Taylor series for K
+from the remainder, which carries a leading factor of (s − s′)4 that cancels the denominator in the
+limit ǫ → 0+. Similarly, the O(z) part of the Taylor series partly cancels the denominator. Thus,
+T(s, s′) = Tsing(s − s′) + Treg(s − s′) with
+Tsing(s) = − 1
+4π2 lim
+ǫ→0+
+K(0)
+(s − iǫ)4 −
+1
+4π2 lim
+ǫ→0+
+κ2K′(0)
+(s − iǫ)2 ,
+(4.15)
+and
+Treg(s) = − κ4
+4π2 J((κs)2),
+where
+J(z) = K(z) − K(0) − K′(0)z
+z2
+(4.16)
+is analytic on a neighbourhood of the positive real axis, so J((κs)2) is smooth for s ∈ R.
+Using the Lemma, we may read off that K(0) = −6, thus establishing (3.15), with A = κ2K′(0).
+Meanwhile, Treg(s) is smooth, even, and real-valued for s ∈ R. Provided that K(z) = O(z) as z → ∞
+on the real axis, we find that Treg(s) = O(s−2) as s → ∞, which completes the properties needed in
+Section 3. Furthermore,
+Treg(0) = −J(0)κ4
+4π2
+= −K′′(0)κ4
+8π2
+.
+(4.17)
+Note that if we had used the tetrad ˜e instead, the function K would be unchanged, owing to the
+remarks before the Lemma. Thus the QEIs obtained from ˜ea and ea are identical.
+These results now provide a calculational method to determine the QEI along stationary worldlines.
+Starting from the generator M ∈ so(1, 3) and the initial 4-velocity u(0), choose a tetrad as described
+at the start of this section, and compute the proper acceleration κ =
+�
+−η(Mu(0), Mu(0)). The
+translational invariance of Ca and Da means that they can be calculated conveniently as
+Ca(s, s′) = ηµνeµ
+a(s − s′)eν
+a(0),
+Da(s, s′) = −(γ(s′ − s) − γ(0))µeµ
+a(0),
+(4.18)
+from which Ga and Ha are easily obtained. The function Υ is computed directly from the Lorentz
+interval between γ(0) and γ(s). Then construct K(z) according to (4.14) and check that K(z) = O(z)
+as z → ∞. Then the QEI along γ is given by (3.19), with constants
+A = κ2K′(0),
+B = −2κ4K′′(0).
+(4.19)
+The constants A and B can be computed from the first few terms of the Taylor expansions of
+�
+a Ga, �
+a Ha and Υ, given in (4.9), (4.10) and (4.12) respectively. After a calculation, one finds
+K(z) = −6 + z κ2 + τ 2 + υ2
+κ2
+− z2
+1
+360κ4
+�
+3κ4 + 62κ2τ 2 + 30(κ2 + τ 2 + υ2)2�
++ O(z3),
+(4.20)
+from which the formulae (1.9) and (1.10) follow immediately. Nonetheless, this is perhaps not the
+most illuminating calculation and also does not provide a check that K(z) = O(z) for large real z,
+which was assumed above. For these reasons, and their own intrinsic interest, we will also provide
+explicit calculations in Section 5 that together cover all possible stationary worldlines.
+5
+QEIs for the prototypical stationary worldlines
+We have now established the general QEI for stationary worldlines in Minkowski spacetime, assuming
+a technical condition on the growth of K. In this section, we reduce the problem of computing the
+QEI for a general stationary worldline to six prototypical cases, which will be treated in turn. These
+calculations follow the method of Section 4 and result in explicit formulae for K. In this way it is seen
+that the growth condition holds in all cases and we also obtain a check on the Taylor series calculations
+in Appendix B.
+We have already discussed the fact that any stationary worldline may be brought into one of the
+six standard forms by a Poincar´e transformation, without changing the curvature invariants. Owing
+to Poincar´e invariance of the vacuum state, and because Poincar´e invariance maps an adapted tetrad
+of the form ea(s) = exp(sM)ea(0) along the original curve to a tetrad with the same properties on
+11
+
+the new one, the point-split energy density obtained by the method of Section 4 is exactly the same
+for the two worldlines, which accordingly share the same QEI bound.
+The QEIs for the prototypical stationary worldlines are now given in turn. Most of the computa-
+tions that follow were conducted using the computer algebra system Maple.
+5.1
+Trivial subgroup: inertial motion
+For the inertial worldline γ(s) = (s, 0, 0, 0), we employ the adapted tetrad ∂t, ∂x, ∂y, ∂z, which is
+constant along γ, leading immediately to the relations C0(s, s′) = 1, Ci(s, s′) = −1 for i = 1, 2, 3, while
+D0(s, s′) = s−s′, Di(s, s′) = 0 for all s, s′. It follows that G0 = H0 ≡ 1, Gi ≡ −1, Hi ≡ 0. Furthermore,
+Υ ≡ 1 because σ0(γ(s), γ(s′)) = −(s − s′)2. Hence K ≡ −6 and one finds T(s, s′) = Tsing(s − s′) where
+Tsing(s) = lim
+ǫ→0+
+3
+2π2(s − iǫ)4 .
+(5.1)
+Consequently Treg vanishes identically, and we may read off immediately that A = B = 0, reproducing
+QEI (1.4) by substituting into (3.19), and in agreement with (1.9) and (1.10). Of course these results
+are easily obtained by direct differentiation of the two-point function; our purpose here is to show how
+they follow from formulae in Section 4.
+Alternatively, we may proceed by taking the Fourier transform
+ˆTsing(u) = u3Θ(u)/(2π),
+(5.2)
+from which we obtain Q(α) = α4Θ(α)/(16π3) by (1.3), leading to (1.4) as discussed in the introduction.
+5.2
+Hyperbolic subgroups: linear acceleration
+We consider a uniformly linearly accelerated worldline
+γ(s) = (a−1 sinh as, a−1 cosh as, 0, 0),
+(5.3)
+whose velocity evolves under the 1-parameter group of tx-boosts ˙γµ(s) = Hµ
+ν(s)˙γν(0), where
+H•
+•(s) =
+
+
+
+
+cosh as
+sinh as
+0
+0
+sinh as
+cosh as
+0
+0
+0
+0
+1
+0
+0
+0
+0
+1
+
+
+
+ = exp
+
+
+
+
+0
+as
+0
+0
+as
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+
+
+
+
+(5.4)
+and 0 ̸= a ∈ R is fixed. Noting that the initial velocity and its first two derivatives are ˙γ(0) = ∂t,
+¨γ(0) = a∂x, ¨γ(0) = a2∂t, we obtain an adapted tetrad by choosing the tetrad ∂t, ∂x, ∂y, ∂z at s = 0,
+and applying the prescription eµ
+a(s) = Hµ
+ν(s)eν
+a(0) to find
+e0(s) = cosh as���t + sinh as∂x,
+e1(s) = sinh as∂t + cosh as∂x,
+e2(s) = ∂y,
+e3(s) = ∂z.
+(5.5)
+Straightforward calculation, following the method of Section 4, gives
+K(a2s2) = −
+3(as)4
+8 sinh4(as/2)
+(5.6)
+and hence
+T(s, s′) = lim
+ǫ→0+
+3a4(s − s′)4 cosech4(a(s − s′)/2)
+32π2(s − s′ − iǫ)4
+,
+(5.7)
+which may be simplified to
+T(s, s′) = lim
+ǫ→0+
+3a4
+32π2 cosech4 �
+a(s − s′ − iǫ)/2
+�
+.
+(5.8)
+Here, we have used the general fact that limǫ→0+ g(x)f(x − iǫ) = limǫ→0+ g(x − iǫ)f(x − iǫ) in the
+sense of distributions, when f is analytic in a strip Z = {x − iy : x ∈ R, 0 < y < y0} ⊂ C with
+supz∈Z |f(z)(Im z)N| < ∞ for some N > 0 and g is analytic on Z and continuous on Z ∪ R.
+12
+
+As the function K(z) evidently decays rapidly as z → ∞ on the real axis, the method of Section 4
+allows us to read off the QEI from the derivatives of K(z) at z = 0 according to (4.19). Using
+K(z) =
+3z2
+8 sinh4(√z/2) = −6 + z − 11
+120z2 + O(z3),
+(5.9)
+we find A = a2 and B = 11a4/30, in agreement with (1.9) and (1.10) using the invariants from Table 1
+and reproducing the result (1.5) from [13]. In that reference, the point-split energy density (5.8) was
+found by a direct calculation. Writing T(s, s′) = T(s − s′), the Fourier transform yields
+ˆT(u) =
+u3 − a2u
+2π(1 − e−2πu/a)
+(5.10)
+and by using the last expression in (3.13), a calculation gives
+Qeven(α) =
+1
+32π3
+�
+α4 + 2a2α2 + 11
+30a4
+�
+,
+(5.11)
+from which (1.5) follows on inserting the above expression into (3.12) and using Parseval’s theorem.
+5.3
+Hyperbolic subgroups: the catenary
+Now consider the catenary
+γ(s) = (a−1 cosh χ sinh as, a−1 cosh χ cosh as, −s sinh χ, 0),
+(5.12)
+for constant a ̸= 0, with initial velocity
+˙γ•(0) = (cosh χ, 0, − sinh χ, 0),
+(5.13)
+and second and third derivatives
+¨γ•(0) = (0, a cosh χ, 0, 0),
+...γ •(0) = (a2 cosh χ, 0, 0, 0).
+(5.14)
+The velocity evolves under the hyperbolic subgroup (5.4). Writing C = cosh χ and V = sinh χ, the
+tetrad
+e•
+0(s) = (C cosh as, C sinh as, −V, 0),
+e•
+1(s) = (sinh as, cosh as, 0, 0),
+e•
+2(s) = (−V cosh as, −V sinh as, C, 0),
+e•
+3(s) = (0, 0, 0, 1)
+(5.15)
+is adapted to γ with eµ
+a(s) = Hµ
+ν(s)eν
+a(0). A calculation results in the formula
+K(z) = −4V 2(sinhc2(r) + v2) sinh2(r) + 2(4C2 − 1) sinhc2(r) − 16V 2 sinhc(2r) + 2v2(4C2 − 3)
+C4(sinhc2(r) − v2)3
+(5.16)
+where v = tanh χ, r = √z/(2 cosh χ) and sinhc(x) = sinh(x)/x is the hyperbolic version of the sinc
+function. Note that we need not specify a branch for the square root as it always appears in the
+argument of an even entire function, and also that K(z) → 0 as z → ∞ in R. The series expansion is
+K(z) = −6 + 2C2 − 1
+C2
+z − 185C4 − 182C2 + 30
+360C4
+z2 + O(z3)
+(5.17)
+and as κ = aC we may read off A = a2(2C2 − 1) = a2 cosh 2χ and B = (185C4 − 182C2 + 30)a4/90.
+It is straightforward that these values agree with (1.9) and (1.10) using the curvature invariants for
+this case. In particular, the resulting QEI is compatible with a constant negative energy density of
+− Treg(0) = −(185 cosh4 χ − 182 cosh2 χ + 30)a4
+1440π2
+(5.18)
+along the worldline (5.12). As would be expected, the QEI for linear acceleration is obtained in the
+limit χ → 0, but for χ ̸= 0, we have −Treg(0) < −11a4/480π2, and the QEI bound is consistent with
+a strictly more negative constant energy density than is the case for the linearly accelerated worldline
+with the same value of a.
+13
+
+5.4
+Parabolic subgroups: the semicubical parabola
+We now consider the semicubical parabola
+γ(s) =
+�
+s + 1
+6a2s3, 1
+6a2s3, 0, 1
+2as2
+�
+,
+(5.19)
+for constant a ̸= 0, whose velocity evolves as ˙γµ(s) = P µ
+ν(s)˙γ(0) with ˙γ(0) = ∂t, where P µ
+ν was
+defined in (2.14). From the initial derivatives ˙γ(0) = ∂t, ¨γ(0) = a∂z, ...γ (0) = a2(∂t + ∂x) one sees that
+the initial tetrad e0(0) = ∂t, e1(0) = ∂z, e2(0) = ∂x, e3(0) = ∂y determines an adapted tetrad
+e•
+0(s) =
+�
+1 + 1
+2(as)2, 1
+2(as)2, 0, as
+�
+,
+e•
+1(s) = (as, as, 0, 1) ,
+e•
+2(s) =
+�
+− 1
+2(as)2, 1 − 1
+2(as)2, 0, −as
+�
+,
+e•
+3(s) = (0, 0, 1, 0),
+(5.20)
+at general proper time s obeying eµ
+a(s) = P µ
+ν(s)eν
+a(0).
+Straightforward calculation now gives
+K(z) = −6 − z/2 + 5z2/36
+(1 + z/12)3
+,
+(5.21)
+with
+K(z) = −6 + 2z − 37
+72z2 + O(z3)
+(5.22)
+as z → 0 and K(z) = O(z−1) for z → ∞. Thus, the point-split energy density is
+T(s, s′) = lim
+ǫ→0+
+3 − a2(s − s′)2/4 + 5a4(s − s′)4/72
+π2(s − s′ − iǫ)4(1 + a2(s − s′)2/12)3
+(5.23)
+and (4.19) gives A = 2a2 and B = 37a4/18, in agreement with (1.9) and (1.10). Thus the QEI along
+a semicubical parabola is
+�
+ds|g(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ −
+1
+16π2
+� ∞
+−∞
+ds
+�
+|g′′(s)|2 + 4a2|g′(s)|2 + 37
+18a4|g(s)|2
+�
+,
+(5.24)
+for any Hadamard state ω. The long-time scaling limit of the above QEI is then
+lim inf
+λ−→∞
+� ∞
+−∞
+ds|gλ(s)|2⟨:Tµν ˙γµ ˙γν:⟩ω(γ(s)) ≥ −
+37
+288π2 a4,
+(5.25)
+where as usual we choose g with unit L2-norm. The QEI is therefore compatible with a constant
+negative energy density −37a4/(288π2) along the semicubical parabola. As one would expect, the
+QEI reduces to the inertial case as a → 0.
+In fact the QEI (5.24) can also be obtained by a different method. Writing T(s, s′) = T(s − s′),
+the Fourier transform may be computed by contour methods as
+ˆT(u) = 1
+2π
+�� 2u2
+√
+12 + 7|u|
+8 a +
+15
+8
+√
+12a2
+�
+ae−|u|
+√
+12/a +
+�
+u3 + 2ua2�
+Θ(u)
+�
+.
+(5.26)
+The calculation is considerably simplified if one first replaces powers of s − s′ in (5.23) by powers of
+s − s′ − iǫ. To find Qeven(α), we note that ˆTodd(u) = (u3 + 2ua2)/(4π), and also that the integral of ˆT
+over (−∞, 0] may be evaluated in terms of Γ-functions. After manipulation, the formula (3.13) gives
+Qeven(α) =
+1
+4π3
+� 0
+−∞
+� 2u2
+√
+12 + 7|u|
+8 a +
+15
+8
+√
+12a2
+�
+ae−|u|
+√
+12/a du +
+1
+8π3
+� α
+0
+�
+u3 + 2ua2�
+du
+=
+1
+32π3 α4 + a2
+8π3 α2 + 37a4
+576π3 .
+(5.27)
+Inserting this expression in (3.12) and using Parseval’s theorem we reproduce (5.24).
+14
+
+5.5
+Elliptic subgroups: uniform rotation
+Next, consider the uniformly rotating worldline
+γ(s) = (s cosh χ, 0, r cos ωs, r sin ωs) ,
+(5.28)
+where the radius r > 0 and proper angular velocity ω ̸= 0 together fix the rapidity χ = sinh−1(rω).
+In this case, the velocity evolves under rotations in the yz-plane as ˙γµ(s) = Rµ
+ν(s)˙γν(0), where
+R•
+•(s) =
+
+
+
+
+1
+0
+0
+0
+0
+1
+0
+0
+0
+0
+cos ωs
+− sin ωs
+0
+0
+sin ωs
+cos ωs
+
+
+
+ = exp
+
+
+
+
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−ωs
+0
+0
+ωs
+0
+
+
+
+ .
+(5.29)
+Meanwhile, the initial velocity and its first two derivatives are
+˙γ•(0) = (C, 0, 0, V )
+¨γ•(0) = (0, 0, −V ω, 0)
+...γ •(0) =
+�
+0, 0, 0, −V ω2�
+,
+where we have written C = cosh χ and V = rω = sinh χ.
+Then e•
+0(0) = (C, 0, 0, V ), e•
+1(0) =
+(0, 0, −1, 0), e•
+2(0) = (−V, 0, 0, −C), e•
+3(0) = (0, 1, 0, 0), defines an adapted tetrad at s = 0, which
+can be extended along γ by eµ
+a(s) = Rµ
+ν(ωs)eν
+a(0) to give
+e•
+0(s) = (C, 0, −V sin ωs, V cos ωs),
+e•
+1(s) = (0, 0, − cos ωs, − sin ωs),
+e•
+2(s) = (−V, 0, C sin ωs, −C cos ωs),
+e•
+3(s) = (0, 1, 0, 0).
+(5.30)
+A calculation gives
+K(z) = 4C2 sin2(θ)(1 + v2 sinc2(θ)) − 2(4C2 − 3)v2 sinc2(θ) + 16V 2 sinc(2θ) + 2(4C2 − 1)
+C4(1 − v2 sinc2(θ))3
+,
+(5.31)
+where θ = √z/(2 sinh(χ)), with series expansion
+K(z) = −6 + 2 cosh2 χ − 1
+sinh2 χ
+z − 185 cosh4 χ − 188 cosh2 χ + 33
+360 sinh4 χ
+z2 + O(z3).
+(5.32)
+As κ = rω2 = ω sinh χ we read off A = ω2 cosh(2χ) = (2(rω)2 + 1)ω2 and
+B = ω4(185 cosh4 χ − 188 cosh2 χ + 33)
+90
+= ω4(30 + 182(rω)2 + 185(rω)4)
+90
+,
+(5.33)
+which may be substituted into (1.8) to obtain the QEI in this case. In particular, the QEI is compatible
+with a constant negative energy density of
+− Treg(0) = −ω4(30 + 182(rω)2 + 185(rω)4)
+1440π2
+(5.34)
+along the worldline. While the point-split energy density may be written down in terms of K, we do
+not know of any closed-form expression for its transform. Thus the method of Sections 3 and 4 is the
+only available way to compute this QEI.
+Note that the QEI reduces to the inertial case if ω → 0 with r fixed – indeed, even if r = o(ω−2).
+One might initially be surprised that it does not reduce in the same way when r → 0+ with ω fixed.
+The explanation is that the torsion of the curve does not vanish in this limit, even though the curvature
+κ does. This neatly illustrates the influence of higher curvature invariants on the QEI bound.
+15
+
+5.6
+Loxodromic subgroups
+Finally, we study the loxodromic worldline
+γ•(s) = (Ca−1 sinh(as), Ca−1 cosh(as), V ω−1 cos(ωs), V ω−1 sin(ωs)),
+(5.35)
+where C = cosh χ, V = sinh χ for fixed χ ̸= 0, a ̸= 0 and ω ̸= 0. This worldline undergoes both rotation
+in the yz-plane at constant proper angular velocity ω and constant distance |V/ω| from the x-axis, while
+undergoing uniform acceleration in the x-direction. The velocity evolves as ˙γµ(s) = La,ω
+µ
+ν(s)˙γν(0),
+where
+La,ω•
+•(s) =
+
+
+
+
+cosh as
+sinh as
+0
+0
+sinh as
+cosh as
+0
+0
+0
+0
+cos ωs
+− sin ωs
+0
+0
+sin ωs
+cos ωs
+
+
+
+ = exp
+
+
+
+
+0
+as
+0
+0
+as
+0
+0
+0
+0
+0
+0
+−ωs
+0
+0
+ωs
+0
+
+
+
+ .
+(5.36)
+It can be checked that
+e•
+0(s) = (C cosh as, C sinh as, −V sin ωs, V cos ωs),
+e•
+1(s) = (Caκ−1 sinh as, Caκ−1 cosh as, −V ωκ−1 cos ωs, −V ωκ−1 sin ωs),
+e•
+2(s) = (−V cosh as, −V sinh as, C sin ωs, −C cos ωs),
+e•
+3(s) = (V ωκ−1 sinh as, V ωκ−1 cosh as, Caκ−1 cos ωs, Caκ−1 sin ωs)
+(5.37)
+defines an adapted tetrad for γ, obeying eµ
+a(s) = La,ω
+µ
+ν(s)eν
+a(0), while the calculation of K by computer
+algebra produces
+K(z) =
+1
+(C2 sinhc2(ar) − V 2 sinc2(ωr))3
+�
+16C2V 2 sinc(2ωr) sinhc(2ar)
++4(C2 sin2(ωr) − V 2 sinh2(ar))(V 2 sinc2(ωr) + C2 sinhc2(ar))
+−2V 2(C2 + 3V 2) sinc2(ωr) − 2C2(3C2 + V 2) sinhc2(ar)
+�
+,
+(5.38)
+where r = √z/(2
+√
+C2a2 + V 2ω2). For large real z, it is easily seen that
+K(z) ∼ −4V 2(ar)2/(C4 sinhc2(ar)) → 0
+(5.39)
+as z → ∞ in R. Meanwhile, the Taylor expansion about z = 0 reads
+K(z) = −6 + (a2 + ω2)(C2 + V 2)
+C2a2 + V 2ω2
+z −
+z2
+360(C2a2 + V 2ω2)2
+�
+(33a4 + 60a2ω2 + 30ω4)C4
++ (122a4 + 250a2ω2 + 122ω4)(CV )2 + (33ω4 + 60a2ω2 + 30a4)V 4�
++ O(z3)
+so A = (C2 + V 2)(a2 + ω2), while B is given by
+90B = (3a4 + 30(a2 + ω2)2)C4 + (3ω4 + (30(a2 + ω2)2)V 4 + (122(a2 + ω2)2 + 6a2ω2)C2V 2
+= 3(C2a2 + V 2ω2)2 + 62(a2 + ω2)2(CV )2 + 30(a2 + ω2)2(C2 + V 2)2,
+(5.40)
+in which the last term is 30A2. These values are easily expressed in terms of curvature invariants.
+Using (2.19) and C2 − V 2 = 1 one has
+κ2(τ 2 + υ2) = (a2 + ω2)2(CV )2 + (aω)2 = (V 2a2 + C2ω2)(C2a2 + V 2ω2) = κ2(V 2a2 + C2ω2), (5.41)
+from which the identity
+κ2 + τ 2 + υ2 = (a2 + ω2)(C2 + V 2) = A
+(5.42)
+follows directly, in agreement with (1.9). Using this in (5.40) together with (2.19) we see that B
+takes the form (1.10). We see that the QEI is compatible with a constant negative energy density of
+−Treg(0) along the worldline (2.18), where
+Treg(0) = 185(a2 + ω2)2C4 − (182a4 + 370a2ω2 + 188ω4)C2 + 33ω4 + 60a2ω2 + 30a4
+1440π2
+(5.43)
+16
+
+and we have used V 2 = C2 − 1. Note that the QEI does not reduce to the hyperbolic QEI in the limit
+χ → 0 with a and ω fixed. This is because the hypertorsion has a nonzero limit υ → sgn(a)ω, even
+though the torsion vanishes and the curvature tends to a. Nonetheless, it is easily seen from (5.40) that
+90B ≥ 33a4 and hence that −Treg(0) < −11a4/(480π2), so that the QEI for loxodromic worldlines can
+be consistent with a more negative constant value of the energy density than the linearly accelerated
+worldline with the same value of a.
+6
+Summary and discussion
+In this paper we have succeeded in giving an exact closed form expression (1.8)–(1.10) for the QEI
+for the massless scalar field on any stationary worldline in four-dimensional Minkowski spacetime.
+This was achieved by a novel method that circumvented the need to take Fourier transforms of the
+point-split energy density along the worldline, and which reduced the problem to the computation of
+certain Taylor coefficients of functions determined by a tetrad adapted to the worldline. In addition,
+we have given explicit calculations for the six prototypical classes of stationary trajectory, obtaining
+agreement with our general result (and also verifying a technical condition needed for the general
+analysis). The resulting QEI bound depends only on the curvature, torsion and hypertorsion of the
+worldline. We have also conducted – in Appendix C – a parallel exercise for a quantum inequality
+on the Wick square. A scaling analysis (see (1.11)) shows how these bounds take a universal form on
+timescales short in relation to the curvature scales, from which they then deviate at longer timescales.
+In the infinite time limit, they would all allow the field to exhibit a constant negative energy density
+(or zero in the inertial case).
+Our results complement those of Kontou and Olum [30, 31], who computed an absolute QEI [14] in
+an approximation of spacetimes where the curvature was weak. There, the worldline was taken to be
+a geodesic. Our present results indicate the corrections that should enter at leading order when that
+assumption is dropped. (We reemphasise that our results are exact for massless fields in Minkowski
+spacetime on stationary trajectories.)
+To conclude, we first mention various potential extensions of our work and then return to the
+question of whether the long-time limits of the QEI are saturated by physical states of the field.
+Starting with extensions, we expect that our general method would extend fairly directly to station-
+ary worldlines in any even-dimensional Minkowski spacetimes, leading to closed form results in terms
+of the appropriate curvature invariants. In odd dimensions, the vacuum two-point function involves
+noninteger powers of the geodesic separation, which adds an extra complication. It would be interest-
+ing to investigate this case in more detail. (For higher-dimensional treatment of the Unruh detector
+response in higher dimensions, which would be related to the Wick QI in these cases, see [26], and for
+specific calculations relating to the detailed balance definition of Unruh temperature along stationary
+worldlines in 4-dimensions, see [23].) Next, massive fields typically have QEI bounds that are expo-
+nentially suppressed relative to the massless ones. Here, we do not expect that our method would
+easily produce closed-form results, but again, it would be worth investigating, as would the situation
+for higher spin fields.
+Finally, we consider the extent to which the long term average bounds can be attained. In the case
+of inertial worldlines this is obvious: the long-term average value of zero is attained in the Minkowski
+vacuum state. For uniformly accelerated curves it was noted in [13] that the bound (1.6) is attained
+by the Rindler vacuum for the right wedge x > |t| in Minkowski spacetime. It is useful to put this
+in a broader context. Adopting coordinates t = ξ sinh χ, x = ξ cosh χ, the Rindler wedge x > |t| of
+Minkowski spacetime has metric ξ2 dχ2 − dξ2 − dy2 − dz2, and any curve χ �→ (aχ, 1/a, y0, z0) with
+a > 0 is a curve of proper acceleration a in proper time parameterisation.
+Moreover, the energy
+density measured by an observer moving on a curve of constant ξ, in the thermal state of temperature
+β−1 with respect to the coordinate χ, is
+⟨:Tµνuµuν:⟩β = (4π2 − β2)(33β2 + 12π2)
+1440π2β4ξ4
+,
+(6.1)
+17
+
+β
+ρ
+−11
+0
+2π
+4π
+6π
+8π
+10π
+12π
+14π
+Figure 1: Plot of ρ = (480π2ξ4)⟨:Tµνuµuν:⟩β on a curve of constant ξ, against β. The dotted line
+corresponds to the QEI bound (1.6), which is attained as β → ∞, corresponding to the Rindler
+ground state.
+reducing to
+⟨:Tµνuµuν:⟩∞ = −
+11
+480π2ξ4
+(6.2)
+for the Rindler ground state. At β = 2π, the thermal state on Rindler spacetime is precisely the
+restriction of the Minkowski vacuum to the right wedge, which is why the energy density vanishes.
+Because most references (e.g., [5, 6, 2]) only discuss the conformally coupled stress-energy tensor (the
+‘new improved’ stress tensor) and [13] only considered the ground state without giving details, the
+relevant calculations are briefly reviewed in Appendix D. On restriction to the curve ξ = 1/a we see
+that all these states have constant energy density consistent with (1.6) (see Fig. 1) and that this bound
+is attained by the Rindler ground state.
+One should note that the Rindler ground state (and indeed all the β-KMS states other than the
+special case β = 2π) is not defined on all of Minkowski, but just on the wedge x > |t|. The obvious
+divergence of the stress-energy tensor as ξ → 0+ shows that the state cannot be extended as a
+Hadamard state beyond the wedge. The reason they satisfy the Minkowski QEI is because this QEI
+is local and covariant – see [13] for a discussion and many similar calculations, and [10] for a more
+abstract viewpoint inspired by [3]. Nonetheless, it remains open as to whether equality in (1.6) can be
+attained by a Hadamard state defined on all of Minkowski; our conjecture is that one can find global
+Hadamard states that approximate the Rindler ground state sufficiently well that the bound (1.6) is
+satisfied in a limiting sense. These issues will be addressed elsewhere.
+Turning to the remaining stationary worldlines, the QEI is again consistent with a constant strictly
+negative energy density and we can again ask whether the bound is attained in any sense. Letaw
+and Pfautsch [35] considered the problem of quantising the field in coordinates associated with the
+various stationary worldlines and seeking an appropriate ground state. For the inertial, uniformly
+rotating, and semicubical parabolic worldlines, they concluded that the resulting state was precisely
+the Minkowski vacuum state. This means that we have no obvious candidate state associated with the
+uniformly rotating and semicubical parabolic worldlines with negative energy density. On the other
+hand, the catenary (5.12) and loxodromic worldlines (2.18) both result in a Rindler vacuum state
+on the x > |t| wedge, which is the causal hull of the worldline in question. One may compute the
+energy density along these curves in the Rindler vacuum, using the renormalised stress energy tensor
+given in Appendix D, yielding constant energy densities −(14 cosh2 χ+19)a4/(1440π2 cosh4 χ) in each
+18
+
+case. This value is strictly greater than −11a4/(480π2) for χ ̸= 0, which is greater than the most
+negative constant energy density consistent with the QEIs in these cases (see the remarks at the end
+of sections 5.3 and 5.6). Thus they are are consistent with the QEIs but do not saturate them.
+It therefore remains an open and intriguing question, whether there are (sequences of) Hadamard
+states that attain these QEI bounds (in a limiting sense). Resolving this question, and its analogues in
+2+1 dimensions, may have relevance to proposed experiments to detect the Unruh effect using a laser
+beam whose intersection with a Bose-Einstein condensate follows a uniformly rotating worldline [24].
+Acknowledgements CJF thanks Alexander Strohmaier and Valter Moretti for useful conversations
+concerning the H¨ormander pseudo-topologies, and Aron Wall for posing an interesting direction for
+further study. The work of JT was in part funded by an EPSRC studentship at the University of
+Sheffield and a summer studentship from the University of York. We thank Elizabeth Winstanley for
+a reading of the manuscript and some helpful suggestions.
+A
+Details on the method
+We give further details on the method described in Section 4 and prove the Lemma stated there. Some
+aspects are treated using techniques of microlocal analysis – we will be rather brief on those details,
+referring the reader to appropriate literature, while indicating the structure of the argument.
+To start, we observe that, for ǫ > 0, F(σǫ(x, 0)) can be written
+F(σǫ(x, 0)) =
+�
+d3k
+(2π)3
+e−∥k∥ǫ−ik·x
+2∥k∥
+,
+(A.1)
+where k• = (∥k∥, k), x• = (t, x). Thus for any ϕ ∈ C∞
+0 (R4), the distribution uǫ(x) = ϕ(x)F(σǫ(x, 0))
+has Fourier transform
+ˆuǫ(k′) =
+�
+d3k
+(2π)3
+e−ǫ∥k∥
+2∥k∥ ˆϕ(k′ − k).
+(A.2)
+As ˆϕ decays faster than inverse polynomials and k ∈ N +, where N +/− is the bundle of future/past-
+pointing null covectors, it may be shown that F(σǫ(x, 0)) converges in D′
+N +(R4) with respect to the
+H¨ormander pseudo-topology [28]. It follows from this that the vacuum 2-point function G0(x, x′) is
+the limit of F(σǫ(x, x′)) = F(σǫ(x − x′, 0)) in D′
+N +×N −(R4 × R4) and has wavefront set WF(G0) ⊂
+N + × N −, as is also known on general grounds because the state is Hadamard [39].
+These facts have various consequences. First, the pull-back of (any derivative operator acting on)
+G0 by ϕ : (s, s′) �→ (γ(s), γ(s′)) is well-defined because the set of normals to ϕ does not intersect
+WF(G0), essentially because timelike and null vectors cannot be orthogonal – see [8] for details.
+Consequently the pull-back is well-defined by standard results explained in Chapter 8 of [28] and has
+wavefront set contained in ϕ∗ WF(G0) ⊂ ϕ∗(N + × N −) = Γ × (−Γ), where Γ = R × (0, ∞) ⊂ T ∗R.
+Moreover, ϕ∗G0 is the limit in D′
+Γ×(−Γ)(R × R) of ϕ∗Fǫ ◦ σǫ as ǫ → 0+, which justifies taking the
+pull-back under the ǫ → 0+ limits in (4.6). Similar arguments apply to the convergence of F ′(σǫ(x, x′))
+and F ′′(σǫ(x, x′)) as ǫ → 0+.
+Next, recall that the stationary worldline γ has velocity u = ˙γ evolving according to u(s) =
+exp(sM)u(0), for M ∈ so(1, 3) with dimensions of inverse time, and that the right-handed tetrad ea(s)
+obeys ea(s) = exp(sM)ea(0), with u(s) = e0(s), ˙u(s) ∈ span{e1(s)}, and ¨u(s) ∈ span{e0(s), e1(s),
+e2(s)}. The Cartesian coordinates of γ(s), and components of ea(s) are evidently real analytic in s.
+We extend ea to a smooth tetrad in a neighbourhood of γ in an arbitrary fashion. Recall that the
+functions Ca and Da are defined, in index-free notation, by
+Ca(s, s′) = η(ea(s), ea(s′)),
+Da(s, s′) = η(γ(s) − γ(s′), ea(s)).
+(A.3)
+We now prove the lemma needed in Section 4, which we restate for convenience.
+Lemma. (a) With the choice of tetrad just described, Ca(s, s′) and Da(s, s′) are translationally in-
+variant, depending only on s − s′. There are entire analytic functions Ga and Ha such that
+Ca(s, s′) = Ga(κ2(s − s′)2),
+Da(s, s′)Da(s′, s) = −(s − s′)2Ha(κ2(s − s′)2),
+(A.4)
+19
+
+where in the limit z → 0,
+3
+�
+a=0
+Ga(z) = −2 + τ 2 + υ2
+κ2
+z + (κτ)2 − (τ 2 + υ2)2
+κ4
+z2 + O(z3),
+(A.5)
+and
+3
+�
+a=0
+Ha(z) = 1 + z
+12 + κ2 + 19τ 2
+360κ2
+z2 + O(z3).
+(A.6)
+(b) The signed square geodesic separation of points along γ obeys
+σ0(γ(s), γ(s′)) = −(s − s′)2Υ(κ2(s − s′)2),
+(A.7)
+where Υ is entire analytic with Υ(z) = 1 + 1
+12z +
+1
+360(1 − τ 2/κ2)z2 + O(z3) as z → 0. Furthermore,
+for z ∈ [0, ∞), Υ(z) is real with Υ(z) ≥ 1.
+Proof. (a) For inertial worldlines, ea(s) is constant and the result holds trivially with G0(z) ≡ 1,
+Gi(z) ≡ 1, H0(z) ≡ −1, Hi(z) ≡ 0. From now on we may assume that κ is nonzero.
+It follows from (4.1) that
+Ca(s, s′) = η(exp
+�
+s′M
+�
+ea(0), exp(sM)ea(0)) = η(ea(0), exp
+�
+(s − s′)M
+�
+ea(0)),
+(A.8)
+so Ca depends only on s − s′. As every component of the matrix exp(sM) is analytic, and because
+Ca(s, s′) = Ca(s′, s), we deduce that Ca(s, s′) = Ga(κ2(s − s′)2) for dimensionless entire analytic
+functions Ga.
+Next, observe that
+∂
+∂s′ Da(s, s′) = −η(e0(s′), ea(s)) = −η(e0(0), exp
+�
+(s − s′)M
+�
+ea(0)).
+(A.9)
+Integrating with respect to s′ and using Da(s, s) = 0 we may deduce that κDa(s, s′) is a dimensionless
+entire analytic function of (s − s′)κ. Again using Da(s, s) = 0 and because (A.9) gives ∂D0/∂s′|s′=s =
+−1 and ∂Di/∂s′|s′=s = 0 for i = 1, 2, 3, we have
+D0(s, s′) = (s − s′)
+�
+1 + O((κ(s − s′))2)
+�
+,
+Di(s, s′) = κ−1O((κ(s − s′))2),
+(A.10)
+where we have also used the fact that D0(s, s′) = −D0(s′, s) as a consequence of (A.9). Because
+Da(s, s′)Da(s′, s) is invariant under interchange of s and s′, we now have
+Da(s, s′)Da(s′, s) = −(s − s′)2Ha(κ2(s − s′)2)
+(A.11)
+for dimensionless entire analytic functions Ha.
+The Taylor series of Ga, Ha and their sums, are
+computed up to second order in Appendix B.
+(b) Next, we study the geodesic separation between γ(s) and γ(s′). We note that
+∂
+∂sσ0(γ(s), γ(s′)) = −2D0(s, s′)
+(A.12)
+depends only on s − s′, so σ0(γ(s), γ(s′)) = Σ(s − s′) + f(s′) and on considering s = s′ we find that f
+is constant and may be absorbed into Σ, which is also seen to be even. The first terms in its Taylor
+expansion are easily found: Σ(0) = 0, while
+Σ′′(s − s′) = −2η(u(s), u(s′)),
+Σ(4)(s − s′) = 2η( ˙u(s), ˙u(s′)),
+Σ(6)(s − s′) = −2η(¨u(s), ¨u(s′))
+(A.13)
+giving
+Σ′′(0) = −2,
+Σ(4)(0) = −2κ2,
+Σ(6)(0) = −2κ2(κ2 − τ 2)
+(A.14)
+using (2.9). Accordingly, we have established (A.7), the analyticity of Υ, and also the expansion
+Υ(z) = 1 + z
+12 + κ2 − τ 2
+360κ2 z2 + O(z3)
+(A.15)
+as z → 0. Finally, as γ(0) and γ(s) are connected by a smooth timelike curve, the timelike geodesic
+that connects them maximises proper time. Thus −σ0(γ(s), γ(0)) ≥ s2 for all s ∈ R and consequently,
+Υ(z) ≥ 1 for z ∈ [0, ∞), which concludes the proof.
+20
+
+Finally, we explain how the identity (4.13) may be proved. First note that
+σǫ(γ(s), γ(s′)) = σ0(γ(s), γ(s′)) + 2iǫ(γ0(s) − γ0(s′)) + ǫ2
+= −(s − s′)2Υ(κ2(s − s′)2) + 2iǫ(γ0(s) − γ0(s′)) + ǫ2
+= −(s − s′ − iǫ)2Υ(κ2(s − s′)2) + ǫΨ(s, s′) + ǫ2Ξ(s, s′)
+for smooth (indeed analytic) functions Ψ and Ξ. Let S be the difference between the distribution on
+the left-hand side of (4.13) and the distribution on the right-hand side. Then, using the fact that Υ
+is nonvanishing on the real axis, S takes the form
+S(s, s′) = lim
+ǫ→0+
+2k
+�
+r=1
+ǫrSr(s, s′)
+σǫ(γ(s), γ(s′))k(s − s′ − iǫ)2k
+(A.16)
+for smooth functions Sr ∈ C∞(R2) (1 ≤ r ≤ 2k).
+All that is needed now is to show that the
+distributional limit
+lim
+ǫ→0+
+1
+σǫ(γ(s), γ(s′))k(s − s′ − iǫ)2k
+(A.17)
+exists, whereupon S must vanish due to the strictly positive powers of ǫ in (A.16). The required result
+now follows from the sequential continuity of the distributional product with respect to the H¨ormander
+pseudo-topology (Theorem 2.5.10 in [27]), and the fact that both 1/σǫ(γ(s), γ(s′)) and
+1
+s − s′ − iǫ = i
+� ∞
+0
+dk e−ik(s−s′−iǫ)
+(A.18)
+have limits as ǫ → 0+ in D′
+Γ×(−Γ)(R2), where, as before, Γ = R × (0, ∞) ⊂ ˙T ∗R.
+B
+Taylor series calculation
+We compute the Taylor series of both Ga and Ha up to second order, using equations (4.7), (4.8) and
+(4.18). Recalling that Ca(s, s′) = Ga(κ2(s − s′)2), one can expand the right hand side into a Taylor
+series in s − s′ about the point s − s′ = 0 and then differentiate to yield
+− 1
+2κ2
+∂2Ca
+∂s∂s′ = G′
+a(0) + 3κ2(s − s′)2G′′
+a(0) + O((s − s′)4)
+(B.1)
+1
+12κ4
+∂4Ca
+∂2s∂2s′ = G′′
+a(0) + O((s − s′)2)
+(B.2)
+as s − s′ → 0. Differentiating equation (4.7) and setting s = s′ = 0, one easily finds
+G′
+a(0) = −η( ˙ea(0), ˙ea(0))
+2κ2
+,
+G′′
+a(0) = η(¨ea(0), ¨ea(0))
+12κ4
+(B.3)
+by equating powers of s − s′. The derivatives of the ea can be read off from the generalized Frenet-
+Serret equations (2.5) and its derivatives (2.8), allowing us to express G′
+a(0) and G′′
+a(0) in terms of
+curvature invariants.
+An easy computation shows that
+G′
+a(0) = 1
+2η0a + κ2 − τ 2
+2κ2
+η1a − τ 2 + υ2
+2κ2
+η2a − υ2
+2κ2 η3a
+(B.4)
+and
+G′′
+a(0) = κ2 − τ 2
+12κ2 η0a + τ 2υ2 + (κ2 − τ 2)2
+12κ4
+η1a − κ2τ 2 − (τ 2 + υ2)2
+12κ4
+η2a + υ2 τ 2 + υ2
+12κ4 η3a,
+(B.5)
+21
+
+where η(ea(0), eb(0)) = ηab by orthogonality of the tetrad field. Reconstructing Ga using a Taylor
+series therefore yields
+Ga(z) = ηaa +
+1
+2κ2 z
+�
+η0aκ2 − η1a(τ 2 − κ2) − η2a(υ2 + τ 2) − η3aυ2�
++
+z2
+24κ4
+�
+η0aκ2(κ2 − τ 2) + η1a(τ 2υ2 + (κ2 − τ 2)2) − η2a(κ2τ 2 − (τ 2 + υ2)2) + η3aυ2(τ 2 + υ2)
+�
++ O(z3).
+(B.6)
+Summing, we obtain
+3
+�
+a=0
+Ga(z) = −2 + τ 2 + υ2
+κ2
+z + (κτ)2 − (τ 2 + υ2)2
+κ4
+z2 + O(z3)
+(B.7)
+as z → 0.
+Applying exactly the same methodology to Ha, one writes Ea(s, s′) = Da(s, s′)Da(s′, s) so that
+Ea(s, s′) = −(s − s′)2Ha(κ2(s − s′)2)
+= −(s − s′)2Ha(0) − κ2(s − s′)4H′
+a(0) − 1
+2κ4(s − s′)6H′′
+a(0) + O((s − s′)8).
+(B.8)
+Differentiation yields
+∂2Ea
+∂s∂s′ = 2Ha(0) + 12κ2(s − s′)2H′
+a(0) + 15κ4(s − s′)4H′′
+a(0) + O((s − s′)6)
+(B.9)
+∂4Ea
+∂2s∂2s′ = −24κ2H′
+a(0) − 180κ4(s − s′)2H′′
+a(0) + O((s − s′)4)
+(B.10)
+∂6Ea
+∂3s∂3s′ = 360κ4H′′
+a(0) + O((s − s′)4),
+(B.11)
+from which Ha(0), H′
+a(0) and H′′
+a(0) can be obtained differentiating equation (4.18) using Leibniz’
+rule and subsequently setting s = s′ = 0. It is easily verifiable that this yields
+Ha(0) = [η(˙γ(0), ea(0))]2 = [η(e0(0), ea(0))]2 ,
+(B.12)
+H′
+a(0) = − 1
+4κ2 [η(¨γ(0), ea(0))]2 +
+1
+3κ2 η(˙γ(0), ea(0))η(...γ (0), ea(0)),
+(B.13)
+H′′
+a(0) =
+1
+18κ4 [η(...γ (0), ea(0))]2 −
+1
+12κ4 η(¨γ(0), ea(0))η(γ(4)(0), ea(0))
++
+1
+30κ4 η(˙γ(0), ea(0))η(γ(5)(0), ea(0)),
+(B.14)
+and after some straightforward computation,
+Ha(0) = η0a
+(B.15)
+H′
+a(0) = 1
+3η0a + 1
+4η1a
+(B.16)
+H′′
+a(0) = (η0a)2
+� 1
+18 + κ2 − τ 2
+30κ2
+�
+− κ2 − τ 2
+12κ2 (η1a)2 +
+τ 2
+18κ2 (η2a)2
+= η0a
+� 1
+18 + κ2 − τ 2
+30κ2
+�
++ κ2 − τ 2
+12κ2 η1a −
+τ 2
+18κ2 η2a
+(B.17)
+using the fact that (η0a)2 = η0a and (ηia)2 = −ηia for i = 1, 2, 3, as can be explicitly seen in the
+calculation of H′′
+a(0). Reconstructing Ha using a Taylor series, one obtains
+Ha(z) = η0a + 1
+12z (4η0a + 3η1a)
++
+1
+360κ2 z2 �
+η0a(10κ2 + 6(κ2 − τ 2)) + 15η1a(κ2 − τ 2) − 10η2aτ 2�
++ O(z3),
+(B.18)
+and summing,
+3
+�
+a=0
+Ha(z) = 1 + z
+12 + κ2 + 19τ 2
+360κ2
+z2 + O(z3).
+(B.19)
+22
+
+C
+Wick square
+In this Appendix we show how a quantum inequality for the Wick square can be obtained along
+stationary trajectories. This is a simpler calculation than the one used for the energy density and we
+shall be relatively brief.
+Recall that the general QEI involves a (sum of) pull-backs of a suitable differential operator acting
+on the two-point function,
+T(s, s′) = ⟨Qφ(γ(s))Qφ(γ(s′))⟩ω0 = ((Q ⊗ Q)G0)(γ(s), γ(s′)).
+(C.1)
+For a quantum inequality on the Wick square, the operator Q can be simply identified as the identity,
+so T(s, s′) can be written in this case as
+T(s, s′) = G0(γ(s), γ(s′)).
+(C.2)
+Using the results of Section 4 and in particular, equation (4.13), the two-point function can be neatly
+expressed as
+T(s, s′) = lim
+ǫ→0+
+1
+4π2σǫ(γ(s), γ(s′)) = − lim
+ǫ→0+
+1
+4π2(s − s′ − iǫ)2
+�
+Υ
+�
+κ2(s − s′)2��−1 .
+(C.3)
+As Υ(κ2s2) ≥ 1 for s ∈ R by the Lemma, the entire function Υ(z) is nonvanishing on the real axis,
+and Υ(z)−1 is therefore analytic in a neighbourhood of the real axis.
+Using (4.12) we may write
+Υ(z)−1 = 1 + zJ(z), where J is also analytic in a neighbourhood of the real axis, with J(0) = −1/12.
+Because 0 < 1 + zJ(z) ≤ 1 for z ≥ 0, we may deduce that 0 ≤ −J(z) < 1/z for z > 0.
+We can now split the pulled back two-point function into its singular and regular parts as T(s, s′) =
+Tsing(s − s′) + Treg(s − s′), where
+Tsing(s) = − 1
+4π2 lim
+ǫ→0+
+1
+(s − iǫ)2 ,
+(C.4)
+and
+Treg(s) = −J(κ2s2)
+4π2
+lim
+ǫ→0+
+κ2s2
+(s − iǫ)2 = −κ2J(κ2s2)
+4π2
+,
+(C.5)
+with Treg(0) = κ2/(48π2). Here we used the identity limǫ→0+ x2/(x−iǫ)2 = limǫ→0+(x−iǫ)2/(x−iǫ)2 =
+1 of distributional limits, because g(z) = z2 is entire, while f(z) = z−2 is analytic in the open lower
+half-plane Z ⊂ C and obeys supz∈Z|f(z)(Im z)2| = 1 (see the argument below equation (5.8)).
+Observing that the two-point function given above is translationally invariant, we can use the
+bound given by (1.2) and (1.3) and thus write
+�
+ds|g(s)|2⟨:(Qφ)2:⟩ω(γ(s)) ≥ −
+� ∞
+−∞
+dα|ˆg(α)|2Qeven(α)
+(C.6)
+where
+Qeven(α) =
+1
+2π2
+�� 0
+−∞
+ˆT(u) du +
+� α
+0
+ˆTodd(u) du
+�
+.
+(C.7)
+The Fourier transform of Tsing is easily shown to be ˆTsing(u) =
+u
+2πΘ(u). Again, Treg is smooth, real
+and even on R, decaying like O(s−2) as |s| → ∞ because of the decay of J. Evidently Treg does not
+contribute to ˆTodd as Treg is absolutely integrable and has a well defined, continuous, real and even
+Fourier transform. In this case, Tsing is actually universal; the information relating to the specific
+worldline is encoded in Treg, as can also be seen below in Eq. (C.10). Clearly, ˆTsing does not contribute
+to the first term in (C.7) and, recalling that Treg is even, the odd part of ˆT is
+ˆTodd(u) = u
+4π,
+(C.8)
+and so Qeven is given in the form
+Qeven(α) =
+1
+2π2
+�� 0
+−∞
+du ˆTreg(u) + 1
+4π
+� α
+0
+du u
+�
+=
+1
+16π3 α2 + Treg(0)
+2π
+.
+(C.9)
+23
+
+In direct analogy to the analysis of the energy density, the evenness of Treg and the Fourier inversion
+formula have been used. Inserting this into (C.6) gives the QI bound
+�
+ds|g(s)|2⟨:φ2:⟩ω(γ(s)) ≥ − 1
+8π2
+� ∞
+−∞
+ds
+�
+|g′(s)|2 + C|g(s)|2�
+.
+(C.10)
+where C = 8π2Treg(0) = κ2/6.
+Considering the scaling behaviour, using the same test function gλ(s) = λ−1/2g(λ/s) as in the case
+for the QEI (1.11), one can easily verify that
+�
+ds|gλ(s)|2⟨:φ2:⟩ω(γ(s)) ≥ − ∥g′∥2
+8π2λ2 − κ2∥g∥2
+48π2 ,
+(C.11)
+where again ∥g∥2 denotes the L2-norm of the function g. Taking the limit λ → ∞ yields the following
+formula,
+lim inf
+λ−→∞
+� ∞
+−∞
+ds|gλ(s)|2⟨:φ2:⟩ω(γ(s)) ≥ − κ2
+48π2
+(C.12)
+when considering the functions g such that ∥g∥2 = 1. Physically, since one can interpret 12⟨:φ2:⟩ as
+the square of a local temperature [4], states with negative expected Wick square are regarded as being
+locally out of equilibrium. The above bound therefore quantifies the extent to which the thermal
+interpretation may fail uniformly along these worldlines, in terms of their proper acceleration. This
+raises an intriguing question as to whether there are states that would saturate this bound – something
+quite relevant to the Unruh experiments discussed in Section 6.
+In relation to the Unruh effect, a study of the detailed balance temperature obtained from the
+excitation of an Unruh-DeWitt detector carried along stationary worldlines can be found in [23]. Here
+the quantum field is assumed to be in the vacuum state, and the temperature depends not only on
+the curvature invariants but also on the energy gap of the detector. Although this is a different focus
+from our results, which concern averages of the Wick square in arbitrary Hadamard states, there are
+technical similarities, because the pulled back vacuum Wightman function plays a key role in both.
+It would be interesting to understand whether some of the methods described here can be used to
+corroborate the numerical results of [23].
+D
+Computation of the renormalised stress-tensor for thermal and
+ground states on Rindler spacetime
+The Feynman propagator for a thermal state at inverse temperature β of the massless scalar field
+in Minkowski spacetime was given by Dowker [6] and the Wightman functions (including for higher
+spin) can be found in [38]. Adopting coordinates t = ξ sinh χ, x = ξ cosh χ, the Rindler wedge x > |t|
+of Minkowski spacetime has metric ξ2 dχ2 − dξ2 − dy2 − dz2, and any curve χ �→ (aχ, 1/a, y0, z0)
+with a > 0 is a curve of proper acceleration a in proper time parameterisation. Given two points
+x = (χ, ξ, y, z) and x′ = (χ′, ξ′, y′, z′), write
+α(x, x′) = cosh−1
+�ξ2 + (ξ′)2 + (y − y′)2 + (z − z′)2
+2ξξ′
+�
+,
+(D.1)
+whereupon the Wightman function Gβ(x, x′) = ⟨φ(x)φ(x′)⟩β for the temperature β−1 KMS state with
+respect to the coordinate χ is
+Gβ(x, x′) =
+1
+4πβξξ′ sinh α(x, x′)
+�
+sinh(2πα(x, x′)/β)
+cosh(2πα(x, x′)/β) − cosh(2π(χ − χ′ − iǫ)/β)
+�
+.
+(D.2)
+The β = 2π case coincides with the restriction of the Minkowski vacuum state to the wedge, while the
+zero temperature limit has Wightman function
+G∞(x, x′) = −
+α(x, x′)
+4π2ξξ′ sinh α(x, x′)(α(x, x′)2 − (χ − χ′ − iǫ)2).
+(D.3)
+24
+
+To obtain the renormalised (minimally coupled) stress-energy tensor, we first apply suitable derivatives
+to Gβ − G2π and take the limit x′ → x, obtaining
+⟨:(∇µφ)(x)(∇νφ)(x):⟩β =
+4π2 − β2
+1440π2β4ξ4
+�
+(16π2 + 14β2)ˆuµˆuν + 30β2ˆaµˆaν − (4π2 + 11β2)ηµν
+�
+,
+(D.4)
+where, at spacetime position x, ˆuµ = ξ−1(∂χ)µ is the 4-velocity of the curve through x with constant
+ξ, y and z, and ˆaµ = (∂ξ)µ is the unit spacelike vector parallel to the 4-acceleration of this curve.
+Consequently,
+⟨:Tµν:⟩β =
+4π2 − β2
+1440π2β4ξ4
+�
+(16π2 + 14β2)ˆuµˆuν + 30β2ˆaµˆaν − (4π2 − 19β2)ηµν
+�
+(D.5)
+and the result for Rindler ground state is obtained by taking β → ∞, giving
+⟨:Tµν:⟩∞ = −
+1
+1440π2ξ4 (14ˆuµˆuν + 30ˆaµˆaν + 19ηµν) .
+(D.6)
+Computing the energy density on curves of constant ξ yields (6.1).
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+

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+arXiv:2301.02158v1  [quant-ph]  5 Jan 2023
+Limits of Fault-Tolerance on Resource-Constrained
+Quantum Circuits for Classical Problems
+Uthirakalyani G†, Anuj K. Nayak†, Avhishek Chatterjee, and Lav R. Varshney, Senior Member IEEE
+Abstract—Existing lower bounds on redundancy in fault-
+tolerant quantum circuits are applicable when both the input
+and the intended output are quantum states. These bounds may
+not necessarily hold, however, when the input and the intended
+output are classical bits, as in the Deutsch-Jozsa, Grover, or Shor
+algorithms. Here we show that indeed, noise thresholds obtained
+from existing bounds do not apply to a simple fault-tolerant
+implementation of the Deutsch-Jozsa algorithm. Then we obtain
+the first lower bound on the minimum required redundancy for
+fault-tolerant quantum circuits with classical inputs and outputs.
+Recent results show that due to physical resource constraints
+in quantum circuits, increasing redundancy can increase noise,
+which in turn may render many fault-tolerance schemes useless.
+So it is of both practical and theoretical interest to characterize
+the effect of resource constraints on the fundamental limits of
+fault-tolerant quantum circuits. Thus as an application of our
+lower bound, we characterize the fundamental limit of fault-
+tolerant quantum circuits with classical inputs and outputs under
+resource constraint-induced noise models.
+Keywords—fault-tolerant
+computing,
+redundancy,
+resource
+constraints
+I. INTRODUCTION
+Initial ideas [1], [2], and especially mathematical demon-
+strations of advantages of quantum computing over classical
+computing [3], [4], have spurred considerable interest. How-
+ever, noise in quantum circuits heavily restricts the class of
+problems that can be solved using quantum hardware. Indeed,
+the formal term NISQ (Noisy Intermediate Scale Quantum)
+has been introduced to describe the current era where quantum
+processors are noise-limited [5].
+To limit the corruption of quantum states due to noise,
+the pursuit of fault-tolerant quantum circuits has led to a
+large literature in quantum error correction. Early papers
+demonstrated that one can achieve arbitrary computational
+accuracy when physical noise is below a certain threshold.
+Achievability of any desired fault tolerance in these initial
+works required a poly-logarithmic redundancy with respect to
+the size of the quantum circuit [6]–[9], but more recent works
+extend such threshold theorems to require only a constant
+overhead [10], [11], reminiscent of work in classical fault-
+tolerant computing [12], [13].
+† The student authors contributed equally.
+Uthirakalyani. G and A. Chatterjee are with the Department of Electrical
+Engineering, Indian Institute of Technology, Madras, Chennai 600036, India
+(emails:{ee19d404@smail,avhishek@ee}.iitm.ac.in).
+A. K. Nayak and L. R. Varshney are with Coordinated Science Labora-
+tory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
+(emails:{anujk4, varshney}@illinois.edu).
+This work was supported in part by National Science Foundation grant
+PHY-2112890.
+In this direction, some works provide fundamental limits
+(lower bounds) on redundancy for arbitrarily accurate compu-
+tation [14]–[18]. However, all of these lower bounds are for
+quantum input/output, rather than classical input/output which
+is common for a large class of algorithms, such as those due
+to Deutsch-Jozsa [4], Shor [19], and Grover [20]. Here, we
+demonstrate by example that lower bounds obtained so far in
+quantum fault tolerance are not applicable for quantum circuits
+with classical input/output, and provide a general alternate
+bound. As far as we know, this is the first lower bound on fault
+tolerance for quantum circuits with classical input/output.
+The effects of noise on computational accuracy of quantum
+circuits are typically studied assuming the noise per physical
+qubit is constant with respect to the the size of the circuit. Un-
+fortunately, this is not true in many quantum devices today—
+often due to limited physical resources such as energy [21],
+volume [22], or available bandwidth [23]—that have physical
+noise levels that grow as the quantum computer grows [24].
+Fellous-Asiani, et al. introduce physical models of such scale-
+dependent noise and also aim to extend threshold theorems
+to this setting. However, the characterization of computational
+error (per logical qubit error) is empirical in nature, lacking
+precise mathematical treatment. Moreover, the characterization
+depends on specific implementation and is restricted to con-
+catenated codes. Here, using our new redundancy lower bound
+and tools from optimization theory, we characterize the limits
+of scale-dependence on fault-tolerant quantum circuits with
+classical input/output, agnostic to specific implementation and
+error correction methods.
+The two motivations for the present work are therefore to
+obtain lower bounds on the required redundancy of a quantum
+circuit for computation with classical input/output, and to
+investigate the effect of resource constraints (like energy or
+volume) on this bound.
+The distance between the distributions of the output clas-
+sical bit corresponding to two different quantum input states
+vanishes exponentially with the depth of the circuit when noise
+is above a threshold [16]. Similarly for the trace distance be-
+tween the output quantum states [15]. These results, however,
+do not apply when the depth of the circuit is small; they also
+do not provide lower bounds on required redundancy for sub-
+threshold noise when fault-tolerant computation is possible.
+This work focuses on shallow quantum circuits whose
+input and output are classical, aiming for converse results for
+classical computation using quantum circuits that yield lower
+bounds on required redundancy. In [14], [18], lower bounds on
+required redundancy that also led to improved noise thresholds
+
+were obtained. However, the fault-tolerance criteria in [14],
+[18] are not appropriate for our setting, as Section II argues
+using the example of the well-known Deutsch-Jozsa algorithm.
+The experimental finding that noise increases with more
+redundancy under resource constraints implies that simple per
+(logical) qubit redundancy cannot achieve arbitrary computa-
+tional accuracy even if noise per physical qubit is below the
+fault-tolerance threshold, in contrast to conventional threshold
+theorems [24]. This limitation is due to two opposing forces:
+improvement in accuracy due to increased redundancy and
+worse overall noise with redundancy due to scale-dependence.
+In this regard, we aim to find the sweet spot on redundancy
+for a desired computational accuracy.
+The remainder of the paper is organized as follows. Sec-
+tion II motivates our work through a counterexample that illus-
+trates the need for a new redundancy lower bound. Section III
+then gives the mathematical models of computation, noise, and
+resource constraints that form the basis of our analysis. Then,
+the primary contributions follow:
+• Section IV proves a converse bound on redundancy
+required for classical computation on quantum circuits,
+drawing on one-shot capacity of classical-quantum chan-
+nels (Theorem 1).
+• Section V through VII analyze the converse results on the
+limits of scale-dependence for fault-tolerant computation,
+including closed-form and numerical solutions for some
+canonical quantum device models.
+Finally, Section VIII concludes.
+II. NEED FOR A NEW REDUNDANCY LOWER BOUND:
+A COUNTEREXAMPLE
+Consider a quantum circuit that suffers from erasure noise
+(with erasure probability p) right before the final measurement.
+From one of the best known noise threshold bounds [14]
+and the capacity results for erasure channels, it follows that
+for an erasure noise per physical qubit p > 1
+2, fault-tolerant
+computation is not possible (i.e., the required redundancy is
+not finite). However, we demonstrate that a simple adaptation
+of the Deutsch-Jozsa algorithm on this quantum circuit (with
+erasure noise before the final measurement) can have a prob-
+ability of error less than any ǫ > 0 even if p > 1
+2.
+Fig. 1. A schema of quantum circuit that implements Deutsch-Jozsa algorithm
+with erasure noise, to demonstrate the need for a new redundancy bound for
+fault-tolerant quantum computation with classical I/Os.
+The Deutsch-Jozsa algorithm is used to determine if the
+given function oracle, f
+: {0, 1}n → {0, 1} is constant
+(0 or 1 for all input strings) or balanced (0 for half the
+input strings and 1 for the rest). From [25, Eq. 1.51], the
+quantum state before measurement under the absence of noise
+is ψ1 = �
+z,x∈{0,1}n
+(−1)x.z+f(x)
+2n
+|z⟩ |y⟩ , where |y⟩ = |0⟩−|1⟩
+√
+2
+.
+Measuring the first n qubits yields either |0⟩⊗n if f(·) is
+constant, or an n-qubit state from {|0⟩ , |1⟩}⊗n \ {|0⟩⊗n} if
+f(·) is balanced. Suppose the quantum states are corrupted by
+erasure right before measurement as in Fig. 1; then the state
+of the circuit becomes
+ψ2 =
+�
+z,x∈{0,1}n
+(−1)x.z+f(x)
+2n
+|z⟩(e) |y⟩(e) ,
+where |z⟩(e) and |y⟩(e) are the corrupted (i.i.d. erased) versions
+of |z⟩ and |y⟩, respectively. For example, if f(·) is constant,
+|z⟩(e) = |e00e00 · · ·e0⟩, i.e., each qubit state |0⟩ is replaced
+i.i.d. with probability p by qubit state |e⟩. Now, consider our
+modified algorithm:
+1) Run Deutsch-Jozsa algorithm T times.
+2) If no |1⟩ state was measured in any run, declare function
+oracle f(·) a constant.
+When f(·) is balanced, the measurement in the no-erasure
+case must have one or more |1⟩ states. Such an oracle can be
+incorrectly declared as constant when all of these |1⟩ states
+are erased. So the probability of error is:
+Pe = P{f(·) is declared constant |f(·) is balanced},
+= P
+
+
+
+�
+j,t
+|zj⟩(e)
+t
+̸= |1⟩
+���f(·) is balanced
+
+
+ ,
+≤ P
+��
+t
+|zj⟩(e)
+t
+= |e⟩
+��� |zj⟩ = |1⟩
+�
+,
+for some j ∈ {1, 2, . . ., n}. Since erasures are independent,
+Pe ≤
+T
+�
+t=1
+P
+�
+|zj⟩(e)
+t
+= |e⟩
+��� |zj⟩ = |1⟩
+�
+= pT .
+Choosing T
+≥
+��� ln ǫ
+ln p
+���, one can achieve Pe ≤ ǫ for any
+p ∈ [0, 1). This counterexample proves that the bound for
+redundancy N ≥
+n
+Q(N) proposed in [14] does not hold for
+quantum computation with classical input/output, since for an
+erasure channel, the quantum capacity Q(N) = max{0, 1 −
+2p} = 0 as p >
+1
+2. This motivates the need for a different
+bound which holds for classical I/O.
+Note that this does not imply the prior bounds are incorrect;
+the apparent contradiction is due differences in the definition
+of accuracy. Prior work [14] uses a notion of distance (or
+similarity) between the output quantum states of noiseless
+and noisy circuits to quantify accuracy. This requirement is
+too stringent when the output bits are classical and error
+probability is a more suitable performance criterion [26], [27].
+As such, we obtain a lower bound on the redundancy under the
+error probability criterion and then study the effect of resource
+constraints.
+
+III. MODEL
+A. Model of Computation
+Consider the computational model in Fig. 2, which is a
+quantum circuit with classical inputs and classical outputs.
+This is denoted by CQC : {0, 1}n → {0, 1}n or equivalently
+CQC(x) for x ∈ {0, 1}n, where n is the input size. The goal
+of the circuit is to realize a function f : {0, 1}n → {0, 1}n.
+The circuit consists of l layers. The first layer takes n clas-
+sical inputs (x) as orthogonal quantum states |0⟩ and |1⟩ along
+with N − n ancillas. It maps the input to a density operator
+of dimension 2N. Any subsequent layer i, for 2 ≤ i ≤ l − 1,
+takes the output of the previous layer, layer i − 1 as input.
+The output of any layer i, 1 ≤ i ≤ l − 1, is a density operator
+of dimension 2N. The final layer, layer l, performs a POVM
+measurement and obtains classical output CQC(x).
+Each layer i, with i ∈ {1, 2, . . ., l − 1} is a noisy quantum
+operation. This is modeled as a noiseless quantum operation
+Li on density operators of dimensions 2N followed by N
+i.i.d. quantum channels N (Fig. 2). Finally, the last layer,
+layer l, performs a measurement (POVM), which yields a
+classical output. Thus the quantum circuit can be represented
+as a composition of quantum operations as CQC(x) = Ll ◦
+N ⊗N ◦ Ll−1 ◦ · · · ◦ L2 ◦ N ⊗N ◦ L1(x), where ◦ has the usual
+meaning of function composition.
+We use the notation QCl−1 to denote the combined opera-
+tions of layers 2 to l, given by Ll◦N ⊗N ◦Ll−1◦· · ·◦N ⊗N ◦L2.
+B. Noise models
+Here, we consider only Holevo-additive channels charac-
+terized by a single parameter p ∈ [0, 1] and whose Holevo
+capacity is monotonically decreasing in p. We use the generic
+notation Np for such a channel with parameter p. Examples
+include erasure, depolarizing, and symmetric GAD channels.
+a) Erasure Channel:
+In a quantum erasure channel
+(QEC), each qubit flips to |e⟩⟨e|, which is orthogonal to every
+ρ ∈ L(Cd), with probability p. Therefore, whenever a qubit
+gets corrupted, the location of corruption is known.
+Np(ρ) = (1 − p)ρ + ρTr[ρ] |e⟩⟨e| .
+The classical capacity is [28]:
+χ(Np) = 1 − p.
+(1)
+b) Depolarizing Channel: When a qubit undergoes de-
+polarizing noise, it is replaced by a maximally mixed state
+I/2 with probability p [28]:
+Np(ρ) = (1 − p)ρ + p
+2I.
+In contrast to the erasure channel, the receiver (or the decoder)
+is not aware of the location of the error. The Holevo informa-
+tion of the depolarizing channel is:
+χ(Np) = 1 − h2
+� p
+2
+�
+,
+(2)
+where h2(·) is the binary entropy function. Note that the
+Holevo information is similar to the capacity of a binary
+symmetric channel with crossover probability p/2.
+c) Generalized Amplitude Damping Channel (GADC):
+Amplitude damping channels model the transformation of
+an excited atom to ground state by spontaneous emission
+of photons. The changes are expressed using |0⟩ for the
+ground (no photon) state and |1⟩ for the excited state. If
+the initial state of the environment |0⟩⟨0|, is replaced by the
+state θµ ≜ (1 − µ) |0⟩⟨0| + µ |1⟩⟨1| , µ ∈ [0, 1] where, µ is
+thermal noise, we get the generalized ADC described using
+the following four Kraus operators [29]:
+A1 =
+�
+1 − µ
+�1
+0
+0
+√1 − p
+�
+,
+A2 =
+�
+1 − µ
+�0
+√p
+0
+0
+�
+,
+A3 = õ
+�√1 − p
+0
+0
+1
+�
+,
+A4 = õ
+� 0
+0
+√p
+0
+�
+.
+GADC is not additive in general (for arbitrary µ). However, in
+the special case of symmetric generalized amplitude damping,
+i.e., generalized amplitude damping with µ = 1/2, it is a
+Holevo additive channel. The classical capacity of symmetric
+GADC (µ = 1/2) is [29]:
+χ(Np) = 1 − h2
+�
+1−√1−p
+2
+�
+,
+(3)
+where p is the probability an atom decays from excited to
+ground state.
+Remark 1. Note that we have used p to describe different
+impairments in different channels, so p must be interpreted
+appropriately based on context.
+C. Resource Constraints and Scale-Dependent Noise
+In [24], it was shown that resource constraints can lead
+to an increase in noise with increase in redundancy, scale-
+dependent noise. A few models of scale-dependent noise have
+been studied in [24].
+Let k ≜ N/n ≥ 1 be the redundancy and p(k) be the noise
+strength when the redundancy is k. (Recall that we consider
+Holevo-additive noise models that can be characterized by
+a single parameter 0 ≤ p ≤ 1.) In the polynomial model,
+p(k) = min(p0(1 + α(k − 1))γ, 1) and in the exponential
+model, p(k) = min(p0 exp(α(k − 1)γ), 1). Here, p0 ∈ [0, 1]
+is the noise strength in the absence of any redundancy, i.e.,
+k = 1, and α and γ are positive parameters.
+Motivated by practically useful noise models like erasure,
+depolarization, and models for scale dependence in [24], we
+consider the following generic scale-dependent noise model.
+Definition 1. Noise Np is parameterized by a single parameter
+p ∈ [0, 1] and the Holevo information χ(Np) is non-increasing
+in p. The parameter p is a function of redundancy k, given
+by min(p(k; p0, θ), 1), where θ is a tuple of non-negative
+parameters from the set K, and
+1) p0 = p(1; p0, θ) for all θ,
+2) for any k ≥ 1, p(k; p0, θ) is non-decreasing in any
+component of θ and in p0, given the other parameters
+are fixed.
+Here, p0 represents the noise without redundancy, i.e., the
+initial noise without any resource constraint arising due to
+
+Fig. 2. CQC model of computation: classical input, quantum computation, and classical output.
+redundancy. Clearly, the polynomial and exponential models
+are special cases with θ = (α, γ) ∈ K = R2
+≥0.
+The threshold for p0, i.e., the minimum p0 beyond which
+reliable quantum computation is not possible, was studied in
+[24] assuming concatenated codes for error correction. Here,
+we obtain a universal threshold for all fault tolerance schemes.
+IV. LOWER BOUND ON REQUIRED REDUNDANCY
+We first define the accuracy criterion for computation using
+quantum circuits with classical input/output. Then we show
+how to convert the noisy computation problem to a communi-
+cation problem over i.i.d. quantum channels. This finally leads
+to the redundancy bound in Theorem 1, which we use to obtain
+thresholds for p0 under resource constraints.
+Definition 2. Suppose f(·) is a classical function realized by
+a quantum circuit CQC(·) as defined in Sec. III. Then the
+ǫ-accuracy is:
+P{CQC(x) ̸= f(x)} < ǫ, for all x ∈ Zn
+2 .
+(4)
+Eq. (4) holds for all x ∈ Zn
+2 . Therefore, the ǫ-accuracy
+condition holds for any subset of Zn
+2. The following lemma
+states a necessary condition for ǫ-accuracy.
+Lemma 1. Consider x(1), x(2), . . . , x(R) ∈ Zn
+2 s.t. |{f(x(i)) :
+1 ≤ i ≤ R}| = R. Then a necessary condition for ǫ-accuracy
+condition (4) to hold is
+P{CQC(x(i)) ̸= f(x(i))} < ǫ, for all i = 1, 2, . . ., R.
+Note that the domain is restricted to R inputs, such that
+the restricted mapping is bijective. Using this bijectivity, we
+obtain the following simpler lemma, which connects ǫ-accurate
+computation with finite blocklength communication.
+Lemma
+2.
+Suppose
+there
+exists
+a
+CQC(x)
+s.t.
+P{CQC(x(i))
+̸=
+f(x(i))}
+<
+ǫ for all 1
+≤
+i
+≤
+R.
+Then there exists a classical circuit C(·) s.t.
+P{C(CQC(x(i))) ̸= x(i)} < ǫ,
+for all 1 ≤ i ≤ R.
+(5)
+Proof: Suppose ˆf(·) is a restriction of f(·) such that the
+mapping ˆf : {x(i), 1 ≤ i ≤ R} → {f(x(i)), 1 ≤ i ≤ R}, is a
+bijection. Then the inverse map ˆf −1(·) is unique. Choosing a
+C(·) that implements ˆf −1(·), the probability of error can be
+equivalently expressed as
+P{C(CQC(x(i))) ̸= x(i)} < ǫ,
+for all 1 ≤ i ≤ R.
+We define ˆx(i) to be the output of C(CQC(x(i))). Then the
+condition in (5) is equivalent to
+max
+x(i) P{x(i) ̸= ˆx(i)} < ǫ,
+1 ≤ i ≤ R.
+This implies that a necessary condition to satisfy accuracy
+condition (4) is
+inf
+L1,C,QCl−1 max
+x(i) P{x(i) ̸= ˆx(i)} ≤ max
+x(i) P{x(i) ̸= ˆx(i)} < ǫ.
+Note that L1 is an encoding of classical bits into a quantum
+state, and QCl−1 followed by C(·) can be interpreted as the
+decoding of the noisy version of the same quantum state
+(depicted in Fig. 3). Hence, infL1,C,QCl−1 maxx(i) P{x(i) ̸=
+ˆx(i)} is equivalent to the maximum probability of error for
+transmitting message x(i), i ∈ {1, . . . , R} over the channel
+N ⊗N. Using this reduction, we lower-bound the redundancy
+for any classical computation using a quantum circuit.
+Theorem 1. Let f : {0, 1}n → {0, 1}n be a classical function
+and Rf = |{f(x) : x ∈ {0, 1}n}| be the cardinality of the
+range of f. Then, for computing a classical function f with
+
+Fig. 3. Reduction of noisy computation model in Fig. 2 to noisy communi-
+cation model.
+ǫ-accuracy using a quantum circuit corrupted by i.i.d. Holevo-
+additive noise, the required number of physical qubits N is
+bounded as
+N > (1 − ǫ) log2(Rf) − h2(ǫ)
+χ(N)
+for all ǫ ∈ [0, 1
+2].
+Proof: For any additive quantum channel N, an upper
+bound for classical communication over a quantum channel
+using an (M, N, ǫ) code is [28]:
+log2(|M|) ≤ χ(N ⊗N) + h2(ǫ)
+1 − ǫ
+,
+where M is the message alphabet. Assigning |M| = Rf
+yields
+ǫ > Pe ≥ 1 − χ(N ⊗N) + h2(Pe)
+log2 Rf
+,
+(6)
+≥ 1 − χ(N ⊗N) + h2(ǫ)
+log2 Rf
+.
+The last inequality holds, since h2(·) is increasing in [0, 1
+2].
+Rearranging, we obtain
+χ(N ⊗N) > (1 − ǫ) log2 Rf − h2(ǫ).
+Noting that Holevo information is sub-additive,
+Nχ(N) > (1 − ǫ) log2 Rf − h2(ǫ),
+N > (1 − ǫ) log2 Rf − h2(ǫ)
+χ(N)
+.
+(7)
+The bound in Theorem 1 states that the number of quantum
+buffers, N, needed for accurate computation of an n-bit
+function f is lower bounded. As given in Sec. III, k ≜
+N
+n
+is the redundancy of the quantum circuit. Thus, Theorem 1
+can be seen as a lower bound on the required redundancy.
+The quantity log2 Rf is the number of bits needed to encode
+the output. We define η ≜ log2 Rf
+n
+as the compression factor of
+f. To understand the impact of scale-dependent noise, we will
+use the following corollary of Theorem 1 that gives a lower
+bound on the redundancy k.
+Corollary 1. The condition for ǫ-accuracy in Theorem 1 is
+alternatively
+k > c(ǫ, η, n)
+χ(N) ,
+(8)
+where c(ǫ, η, n) ≜ (1 − ǫ)η − h2(ǫ)
+n .
+Proof: Substituting log2 Rf = ηn and k = N/n in (7),
+and rearranging we obtain
+N > (1 − ǫ)ηn − h2(ǫ)
+χ(N)
+,
+k > c(ǫ, η, n)
+χ(N) .
+V. SCALE-DEPENDENT NOISE: CONVERSE REGIONS
+Let Np denote the channel parameterized by a noise in-
+tensity term p. Examples include the probability of erasure,
+p, for erasure channels; the probability of a quantum state
+being replaced by a maximally mixed state, p, for depolarizing
+channels; and the amplitude decay parameter, p, for symmetric
+GAD channels. If the error per physical qubit p is a constant
+w.r.t. k, then χ(Np) is constant. Therefore, one can ensure
+that the necessary condition for ǫ-accuracy in (8) is satisfied
+by sufficiently increasing redundancy (choosing large k).
+On the other hand, if p scales (increases) with redundancy,
+then χ(N) decreases with k (we denote the dependency on
+p(k) as χ(Np(k))). Therefore, satisfying ǫ-accuracy condition
+k > c(ǫ, η, n)/χ(Np(k)) is not guaranteed. In fact, Fig. 4
+plots the error probability lower bound (6) with both scale-
+independent and scale-dependent erasure. Notice that when
+the physical noise is independent of k, the Pe lower bound
+rapidly decreases with an increase in redundancy, whereas
+when the noise is scale dependent, the probability of error
+initially decreases with increasing redundancy k, but then
+grows beyond a certain optimum k. With this motivation, we
+explore the limitations of ǫ-accurate computation under scale-
+dependent noise.
+We specifically aim to characterize the set of (p0, θ) for
+which ǫ-accurate computation is not possible. This is equiv-
+alent to the noise threshold in traditional models, with scale-
+independent noise. The following corollary to Theorem 1
+provides a converse in terms of θ.
+Corollary 2. Suppose we have,
+¯Θ ≜
+�
+(p0, θ) ∈ K
+���min
+k≥1 g(k, p0, θ, ǫ) ≥ 0
+�
+,
+where
+g(k, p0, θ, ǫ) ≜ c(ǫ, η, n)
+k
+− χ(Np(k)).
+Then ǫ-accurate computation is not possible for θ ∈ ¯Θ. Also,
+if (p0, θ) ∈ ¯Θ then (p′
+0, θ′) ∈ ¯Θ if (p′
+0, θ′) ≥ (p0, θ) in a
+component-wise sense.
+
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+2.50
+2.75
+Redundancy (k)
+10−2
+10−1
+100
+Pe (lower bound)
+p0 = 0.15
+p0 = 0.20
+p0 = 0.30
+Fig. 4. Comparison between Pe lower bound with (solid lines) and without
+(dashed-lines) scale-dependent physical noise for erasure channel.
+Proof: From Definition 2, we must prove that if Pe < ǫ,
+then (p0, θ) /∈ ¯Θ. Considering the scale-dependent noise in
+Corollary 1, we have if Pe < ǫ, then
+k > c(ǫ, η, n)
+χ(Np((k)),
+g(k, p0, θ, ǫ) = c(ǫ, η, n)
+k
+− χ(Np(k)) < 0.
+(9)
+For any θ ∈ K, (9) is satisfied only if
+min
+k≥1 g(k, p0, θ, ǫ) < 0.
+In other words, (p0, θ) /∈ ¯Θ.
+As χ(Np) is non-increasing in p and p(k; p0, θ) is
+non-decreasing in each component, (p′
+0, θ′) ≥ (p0, θ) in
+a component-wise sense implies (p′
+0, θ′) ∈
+¯Θ whenever
+(p0, θ) ∈ ¯Θ.
+We refer to ¯Θ as the converse region since ǫ-accurate
+classical computation on quantum circuits is not possible if the
+parameters of the scale-dependent noise are in ¯Θ. As any fault-
+tolerant implementation has to avoid this region, characterizing
+¯Θ is of particular interest. By Corollary 2, for characterizing
+¯Θ, it is enough to find the minimum p0 for each θ such that
+(p0, θ) ∈ ¯Θ. More precisely, for a fixed θ, the threshold pth(θ)
+can be defined as:
+pth(θ) := inf{p0 | (p0, θ) ∈ ¯Θ}.
+(10)
+The threshold pth(θ) (or pth for brevity) can be obtained
+by solving the following optimization problem.
+minimize p0
+s.t.
+min
+k≥1, 0≤p(k)≤1 gθ(k, p0) ≥ 0,
+(11)
+where, gθ(k, p0) := g(k, p0, θ, ǫ).
+Consider the following optimization problem
+PL :
+min
+k≥1, 0≤p(k)≤1 gθ(k, p0).
+Clearly, (11) has the optimization problem PL, which we refer
+to as the lower-level optimization problem, as a constraint.
+Thus, (11) is a bi-level optimization problem. For a given set
+of θ the solution to PL is a function of p0, which we denote
+as g∗
+θ(p0). Thus, the bi-level optimization problem in (11) can
+also be written as
+min p0
+s.t. g∗
+θ(p0) ≥ 0.
+(12)
+In general, to compute the threshold pth one needs to solve
+(11). However, for erasure noise and some special classes of
+p(k; p0, θ), closed-form expressions for pth can be obtained.
+Theorem 2. For erasure noise, thresholds are as follows.
+1) If p(k; p0) = p0 (constant), then pth = 1.
+2) If p(k; p0, α) = p0(1 + α(k − 1)), then
+pth =
+
+
+
+1 − c,
+if α ≥
+c
+1−c, and
+(
+√cα−√cα−α+1)
+2
+(α−1)2
+,
+otherwise.
+3) If p(k; p0, γ) = p0kγ, then
+pth =
+
+
+
+1 − c,
+if γ ≥
+c
+1−c, and
+( γ
+c )
+γ
+(γ+1)γ+1 ,
+otherwise.
+Here c = c(ǫ, η, n), defined in Corollary 1. Note that θ = ∅, α
+and γ in cases 1), 2) and 3), respectively.
+Proof: Consider a procedure to find a closed-form expres-
+sion for pth as follows.
+1) Minimize gθ(k, p0) over k. Since p(k; p0, θ) is non-
+decreasing in k, it is enough to minimize gθ(k, p0) over
+[1, kmax], where kmax = max{k | p(k; p0, θ) ≤ 1} (see
+Appendix B for more details). The minimum occurs at
+either k = 1, k = kmax or a stationary point of gθ(k, p0)
+in [1, kmax].
+2) Substitute the minimizer k into gθ(k, p0) ≥ 0, which
+yields an equation in p0, θ.
+3) Solving the equation for p0 yields a closed-form expres-
+sion for pth.
+The derivation of pth for corresponding p(k; p0, θ) is given in
+Appendix A.
+For a general p(k; p0, θ), however, a closed-form expression
+for pth in terms of θ cannot be obtained, and therefore, pth
+must be computed numerically.
+We develop Algorithm 1 to obtain pth by solving bi-level
+optimization problem (11). In Algorithm 1, we solve the
+alternate formulation (12) using the bisection method, while
+assuming access to an oracle that computes g∗
+θ(p0) for any p0.
+Later, we also develop efficient algorithms that solve PL and
+obtain g∗
+θ(p0) for any p0.
+Algorithm 1 computes the threshold pth (up to an error of
+δp0), for a pre-determined set of θ. First, a channel-specific
+Lipschitz constant L is computed using Eqs. (15), (17), or
+(19) for a given (p0, θ), which determines how quickly PL is
+solved. Lines 7–17 describe the bisection method to compute
+pth. Depending on whether PL is convex or non-convex,
+Algorithm 2 or Algorithm 3 is used to compute g∗
+θ(p0),
+respectively.
+
+The following theorem provides a proof of global conver-
+gence of Algorithm 1, with only a monotonicity assumption
+in θ (note that continuity in θ is not needed).
+Theorem 3. Suppose a quantum circuit is corrupted by a scale-
+dependent noise-per-physical qubit, p(k; θ) that is monotonic
+in θ. Then for any given θ ∈ K, the sequence {p0i} generated
+using Algorithm 1 converges to the threshold pth in (10).
+Proof: Algorithm 1 generates a non-increasing sequence
+{p+
+0i} and a non-decreasing sequence {p−
+0i}, which at every
+iteration yields g∗(p+
+0i) ≥ 0 and g∗(p−
+0i) < 0, with p0i =
+p+
+0i +p−
+0i
+2
+. Since the bisection method halves the difference
+between p+
+0i and p−
+0i at every iteration (i.e., p+
+0i+1 − p−
+0i+1 =
+p+
+0i −p−
+0i
+2
+), we have that for all ǫ > 0, there exists an i0 such
+that for all i ≥ i0, we get p+
+0i −p−
+0i < ǫ. Also, since both {p+
+0i}
+and {p−
+0i} are bounded, they converge, and since for all i ≥ i0,
+p+
+0i −p−
+0i < ǫ, they converge to a common limit point (say p∗).
+Since, K is closed, (p∗
+0, α, γ) ∈ K. Due to the monotonicity
+of g∗
+θ(p0) (non-decreasing with p0), the following inequality
+holds: g∗
+θ(p−
+0i) ≤ g∗
+θ(p∗
+0) ≤ g∗
+θ(p+
+0i). Therefore, g∗
+θ(p0) < 0,
+for all p0 < p∗, and g∗
+θ(p0) ≥ 0, for all p0 > p∗, which is by
+definition p∗ = pth.
+Obtaining g∗
+θ(·) requires solving PL. Next, we present
+efficient algorithms for solving PL for erasure, depolarizing,
+and symmetric GAD channels, and numerically obtain the
+converse surface for those noise models.
+Algorithm 1 Algorithm to obtain pth/¯Θs numerically.
+1: Initialize the set K′ ⊆ K.
+2: Initialize max iters, δp0, δ, ¯Θs = {}, k ← 1.
+3: for each θ ∈ K′ do
+4:
+Initialize i ← 0, ∆p0 ← 1, p0 ← 0.5,
+5:
+p−
+0 ← 0, p+
+0 ← 1.
+6:
+% BISECTION METHOD
+7:
+while ∆p0 > δp0 and i <max iters do
+8:
+p−1 ← p0.
+9:
+p0 ← (p−
+0 + p+
+0 )/2.
+10:
+kmax ← 1 + α−1((p0)− 1
+γ − 1).
+11:
+Solve PL to obtain g∗
+θ(p0) = mink≥1 gθ(k, p0).
+12:
+if g∗
+θ(p0) > 0 then p+
+0 ← p0,
+13:
+else p−
+0 ← p0.
+14:
+end if
+15:
+∆p0 ← |p0 − p−1|.
+16:
+i ← i + 1.
+17:
+end while
+18:
+¯Θs = ¯Θs
+�{(p0, α, γ)}.
+19: end for
+VI. CONVERSE REGION FOR ERASURE
+In this section, we derive necessary conditions for ǫ-accurate
+computation when the source of corruption of quantum states
+is erasure. Substituting for the classical capacity of QEC from
+(1) in (8) yields
+g(k, p0, θ, ǫ) = gθ(k, p0) = c(ǫ, η, n)
+k
++ p(k) − 1 < 0. (13)
+Remark 2. One can equivalently solve PL by restricting the
+range of k to [1, kmax], where kmax = max{k | p(k; p0, θ) ≤
+1}. Also, kmax is finite and hence [1, kmax] is compact, which
+makes it convenient to solve (11). Therefore, one can replace
+line 11 with g∗
+θ(p0) = mink∈[1,kmax] gθ(k, p0) to obtain the
+same value of threshold pth. See Appendix B for more details.
+A. Physical Noise p(k; p0, θ) is Convex in Redundancy k
+For the erasure channel, if p(k; p0, θ) is convex, then
+g(k, p0, θ, ǫ) is convex in [1, kmax], since the Holevo informa-
+tion χ(Np) is affine in p. Therefore, the problem PL in (11),
+is convex, and from Remark 2, the feasible set is compact.
+A convex function over a compact set can be optimized
+using a gradient projection method given in [30]. There are
+many algorithms to solve general gradient projection prob-
+lems such as sequential quadratic programming (SQP) and
+augmented Lagrangian methods that can be directly applied
+to solve PL. Since, our problem is a one-dimensional convex
+problem (with only a Lipschitz gradient constraint) over a
+finite range [1, kmax], we provide a simple constant step-size
+gradient projection algorithm (Algorithm 2).
+Algorithm 2 Projected gradient descent routine.
+1: function PROJGD(gθ, p0, kin, kmax, L, ζ)
+2:
+Initialize j ← 0, kj ← kin, ξ = 1
+L, ∆g = 2ζ.
+3:
+while ∆g ≥ ζ do
+4:
+if gθ(·, p0) is convex then dj ← g′
+θ(kj, p0)
+5:
+else dj ← |g′
+θ(kj, p0)|
+6:
+end if
+7:
+kj+1 ← min{kmax, max{1, kj + ξdj}}.
+8:
+˜g ← gθ(kj+1, p0).
+9:
+∆g ← |gθ(kj, p0) − ˜g|.
+10:
+j ← j + 1.
+11:
+end while
+12:
+return ˜g, kj.
+13: end function
+Algorithm 2 solves PL optimally if step size (ξ) and
+stopping criterion (ζ) are chosen appropriately. Sufficient
+conditions for convergence are: 1) ξ ∈ (0, 1
+L], if g′
+θ(k, p0) ≜
+∂
+∂kgθ(k, p0) is L-Lipschitz over [1, kmax], and 2) stopping
+criterion provided in Definition 3. In all our computations,
+we choose ξ = 1
+L as the step size for fast convergence.
+Definition 3. Stopping criterion 1: Let {kj} be the iterates
+generated by the projected gradient descent algorithm (Algo-
+rithm 2), we use the following stopping criterion for projected
+gradient descent algorithm:
+|gθ(kj, p0) − gθ(kj+1, p0)| <
+δ2
+2Lk2max
+=: ζ.
+(14)
+Then, it follows from the convexity and L-Lipschitz prop-
+erty of gθ(·, p0) that stopping criterion (14) is a sufficient
+condition for convergence, which is gθ(kj+1, p0)−g∗
+θ(p0) ≤ δ.
+The following theorem provides proof of convergence of
+Algorithm 2. For better readability, the associated lemmas used
+in the proof are included in Appendix D.
+
+Theorem 4. Convergence of Algorithm 2: Suppose g∗
+θ(p0) =
+min
+k≥1 gθ(k, p0), which is convex in k. Then Algorithm 2 yields
+˜g arbitrarily close to g∗
+θ(p0), i.e., for any pre-determined δ > 0,
+|˜g − g∗
+θ(p0)| ≤ δ.
+Proof: Let {1, . . . , kl} be a sequence generated by
+projected gradient descent, PROJGD, where kl satisfies the
+stopping criterion. Note that PROJGD does not cross any
+stationary point if the step-size ξ ≤
+1
+L (from Lemma 8).
+So, kl = 1 if and only if ˜g = g∗
+θ(p0) = gθ(1, p0), and
+similarly kl = kmax if and only if ˜g = g∗
+θ(p0) = gθ(kmax, p0).
+Otherwise kl ∈ (1, kmax) and gθ(1, p0) < 0, which implies
+from Lemma 8 that gθ(kl, p0) ≤ 0. From Lemmas 6 and 7,
+kl satisfying the stopping criterion in Def. 3 is sufficient for
+convergence, i.e., ˜g = gθ(kl, p0) and |˜g − g∗
+θ(p0)| ≤ δ.
+The following lemma shows g′
+θ(k, p0) is indeed L-Lipschitz
+over [1, kmax] for a general polynomial noise model and gives
+a closed-form expression for L.
+Lemma 3. Computing Lipschitz constant L: g′
+θ(k, p0) is L-
+Lipschitz over [1, kmax] for scale-dependent erasure noise
+p(k; p0, θ) = p0(1 + α(k − 1))γ with γ ≥ 1 where, for
+c = c(ǫ, η, n),
+L = 2c + α2γ(γ − 1)p
+1
+γ
+0 .
+(15)
+Proof: The proof is given in Appendix C-A.
+Next, we provide a closed-form upper bound for p0 for the
+polynomial noise model; however, the bound is looser than
+pth computed in closed-form in Corollary 2 and numerically
+using Algorithm 1.
+Claim 1. If p(k) = p0(1 + α(k − 1))γ for α > 0 and γ ≥ 1,
+then for any p0 with
+p0 ≥ max
+�c(ǫ, η, n)
+γα
+, 1 − c(ǫ, η, n)
+�
+,
+ǫ-accurate computation is not possible.
+Proof: Since p(k) is convex, so is the following function:
+gθ(k, p0) = p(k) − 1 + c(ǫ, η, n)
+k
+.
+Notice that if the slope g′
+θ(k0, p0) = 0, for some k0 ≥ 1,
+then gθ(k, p0) is increasing in k ≥ k0 due to its convexity
+over k. Therefore, if gθ(k, p0) ≥ 0 and g′
+θ(k, p0) = 0 at
+k = 1, then gθ(k, p0) ≥ 0 is always satisfied ∀k ≥ 1, and
+such (p0, α, γ) ∈ ¯Θ. Solving the following
+∂
+∂k
+�
+p(k) − 1 + c(ǫ, η, n)
+k
+�����
+k=1
+≥ 0,
+yields:
+p0γα ≥ c(ǫ, η, n).
+Noting that p0 ≥ 1 − c(ǫ, η, n) ≥ 0 (at k = 1), we obtain
+p0 ≥ max
+�c(ǫ, η, n)
+γα
+, 1 − c(ǫ, η, n)
+�
+= pth.
+■
+A comparison of the looser bound with that obtained by
+Algorithm 1 is shown in Fig. 5.
+Fig. 5. Comparison of converse surfaces, ¯Θs between numerical optimization
+(Algorithm 1) and the derivative approach (Claim 1). Evidently, the converse
+bound obtained using the algorithm is tighter.
+B. Physical Noise p(k; p0, θ) is Non-convex in Redundancy k
+Suppose p(k) is non-convex, then gθ(k; θ, ǫ) is also non-
+convex. Hence, the lower-level problem PL cannot be solved
+using Algorithm 2 (PROJGD). Therefore, we provide a line-
+search algorithm (Algorithm 3) to compute solution for a non-
+convex problem PL.
+In Algorithm 3, the compact set [1, kmax] is traversed by
+successive gradient descent (or ascent) and perturbation over a
+one-dimensional non-convex function using an iterate starting
+from k = 1 (w.l.o.g.) and moving in the positive k direction.
+Lines 4–9 include one iteration of Algorithm 3, which contains
+calls to PROJGD and PERTURB as subroutines. The variable
+˜g keeps track of the minimum value of gθ(·, p0) encountered
+thus far with an error of δ > 0.
+In Algorithm 3 we reuse the PROJGD routine for gradient
+ascent/descent but with a different (more relaxed) stopping
+criterion than in Def. 3.
+Definition 4. Stopping criterion 2: Let {kj} be the iterates
+generated by the projected gradient descent algorithm (Algo-
+rithm 2). We use the following stopping criterion for projected
+gradient descent algorithm:
+|gθ(kj, p0) − gθ(kj+1, p0)| < δ =: ζ.
+Definition 5. Stopping criterion for PERTURB: Let {kj} be a
+sequence generated by PERTURB routine.
+|gθ(kj, p0) − gθ(k′
+j, p0)| ≥ δ
+L.
+where k′
+j
+= min{kmax, k + ξ|g′
+θ(kj, p0)|} in line 17 of
+Algorithm 3, and g′
+θ(z, p0) = ∂g′
+θ(k,p0)
+∂k
+���
+k=z.
+Remark 3. Note that the stopping criterion for PERTURB
+is similar to Definition 4, but with the inequality reversed.
+Since the stopping criteria of PROJGD and PERTURB are
+complementary, only one of the routines will be active during
+the execution of Algorithm 3.
+Next, Theorem 5 proves convergence of Algorithm 3. Re-
+quired lemmas are in Appendix E.
+
+ure Channel)
+Optimization
+Looser BoundConverse Regions (Eras
+0.84
+3
+2
+20.6
+0
+p
+0.4
+0.2
+0
+0
+2
+3
+1
+aAlgorithm 3 Line search algorithm to find mink≥1 gθ(k, p0),
+when gθ(k, p0) is non-convex w.r.t. k.
+1: function LINESEARCH(gθ, p0, kmax, L, δ)
+2:
+Initialize i ← 0, ki ← 1.
+3:
+˜g ←
+min
+k∈{1,kmax}gθ(k, p0).
+4:
+while i <max iters and ki < kmax do
+5:
+gi, k−
+i ← PROJGD(gθ, p0, ki, kmax, L, δ).
+6:
+ˆgi, ˆki, ki+1 ← PERTURB(gθ, p0, k−
+i , kmax, δ).
+7:
+˜g ← min{˜g, ˆgi}.
+8:
+i ← i + 1.
+9:
+end while
+10:
+return g∗.
+11: end function
+12: function PERTURB(gθ, p0, k, kmax, L, δ)
+13:
+∆k ←
+�
+2δ
+L , ξ ← 1
+L, ˆg ← gθ(k, p0), ˆk ← k.
+14:
+k′ ← min{kmax, k + ξ|g′
+θ(k, p0)|}.
+15:
+while |gθ(k, p0) − gθ(k′, p0)| < δ and k < kmax do
+16:
+k ← min{kmax, k + ∆k}.
+17:
+k′ ← min{kmax, k + ξ|g′
+θ(k, p0)|}.
+18:
+ˆg ← min{ˆg, gθ(k, p0)}.
+19:
+ˆk ← argmin
+k∈{ˆk,k}
+{ˆg, gθ(k, p0)}.
+20:
+end while
+21:
+return ˆg, ˆk, k.
+22: end function
+Theorem 5. Proof of convergence of Algorithm 3: Algo-
+rithm 3 yields ˜g, which is arbitrarily close to g∗
+=
+mink∈[1,kmax]gθ(k, p0), i.e., |˜g−g∗| ≤ δ, for a pre-determined
+δ > 0.
+Proof: Suppose {. . . , ki, k−
+i , ki+1, k−
+i+1, . . .} is the se-
+quence generated by Algorithm 3. From Lemma 8 there are
+no stationary points in (ki, k−
+i ). Then, the PERTURB routine
+keeps track of the minimum value of gθ(·, p0) in [k−
+i , ki+1] at
+discrete increments: ˆgi = mink∈{k−
+i ,k−
+i +∆k,...,ki+1}gθ(k, p0).
+This is followed by executing PROJGD again from ki+1 to
+k−
+i+1, and so on. In every call to the PERTURB routine, ˜g tracks
+the minimum of ˆgi until the ith iteration. From Lemma 10, ˆgi
+differs from mink∈[k−
+i ,ki+1]gθ(k, p0) by at most δ. In line 3
+of Algorithm 3, ˜g is initialized with minimum at boundary
+points k = {1, kmax}. Therefore, ˜g − g∗ ≤ δ. Finally,
+from Lemma 11, Algorithm 3 terminates in finite steps when
+kj = kmax or k−
+j = kmax for some j ≥ i + 1.
+VII. CONVERSE REGION FOR SYMMETRIC GAD AND
+DEPOLARIZING CHANNELS
+A. Converse Region for Symmetric GAD Channel
+Let us compute converse regions when quantum states are
+corrupted by GADCs. We only consider symmetric GADC
+(with µ = 1/2), since its classical capacity is additive; for
+µ ̸= 1/2, the additivity of classical capacity is not known.
+Substituting classical capacity of symmetric GADC from (3)
+in the necessary condition for ǫ-accuracy in (8) yields:
+gθ(k, p0) = c(ǫ, η, n)
+k
+−1+h2
+�
+1 −
+�
+1 − p(k)
+2
+�
+≤ 0. (16)
+In (16), the last term is monotonic (increasing) in p, and
+p(k; p0, θ) is monotonic (increasing) in θ. Therefore, Corol-
+lary 2 also holds for symmetric GAD channel. Therefore, for
+a given θ, the threshold pth can be computed by solving bi-
+level optimization problem (11). However, we cannot obtain
+closed-form expressions like for the erasure channel due to
+the challenge from the binary entropy term in (16); therefore,
+the threshold pth must be computed numerically. Since, sym-
+metric GAD channel is additive, and scale-dependent noise
+p(k; p0, θ) is monotonic in θ component-wise, the threshold
+pth can be computed using Algorithm 1 (Theorem 3 holds).
+However, since Holevo information of symmetric GADC
+is concave in p, even if p(k; p0, θ) is convex in k, unlike
+the erasure case, gθ(k, p0) is not convex in k. Therefore, the
+lower-level problem PL can be solved using Algorithm 3 to
+obtain the threshold pth for a given θ. For a polynomial noise
+model described in Section III-C, we can compute Lipschitz
+constant L in closed form for a given θ as follows.
+Lemma 4. Computing Lipschitz constant L: g′
+θ(k, p0) is L-
+Lipschitz over [1, kmax] for a polynomial scale-dependent
+symmetric GAD noise p(k; p0, θ) = p0(1+α(k−1))γ, where
+L := 2c+α2γ2p
+min{1, 2
+γ }
+0
+4(1 − p0)
+�
+1
+p0 ln 2 + 1 + 2(1 − p0)
+2√1 − p0
+log2 P
+�
+,
+(17)
+with P = 1+√1−p0
+1−√1−p0 .
+Proof: See Appendix C-B.
+B. Converse Region for Depolarizing Channel
+In this section, we compute the converse region when
+computational states are corrupted by depolarizing noise. Sub-
+stituting for the classical capacity of the depolarizing channel
+from (2) in (8), we obtain
+gθ(k, p0) = h2
+�p(k)
+2
+�
+− 1 + c(ǫ, η, n)
+k
+≤ 0.
+(18)
+Similar to the symmetric GAD channel, the first term is
+increasing in p, and p(k; p0, θ) is non-decreasing in θ. There-
+fore, Corollary 2 and computation of threshold pth by solving
+bi-level optimization problem (11) also hold. Also, similar to
+symmetric GADC, since obtaining closed-form expressions for
+pth is not possible, it can be computed using Algorithm 1.
+Since gθ(k, p0) is non-convex (due to h2(·) in (18) being
+concave), the threshold pth can be computed using line-search
+(Algorithm 3). Again, similar to the symmetric GAD channel,
+Lipschitz constant L can be computed in closed form for a
+given θ.
+
+Lemma 5. Computing Lipschitz constant L: g′
+θ(k, p0) is L-
+Lipschitz over [1, kmax] for a polynomial scale-dependent
+depolarizing noise p(k; p0, θ) = p0(1 + α(k − 1))γ, where
+L = 2c + α2γ2p
+min{1, 2
+γ }
+0
+�
+log2(e)
+p0(2 − p0) + 1
+2 log2
+�2 − p0
+p0
+��
+.
+(19)
+Proof: Refer to Appendix C-C.
+C. Comparing Converse Surfaces of Different Channels
+Fig. 6 shows the converse surfaces ¯Θs when quantum
+computation is affected by erasure, depolarizing, and general-
+ized amplitude damping noise. For a given θ = (α, γ) the
+thresholds are related as p(e)
+th
+≥ p(g)
+th
+≥ p(d)
+th
+(point-wise),
+where the superscripts stand for erasure, symmetric GAD, and
+depolarizing channels, respectively. This relation is expected
+since Holevo information of the channels are related for a
+given p ∈ (0, 1) as χ(e)(Np) ≥ χ(g)(Np) ≥ χ(d)(Np) (point-
+wise).
+Fig. 6.
+Comparison of converse surfaces ¯Θs for erasure, depolarizing and
+GADC with ǫ = 0.1, n = 128, and η = 1. The probability of error per
+physical qubit is assumed to scale with redundancy k as p(k; p0, θ) = p0(1+
+α(k − 1))γ.
+VIII. CONCLUSION
+We considered a model of quantum circuits where inputs
+and outputs are classical, which includes a large class of
+quantum circuits that are used to efficiently solve classical
+problems, such as algorithms due to Deutsch-Jozsa, Grover,
+and Shor. We demonstrated that the currently best-known
+redundancy bound for quantum computation is not applicable
+for quantum circuits with classical input and output. We
+considered the scenario where quantum states are corrupted
+by i.i.d. additive quantum channels. We reduced the problem
+of noisy computation to noisy classical communication over
+a quantum channel and used one-shot classical capacity of
+quantum channels to obtain a lower bound on redundancy.
+We also considered a problem of practical interest, namely,
+fault-tolerant quantum computation under resource constraints,
+which results in physical noise per qubit being scale-
+dependent. We cast determining limits of scale dependence
+on computational accuracy as an optimization problem, and
+derived closed-form expressions whenever possible, and for
+other cases we solved the optimization problem numerically.
+APPENDIX A
+CLOSED-FORM EXPRESSIONS OF THRESHOLDS FOR
+COMPUTATIONS CORRUPTED BY ERASURES
+The Holevo capacity of erasure channel is χ(Np) = 1 − p.
+Therefore,
+g(k, p0, θ, ǫ) = c(ǫ, η, n)
+k
++ p(k) − 1.
+If θ ∈ ¯Θ, then from Corollary (2) the following holds:
+g(k, p0, θ, ǫ) = c(ǫ, η, n)
+k
++ p(k) − 1 ≥ 0,
+∀k ≥ 1
+(20)
+Differentiating w.r.t. k and equate to 0 (to find stationary
+point),
+g′(k, p0, θ, ǫ) = −c(ǫ, η, n)
+k2
++ p′(k; p0, θ) = 0
+(21)
+Henceforth, we shall use c = c(ǫ, η, n) for brevity. For a fixed
+θ = ∅, α and γ (respectively), the thresholds pth are derived
+for some well-behaved p(k; p0, θ) as follows:
+1) p(k; p0, θ) = p0 (scale-independent noise, θ = ∅):
+In this case, g′(k, p0, θ, ǫ) = 0 as k → ∞. Substituting
+in (20), we obtain
+p0 − 1 ≥ 0 =⇒ pth = 1.
+2) p(k; p0, θ) = p0(1 + α(k − 1)):
+Suppose (20) holds for some p0, then:
+g(k, p0, θ, ǫ) ≥ 0,
+for k = 1,
+c + p0 − 1 ≥ 0,
+p0 ≥ 1 − c.
+(22)
+Note that kmax = 1 + α−1((p0)−1 − 1) (obtained by
+solving for k in p(k; p0, θ) = 1). Also since p(·; p0, θ)
+is linear, g(·, p0, θ, ǫ) is convex in [1, kmax]. Therefore,
+g(k, p0, θ, ǫ) is minimized at any one of k = 1, k =
+kmax or a stationary point in (1, kmax). Substituting
+p′(k; p0, θ) = p0α in (21), the stationary point is:
+k =
+� c
+p0α.
+(23)
+(a) Note that for k ∈ (1, kmax) to be the minimum,
+g′(k, p0, θ, ǫ)|k=1 < 0. Also, noting that p0 ≥ 1 − c
+(from Eq. (22)), we obtain
+(1 − c)α ≤ p0α < c.
+Therefore, α <
+c
+1−c. Substituting (23) in (20):
+p0 ≥
+�√cα − √cα − α + 1
+�2
+(α − 1)2
+,
+pth =
+�√cα − √cα − α + 1
+�2
+(α − 1)2
+.
+
+onsConverse regi
+Erasure
+0.6
+GADC
+Depolarizin
+0.53
+2
+I
+人0.4
+0.3
+0.2
+0.1
+0
+0
+I
+2
+3
+aNote that since α <
+c
+1−c, the second term in the
+numerator, cα − α + 1 = 1 − c ≥ 0. Therefore, the
+threshold pth exists.
+(b) If α ≥
+c
+1−c, then
+k =
+� c
+p0α ≥
+�
+1 − c
+p0
+≥ 1,
+p0 ≤ 1 − c.
+However, p0 ≥ 1−c from (23). Therefore, pth = 1−c.
+3) p(k; p0, θ) = p0kγ:
+Here, θ = γ. The value of k ranges from 1 ≤ k ≤
+�
+1
+p0
+� 1
+γ . For the choice of pth, (20) must hold for all k in
+this range. Similar to linear case, for this choice of p(k)
+and range of k, (20) is convex. Hence for pth,
+g(k, p0, θ, ǫ) |k=1 ≥ 0
+c + p0 − 1 ≥ 0
+p0 ≥ 1 − c.
+Substituting p′(k; p0, θ) = p0γkγ−1 in (21), we obtain
+the stationary point as:
+k =
+� c
+p0γ
+�
+1
+γ+1
+.
+(24)
+Similar to linear p(k; p0, θ), there are two cases:
+(a) If γ <
+c
+1−c,
+Substituting the stationary point computed in (24) in
+(20), the threshold pth can be computed as:
+p0 ≥
+� γ
+c
+�γ
+(γ + 1)γ+1 ,
+pth =
+� γ
+c
+�γ
+(γ + 1)γ+1 .
+(b) If γ ≥
+c
+1−c, then,
+k =
+� c
+p0γ
+�
+1
+γ+1
+≥
+�1 − c
+p0
+�
+1
+γ+1
+≥ 1,
+p0 ≤ 1 − c.
+However, p0 ≥ 1−c from (23). Therefore, pth = 1−c.
+APPENDIX B
+RESTRICTION OF THE FEASIBLE SET OF PL TO [1, kmax]
+Let kmax = max{k | p(k; p0, θ) ≤ 1}. If kmax = ∞, then
+solving (11) yields pth = 1. Therefore, (11) is non-trivial only
+if kmax is finite. Let g1(p0) = mink≥1 gθ(k, p0) and g2(p0) =
+mink∈[1,kmax] gθ(k, p0). From (13), it can be observed that
+g1(p0) = 0 whenever g2(p0) > 0, and g1(p0) = g2(p0)
+whenever g2(p0) ≤ 0. Hence, the threshold pth obtained using
+g1(·) and g2(·) as a solution to PL in (11) are identical.
+Therefore, (11) can be equivalently solved by restricting the
+domain of gθ(·, p0) in PL to [1, kmax]. In other words, one
+can replace line 11 with g∗
+θ(p0) = mink∈[1,kmax] gθ(k, p0)
+to obtain the same value of threshold pth. Additionally, this
+restriction makes the feasible set compact. Moreover, notice
+that the restriction and equivalence hold for all channels (not
+just erasure) as long as χ(Np) = 0 whenever p = 1.
+APPENDIX C
+DERIVATION OF LIPSCHITZ CONSTANTS
+A. Proof of Lemma 3: Erasure Channel
+Let g′
+θ and g′′
+θ denote the partial derivatives
+∂2
+∂k2 gθ(k, p0)
+and
+∂2
+∂k2 gθ(k, p0), respectively. The magnitude of the second
+order partial derivative is bounded above as:
+|g′′
+θ| ≤ max
+k
+����
+2c
+k3
+���� + max
+k
+|p′′(k; p0, θ)| ,
+where the inequality follows from triangle inequality and
+maximizing each summand. Observe that the first summand
+is maximized when k = 1, and the second term is bounded
+above as
+p′′(k; p0, θ) ≤
+�
+α2γ(γ − 1)p0,
+1 ≤ γ < 2, k = 1, and
+α2γ(γ − 1)p2/γ
+0
+,
+γ ≥ 2, k = kmax,
+≤ α2γ2p
+1
+γ
+0 ,
+γ > 0,
+where kmax = 1 + α−1(p−(1/γ)
+0
+− 1). Therefore,
+g′′
+θ ≤ 2c + α2γ2p
+1
+γ
+0 =: L.
+B. Proof of Lemma 4: Symmetric GADC
+Let q(p) = q(p(k)) =
+1−√
+1−p(k)
+2
+; the magnitude of the
+second-order derivative of gθ(k, p0) is bounded above as:
+|g′′
+θ| ≤ max
+k
+����
+2c
+k3
+���� + max
+k
+|h′′
+2 (q(k))| ,
+≤ 2c + max
+k {|h′′
+2(q)|q′(p)2p′(k; p0, θ)2
++ h′
+2(q)q′′(p)p′(k; p0, θ)2 + h′
+2(q)q′(p)p′′(k; p0, θ)}
+Noting that |h′′(q)| is maximized when k = 1, and remaining
+the terms are maximized when k = kmax, we obtain:
+|g′′
+θ| ≤ 2c+α2γ2p
+min{1, 2
+γ }
+0
+4(1 − p0)
+�
+1
+p0 ln 2
++1 + 2(1 − p0)
+2√1 − p0
+log2 P
+�
+=: L,
+where P = 1+√1−p0
+1−√1−p0 .
+C. Proof of Lemma 5: Depolarizing Channel
+The second partial derivative of gθ(k, p0) is bounded above
+as:
+|g′′
+θ| ≤ max
+k
+����
+2c
+k3
+���� + max
+k
+|h′′
+2(p(k)/2)| ,
+|g′′
+θ| ≤ 2c + max
+k
+�1
+4|h′′
+2(z)|p′(k; p0, θ)2 + 1
+2h′(z)p′′(k; p0, θ)
+�
+,
+
+where z = p(k; p0, θ)/2.
+|g′′
+θ| ≤ 2c+
+α2γ2p2/γ
+0
+p0(2 − p0) ln 2
++α2γ(γ − 1)p
+min{1, 2
+γ }
+0
+2
+log2
+�2 − p0
+p0
+�
+,
+|g′′
+θ| ≤ 2c+α2γ2p
+min{1, 2
+γ }
+0
+�
+log2 e
+p0(2 − p0) + 1
+2 log2
+�2 − p0
+p0
+��
+=: L.
+APPENDIX D
+LEMMAS: PROJECTED GRADIENT DESCENT
+Definition 6. In Appendices D and E, we consider g(·) to be
+of the following form: g : [1, kmax] →
+R : k �→ g(k), where
+g′(·) is L-Lipschitz.
+Lemma 6. Stopping criterion and bounded gradient: Suppose
+a pair of iterates (kj, kj+1), which lie in the interior (1, kmax),
+generated by PROJGD satisfy the stopping criterion g(kj) −
+g(kj+1) <
+δ2
+2Lk2max , then the first order derivative is bounded
+above as |g′(kj)| <
+δ
+kmax .
+Proof: Applying the descent lemma to kj, kj+1, we get
+g(kj+1, p0) ≤ g(kj, p0)
++g′(kj, p0)(kj+1 − kj) + 1
+2L|kj+1 − kj|2.
+(25)
+Substituting kj+1 − kj = −ξgθ(kj) in (25):
+ξ
+�
+1 − ξL
+2
+�
+|g′(kj)|2 ≤ g(kj) − g(kj+1).
+Choosing ξ = 1
+L, we obtain:
+1
+2L|g′(kj)|2 ≤ g(kj) − g(kj+1) <
+δ2
+2Lk2max
+.
+Therefore,
+|g′(kj)| <
+δ
+kmax
+.
+Lemma
+7.
+Suppose
+g(·)
+is
+convex,
+and
+g∗
+=
+mink∈[1,kmax]
+g(k).
+If
+|g′(kj)|
+<
+δ/kmax,
+then
+|g(kj) − g∗| ≤ δ, for any kj ∈ [1, kmax].
+Proof: From the convexity of g(·), we have g(k) ≥
+g′(kj)(k−kj)+g(kj), for any k, kj ∈ [1, kmax]. If g′(kj) < 0,
+then:
+g(k) − g(kj) ≥ g′(kj)(kmax − kj) ≥ g′(kj)(kmax − 1), ∀k.
+Therefore,
+g(kj) − g∗ ≤ |g′(kj)|(kmax − 1) ≤ |g′(kj)|kmax ≤ δ.
+On the other hand, if g′(kj) ≥ 0, then
+g(k) − g(kj) ≥ g′(kj)(k − kj) ≥ g′(kj)(1 − kj), ∀k.
+g(kj) − g∗ ≤ g′(kj)(kmax − 1) ≤ g′(kj)kmax ≤ δ.
+Therefore, combining both cases: if |g′(kj)| ≤
+δ
+kmax , then
+|g(kj) − g∗| ≤ δ.
+Lemma 8. Projected gradient descent (PROJGD) does not
+cross any stationary point: Let kj and kj+1 be the successive
+iterates generated by PROJGD routine for g(·). Suppose, the
+step-size ξ ∈ (0, 1
+L], then g′(kj)g′(kj+1) ≥ 0.
+Proof: From the definition of Lipschitz gradient, we have
+|g′(kj) − g′(kj+1)| ≤ L|kj − kj+1| = Lξ|g′(kj)|, where the
+last equality holds, since kj+1 is generated from PROJGD
+routine. Suppose, g′(kj) ≥ 0, then the following inequalities
+hold:
+−Lξg′(kj) ≤ g′(kj) − g′(kj+1) ≤ Lξg′(kj),
+−Lξg′(kj) ≤ g′(kj+1) − g′(kj) ≤ Lξg′(kj),
+(1 − Lξ)g′(kj) ≤ g′(kj+1) ≤ (1 + Lξ)g′(kj).
+For ξ ≤ 1
+L, we obtain:
+g′(kj+1) ≥ g′(kj)(1 − Lξ) ≥ 0.
+Symmetrically, if g′(kj) ≤ 0, then g′(kj+1) ≤ 0. Combining
+both cases, we obtain g′(kj)g′(kj+1) ≥ 0.
+Lemma 9. Least difference between the successive iterates of
+PROJGD: Let kj and kj+1 be the successive iterates generated
+by PROJGD routine for g(·), with a step size ξ ∈ (0, 1
+L]. If
+|g(kj) − g(kj+1)| > ζ, then |kj − kj+1| >
+�
+2ζ
+L .
+Proof: Using descent lemma [31] on g(·) at kj and kj+1,
+we obtain
+g(kj)−g(kj+1) ≤ g′(kj+1)(kj−kj+1)+ L
+2 |kj−kj+1|2. (26)
+Suppose we g′(kj) ≤ 0 and g(kj) − g(kj+1) > ζ, then
+g′(kj+1) ≤ 0 (from Lemma 8), and noting that (kj−kj+1) ≥ 0
+we obtain:
+ζ < L
+2 |kj − kj+1|2,
+(27)
+Therefore,
+|kj+1 − kj| >
+��
+2ζ
+L ,
+if kj+1 < kmax,
+2
+L(g(kj) − g(kmax)),
+if kj+1 = kmax
+The second case should be handled separately since (27) may
+not hold when kj+1 = kmax.
+Corollary 3. Suppose PROJGD is used with stopping criterion
+1 (Definition 3). If |kj − kj+1| ≤
+δ
+Lkmax , then |g(kj) −
+g(kj+1)| ≤
+δ2
+2Lk2max , when kj+1 < kmax.
+Proof: The result follows by substituting ζ =
+δ2
+2Lk2max in
+Lemma 9.
+Corollary 4. Suppose PROJGD is used with stopping criterion
+2 (Definition 4). If |kj − kj+1| ≤
+�
+2δ
+L , then |g(kj) −
+g(kj+1)| ≤ δ, when kj+1 < kmax.
+Proof: The result follows by substituting ζ
+= δ in
+Lemma 9.
+
+APPENDIX E
+LEMMAS: PERTURBATION
+Lemma 10. PERTURB routine does not miss stationary points:
+Suppose kj meets the stopping criterion 2 (Definition 4). If the
+perturbation ∆k ≤
+�
+2δ
+L , then the PERTURB routine does not
+miss any stationary points with an error greater than δ.
+Proof: Let kj, kj+1 be any two points in [1, kmax]. Since
+g′(·) is L-Lipschitz, (26) holds. Let kj+1 be the closest
+stationary point to kj, then:
+|g(kj) − g(kj+1)| ≤ L
+2 |kj − kj+1|2.
+Therefore, the following condition is necessary for the stop-
+ping criterion 2, i.e., |g(kj) − g(kj+1)| ≥ δ (Definition 4) to
+hold:
+∆k = |kj − kj+1| ≥
+�
+2δ
+L .
+Lemma 11. Upper bound on the number of stationary points:
+Consider a set of stationary points {ks} of g(·) in [1, kmax]
+such that for every ks, the adjacent stationary point ks+1,
+|g(ks) − g(ks+1)| ≥ δ. The number of such stationary points
+is finite and bounded above as ⌈kmaxL/δ⌉.
+Proof: From the proof of Lemma 10, it follows that if
+|g(ks)−g(ks+1)| ≥ δ, then the stationary points are separated
+by at least |ks − ks+1| ≥
+�
+2δ
+L . Therefore, the number of
+stationary points in [1, kmax] is at most ⌈kmax
+�
+L
+2δ ⌉.
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+

6NE2T4oBgHgl3EQfkgd1/content/tmp_files/2301.03979v1.pdf.txt

@@ -0,0 +1,1032 @@
+Apodized photonic crystals: A non-dissipative system hosting multiple exceptional
+points
+Abhishek Mondal, Shailja Sharma and Ritwick Das∗
+School of Physical Sciences, National Institute of Science Education and Research,
+An OCC of Homi Bhabha National Institute, Jatni - 752050, Odisha, India
+(Dated: January 11, 2023)
+Optical systems obeying non-Hermitian dynamics have been the subject of intense and concerted
+investigation over the last two decades owing to their broad implications in photonics, acoustics,
+electronics as well as atomic physics. A vast majority of such investigations rely on a dissipative,
+balanced loss-gain system which introduces unavoidable noise and consequently, this limits the
+coherent control of propagation dynamics.
+Here, we show that an all-dielectric, non-dissipative
+photonic crystal (PC) could host, at least two exceptional points in its eigenvalue spectrum. By
+introducing optimum apodization in the PC architecture, namely 1D-APC, we show that such
+a configuration supports a spectrum of exceptional points which distinctly demarcates the PT -
+symmetric region from the region where PT -symmetry is broken in the parameter space.
+The
+analytical framework allows us to estimate the geometric phase of the reflected beam and derive
+the constraint that governs the excitation of topologically-protected optical Tamm-plasmon modes
+in 1D-APCs.
+I.
+INTRODUCTION
+Optical systems which are governed by non-Hermitian
+Hamiltonian dynamics through an engineered gain and
+dissipation mechanism, provide a route to overcome the
+limitations imposed by closed optical systems that obey
+the Hermitian-Hamiltonian led dynamics.
+Such non-
+Hermitian systems give rise to a real eigenvalue spec-
+trum when the Hamiltonian commutes with the parity-
+time (PT ) operator.
+A continuous change in the pa-
+rameter governing the Hermiticity (of the Hamiltonian)
+breaks the PT symmetry which manifests in the form
+of complex eigenvalues for the system.
+In the phase
+space, such points where the real and complex eigenval-
+ues coalesce are termed as exceptional points (EPs) [1, 2].
+This spontaneous PT -symmetry breaking has catalyzed
+a plethora of non-intuitive outcomes such as directional
+invisibility [3, 4], coherent perfect lasing and absorption
+[5–9], negative refraction [10], single-particle based sens-
+ing [11–13], distortion-free wireless optical power trans-
+fer [14] and a few more [15–19]. It is, however, worth
+noting that the incommensurate gain and loss distribu-
+tion in non-Hermitian systems impose the primary limi-
+tation on the practical applications due to unpredictable
+signal-to-noise ratio near EP [20–23].
+In order to cir-
+cumvent such bottlenecks, a few possibilities have been
+explored. One such promising route is to create an asym-
+metric loss in the system (without gain) whose dynamics
+could be explored using a non-Hermitian Hamiltonian
+with a uniform background loss [20, 24, 25]. Such a con-
+figuration would exhibit PT -symmetry which could be
+broken through scaling up the loss asymmetry. In a dif-
+ferent scheme, a pseudo-Hermitian system was explored
+which allowed strong coupling between a large number
+∗ [email protected]
+of modes via manipulation of the parameters governing
+the Hamiltonian [24]. This led to the existence of EPs of
+multiple order and the interaction of eigenvalues around
+each EP provides a robust control on the propagation
+dynamics [26, 27]. In spite of the aforementioned devel-
+opments, a useful and practical proposition would be to
+devise a configuration hosting a multitude of EPs with
+the constraint that the electromagnetic (EM) energy lost
+due to the non-Hermitian dynamics is stored in a reser-
+voir. This essentially implies that the dissipative channel
+associated with a non-Hermitian system drives a separate
+Hermitian system which could allow reverse flow of EM
+energy by virtue of cyclical dynamics. Such systems have
+been explored in the area of parametric frequency con-
+version processes where the EM energy lost in one of the
+parametric processes (obeying non-Hermitian dynamics)
+is coherently added to the other parametric process that
+follows a Hermitian dynamics [28]. A plausible transla-
+tion of such an idea in the non-absorptive linear systems
+would be to introduce a virtual loss in an intermodal
+interaction process thereby generating multiple EPs in
+the parameter space. One of the simplest configurations
+imitating such a process is a multimodal interaction in
+an all-dielectric one-dimensional (1D) photonic-crystal
+(PC) with a gradually varying duty cycle (for each unit
+cell). In such an apodized 1D-PC, the forward (source)
+to backward (sink) mode-coupling dynamics is essentially
+governed by a pseudo-Hermitian Hamiltonian whose Her-
+miticity is determined by the apodization along the prop-
+agation direction. In the present work, we show the exis-
+tence of multiple EPs in an apodized 1D-PC and develop
+an analytical framework for ascertaining the possibility
+of exciting topologically-protected optical edge modes in
+such aperiodically stratified configurations.
+arXiv:2301.03979v1  [physics.optics]  10 Jan 2023
+
+2
+II.
+THEORETICAL FRAMEWORK AND
+COUPLED-MODE FORMALISM
+We consider a 1D-PC comprised of periodic bilayers
+with refractive indices n1 and n2 with thicknesses d1
+and d2. Such conventional 1D-PCs or alternatively, dis-
+tributed Bragg reflectors (DBRs) are usually character-
+ized by photonic bandgaps (PBGs) which are separated
+from each other by high transmission (or pass) bands. In
+order to appreciate the EM wave propagation dynamics,
+we consider the coupling between pth-mode (|p⟩) with
+qth-mode (|q⟩) which could be represented employing the
+coupled-amplitude equations given by [29]
+dAq
+dz = −i βq
+|βq|
+�
+p
+�
+m
+˜κ(m)
+qp Ape−i(βq−βp−m 2π
+Λ )z
+(1)
+where βp and βq are the longitudinal (z) components
+of wavevector kp and kq respectively.
+˜κ(m)
+qp
+defines the
+strength of coupling (or coupling coefficient) between
+the pth and qth mode that is coupled through the mth
+Fourier component of the periodic dielectric distribution
+( Λ = d1 + d2). The factor ∆β = βq − βp − m 2π
+Λ (known
+as the phase-mismatch) is one of the dynamical variables
+(along with κqp) which dictate the measure of optical
+power transferred from one mode to the other. For the
+present work, we consider a contra-directional coupling
+set-up where a forward (along +z) propagating mode
+(|p⟩ ≡ |f⟩) is coupled to a backward (along −z) prop-
+agating mode (|q⟩ ≡ |b⟩). Accordingly, it could asserted
+that βb = −βf or alternatively ∆β = 2βf − 2π
+Λ and there-
+fore, Eq. (1) could be simplified to [29]
+dAb
+dz = i˜κAfe−i∆βz
+(2)
+dAf
+dz
+= −i˜κ∗Abei∆βz
+(3)
+where ˜κ
+=
+i(1−cos 2πζ)
+2λ
+(n1
+2−n2
+2)
+¯n
+= iκ and ζ is the di-
+electric filling fraction of layer with refractive index n1
+in the unit cell i.e.
+d1
+Λ .
+The mean refractive index
+for an unit cell of thickness Λ is ¯n
+=
+�
+d1n12+d2n22
+Λ
+.
+By using a Gauge transformation given by [Af, Ab] →
+[ ˜Af, ˜Ab]ei/2[∆β0z−
+�
+0
+zq(z′)dz′], we obtain [30]
+i d
+dz
+� ˜Ab
+˜Af
+�
+=
+�
+−∆k −˜κ
+˜κ∗
+∆k
+� � ˜Ab
+˜Af
+�
+(4)
+Equation
+(4)
+is
+analogous
+to
+time-dependent
+Schr¨odinger’s equation with t-coordinate being replaced
+by z-coordinate. Here, ∆k (= ∆β
+2 ) and q(z) = 0 remains
+constant (for a given frequency) across the 1D-PC which
+has a fixed duty cycle.
+The autonomous Hamiltonian
+ˆH = −⃗σ · ⃗B with ⃗σ ≡ [σx, σy, σz] are the Pauli’s spin
+matrices and ⃗B ≡ [0, κ, ∆k] (magnetic field analog) rep-
+resents a pseudo-Hermitian evolution dynamics. In order
+to appreciate this point, we note that the eigenvalues of
+ˆH which are given by e1,2
+=
+±
+�
+∆k2 − κ2 whereas
+the eigenfunctions are |ψ1⟩
+=
+�
+−i (∆k+√
+∆k2−κ2)
+κ
+1
+�
+and |ψ2⟩
+=
+�
++i (−∆k+√
+∆k2−κ2)
+κ
+1
+�
+. Here, ˜κ = iκ. A
+closer look into the eigenvectors reveals that the equality
+κ
+=
+± ∆k manifests as coalescing of eigenvectors
+accompanied by vanishing eigenvalues.
+Such points in
+parameter space where κ equals ±∆k are termed as
+exceptional points (EPs) and they distinctly demarcate
+the regions exhibiting Hermitian (PT -symmetric phase)
+and
+non-Hermitian
+(PT -broken
+phase)
+dynamical
+evolution of states (or modes).
+In order to appreciate the aforementioned idea, we
+consider a practical 1D-PC with n1
+≡
+TiO2 layer
+and n2
+≡
+SiO2 layer.
+The layer thicknesses are
+d1 = d2 = 150 nm. The reflection spectrum for N = 20
+unit cells is plotted in Fig. 1(a) which exhibits a high
+reflection band (or PBG) spreading over a 75 THz band-
+width. In order to obtain the reflection spectrum, finite
+element method (FEM) based simulations were carried
+out using the commercially available computational tool
+(COMSOL Multiphysics). In the simulations, the peri-
+odic boundary condition is imposed along the transverse
+direction and a mesh size of 5 nm is considered.
+We
+ignore the material dispersion for the simulations and
+assume n1 = 2.5 (≡ TiO2) and n2 = 1.5 (≡ SiO2) across
+the entire spectrum. For this 1D-PC, we also plotted the
+eigenvalues e1 and e2 (see Fig. 1(b)) as a function of the
+frequency of the incident electromagnetic wave. It is ap-
+parent that the eigenvalues vanish at ν1 ≈ 210 THz and
+ν2 ≈ 285 THz. These two frequencies (ν1 and ν2) define
+the EPs (κ =
++ ∆k and κ =
+− ∆k) for the periodic
+1D-PC. A closer look would also reveal that the eigenval-
+ues are purely imaginary within the PBG and the band
+edges (Fig. 1 (a)) coincide with ν1 and ν2. The mode
+fields for frequencies lying inside the PBG (240 THz)
+and outside the PBG (310 THz) are presented in Figs.
+1(c) and (d) respectively.
+It is worth noting that the
+investigations on systems exhibiting PT -symmetry (or
+PT -broken symmetry) led dynamics in photonics essen-
+tially involve optimally balanced gain-loss architectures
+such as segmented waveguides and photonic crystals. In
+such systems, a complex relative permittivity in different
+sections depicting actual gain or loss for the propagat-
+ing light beam gives rise to the PT -symmetry (or PT -
+broken symmetry). The present configuration involving
+1D-PC does not include an actual dissipative component
+for achieving the PT -symmetric to PT -symmetry bro-
+ken phase transition. Alternatively, the coupling of opti-
+cal power to the backscattered mode |b⟩ is analogous to
+a virtual loss for a forward propagating |f⟩ mode. When
+this coupling is relatively weak i.e. ∆k > ˜κ, |f⟩ and |b⟩
+exhibits cyclic exchange of optical power (as a function
+of z) which is a primitive outcome for a PT -symmetric
+dynamics. On the other hand, a strong coupling regime
+
+3
+FIG. 1. a) Shows the reflection spectrum of a conventional
+(periodic) 1D-PC. b) Shows the variation in Re(e1) (dotted
+black curve), Im(e1) (dotted maroon curve), Re(e2) (solid
+black curve) and Im(e2) (solid maroon curve) as a function
+of frequency (ν). c) and d) Shows the mode-field intensity for
+frequencies within the PBG 240 THz and that outside the
+PBG 310 THz respectively. The solid red arrow represents
+the direction of incidence of light.
+where ∆k < ˜κ manifests through a monotonic growth
+of backscattered mode (|b⟩) that is a signature of PT -
+symmetry broken phase. It is worthwhile to reiterate the
+point that the two regimes depicted by the inequality
+of ∆k and ˜κ (in the parameter space) could be mapped
+onto the PBG and pass or transmission band (s) in the
+reflected spectrum. Subsequently, each PBG is necessar-
+ily bounded by two EPs in this framework. Additionally,
+these two EPs are fixed and could not be tailored for
+a given 1D-PC with a fixed duty cycle and fixed period.
+Also, the conventional 1D-PC geometry excludes the pos-
+sibility of realizing higher-order exceptional points [31].
+Taking a cue from this critical viewpoint, we note that
+a small apodization or gradual change in dielectric fill-
+ing fraction (ζ) of each unit cell of the 1D-PC would
+allow us to realize discretely spaced (multiple) EPs at
+different optical frequencies (or wavelengths). In order
+to elucidate this point, we recall that ∆k as well as ˜κ is
+a function of ζ. An optimum spatial variation in ζ could
+essentially give rise to the possibility of EPs at different
+physical locations (along z) in a 1D-PC. As an example,
+we show below that an optimally apodized 1D-PC (1D-
+APC) which satisfies the adiabatic constraints enables us
+to observe EPs at discreetly separated points along z.
+A.
+Design of an 1D apodized PC and intermodal
+coupling
+We consider a 1D-PC configuration that exhibits
+varying dielectric filling fraction (ζ) in each unit cell.
+This variation is essentially dictated through the relation
+d1M = d1 −Mδ and d2M = Λ−d1M. Here, d1M and d2M
+are the thickness of TiO2 and SiO2 layers respectively
+in M th unit cell (M = 0, 1, 2, 3, ..., (N − 1) for N number
+of unit cells). The unit cell period, however remains un-
+changed i.e. Λ = d1M +d2M = d1 +d2. This apodization
+FIG. 2. a) Shows the reflection spectrum for designed 1D-
+APC. (b) and (c) Shows the mode-field intensities for two
+different frequencies νa = 250 THz and νb = 300 THz which
+are within the PBG of 1D-APC. (d) and (e) Shows the vari-
+ation in Re(e1) (dotted black curve), Im(e1) (dotted maroon
+curve), Re(e2) (solid black curve) and Re(e2) (solid maroon
+curve) as a function of TiO2 layer thickness for each unit cell
+(i.e. d1M) at frequencies νa = 250 THz and νb = 300 THz
+respectively.
+in 1D-PC could be visualized through a longitudinal
+variation in ∆k as well as ˜κ by virtue of a monotonic
+change in average refractive index (¯n) for an unit cell.
+This variation in ∆k and ˜κ in a 1D-APC geometry
+leads to an adiabatic evolution of the Stokes vector
+along the propagation direction and manifests through
+a broader PBG (≈ 140 THz) in comparison with a
+conventional (periodic) 1D-PC [30].
+This is presented
+in Fig. 2(a) which shows a broader reflection spectrum
+for the 1D-APC in comparison with the conventional
+1D-PC (Fig.1(a)). In addition, a flat transmission band
+and the absence of sharp transmission resonances is
+a distinct feature of 1D-APC. The mode-propagation
+characteristics for the frequencies within the PBG (of
+1D-APC) is explored by drawing a comparison with the
+mode-field distributions for the equivalent modes within
+the PBG of a conventional 1D-PC. Figures 2(b) and (c)
+shows the mode-field distribution for two frequencies
+νa = 250 THz and νb = 300 THz which are within the
+PBG of 1D-APC. In comparison with the mode-field
+distribution shown in Fig. 1(c), it could be observed that
+different modes are reflected from spatially separated
+z values.
+The smaller frequency (νa = 250 THz) is
+reflected from the regions which are closer to z = 0 edge
+of the 1D-APC in comparison to that for νb = 300 THz.
+This variation is indicative of the fact that the field
+is localized and exhibits instantaneous localization in
+different 1D-APC sections. From a different perspective,
+it is apparent that the variation in dielectric filling
+fraction (ζ) would result in different eigenvalues (and
+corresponding eigenvectors) for each unit cell. Accord-
+ingly, we plot the eigenvalues e1 and e2 as a function of
+d1M for two frequencies νa = 250 THz (Fig. 2(d)) and
+νb = 300 THz (Fig. 2(e)) which are within the PBG of
+1D-APC. Each one of the figures shows that the eigen-
+
+1
+0.15
+a)
+b)
+0.1
+0.8
+0.05
+0.05
+0.6
+R
+1,2′
+0
+0
+0.4
+e
+-0.05
+Re(
+-0.05
+0.2
+-0.1
+-0.15
+-0.1
+0
+175
+200
+225
+250
+175
+200
+225
+250
+275
+300
+325
+275
+300
+325
+v (THz)
+v (THz)
+C
+310THz
+240THza)
+0.8
+250THz
+0.6
+R
+0.4
+c)
+300THz
+0.2
+0
+160
+200
+250
+300
+350
+380
+V (THz)
+0.15
+0.3
+Re(e,)... Ree,)Im(e,.m(e,)
+0.04
+d)
+e)
+0.1
+0.2
+(V/) (
+0.05
+(V /z)
+(V /Z)
+0.02
+(V /)
+0.05
+0.1
+0
+0
+0
+0
+?
+1
+Re(
+-0.05
+-0.1
+-0.1
+-0.2
+250THz
+300THz
+-0.15
+-0.1
+-0.3
+-0.04
+0
+50
+100
+150
+200
+250
+300
+0
+50
+100
+150
+200
+250
+300
+d
+(nm)
+d
+(nm)
+1M
+1M4
+FIG. 3.
+(a) Shows the variation of ⃗B in parameter space
+(spanned by κ and ∆k) at different operating frequencies
+(ν1
+=
+400 THz, ν2
+=
+250 THz, ν3
+=
+160 THz) for
+the designed 1D-APC. The blue and green solid lines repre-
+sent the ∆k = κ and ∆k = −κ curves. (b) Shows the location
+of EPs in different unit cells (with different filling fraction ζ)
+as a function of frequency (ν).
+values (e1 and e2) vanish at two different values of d1M
+i.e. at the location of two different unit cells. Therefore,
+the 1D-APC geometry hosts two EPs for every d1M.
+Consequently, for a multitude of ζ, there would be
+multiple EPs in the 1D-APC for a forward-propagating
+mode to a backscattered mode-coupling process.
+As
+discussed before, the regions where ℜe1 and ℜe2 are
+non-zero in Figs. 2(d) and 2(e) exhibit a PT -symmetric
+coupling dynamics between the forward-propagating and
+backscattered modes. On the other hand, in the regions
+where e1 and e2 are purely imaginary, the mode-coupling
+process exhibits PT -symmetry broken manifolds.
+The
+illustrations presented in Figs. 2(d) and 2(e) show that
+for each frequency within the PBG, the 1D-APC hosts
+two EPs at two different d1M. This essentially implies
+that there exists one or more than one EPs hosted by
+each unit cell of the 1D-APC. Therefore, an 1D-APC
+is expected to host multiple EPs which are spectrally
+as well as spatially separated from each other. In order
+ascertain the spectral location of EPs in the 1D-APC, we
+plot the evolution of ⃗B in the parameter space for three
+different frequencies ν1 = 400 THz, ν2 = 250 THz,
+and ν3
+= 160 THz as shown in Fig.3(a). It could be
+noted at ν1 and ν3 are situated outside PBG of 1D-APC
+(see Fig.
+2(a)).
+Since, the EPs are depicted by the
+condition ∆k = |κ|, Fig.3(a) also contains the curve
+∆k = ±κ (solid blue and green curves). It is apparent
+that ∆k = ±κ curve intersects ⃗Bν2 at two points and
+it does not intersect the ⃗Bν1 curve as well as the ⃗Bν3
+curve in the parameter space. For frequencies close to
+the band-edge of 1D-APC (say 200 THz or 350 THz),
+it could be ascertained that there exists only one EP in
+the eigenvalue spectrum.
+This is primarily due to the
+adiabatic constraints followed by the 1D-APC design. In
+other words, for the band-edge frequencies, the forward
+and backward propagating modes are decoupled (˜κ)
+at entry (z = 0) and exit (z = L) face of the crystal.
+Additionally, d1M = Λ for m = 0 (or d2M = Λ for
+m = N) in case of band-edge frequencies that leads to
+∆k = 0 for ζ = (or ζ = 1). Therefore, ˜κ = ∆k = 0
+depicts the only EP for the band-edge frequencies.
+In order to elucidate the aforementioned point, we
+present the spectral location of EPs as a function of di-
+electric filling fraction (ζ) or propagation direction (z)
+in Fig. 3(b). It could be observed that there exists two
+(2) EPs (at different ζ or z) for all the frequencies well
+within the PBG of 1D-APC. However, for the band-edge
+frequencies (νl = 200 THz and νh = 330 THz), the 1D-
+APC hosts one EP only. Nevertheless, the area enclosed
+by the EPs in Fig.
+3(b) represents the region of PT -
+symmetry broken phase for the 1D-APC. It is interesting
+to note that the separation between the two EPs for fre-
+quencies closer to the band-edges (say ν ≤ 210 THz or
+ν ≥ 310 THz) very less and they tend to overlap at the
+same filling fraction. It is important to note that these
+EPs are physically positioned close to the entry (z = 0)
+and exit (z = L) face of the 1D-APC where ˜κ is very
+small. By virtue of this, the PBG corresponding to that
+unit cell of 1D-APC is relatively smaller in comparison
+with the PBG for a unit cell close to the center (z ≈ L
+2 )
+of 1D-APC. Due to the fact that the EPs exist at the
+band-edges of PBG for each unit cell of APC, a smaller
+PBG would essentially imply closely spaced EPs near the
+band-edges (see Fig. 3(b)).
+B.
+Geometric phase estimation of reflection band
+It is well known that the geometric phase of a pass-
+band (or transmission band) for a one-dimensional con-
+ventional photonic crystal is quantized (0 or π) and it
+is known as the ‘Zak’ phase.
+However, the geomet-
+ric interpretation of backscattered (or reflection) phase
+from a 1D-PC remains irrelevant. However, in case of
+1D-APC, the reflection of different spectral components
+(within the PBG) takes place from different unit cells
+(or z) along the propagation direction [30]. For exam-
+ple, the adiabatic following constraint leads to conver-
+sion of optical power from the forward-propagating to
+the backscattered mode predominantly towards the exit
+face of 1D-APC for frequency ν = 250 THz which could
+be seen in Fig. 4(a). Through a similar route, it could
+be shown that different spectral components within the
+PBG are reflected strongly from different unit cells of
+1D-APC [30]. The primary underlying reason could be
+traced to the variation in ˜κ and ∆k for each spectral
+component in the PBG which are non-identical. Conse-
+quently, the estimation of geometric phase acquired by
+different backscattered modes is expected to be differ-
+ent and must play a crucial role in establishing the bulk-
+boundary correspondence in case of 1D-APC. In order to
+obtain the geometric phase γ, we consider a triad defining
+the state vector ⃗S (≡ [u, v, w]) where u = ˜Ai ˜A∗
+r + ˜Ar ˜A∗
+i ,
+v = −i[ ˜Ai ˜A∗
+r − ˜Ar ˜A∗
+i ] and w = | ˜Ar|
+2 − | ˜Ai|
+2 [30].
+The z-component of the state-vector (w) represents the
+conversion efficiency of optical power from a forward-
+propagating to a backscattered mode [30]. It is also worth
+
+10
+1
+B(v/)
+a)
+b)
+0.8
+5
+△k (μm"
+0.6
+米米
+△k= 
+S
+0.4
+米
+B(v)
+0
+0.2
+B(v)
+Ak= - k
+0
+-5
+0.5
+2
+2.5
+190
+210
+230
+250
+270
+290
+310
+330
+0
+1
+350
+-11.5
+k(um
+v (THz)5
+FIG. 4. a) Shows the variation in conversion efficiency ( w+1
+2 )
+for optical power transfer between a forward-propagating
+mode to a backscattered mode as a function of 1D-APC length
+(z) for a frequency ν2 = 250 THz which is within the PBG.
+(b) Presents the state-vector (⃗S = [u, v, w]) trajectory on
+the Bloch sphere for ν2 = 250 THz.
+noting that the trajectory of state-vector (⃗S) correspond-
+ing to the frequencies within the PBG is non-closed. Al-
+ternatively, the geometric phase is not a conserved quan-
+tity during the dynamical evolution of states owing to the
+PT -symmetry broken phase. In general, the solid angle
+subtended by the state-vector trajectory at the center
+of the Bloch sphere is used for computing the geomet-
+ric phase.
+However, in case of an adiabatic evolution,
+the state-vector trajectory could be very complicated. In
+Fig.
+4(b), we have plotted such a state-vector trajec-
+tory (on the Bloch sphere) corresponding to a frequency
+ν = 250 THz (which is within the PBG of 1D-APC). It
+is important to note that ⃗S = [0, 0, −1] and ⃗S = [0, 0, 1]
+represent states in which all the optical power (∝ | ˜Af,b|2)
+is present in the forward-propagating and backscattered
+mode respectively.
+Although, the adiabatic evolution
+of state-vector results in complete optical power trans-
+fer from the forward to backward-propagating mode i.e.
+w = −1 to w = 1, the estimation of acquired geometric
+phase is quite complicated owing to the spiralling trajec-
+tory of ⃗S on the Bloch-sphere. However, it is interest-
+ing to note that ⃗S goes from [0, 0, −1] to [0, 0, 1] for all
+the frequencies within the PBG of 1D-APC by virtue of
+satisfying the adiabatic following constraints. The most
+important point is to note that the conversion efficiency
+(or reflectivity) is ‘unity’ for all the frequencies within
+the PBG of 1D-APC [30]. In other words, ⃗B goes from
+[0, 0, −∆k] to [0, 0, ∆k] in the parameter space for all the
+PBG frequencies (through any trajectory) when the adi-
+abatic following constraints are satisfied [30].
+By virtue of the fact that the state-vector ⃗S adiabat-
+ically follows ⃗B (as per the Bloch equation), the initial
+and the final value of ⃗B could also yield the geometric
+phase (γ). It is known that γ is estimated from angle
+φ (subtended by ⃗B at the origin ∆k = ˜κ = 0) through
+the relation γ =
+φ
+2 . In that case, the geometric phase
+for each spectral component within the PBG is
+π
+2 . In
+order to elucidate this point, we plot ⃗B at different z
+of 1D-APC in the parameter space for ν = 250 THz as
+FIG. 5.
+Represents the evolution of ⃗B as a function of
+length (L) of 1D-APC in parameter (∆k − κ) space for a)
+ν2 = 250 THz and b) ν4 = 180 THz. φ represents the angle
+subtended by curve ⃗B at the origin.
+shown in Fig. 5(a). At the entry face of 1D-APC (z = 0),
+⃗B(z = 0) = [0, 0, −2.7 µm−1] (black arrow) and gradu-
+ally goes to ⃗B(z = L) = [0, 0, +2.7 µm−1] (red arrow)
+at z = L. At z = L
+2 , ∆k = 0 and ˜κ is maximum (green
+arrow in Fig. 5(a)) The evolution of ⃗B in Fig. 5(a) yields
+φ = π and consequently, γ = π
+2 . In a similar manner, γ
+for all the frequencies within the PBG would be π
+2 by
+virtue of adhering to the constraints imposed by adia-
+batic following. Hence, it could be asserted that a geo-
+metric phase of π
+2 is acquired by a reflected beam in a 1D-
+APC for the values of parameters which results in PT -
+symmetry broken phase. On the contrary, the variation
+in ⃗B is plotted as a function of z for ν = 180 THz which
+is outside the PBG of 1D-APC (see Fig. 5(b)). ⃗B(z = 0)
+(black arrow) and ⃗B(z = L) (red dashed arrow) are both
+negative as well as co-parallel in this case. Consequently,
+the geometric phase γ = φ
+2 = 0 for ν = 180 THz. In
+addition, it is apparent that ∆k ̸= 0 at any point (or any
+z) in the 1D-APC.
+C.
+Tamm-plasmon excitations in 1D-APC and
+topological connection
+The presence of a plasmon-active layer adjacent to
+the all-dielectric 1D-APC results in excitation of mul-
+tiple Tamm-plasmon modes which are non-degenerate.
+As an example, we consider a thin (dAu = 30 nm) layer
+of gold placed in contact with high index layer (TiO2)
+of 1D-APC (see Fig.6(a)). The simulated reflection spec-
+trum (using transfer matrix method) exhibits a sharp res-
+onance within the PBG as shown in Fig.6(b). These res-
+onances are essentially due to Tamm-plasmon mode exci-
+tations which are highly localized electromagnetic states.
+Figure 6(b) depicts the existence of 10 Tamm-plasmon
+modes within the PBG of 1D-APC. Although there are a
+few sharp resonances outside the PBG, their mode-field
+signatures do not resemble that for a Tamm-plasmon
+mode [32]. In general, the existence of Tamm-plasmon
+modes is governed by the condition φAP C + φAu = 2sπ
+where s
+=
+0, 1, 2, 3.... is an integer [33–35]. Here,
+φAP C is the total phase acquired by the reflected beam
+from the 1D-APC (light incident from Au side), and φAu
+
+a)
+b)
+(0,0,1)
+0.8
+0.6
+0.4
+0.2
+(0,0,-1)
+0
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9.3
+z (μm)3
+B(z = L)
+0
+B (z = 0)
+a)
+b)
+2
+1
+B (z = L/2)
+)
+.2
+0
+B(z = L/2)
+-3
+.4
+-2
+250 THz
+B (z = L)
+180 THz
+B(z = 0)
+-3
+-5
+0
+0.5
+1.5
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+k (μm"1)
+k (um=1)6
+FIG. 6.
+a) Shows the schematic of the Au-1D-APC het-
+erostructure.
+The Au-layer is placed adjacent to the high-
+index TiO2 layer. The thick brown arrow depicts the direction
+of light incidence on the Au-1D-APC b) Shows the simulated
+reflection spectrum of 1D-APC without Au (black solid curve)
+and that of Au-1D-APC (maroon solid curve).
+is the phase acquired by reflected beam at the Au−TiO2
+interface. It is worthwhile to reiterate that the dielec-
+tric layer (of 1D-APC) adjacent to the Au-film is TiO2
+which is the high index layer.
+In the present context
+φAP C = γ + α, where α is the dynamic phase acquired
+by the reflected beam [30]. This could be estimated by
+noting the fact that the EPs (for a given frequency) are
+situated in different unit cells (or ζ) of the 1D-APC. For
+a frequency ν, if the nearest EP (with respect to z = 0)
+is present in the pth-unit cell of 1D-APC, then α could
+be determined using
+α = 2πν
+c
+p
+�
+M=0
+[n1d1M + n2d2M]
+(5)
+The knowledge of location for EPs in the 1D-APC (ob-
+tained from the eigenvalue spectrum of ˆH) would accu-
+rately yield the dynamic phase (α) for any frequency of
+operation (ν).
+In conjunction with the estimate of γ,
+this information would allow us to determine the Tamm-
+plasmon mode resonance frequencies (νr).
+This recipe
+provides a flexibility in terms of designing an 1D-APC
+which would facilitate excitation of Tamm-plasmon mode
+at a target (desirable) frequency (or wavelength) of op-
+eration. One such application could be the generation
+of higher harmonics or frequency downconversion using
+optical surface states [36].
+In this case, the 1D-APC
+could be designed such that the Tamm-plasmon modes
+(localized modes) have resonance frequencies that are
+governed by the energy conservation and phase-matching
+constraints imposed by the frequency conversion process.
+III.
+CONCLUSIONS
+In conclusion, we presented an all-dielectric 1D-APC
+design which hosts multiple exceptional points in its
+eigenvalue spectrum by virtue of exhibiting a non-
+Hermitian dynamics for a mode-coupling process between
+a forward-propagating mode to its backscattered coun-
+terpart.
+Although, the 1D-APC does not include any
+dissipative component, the intermodal coupling mecha-
+nism could be classified in terms of PT -symmetric and
+PT -broken phases which are connected through a quan-
+tum phase-transition. We also showed that the reflected
+beam (within the PBG) acquires a geometric phase of π
+2
+in the PT -symmetry broken phase. As a consequence of
+this outcome, the 1D-APC could be designed for excit-
+ing the optical Tamm-plasmon modes at any desirable
+frequency within the PBG. This design flexibility allows
+us to employ such architectures for quite a few appli-
+cations such as efficiently carrying out optical frequency
+conversion using surface states [36].
+IV.
+DISCLOSURES
+The authors declare that there are no conflicts of in-
+terest related to this article.
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+

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+Fuzzballs and Random Matrices
+Suman DASa, Sumit K. GARGb, Chethan KRISHNANc, Arnab KUNDUa
+aTheory Division, Saha Institute of Nuclear Physics,
+A CI of Homi Bhabha National Institute,
+1/AF, Bidhannagar, Kolkata 700064, India
+Email: [email protected], [email protected]
+bManipal Centre for Natural Sciences,
+Manipal Academy of Higher Education,
+Dr.
+T.M.A. Pai Planetarium Building,
+Manipal-576104, Karnataka, India
+Email:
+[email protected]
+cCenter for High Energy Physics, Indian Institute of Science,
+C.V. Raman Road, Bangalore 560012, India.
+Email:
+[email protected]
+Black holes are believed to have the fast scrambling properties of random matrices. If the fuzzball
+proposal is to be a viable model for quantum black holes, it should reproduce this expectation.
+This is considered challenging, because it is natural for the modes on a fuzzball microstate to
+follow Poisson statistics. In a previous paper, we noted a potential loophole here, thanks to the
+modes depending not just on the n-quantum number, but also on the J-quantum numbers of the
+compact dimensions. For a free scalar field φ, by imposing a Dirichlet boundary condition φ = 0
+at the stretched horizon, we showed that this J-dependence leads to a linear ramp in the Spectral
+Form Factor (SFF). Despite this, the status of level repulsion remained mysterious. In this letter,
+motivated by the profile functions of BPS fuzzballs, we consider a generic profile φ = φ0(θ) instead
+of φ = 0 at the stretched horizon. For various notions of genericity (eg. when the Fourier coefficients
+of φ0(θ) are suitably Gaussian distributed), we find that the J-dependence of the spectrum exhibits
+striking evidence of level repulsion, along with the linear ramp. We also find that varying the profile
+leads to natural interpolations between Poisson and Wigner-Dyson(WD)-like spectra. The linear
+ramp in our previous work can be understood as arising via an extreme version of level repulsion in
+such a limiting spectrum. We also explain how the stretched horizon/fuzzball is different in these
+aspects from simply putting a cut-off in flat space or AdS (ie., without a horizon).
+Introduction: The quest for an understanding of quan-
+tum black holes has been one of the engines driving re-
+search in quantum gravity in the last half century. In
+particular, the recent revival of the black hole informa-
+tion paradox [1, 2] due to the works of Mathur [3] and
+AMPS [4] has raised questions about the smoothness of
+the horizon which are still not fully settled.
+In the context of holography/string theory, there are
+two broad lines along which work on quantum black holes
+has progressed. The first approach, which we will call
+the semi-classical approach following [5], is built on in-
+sights from bulk (often Euclidean) effective field theory,
+toy models of 2D gravity, and holographic entanglement
+entropy. Considerable intuition has been gleaned about
+the quantum nature of black holes from this approach
+(eg. [6–10]) with the crowning achievement being a semi-
+classical reproduction of the Page curve [11, 12]. Despite
+this, the precise status of detailed unitarity and smooth-
+ness are still unclear from this perspective, because the
+calculation is fundamentally Euclidean. The second line
+of approach is the fuzzball program of Mathur and others
+which argues that black hole microstates cap off smoothly
+before the horizon. In our opinion, the operational mean-
+ing of this bulk statement in the full quantum setting is
+not yet completely clear. But the mere existence of large
+classes of such solutions [13–20] in the supergravity limit
+of stringy BPS black holes is surprising. In conventional
+general relativity, they would not exist thanks to the no
+hair theorems. See [5] for a more detailed discussion of
+the pros and cons of the two approaches.
+It was suggested in [5] that one way to make progress
+may be to try and reproduce general lessons of the semi-
+classical approach, from fuzzball-motivated considera-
+tions. The hope is that since many of these expectations
+are generic, this may teach us something about how to
+think about quantum fuzzballs at finite temperature even
+arXiv:2301.11780v1  [hep-th]  27 Jan 2023
+
+2
+though constructing explicit solutions is possible only
+in the supergravity BPS limit. Conversely, if realizing
+these lesson from fuzzball-motivated ideas is impossible
+or highly contrived, that could be viewed as an argument
+against the fuzzball program.
+A particularly sharp setting in which one could explore
+this tension is in the expectation that black holes are fast
+scramblers [6], and that they exhibit dynamical features
+of random matrices [21]. A linear ramp [22] in the spec-
+tral form factor (SFF) and repulsion in the level spacing
+distribution (LSD), are viewed as indicators of chaos in
+random matrix theory (RMT) [23]. However, these RMT
+signatures are generally thought to be challenging to re-
+alize from the fuzzball paradigm, see eg. [25] – we expect
+capped geometries to have roughly evenly spaced levels,
+in loose analogy with the standing waves of a cylindrical
+column. This makes conventional level repulsion and the
+linear ramp, difficult to understand from the fuzzball per-
+spective. Note also that simply declaring that the black
+hole is an ensemble of such spectra, does not solve the
+problem [26] – While this will certainly allow a richer set
+of level spacings in the collective spectrum, there is still
+no mechanism to ensure level repulsion [27]. Instead, an
+ensemble of fuzzballs will give rise to Poisson statistics,
+just as an ensemble of simple harmonic oscillators (SHO)
+would [28].
+These expectations are reasonable, but they are also
+difficult to test. This is because solving wave equations
+in generic fuzzball microstate geometries is both difficult
+(because the metric is complicated) and not immediately
+useful (because explicit metrics in BPS cases are at zero
+temperature). Exploiting the fact that the questions we
+wish to tackle are generic, in [5] it was suggested that
+one may be able to make progress by studying a black
+hole at finite temperature with a stretched horizon. In
+particular, the normal modes of a scalar field were stud-
+ied in [5], by computing the spectrum of modes that re-
+sult from a φ = 0 boundary condition at the stretched
+horizon. The results of [5] showed that the expectations
+listed in the previous paragraph have a major caveat,
+they are true only if one ignored the dependence of the
+spectrum on the angular quantum numbers of the com-
+pact dimensions. Unlike the dependence on the principal
+n-quantum number, the dependence on the J-quantum
+numbers was found not to be (approximately) linear. In-
+stead there was a quasi-degeneracy of levels as a function
+of J for moderately large J. Most strikingly, it was found
+that the SFF computed from the spectrum showed very
+clear evidence of a linear ramp, even though conventional
+level repulsion was not present in the J-direction [29]. It
+should be emphasized here that this is the only case in
+the literature that we are aware of, where a linear ramp
+in the SFF exists without an underlying RMT spectrum
+with Wigner-Dyson (WD) level spacing [31].
+While the results of [5] were a tantalizing hint of RMT
+behavior in fuzzballs, a coherent understanding of them
+could not be found. In particular, the presence of a lin-
+ear ramp together with the absence of conventional level
+repulsion, made a compelling interpretation impossible.
+The purpose of this letter, is to shed some light on this
+mysterious state of affairs. We will place the results of
+[5] in context by finding a more general calculation that
+can interpolate between Poisson and RMT-like spectra.
+The idea (at least in hindsight) is extremely simple, and
+motivated by the fact that the known BPS fuzzball solu-
+tions [13, 15, 17] are described by profile functions that
+are supposed to capture the fluctuations of the cap. This
+suggests that a natural generalization of our simple φ = 0
+boundary conditions of [5] is to consider a generic pro-
+file φ = φ0(θ) at the stretched horizon, where θ is a
+mnemonic for the angular directions of the metric. In
+this paper, we will consider profiles of this type, where
+“genericity” will be implemented via choosing Fourier
+coefficients of φ0(θ) from suitable random distributions.
+This is a natural implementation of the intuitive notion
+of “fluctuation at the horizon”. Remarkably, in this very
+natural set up, we see both level repulsion as well as the
+linear ramp. By tuning the variance of the distribution
+from which φ0(θ) is chosen, we show that the LSD can
+interpolate from Poisson to WD-like spectra. In partic-
+ular, as the variance collapses to zero and the boundary
+condition reduces to φ = 0, we find that the LSD col-
+lapses to a very sharp (almost delta-function-like) peak,
+as found in [5]. It was speculated in [5] that this should
+be viewed as an “extreme” version of level-repulsion, and
+our present paper clarifies the precise sense in which
+this is true. Conversely, as the variance is steadily in-
+creased, the LSD transitions from “extreme” to conven-
+tional Wigner-Dyson spectra and eventually to Poisson
+[32].
+Our results demonstrate that fuzzball/stretched hori-
+zon modes can exhibit the spectral features of RMT and
+late time chaos. We emphasize that this is a bulk cal-
+culation of RMT behavior.
+The expectation of RMT
+behavior and eigenstate thermalization in black hole mi-
+crostates is natural in the dual holographic theory, be-
+cause it is strongly coupled.
+This has been explicitly
+demonstrated in the setting of toy dual theories like SYK
+and tensor models [34]. From the bulk however, while
+early time chaos is captured by out-of-time-ordered cor-
+relators [7, 8], late-time chaos as captured by level repul-
+sion and discreteness of the spectrum are very difficult
+
+3
+to understand. Fuzzballs can exhibit discreteness in the
+spectrum trivially, by virtue of the fact that they do not
+have a horizon. On the other hand as we noted earlier,
+the origin of RMT behavior from fuzzballs is supposedly
+non-trivial to arrange. Our results show on the contrary,
+that there are generic bulk mechanisms that can enable
+fuzzballs to capture RMT features.
+In the following section, we will present our main re-
+sults while relegating the technical details to various Sup-
+plementary Material.
+To give further confidence that
+these results really do have to do with the magic of black
+holes and horizons, we will also discuss some examples
+where there are no horizons. Putting a cut-off in such
+geometries leads to major qualitative differences from
+stretched horizons, which we elaborate. In the Conclu-
+sions section we review and emphasize the salient points
+of our results and extract some lessons. Some related fur-
+ther observations and comments about future directions
+[30], as well as various technical details, are presented in
+various Supplementary Material.
+Main Results: We will solve the massless scalar field
+equation in a black hole geometry with a stretched hori-
+zon, while demanding the boundary condition φ = φ0(θ)
+at the stretched horizon. We will do this for the BTZ
+black hole as well as for the Rindler wedge (times a com-
+pact space); these were the two cases studied in detail in
+[5]. The primary virtue of these choices is that the wave
+equation is solvable in terms of well-known special func-
+tions. We will see that the resulting physics is identical in
+both cases, and we do not expect qualitative changes in
+our conclusions for other black holes, in 2+1 dimensions
+and higher.
+The details of the wave equations and how we obtain
+the normal modes for a general stretched horizon pro-
+file are presented in the Supplementary Material. The
+scalar field boundary condition profile can be described
+in terms of its Fourier coefficients. We will choose each of
+these Fourier coefficients randomly from a suitable Gaus-
+sian distribution (see the discussion in the Supplementary
+Material, for details on how this is done). This means
+that there are two choices we need to make in order to
+fully define the problem – the mean and the variance of
+this Gaussian distribution [35]. To make sure that the
+Fourier series sum converges and leads to a well-defined
+profile, we will also cut-off the sum at some J.
+This
+should be compared to the cut-off in J that is required
+to define the SFF [5]. It turns out that the mean and
+the variance have a heuristic (but suggestive) interpre-
+tation in terms of the location and the fluctuations of
+the stretched horizon, see again the Supplementary Ma-
+terial. To have a natural interpretation as the stretched
+horizon at a Planck length, we will take the mean to be
+very large in tortoise coordinates (and therefore close to
+the horizon). Note that since we are working with a fixed
+background geometry, the Planck length is an arbitrary
+choice.
+Our conclusions are entirely analogous for both BTZ
+and Rindler, so we will discuss BTZ here for concreteness;
+see Figures.
+More plots and discussions are provided
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+s
+p(s)
+FIG. 1: LSD for BTZ black hole normal modes
+ω(n = 1, J), with ⟨λ⟩ = −103, Jmax = 800 and
+σλJ = σ0/J with σ0 = 0.3 . Supplementary Material
+contains definitions and explanations of the notation.
+The blue curve is the GUE level spacing curve.
+β=0
+105
+106
+107
+108
+109
+1010
+1011
+10-7
+10-5
+0.001
+0.100
+t
+g(t)
+FIG. 2: SFF for BTZ black hole normal modes; same
+parameters as above. The slope of the line is unity.
+Together these two figures (and the many others in the
+Supplementary Material) show that we can get both the
+linear ramp as well as level repulsion from “synthetic”
+fuzzball normal modes.
+in the Supplementary Material. To summarize – Our re-
+sults for the SFF and the LSD reduce to those of [5] when
+the variance is zero; the SFF has a linear ramp, but the
+LSD is of the “extreme” delta function-like form. But
+remarkably, for small but non-zero choices of the vari-
+
+4
+ance, one finds LSDs that fit Wigner-Dyson [36], while
+the linear ramp remains intact. Finally, as the variance
+becomes large, the LSD reduces to the Poisson form and
+the ramp goes away.
+These results are qualitatively different from corre-
+sponding results in a geometry where a cut-off is intro-
+duced without a horizon. To demonstrate this, we also
+study flat space and AdS with a cut-off.
+Once again
+the details of the computation and plots are presented
+in the Supplementary Material.
+In flat space, we find
+that there is never a ramp of slope ∼ 1, but for moderate
+variances, there is a clear non-linear ramp of slope ∼ 1.7.
+The level-spacing distribution when there is no variance
+is again a strongly peaked delta-function-like form. But
+the origin of this fact has a simple (and less interesting)
+understanding, as opposed to when there was a horizon.
+In flat space the levels are roughly evenly spaced and
+therefore the spectrum is analogous to that of an SHO
+(which also has a delta function LSD, even though it is
+the farthest thing from RMT). Indeed, we have directly
+checked that the SFF of an SHO with a small amount of
+noise added to its energy levels, reproduces precisely the
+non-linear ramp of slope ∼ 1.7 we noted above. This, and
+some interesting related results are presented in some of
+the Supplementary Material and a follow-up paper [30].
+The bottom line is that the linearity of the ramp is lost
+when we simply put a cut-off in flat space as opposed to
+at a stretched horizon. Loosely similar statements hold
+in AdS as well. We will suppress the details, except to
+mention that one has to take care of two separate cases.
+One where the cut-off size is much larger than the AdS
+scale, and another where it is much smaller. The latter
+turns out to yield a discussion identical to the flat space
+case above (as expected). In the former case, there is
+no well-defined constant slope ramp at all in the log-log
+plot, so it will not be of interest to us here.
+A second distinction between the modes of a horizon-
+less cut-off and a stretched horizon is that the variance
+one introduces in the former case can heuristically be in-
+terpreted as due to macroscopic fluctuations at the cut-
+off. In the stretched horizon case, the fluctuations are
+in the tortoise coordinate, and therefore have a natu-
+ral interpretation as Planckian suppressed. This is again
+very natural from the membrane paradigm/fuzzball per-
+spective.
+These matters are discussed in detail in the
+Supplementary Material.
+Conclusions: Our goal in [5] and this paper has been
+to see whether the fuzzball/stretched horizon paradigm
+can be useful for reproducing some of the successes of
+the semi-classical approach to quantum black holes. As
+pointed out in [5], both approaches have produced inter-
+esting results, yet major open problems remain. While
+the stretched horizon/fuzzball will trivially get rid of
+some aspects of the information paradox, finding hints
+of RMT behavior is considered challenging.
+We demonstrated that we can find both the linear
+ramp and conventional level repulsion (as well as RMT
+level spacing ratios) from a stretched horizon. The linear
+ramp is a direct consequence of a cut-off near the hori-
+zon. In a cut-off geometry without a horizon, the linear
+ramp never exists, and a non-linear ramp when it exists,
+can be understood as related to an SHO spectrum with
+noise. We also found that conventional level repulsion
+is easy to realize, by simply incorporating angular de-
+pendence in the boundary condition. This is interesting,
+because such angle-dependence is generic in BPS fuzzball
+microstates.
+The existence of the linear ramp is usually taken as an
+indicator of rigidity in the spectrum. It is a signature of
+strong chaos. Finding the linear ramp in our previous
+paper [5] was encouraging, but the absence of conven-
+tional level repulsion made the result puzzling. But given
+the ramp, it is natural to suspect that some small per-
+turbation may be able to produce the nearest-neighbor
+correlations [37] as well. The challenge was to identify
+the right kind of perturbation. The fluctuations at the
+stretched horizon that we have included in this paper
+can be viewed as a natural candidate for such a small
+perturbation. The variance in the Fourier modes of the
+fluctuation profile leads to a small noise in the spectrum,
+which leads to the requisite spread in the LSD.
+Our results also strengthen the case that level repulsion
+is a weaker hint of chaos than the linear ramp. This is
+because it is only sensitive to nearest neighbor physics.
+We explicitly demonstrate this using the example of the
+SHO in the Supplementary Material, where it is shown
+that adding a small amount of noise to the SHO energy
+levels is sufficient to produce conventional WD-like LSD
+plots.
+But this is not sufficient to produce the linear
+ramp, which is sensitive to long range correlations within
+the spectrum. This again ties nicely with the observation
+that the linear ramp is present only when the cut-off is
+near the black hole horizon, while level repulsion can be
+realized in a cut-off geometry with or without a horizon
+as long as we are working with a fluctuating profile [38].
+The SFFs of horizonless cases with moderate variance
+have a power law ramp of slope ∼ 1.7 – This is the same
+as that of a moderately noisy SHO.
+A natural proposal that ties together our observations
+then, is as follows – Signatures of robust chaos (in the
+sense of spectral rigidity) emerge when we consider a
+stretched horizon close to the black hole.
+Such signa-
+
+5
+tures are not present when the cut off is in empty space
+or far from the horizon. These statements are indepen-
+dent of the profile choices at the cut-off. But the profiles
+do play a role, when we are discussing nearest neighbor
+physics and level repulsion in the system. A profile with
+non-vanishing variance can lead to nearest-neighbor level
+repulsion both with or without a horizon, but the natural
+length scale associated to the variance has to be macro-
+scopic for this to happen in a horizonless geometry. In
+other words, even if we allow macroscopic fluctuations,
+we can at best see nearest neighbor effects in a horizon-
+less geometry with a cut-off. On the contrary, stretched
+horizon/fuzzball modes automatically carry signatures of
+robust chaos and a linear ramp, with or without a non-
+trivial profile. If the profile is generic in the sense of hav-
+ing a small non-zero variance, they reproduce the correct
+nearest neighbor effects as well.
+Semi-classical
+bulk
+calculations
+involving
+replica
+wormholes (and implicitly, ensemble averages) are known
+to produce a smooth linear ramp without fluctuations.
+The challenge for quantum gravity is to reproduce a lin-
+ear ramp without any ensemble average from a single
+microstate, and with fluctuations. Our calculation, de-
+spite its simplicity has reproduced both these features.
+This may seem surprising because our set up is super-
+ficially (semi-)classical.
+But this is misleading – The
+boundary conditions we are imposing at the stretched
+horizon, while technically simple, are conceptually highly
+non-trivial from the dual CFT. It is clearly of interest to
+understand this boundary condition better from an in-
+trinsically CFT perspective.
+It may be useful to re-visit the many questions about
+(quantum) black holes at finite temperature, armed with
+the perspectives we have added in this paper.
+In this
+section, we have only emphasized black hole physics. A
+more detailed discussion of open questions and questions
+more intrinsic to RMT physics are presented in the Sup-
+plementary Material.
+ACKNOWLEDGMENTS
+We thank A. Preetham Kumar for crucial contribu-
+tions in our previous collaboration [5], and Masanori
+Hanada, Romesh Kaul, Alok Laddha, R. Loganayagam,
+Ayan Mukhopadhyay,
+Onkar Parrikar,
+Ashoke Sen,
+Kostas Skenderis and Amitabh Virmani for discussions
+and/or correspondence.
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+[25] www.youtube.com/watch?v=0BO-p58Pypc&t=3397s
+[26] Even with an ensemble, there are conceptual questions
+on when/how an ensemble should replace a microstate.
+Ensembles arise in physics typically as effective repre-
+sentations of microscopic physics, eg. when an ensemble
+average can stand in for a time average. So it is not clear
+in the first place that one should simply adjoin the nor-
+mal modes of all the separate microstates in order to get
+the “effective” spectrum.
+[27] It is generally expected that level repulsion and linearity
+of the ramp go hand in hand. Our results in [5] and this
+paper demonstrate that this is very far from a theorem.
+Nonetheless the general expectation that RMT behavior
+is connected to level repulsion and linear ramp is broadly
+true.
+[28] See Appendix C of [5].
+[29] It was speculated in [5] that the level spacing found
+there may perhaps be viewed as an “extreme” version of
+a Wigner-Dyson-like distribution. The grounds for this
+speculation were quite scanty, but in this paper we will
+see that there is a systematic sense in which it is true!
+Note that just because a level spacing plot has no support
+at the origin does not guarantee that we are dealing with
+a random matrix. The simplest illustration of this fact is
+the SHO – the LSD of the SHO is a delta function sep-
+arated from the origin. We will have more to say about
+this in the Supplementary Material and also in [30].
+[30] S. Das, S. K. Garg, C. Krishnan and A. Kundu, “Gener-
+alized Random Matrix Spectra”, To Appear.
+[31] We have since been able to construct many examples of
+this type, this will be presented elsewhere [30].
+[32] Even though we do no report the details here, we have
+also studied the level-spacing ratios γ [33] of these spec-
+tra. This is another diagnostic of RMT behavior along
+with SFF and LSD. For small/zero variance, we find
+γ values that are consistent with RMT spectra. But it
+steadily increases with the variance and becomes (very)
+large, matching the expectation that γ = ∞ for Poisson
+systems [33]. γ is a diagnostic defined via nearest neigh-
+bor data and is therefore somewhat redundant with the
+LSD. This is one reason why we do not consider this as
+truly distinct diagnostic, and do not emphasize it in this
+paper. In all the examples we consider, the behavior of
+LSD and LSR are mutually consistent. The LSD and the
+(linear ramp of the) SFF on the other hand, do genuinely
+capture somewhat distinct aspects of random matrix be-
+havior as we will elaborate.
+[33] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux,
+“Distribution of the Ratio of Consecutive Level Spacings
+in Random Matrix Ensembles”, Phys. Rev. Lett. 110,
+084101, arXiv:1212.5611[math-ph].
+[34] Y. Liu,
+M. A. Nowak and I. Zahed,
+“Disorder in
+the Sachdev-Yee-Kitaev Model,” Phys. Lett. B 773,
+647-653
+(2017)
+doi:10.1016/j.physletb.2017.08.054
+[arXiv:1612.05233 [hep-th]]. C. Krishnan,
+S. Sanyal
+and
+P.
+N.
+Bala
+Subramanian,
+“Quantum
+Chaos
+and
+Holographic
+Tensor
+Models,”
+JHEP
+03,
+056
+(2017) doi:10.1007/JHEP03(2017)056 [arXiv:1612.06330
+[hep-th]].
+A.
+del
+Campo,
+J.
+Molina-Vilaplana
+and
+J. Sonner, “Scrambling the spectral form factor: uni-
+tarity constraints and exact results,” Phys. Rev. D 95,
+no.12, 126008 (2017) doi:10.1103/PhysRevD.95.126008
+[arXiv:1702.04350 [hep-th]]. C. Krishnan, K. V. Pa-
+van Kumar and S. Sanyal, “Random Matrices and
+Holographic Tensor Models,” JHEP 06, 036 (2017)
+doi:10.1007/JHEP06(2017)036
+[arXiv:1703.08155
+[hep-th]].
+A.
+Gaikwad
+and
+R.
+Sinha,
+“Spectral
+Form
+Factor
+in
+Non-Gaussian
+Random
+Matrix
+Theories,” Phys. Rev. D 100, no.2, 026017 (2019)
+doi:10.1103/PhysRevD.100.026017
+[arXiv:1706.07439
+[hep-th]]. C. Krishnan, K. V. Pavan Kumar and D. Rosa,
+“Contrasting SYK-like Models,” JHEP 01, 064 (2018)
+doi:10.1007/JHEP01(2018)064 [arXiv:1709.06498 [hep-
+
+7
+th]]. R. Bhattacharya, S. Chakrabarti, D. P. Jatkar and
+A. Kundu, “SYK Model, Chaos and Conserved Charge,”
+JHEP 11,
+180 (2017) doi:10.1007/JHEP11(2017)180
+[arXiv:1709.07613 [hep-th]]. C. V. Johnson, F. Rosso
+and
+A.
+Svesko,
+“Jackiw-Teitelboim
+supergravity
+as
+a
+double-cut
+matrix
+model”,
+Phys.
+Rev.
+D
+104,
+no.8, 086019 (2021) doi:10.1103/PhysRevD.104.086019
+[arXiv:2102.02227 [hep-th]]. Y. Chen, “Spectral form
+factor for free large N gauge theory and strings,”
+JHEP 06,
+137 (2022) doi:10.1007/JHEP06(2022)137
+[arXiv:2202.04741 [hep-th]].
+[35] The precise distribution does not seem too important for
+our results. This is natural because (as noted in our mo-
+tivations), we are looking for results like linear ramp and
+level repulsion, which are semi-qualitative and robust. We
+have checked that similar statements hold also for uni-
+formly distributed Fourier modes, but we will not elabo-
+rate on it here.
+[36] By choosing the variance suitably, we can get good fits
+with GSE, GUE or GOE. We will mostly present GUE
+fits in this paper. A very interesting feature of these re-
+sults is that since they arise by tuning certain continu-
+ous boundary conditions and not the (discrete choice of)
+ensemble from which the Hamiltonian matrix is chosen,
+they seem to allow a continuum of LSDs that naturally
+generalize WD.
+[37] We thank M. Hanada for some encouraging comments on
+this point.
+[38] Let us also re-iterate that the fluctuations should nat-
+urally be viewed as macroscopic (and not Planck sup-
+pressed) if they are to give rise to level repulsion in a
+cut-off geometry without a horizon.
+[39] M. B. Marcus and G. Pisier, “Random Fourier Series with
+Applications to Harmonic Analysis”, (AM-101), Volume
+101 (Annals of Mathematics Studies, 101), Princeton
+University Press.
+[40] B. Bhattacharjee and C. Krishnan, “A General Prescrip-
+tion for Semi-Classical Holography,” [arXiv:1908.04786
+[hep-th]].
+[41] C. Krishnan and V. Mohan, “Hints of gravitational
+ergodicity:
+Berry’s ensemble and the universality of
+the semi-classical Page curve,” JHEP 05, 126 (2021)
+doi:10.1007/JHEP05(2021)126 [arXiv:2102.07703 [hep-
+th]].
+
+8
+Supplementary material
+CASE STUDY: BTZ
+As in [5], we will start by considering a scalar field Φ of mass m in the BTZ background,
+ds2 = −(r2 − r2
+h)
+L2
+dt2 +
+L2
+(r2 − r2
+h)dr2 + r2dψ2
+(1)
+with −∞ < t < ∞, 0 < r < ∞ and 0 ≤ ψ < 2π. In [5] we fixed units by setting L = 1 and worked with the numerical
+choice rh = 1 from the outset. Here, we will present the more general expressions because it is useful in comparisons
+with cut-off empty space. The new boundary conditions and the corresponding results/plots start only after (14).
+So a reader who is familiar with the results of [5] and is willing to believe the slightly more general expressions we
+present here, can skip directly to the discussion after (14).
+The wave equation
+□Φ ≡
+1
+�
+|g|
+∂µ
+��
+|g|∂µΦ
+�
+= m2Φ
+(2)
+can be solved by writing
+Φ =
+1
+√r
+�
+ω,J
+e−iωteiJψφω,J(r)
+(3)
+with integer J. The radial part of (2) satisfies,
+(r2 − r2
+h)2φ
+′′
+ω,J(r) + 2r(r2 − r2
+h)φ
+′
+ω,J(r) + ω2L4φω,J(r) − VJ(r)φω,J(r) = 0
+(4)
+where
+V (r) = (r2 − r2
+h)
+� 1
+r2
+�
+J2L2 + r2
+h
+4
+�
++ ν2 − 1
+4
+�
+,
+ν2 = 1 + m2.
+(5)
+We will generally work with the massless case, ν = 1. The solution1 of this is given in terms of hypergeometric
+functions as
+φ(r) = (r)
+1
+2 − iJL
+rh �
+r2 − r2
+h
+�− iωL2
+2rh
+�
+e− πJL
+rh
+� r
+rh
+� 2iJL
+rh C2H (r) + C1G (r)
+�
+,
+(6)
+where we are suppressing the subscripts ω, J on the LHS as well as on C1 and C2. Here,
+G (r) = 2F1
+�1
+2
+�
+1 − iωL2
+rh
+− iJL
+rh
+− ν
+�
+, 1
+2
+�
+1 − iωL2
+rh
+− iJL
+rh
++ ν
+�
+; 1 − iJL
+rh
+, r2
+r2
+h
+�
+(7)
+H (r) = 2F1
+�1
+2
+�
+1 − iωL2
+rh
++ iJL
+rh
+− ν
+�
+, 1
+2
+�
+1 − iωL2
+rh
++ iJL
+rh
++ ν
+�
+; 1 + iJL
+rh
+, r2
+r2
+h
+�
+.
+(8)
+1We will work with the massless scalar and the J = 0 mode needs special treatment. See footnote 13 of [5].
+
+9
+Near the AdS boundary (r → ∞), the radial solution (6) becomes
+φbdry(r) ≈ −ir
+iωL2
+rh
+−ν− 1
+2 r
+1− iωL2
+rh
+− iJL
+rh +ν
+h
+(r2 − r2
+h)− iωL2
+2rh e− πL(J+ωL)
+2rh
+×
+×
+�
+e−i π
+2 ν�
+γ (J, −ν) C1 + γ (−J, −ν) C2
+�
++ O
+�
+1/r3/2�
++ r2νei π
+2 ν
+r2ν
+h
+�
+γ (J, ν) C1 + γ (−J, ν) C2 + O
+�
+1/r3/2���
+, (9)
+where
+γ (J, ν) ≡
+Γ(1 − iJL
+rh )Γ(ν)
+Γ
+�
+1
+2(1 − iωL2
+rh
+− iJL
+rh + ν)
+�
+Γ
+��
+1
+2(1 + iωL2
+rh
+− iJL
+rh + ν)
+�,
+(10)
+Normalizability at r → ∞ sets the 2nd term of equation (9) to zero, which leads to
+C2 = − γ (J, ν)
+γ (−J, ν)C1,
+(11)
+fixing the constant of integration C2 in terms of C1 or vice versa.
+We will eventually place our boundary condition at a stretched horizon, to be thought of as a Planck length or so
+outside the horizon. Near the horizon, the radial solution can be approximated as
+φhor(r) ≈ C1
+�
+P1 (r/rh − 1)− iωL2
+2rh + Q1 (r/rh − 1)
+iωL2
+2rh
+�
+,
+(12)
+where
+P1 = −
+2− iωL2
+2rh e− πJL
+rh (JπL)
+�
+e
+2πJL
+rh
+− 1
+�
+r
+− 1
+2 − iωL2
+rh
+− iJL
+rh
+h
+csch( πωL2
+rh )Γ(− iJL
+rh )
+�
+e
+πJL
+rh + eπ(iν+ ωL2
+rh )
+�
+Γ(1 − iωL2
+rh )Γ( 1
+2(1 + iωL2
+rh
+− iJL
+rh − ν))Γ( 1
+2(1 + iωL2
+rh
+− iJL
+rh + ν))
+(13)
+Q1 =
+(−1)
+iωL2
+rh 21+ iωL2
+2rh e
+2πωL2
+rh
+π2r
+1
+2 − iωL2
+rh
+− iJL
+rh
+h
+(coth( πωL2
+rh ) − 1)
+�
+eiπν + e
+πL(J+ωL)
+rh
+�
+Γ( iJL
+rh )Γ(1 + iωL2
+rh )Γ( 1
+2(1 − iωL2
+rh
+− iJL
+rh − ν))Γ( 1
+2(1 − iωL2
+rh
+− iJL
+rh + ν))
+.
+(14)
+In [5] we demanded a vanishing condition for the scalar at the stretched horizon r = r0. Motivated by the angle-
+dependent profiles that are found in BPS fuzzballs, we will generalize this in the present paper. We will demand that
+at r = r0 the scalar field takes the form of a given profile φ0(ψ). In terms of the notation introduced in (3), we will
+write
+1
+√r0
+�
+J,ω
+eiJψe−iωtφω,J(r0) = φ0(ψ, t)
+(15)
+Expanding the RHS in terms of the Fourier modes eiJψ and e−iωt and absorbing some constants suitably, we get an
+equation of the form φhor(r = r0) = C0 where on both LHS and RHS we have suppressed the ω and J subscripts.
+Note that ultimately we will get a quantization condition for our ω’s, and this means that an implicit assumption in
+the above approach is that the φ0(ψ, t) can be expanded in terms of these modes. Our explicit boundary conditions
+below and their solution can be viewed as a self-consistent way to do this.
+Concretely, this leads to
+C1
+�
+P1 (r0/rh − 1)− iωL2
+2rh + Q1 (r0/rh − 1)
+iωL2
+2rh
+�
+= C0,
+(16)
+=⇒ P1
+Q1
+=
+C0
+C1Q1
+(r0/rh − 1)
+iωL2
+2rh − (r0/rh − 1)
+iωL2
+rh .
+(17)
+
+10
+As in [5], it is possible to show that |P1| = |Q1|. So by writing P1 = |P1|eiα and Q1 = |Q1|eiβ, (17) can be written as
+ei(α−β) = µJe
+i
+�
+λJ ωL2
+rh
++ θ
+2
+�
+− eiθ
+(18)
+where
+θ = Arg
+�
+(r0/rh − 1)
+iωL2
+rh
+�
+,
+µJ =
+��� C0
+C1Q1
+���,
+and λJωL2
+rh
+= Arg
+� C0
+C1Q1
+�
+(19)
+We have emphasized the J-dependence of µ and λ in the notation, but it should be noted that with these definitions,
+they have an n-dependence as well. The real and imaginary parts of (18) lead to the definition
+µJ = 2 cos
+�λJωL2
+rh
+− θ
+2
+�
+(20)
+as well as the quantization condition on ω,
+cos(α − β) = cos
+�2λJωL2
+rh
+�
+(21)
+This last equation is a key equation for our purposes. Since this is a phase equation, the modes depend on a free
+integer n. It is possible to check that these two equations together reduce to the quantization condition we had in [5]
+when we set µJ = 0. More generally, one can solve the quantization condition by choosing λJ from a distribution,
+which we will usually take to be Gaussian.
+We will take λ for each value of J from the same distribution. Note that heuristically, λJ is comparable to the
+stretched horizon location. One way to see this is to note that (20) implies (if there are no fluctuations, and λ and µ
+are taken to be J-independent constants) that fixing
+λJ = 1
+2 ln
+� r0
+rh
+− 1
+�
+(22)
+fixes µJ. More generally, the fact that the difference between λJ and 1
+2 ln
+�
+r0
+rh − 1
+�
+is what shows up in (20) suggests
+that the natural scale of λJ is the stretched horizon radius in (essentially) tortoise coordinates. Eqn (20) also makes
+it tempting to view the fluctuations in µJ as due not to the fluctuations in λJ but due to the fluctuations of the
+stretched horizon. This last interpretation is of course simply a heuristic, because it is not meaningful to have a
+J-dependent notion of stretched horizon radius. Nonetheless, we view this as highly suggestive, in light of the usual
+claim that the profile functions in fuzzball geometries are supposed to capture the fluctuations of the cap. Indeed,
+our initial motivation for considering the scalar field profile, was as a proxy for this.
+It is worth emphasizing in the above discussion (and elsewhere), that there is some leftover freedom in fixing C1
+in terms of C0 and the rest of the quantities. An analogous freedom existed in [5] as well – our demands do not
+completely fix the boundary conditions, but they fix them enough to determine the normal modes. We can fix this
+extra freedom by setting C1Q1 = 1 so that µJ and λJ have the nice interpretation as (essentially) the magnitude and
+phase of C0. Remember that C0 has J-dependence which we often suppress to avoid notational congestion, it is the
+Fourier coefficient of the scalar profile.
+There is one choice we have made in the above definitions, which may be worth further study. In defining λJ via the
+last equation in (19), we have extracted an ω on the LHS. It may also be natural to define the λ variable without this,
+in which case our quantization conditions should be solved after the replacement λJ → λJ/ω and choosing the new λ’s
+from some suitable distribution. Since the target results we are aiming for are believed to be robust semi-qualitative
+statements like level repulsion and the linear ramp, these choices should not affect them. We have checked that indeed
+this is the case. Ultimately these choices all correspond to how we parametrize the Fourier modes C0 of the profile
+φ0(ψ, t) in (15). Explicitly, the profile should be written as
+φ0(ψ, t) =
+�
+n,J
+C0(n,J)eiJψe−iω(n,J)t
+(23)
+
+11
+and our choice corresponds to the parametrization
+C0(n,J) = µJ,nei λJ ω(n,J)L2
+rh
+(24)
+where we have kept the n and J dependencies, fully explicit. If we absorb the ω into the definition of λ as discussed
+above, then the µ (and therefore the C0) have only J-dependence. (Superficially, this may seem illegal because ω’s
+have an n-dependence. But remember that the ω’s are determined after the definition of λ, so one can check that
+this is perfectly well-defined.) This leads to some nice features in some expressions, but also some compensating
+complications/ugliness in others. So we will stick to the form defined by (18) and (19), or (24). It may be interesting
+to investigate the naturalness of the choices involved here from the perspective of Haar typicality in the phase space
+of the scalar field, but we will not undertake it here.
+With these caveats, one way to get some intuition for the profile is to consider the quantity
+˜φ(ψ) ≡
+Jcut
+�
+J=0
+C0(n=0,J)eiJψ =
+Jcut
+�
+J=0
+µJ,n=0ei λJ ω(n=0,J)L2
+rh
+eiJψ.
+(25)
+This is what we will often call the profile function. It should be emphasized that our quantization condition arises
+essentially as a condition on the phase of the Fourier coefficient. The various arbitrary choices we discussed above
+can be understood as arising from the fact that it does not unambiguously fix C0. In writing the second equality of
+(25) we have fixed C1Q1 = 1 as mentioned above, but this is an ad-hoc choice. Similar statements were true in the
+discussion in [5] as well, where the magnitude information was again not needed to determine the normal modes. One
+way to understand this in the present setting is to note that the last two equations in (19) basically determine the
+phase and the magnitude of the profile C0 via
+µJei λJ ωL2
+rh
+=
+C0
+C1Q1
+.
+(26)
+Once we make a choice of λ (which is a single real variable that captures the phase information) the quantization
+condition is obtained via (21). Then µJ is completely fixed via (20). All of this only fixes the ratio on the RHS of
+(26), while the profile itself is controlled by C0. Fourier series where the phase is suitably random have been studied
+extensively by mathematicians, see eg. the book [39]. It seems significant that this structure naturally arises in our
+discussions; this is clearly worthy of further study.
+In the plots in this section, we have set L = rh = 1, and ⟨λ⟩ = 1
+2 ln
+�
+r0
+rh − 1
+�
+, as we change the variance of the
+Gaussian distribution from which λ is chosen. This choice of ⟨λ⟩ ensures that µJ = 2 in the zero-variance limit. This is
+slightly different from the µJ = 0 condition in [5] but it is natural (and straightforward to check) that the qualitative
+results on LSD and SFF remain identical. One can also in principle treat µJ (instead of λJ) as the quantity chosen from
+a distribution. This is slightly more convenient to connect to the language of [5]. This changes some of our formulas
+in minor ways, but the essential point that there is one real parameter worth of freedom that we are fixing, remains
+intact. We have experimented with various choices of λ-variance as a function of J, eg. σλJ ≡ σ0, σ0/J, σ0/
+√
+J. In
+the plots in this section, we present the σ0/
+√
+J case and we quote the value of σ0. We will sometimes refer to σ0
+loosely as the variance. A suppression of the variance with J is useful because the normal mode level-spacing gets
+smaller as J increases, and therefore too large a variance can completely destabilize the structure of the spectrum
+(and along with it, the linear ramp and level repulsion). Let us also mention that when we juxtapose the plots of an
+SFF and an LSD for some choice of variance, we show it for the same realization that we choose from the Gaussian
+distribution. This statement applies to the Rindler plots of the next section as well.
+For zero variance, we reproduce the “extreme” Wigner-Dyson plots for the level spacing that we found in [5] as well
+as the linear ramp. If we increase the variance slightly, the ramp remains intact, but the level-spacing takes the more
+conventional WD form. We can find fits with GSE, GUE or GOE with minor increments in variance, we present GUE
+in the plots. Eventually, as we increase the variance to very large values, the level spacing degenerates to a Poisson
+form and the ramp is lost.
+
+12
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+0
+1
+2
+3
+4
+5
+s
+p(s)
+β=0
+107
+108
+109
+1010
+1011
+1012
+10-6
+10-5
+10-4
+0.001
+0.010
+0.100
+1
+t
+g(t)
+FIG. 3: LSD (left) and SFF (right) for BTZ with ⟨λ⟩ = −104 and Jmax = 400 with σ0 = 0.0. We are working with
+ω(n = 2, J). These results are a version of the results in [5].
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+s
+p(s)
+β=0
+107
+108
+109
+1010
+1011
+1012
+1013
+10-6
+10-5
+10-4
+0.001
+0.010
+0.100
+1
+t
+g(t)
+FIG. 4: Same as before, but with σ0 = 0.025. The blue curve on the left is GUE.
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+s
+p(s)
+β=0
+107
+108
+109
+1010
+1011
+1012
+10-6
+10-5
+10-4
+0.001
+0.010
+0.100
+1
+t
+g(t)
+FIG. 5: Same as in the previous figures, but with σ0 = 2.0. The red curve on the left is Poisson.
+
+13
+0
+100
+200
+300
+400
+0.00015711
+0.00015712
+0.00015713
+0.00015714
+0.00015715
+0.00015716
+0.00015717
+0.00015718
+J
+ω(2,J)
+0
+100
+200
+300
+400
+0.00015712
+0.00015714
+0.00015716
+0.00015718
+J
+ω(2,J)
+FIG. 6: Spectrum of BTZ with σ0 = 0 (left) vs σ0 = 2.0 (right). ⟨λ⟩ = −104 and Jmax = 400. We show ω(n = 2, J).
+CASE STUDY: RINDLER × COMPACT SPACE
+We will follow the motivations and discussion in section 4.2 of [5] when developing the Rindler case, which the
+reader should consult for notations. We solve the wave equation in the metric
+ds2 = e2aξ(−dη2 + dξ2) + R2dφ2
+(27)
+and introduce A ≡ ω/a and y ≡ eaξ(J/aR) as in [5]. In terms of y variable the position of boundary and horizon are
+given by y → ∞ and y → 0 respectively. In the notations of [5], we require that the field φ(y) vanish at boundary.
+We also demand that it has a profile at some small y0 (or ξ = ξ0). When y → ∞, the relevant equation is [5]
+φ(y) → (C1 + C2)
+ey
+√2πy + (C1eπA + C2e−πA) e−y
+√2πy .
+(28)
+The boundary condition at infinity leads to C1 = −C2, and at y0 implies (in notation that is parallel to the BTZ case
+before):
+C1(I[−iA, y0] − I[iA, y0]) = C0,
+=⇒ I[−iA, y0] − I[iA, y0] = C0
+C1
+(29)
+Near horizon i.e. in the limit y0 → 0 the above expressions can be approximated by
+C1
+�
+y−iA
+2iA
+Γ(1 − iA) − yiA
+2−iA
+Γ(1 + iA)
+�
+= C0
+(30)
+C0
+C1
+� J
+aR
+�−iA �eaξ
+2
+�iA
+−
+�eaξ
+2
+�2iA
+=
+� J
+aR
+�−2iA Γ(iA)
+Γ(−iA)
+(31)
+Now Abs
+�� J
+aR
+�−2iA
+Γ(iA)
+Γ(−iA)
+�
+= 1, so (31) can be written, again in notation that simulates the BTZ case as
+µJeiωλJeiθ/2 − eiθ = eiα
+(32)
+with
+µJ = Abs
+�
+C0
+C1
+� J
+aR
+�−iA�
+,
+ωλJ = Arg
+�
+C0
+C1
+�eaξ
+2
+�iA�
+,
+α = Arg
+�� J
+aR
+�−2iA Γ(iA)
+Γ(−iA)
+�
+,
+θ = Arg
+��eaξ
+2
+�2iA�
+(33)
+
+14
+0
+1
+2
+3
+4
+0
+1
+2
+3
+4
+5
+s
+p(s)
+β=0
+105
+106
+107
+108
+109
+1010
+1011
+10-7
+10-5
+0.001
+0.100
+t
+g(t)
+FIG. 7: LSD (left) and SFF (right) for Rindler with parameters described in the text. σ0 = 0.0. We are working
+with ω(n = 1, J). These results should be compared to the results in [5].
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+s
+p(s)
+β=0
+105
+106
+107
+108
+109
+1010
+1011
+10-7
+10-5
+0.001
+0.100
+t
+g(t)
+FIG. 8: Same as the previous figure, but with σ0 = 0.02. The blue curve on the left is GUE.
+0
+1
+2
+3
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+s
+p(s)
+β=0
+105
+106
+107
+108
+109
+1010
+10-7
+10-5
+0.001
+0.100
+t
+g(t)
+FIG. 9: Same as the two previous figures, but with σ0 = 1. The red curve on the left is Poisson.
+
+15
+0
+100
+200
+300
+400
+500
+600
+700
+0.001561
+0.001562
+0.001563
+0.001564
+0.001565
+0.001566
+0.001567
+J
+ω(1, J)
+0
+100
+200
+300
+400
+500
+600
+700
+0.001560
+0.001561
+0.001562
+0.001563
+0.001564
+0.001565
+0.001566
+0.001567
+J
+ω(1, J)
+FIG. 10: Spectrum of Rindler with σ0 = 0 (left) vs σ0 = 1.0 (right). We show ω(n = 1, J).
+This therefore again leads to similar structures as in BTZ. We find
+µJ = 2 cos(λJω − θ/2)
+(34)
+as well as the quantization condition
+cos(α) = cos(2λJω)
+(35)
+Because the structure is precisely parallel to BTZ, we will not repeat the discussion; it is clear that the normal mode
+calculation proceeds in an identical manner. The mean value of λ can be related to the stretched horizon location.
+Once we choose R, ξ0 and a, the normal modes ω(n, J) can be numerically solved for as a function of J (and an integer
+n). We present the plots in precise parallel to the BTZ case. The qualitative results are identical, despite the fact
+that the special functions that showed up in the wave equations here are different. In the plots we present, we have
+chosen a = 1, R = 2, Jmax = 700, ⟨λ⟩ = −103 and σJ = σo/
+√
+J. The σ0 values are quoted in the plots.
+THE HAIRY HARMONIC OSCILLATOR AND CUT-OFF IN EMPTY SPACE:
+LEVEL REPULSION WITHOUT LINEAR RAMP
+We noted that the linear ramp in the SFF and repulsion in the LSD can both be seen in the stretched horizon
+spectrum if the boundary condition is generic. We also pointed out that the level spacing ratio discussed in [33] is also
+consistent with RMT expectations. Together, these constitute very strong evidence that fuzzball/stretched horizon
+spectra have strong connections to random matrices and chaos.
+In this section, we will ask a slightly more resolved question: which of these is a more robust indicator of chaos? Is
+it the linear ramp or is it level repulsion? Or are both these features always present in systems concomitantly? We
+will present some hints in this section that the linear ramp may be a more robust diagnostic of strong chaos than
+nearest-neighbor data. This is not an entirely new suggestion (the length of the ramp is often viewed as an indicator
+of the “strength” of chaos), but we will give some examples which we feel are instructive.
+We will start (as often in physics) with the simple harmonic oscillator (SHO). For our purposes, the SHO is
+interesting because even though it is the farthest thing from a chaotic system, it exhibits a naive (or extreme) version
+of level repulsion – the levels are equally spaced, and the LSD is a delta function shifted from the origin. Motivated
+by the results of this paper, we can ask if there is a natural way to “perturb” the SHO spectrum so that the level
+spacing becomes a more conventional Wigner-Dyson-like form. It turns out that a simple way to engineer this exists
+– we simply allow a small amount of (Gaussian) noise in the levels of the SHO. We will call this set up a hairy or
+noisy SHO. See Figure 11 right panel, for a typical LSD of an SHO perturbed in this way. We present a GOE fit
+for concreteness. But again, by adjusting the variance, we can find fits with GSE or GUE. We are not aware of a
+previous observation of this simple but striking fact in the literature, but it is easy enough to understand – Random
+noise in the energy levels directly affects the nearest neighbor data, which explains why the delta function in the LSD
+gets spread out.
+
+16
+0
+1
+2
+3
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+s
+p(s)
+0
+1
+2
+3
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+s
+p(s)
+FIG. 11: LSDs of Cut-off flat space with fluctuation profile (left) vs hairy SHO (right). Flat space data:
+Jmax = 300, rcut = 1, λ-variance = 0.0174. We are working with ω(n = 1, J). SHO data: nmax = 600, ω = 1,
+spectral noise variance = 0.36. Both fits are GOE.
+β=0
+0.01
+0.10
+1
+10
+100
+10-6
+10-5
+10-4
+0.001
+0.010
+0.100
+1
+t
+g(t)
+β=0
+0.001
+0.010
+0.100
+1
+10
+100
+10-7
+10-5
+0.001
+0.100
+t
+g(t)
+FIG. 12: SFFs of the same systems (SHO on the right). The yellow line has slope 1.7 (both left and right). In other
+words, this is a power law ramp.
+0
+50
+100
+150
+200
+250
+300
+0
+50
+100
+150
+200
+250
+300
+J
+ω(1,J)
+0
+100
+200
+300
+400
+500
+600
+0
+100
+200
+300
+400
+500
+600
+n
+ω(n)
+FIG. 13: Shapes of spectra, for the same systems as above (SHO again on the right). It is clear that both the
+spectra are approximately evenly spaced. The punchline of the figures in this page is that the spectral features of
+the two systems have crucial similarities.
+On the other hand, strong chaos is characterized by spectral rigidity which is encoded in the linear ramp in the
+SFF. And indeed, if one computes the SFF of the SHO with noise in the spectrum, one finds that the ramp is in fact
+
+17
+non-linear. This is illustrated in Figure 12, right panel. We emphasize that it is remarkable that a well-defined ramp
+exists, even though it is not linear. In fact, we find that on a log-log plot, it has a well-defined slope of ∼ 1.7. In other
+words, a hairy SHO has a power law ramp, at least within the context of our numerical results.
+These SHO results shed light on the distinctions between a black hole with a stretched horizon, and a cut-off in
+empty space. If we impose a simple Dirichlet condition φ = 0 at the cut-off, in the former case we find a linear ramp
+[5], but in empty space there is no clear ramp, certainly nothing of slope ∼ 1. See Figure 14. But as we add variance
+to the profile, we see the emergence of a power law ramp, see Figure 12 left panel. The SHO example above provides
+us a clear understanding of this. A cut-off in flat space leads to eigenvalues that are connected to the zeros of Bessel
+functions (as we will see). These are roughly evenly spaced – so the spectrum looks crudely like that of an SHO.
+Relatedly, the level spacing in the φ = 0 case is essentially a delta function. But this can be made to look like a
+more spread out (WD-like) form by demanding instead that the boundary condition is φ ∼ φ0(θ) where the profile
+has some variance in its Fourier modes. The noise in the spectrum increases when we do this, and as a result (as
+pointed out above for the hairy SHO) we find that the LSD takes a more conventional WD form. Of course, when
+the variance is very large, the spectrum ends up becoming Poisson. Crucially, the slope of the ramp is never ∼ 1 in
+these cases. For moderate values of the variance, it is consistent with the ∼ 1.7 quoted above for the noisy SHO – see
+figures. (Note that when the variance is steadily increased, the ramp gets increasingly washed out. So this statement
+applies only to those values of the variance for which there is a clear ramp.)
+The basic message we extract from these calculations is that the spectrum on a cut-off geometry without a horizon
+is essentially a hairy SHO spectrum. When we have a horizon on the other hand, the spectrum is not that of an SHO
+in any sense (as we saw in previous sections). Together with the striking linearity of the ramp, we are therefore lead
+to conclude that the physics in the latter case is not simply due to nearest-neighbor physics.
+We conclude this section by providing some of the details of the flat space calculation. We will work with 2+1
+dimensions, the physics we aim for is unaffected by increase in dimensions:
+ds2 = −dt2 + dr2 + r2dψ2
+(36)
+Separating the scalar field as (say) in the BTZ case, we find the radial part
+φ
+′′
+ω,J(r) + 1
+r φ
+′
+ω,J(r) + ω2φω,J(r) − V (r)φω,J(r) = 0
+(37)
+with
+V (r) = 1
+r2
+�
+J2 + m2�
+.
+(38)
+We will consider the solution of this equation (37) in the massless case, which is given in terms of Bessel functions:
+φ(r) = C1JJ(ωr) + C2YJ(ωr),
+(39)
+where, JJ and YJ are Bessel functions of first and second kind respectively. We suppress the J and ω (or n) subscripts
+of C1 and C2.
+As before, we need one boundary condition to fix a relationship between C1 and C2 and another condition at a
+cut-off to fix the normal modes. The former role was played by AdS-normalizability in the BTZ case. We could
+likewise demand a suitably chosen bulk condition here as well that relates C1 and C2. By numerical experimentation
+we have found that the qualitative features of the ramp and LSD that we are after, are insensitive to this choice.
+This is unsurprising because the physics we are interested in, is the result of the quantization condition, and not the
+relationship between C1 and C2. In the following, we will simply demand that C2 = 0. Note that this sets the bulk
+source mode (which is singular at the origin) to zero, while retaining the homogeneous mode. It was noted in [40]
+that the bulk source mode is the analogue in flat space, to the non-normalizable mode in AdS. So this choice is a
+natural analogue of the normalizability demand in AdS. But we emphasize that large classes of choices are likely to
+give similar results.
+
+18
+Using this boundary condition, equation (39) becomes
+φ(r) = C1JJ(ωr).
+(40)
+Demanding a profile at the cut-off r = r0 leads to an equations analogous to what we found for BTZ: φ(r = r0) = C0.
+C1JJ(ωr0) = C0 =⇒ JJ(ωr0) = C0
+C1
+≡ λJ.
+(41)
+Note that we could also define the RHS to be ωλJ, which is more analogous to some of our discussions in BTZ and
+Rindler. But as we mentioned, these choices do not affect the semi-qualitative features we are after, so we will stick
+with this simple choice here for concreteness.
+We will take λJ to be Gaussian distributed with mean zero, and adjustable variance. The equation is easy to solve
+numerically, by taking the seed for the root search to be the 1st zero of the J-the Bessel function. When the variance
+is zero, we find an “extreme” delta-function like distribution in the LSD. The ramp of the SFF is not particularly
+well-defined, but we can already see a crude similarity to an SHO with a very small amount of noise – See Figure 14
+below.
+0.01
+1
+100
+104
+10-7
+10-5
+0.001
+0.100
+t
+g(t)
+β=0
+0.01
+1
+100
+104
+10-8
+10-5
+0.01
+t
+g(t)
+FIG. 14: Cut-off flat space with no variance vs SHO with a tiny amount of noise. The precise values are
+unimportant. Our goal here is not to make a detailed comparison, but to observe the crude similarity which
+becomes more striking as we increase the variance/noise, see Figure 12. The two lines are of slope ∼ 1.7 and ∼ 1.
+When we steadily add variance, we find more conventional level repulsion and the emergence of a robust ramp
+of slope ∼ 1.7, which we presented in Figure 12 left panel. As noted above, this is precisely what one finds from a
+noisy SHO as well. Eventually we find a Poisson distributed LSD. The (power law) ramp gets washed out, when the
+variance becomes very large. These features are identical to what we find in the hairy/noisy SHO case.
+To summarize – flat space with a cut-off is qualitatively identical to hairy SHO. Unlike in the case of the stretched
+horizon cut-off, the levels are essentially evenly spaced. We have done a similar calculation in empty AdS as well,
+as discussed in the main body of the paper, and the results are again consistent. These results mean that the linear
+ramp (which is often viewed as an indicator of strong chaos) does not arise from a cut-off in flat space. But for the
+same reason that a hairy SHO can mimic the LSD of an RMT (which in itself is a fact not emphasized previously in
+the literature, to our knowledge), the spectrum of cut-off flat space can also exhibit level repulsion – the variance in
+the boundary condition simply introduces a variance in the nearest neighbor levels. But this is not sufficient to create
+conventional spectral rigidity or robust chaos.
+A further distinction between empty space with cut-off and the stretched horizon is discussed in the next section.
+PLANCK-SCALE HIERARCHY
+We observed that the fluctuations at the cut-off in empty space translate to fluctuations in the energy levels and
+therefore lead to level repulsion. In other words, nearest neighbor effects of chaos can be produced simply by having
+
+19
+fluctuations at the cut-off. We also noted however that the linear ramp (which is a deeper signature of chaos) cannot
+be realized this way, and requires the presence of a horizon.
+In fact there is another interesting distinction between the stretched horizon and a cut-off in empty space. This
+has to do with the fact that the fluctuations at the cut-off needed in the stretched horizon scenario are hierarchically
+suppressed, allowing the interpretation that they are Planck-scale. The fluctuations in the empty space cut-off on the
+other hand are naturally macroscopic. To see this, first note that in (41), the first zero of the J-th Bessel function
+is linearly spaced in J with the scale controlled by r0. The natural scale controlling the fluctuations in the RHS is
+therefore r0 (this dependence is approximately linear if we define the RHS of (41) to be ωλJ instead of λJ). On the
+other hand in the horizon case, the situation is more interesting. To see this in detail, let us work with the concrete
+case of BTZ, and observe that the conventional tortoise coordinate here is defined via
+z = L2
+2 rh
+ln
+�r + rh
+r − rh
+�
+(42)
+This means that the usual radial coordinate of the stretched horizon x ≡ r − rh is approximately
+x = 2 rhe−2rhz/L2,
+(43)
+from which it follows that the fluctuation in the stretched horizon location goes as
+|∆x| ∼ 4 (rh/L)2 e−2rhz/L2|∆z|
+(44)
+where we have instated a magnitude sign because z → ∞ corresponds to the horizon. Now, from (21) it follows that
+e2λ = (x/rh) and therefore
+2 e2λ∆λ = ∆x
+rh
+=⇒ 2 x ∆λ ∼ ∆x.
+(45)
+Using (43) and (44) in this final relation, we get
+L2
+rh
+|∆λ| = |∆z|.
+(46)
+Since the horizon size and AdS length scale are both macroscopic, this means that the fluctuations in λ are naturally
+in tortoise coordinate, implying via (44) that the stretched horizon fluctuations are suppressed by a factor of
+e−2rhz0/L2
+(47)
+where z0 is the mean stretched horizon in tortoise coordinate. We also see that L2/z0 is a natural candidate for the
+Planck length. In units where L = 1, note that this is a small quantity because z0 is very large when the cut-off
+is close to the horizon. Of course, since we are working with a fixed background, these are all somewhat heuristic
+statements.
+To summarize: The variance in both cases (with and without horizon) can be used as a heuristic proxy for fluctu-
+ations of the cut-off surface. But a key distinction in the stretched horizon is that there, the variance captures the
+tortoise coordinate and therefore the fluctuations can naturally be viewed as Planck suppressed.
+OPEN QUESTIONS AND FUTURE DIRECTIONS
+In this section, we discuss some questions that are worth understanding better in the wake of our results. Some of
+these are more conceptual than others.
+• Are there more natural choices for the profile functions? We have considered the most simple-minded notion of
+a “generic” profile – choose some randomly distributed Fourier coefficients. The BPS fuzzball profiles, at least
+in the 2-charge case [14, 16] are known to contain enough phase space to reproduce the entropy of the black
+hole. This suggests that perhaps Haar typicality in some form is a better notion of genericity than our present
+proposal. It will be interesting to incorporate this in some systematic way.
+
+20
+• Despite the simplicity of our calculation, we have managed to find a linear ramp with fluctuations and level
+repulsion in (a heuristic candidate for) a single microstate. The price we have paid is that we have sacrificed a
+(manifestly) smooth horizon. But the emergence of RMT behavior in our calculation suggests that thermality
+(and therefore smoothness) may emerge via a suitable ensemble replacement of the microstate. Understanding
+this operationally is clearly a problem of outstanding interest.
+• In our previous paper [5], the LSD was not one of the conventional RMT distributions, but there was a clear
+linear ramp. Our main point in that paper was that this is a generic feature of normal modes at stretched
+horizons, when the boundary condition φ = 0 was imposed. In this paper, we have seen systems which exhibit
+the opposite behavior – The ramp is non-linear, but one has level spacing that matches well with conventional
+Wigner-Dyson-like statistics. In fact, we noticed that the latter can be arranged very simply via an SHO with a
+noisy spectrum. Together the results of these papers are a very clear demonstration that the folk wisdom that the
+linear ramp is a smoking gun of conventional Wigner-Dyson classes (or their Altland-Zernbauer generalizations)
+is not always true. It will be good to understand the broader setting in which these features arise as special
+cases.
+• We did not have to introduce any form of ensemble average.
+Our profile curve is chosen via a Gaussian
+distribution in the Fourier coefficients, but it should be emphasized that once the curve is chosen, there is
+absolutely nothing “averaged” about the calculation. The emergence of RMT behavior is entirely deterministic.
+It has been suggested in [41] that semi-classical gravity should be viewed as a tool for capturing ergodic averaged
+gravitational dynamics, for evolution that is in bulk local equilibrium. This would give an understanding of the
+surprising utility of Euclidean gravity in each epoch of Hawking radiation in obtaining the Page curve [11]. It
+will be very interesting to connect these two perspectives.
+• In [5] we had observed that there was a kink-like structure at the top of the ramp. A tangential consequence of
+the calculations in the present paper is that we have understood that this kink becomes less and less prominent,
+as we bring the stretched horizon closer and closer to the horizon. This is a strong indication that one of the
+worries expressed in [5] – that the ramp may be an artefact – is very unlikely to be true.
+• Inspired by the results of this paper and [5], we have been able to identify a broader class of spectra which lead
+to interesting ramps and level spacing structures. These results together suggest the notion of a generalized
+RMT spectrum, which will be elaborated elsewhere [30]. A key message is that boundary conditions are often
+a crucial ingredient in quantum chaos. This is true in our black hole problem, but note that the idea is much
+more general. Eg., the Hamiltonian of the hard sphere gas is simply that of a collection of free particles – it is
+the boundary conditions that breathe life (and chaos) into the system.
+• One of the technical features underlying the results of this paper and [5] is the observation that the dependence
+of the spectrum on the angular quantum numbers is not linear. Instead it gets pulled logarithmically along J.
+The resulting quasi-degeneracy was essential for our results. It will be good to get a more mechanical/conceptual
+understanding of this observation as well as to explore its consequences more broadly.
+• We found a clear ramp with slope ∼ 1.7 in our SFF plots for hairy SHO and cut-off flat space. This is an
+extremely simple example of a non-linear ramp, whose slope is a constant (̸= 1) in a log-log plot. It seems
+surprising and interesting that it is closely related to the SHO. Can this shed light on the fact that despite being
+the “ultimate” integrable system, the SHO exhibits an extreme version of level repulsion (ie., its LSD has no
+support at the origin, and has a delta function form)?
+• Relatedly, and more speculatively – does the fact that extreme WD spectra arise from Dirichlet boundary
+conditions at stretched horizons indicate that black holes are the “ultimate” RMT systems? If this is true, black
+holes can be viewed as the natural counterpoint to SHOs from our previous item. Note that the suggestion
+we are making here is distinct from the chaos bound of [8], which is about early time chaos and OTOCs. The
+observation about LSDs that we are making here is related to late time chaos. Black holes may not just be fast
+scramblers [6], they may also be the most robust scramblers. Clearly, more work remains to be done.
+

7dE0T4oBgHgl3EQfwQEF/content/tmp_files/2301.02628v1.pdf.txt

@@ -0,0 +1,1228 @@
+arXiv:2301.02628v1  [math.CO]  6 Jan 2023
+PINNACLE SETS OF SIGNED PERMUTATIONS
+NICOLLE GONZ´ALEZ, PAMELA E. HARRIS, GORDON ROJAS KIRBY, MARIANA SMIT VEGA GARCIA,
+AND BRIDGET EILEEN TENNER
+Abstract. Pinnacle sets record the values of the local maxima for a given family of permutations.
+They were introduced by Davis-Nelson-Petersen-Tenner as a dual concept to that of peaks, previ-
+ously defined by Billey-Burdzy-Sagan. In recent years pinnacles and admissible pinnacles sets for
+the type A symmetric group have been widely studied. In this article we define the pinnacle set
+of signed permutations of types B and D. We give a closed formula for the number of type B/D
+admissible pinnacle sets and answer several other related enumerative questions.
+1. Introduction
+The study of permutation statistics is an active subdiscipline of combinatorics. Given a per-
+mutation w = w(1)w(2) · · · w(n), two particularly well-studied statistics are descents and peaks.
+Respectively, these statistics refer to indices i such that w(i) > w(i + 1), and indices i such that
+w(i − 1) < w(i) > w(i + 1). The collection of a permutation’s descent indices is its descent set,
+with a permutation’s peak set being similarly defined. Two fundamental goals in the study of these
+particular statistics are (1) understanding which subsets can arise as descent sets or peak sets (i.e.,
+which sets are admissible as descent or peak sets), and (2) enumerating the permutations that have
+a given admissible descent or peak set.
+For descents of permutations in the (type A) symmetric group Sn, this question was answered
+by Stanley [15, Ex. 2.2.4] and is well known to give rise to the Eulerian numbers. Inspired by
+Stembridge’s study of peaks in the context of poset partitions [16], Billey, Burdzy, and Sagan [1]
+introduced the study of admissible peak sets for Sn with an interest in probabilistic applications, and
+established that the number of permutations with peak set I is given by 2n−|I|−1p(n), where p(n) is
+a polynomial of degree max(I) − 1. Shortly thereafter, their results were extended to permutations
+in type B by Castro-Velez et al. [2] where it was shown that the number of permutations with a
+given peak set I is 22n−|I|−1p(n), with p(n) the same as in [1] above. The second author and various
+collaborators went further by extending these results to types C and D [7], using peaks to study
+properties of the descent polynomial [6], and then initiating the study of peaks in the context of
+graphs [4].
+A notion that is closely related to peaks is the pinnacle set of a permutation. Pinnacles are the
+set of values held by the permutation at the peak indices. More precisely, given a permutation w =
+w(1)w(2) · · · w(n) with peak set Peak(w), the pinnacle set of w is Pin(w) = {w(i) : i ∈ Peak(w)}.
+Given a subset I ⊆ [n], if there exists a permutation w whose pinnacle set is I, we say that
+I is an admissible pinnacle set.
+In [3], Davis, Nelson, Petersen, and the last author pioneered
+the study of pinnacles for permutations in Sn and gave a complete characterization of admissible
+pinnacle sets. They provided a closed formula for the number of admissible pinnacle sets with a
+given maximum value, as well as a refinement to those appearing in Sn. In particular, Davis et
+Date: January 9, 2023.
+P.E.H. was partially supported through a Karen Uhlenbeck EDGE Fellowship.
+M.S.V.G was partially supported by the NSF grant DMS 2054282.
+B.E.T. was partially supported by the NSF grant DMS-2054436.
+1
+
+al. gave a recursive formula for the number of permutations in Sn with a given pinnacle set p(n)
+and asked whether a more efficient expression could be computed. This paper led to a sequence
+of articles in recent years, many focused on improved and faster formulas for p(n), by realizing
+permutations with given pinnacle sets as invariants under certain modified Sn-actions [5, 9] or via
+more traditional enumerative methods [8, 10, 11]. In related work, Rusu [13] and Rusu-Tenner [14]
+deepened the knowledge of pinnacles in Sn by investigating further properties of these statistics
+and characterizing admissible pinnacle orderings.
+In this article we look beyond type A and study pinnacles and admissible pinnacle sets for the
+type B and type D signed symmetric groups, SB
+n and SD
+n . Our main results are the following,
+where we write APSX
+n to denote the admissible pinnacle sets in SX
+n for X ∈ {A, B, D}:
+(1) Theorem 3.12 gives a closed formula for the number of admissible pinnacle sets in SB
+n ,
+|APSB
+n | =
+⌊ n−1
+2 ⌋
+�
+k=0
+�n
+k
+��n − 1 − k
+� n−1
+2
+�
+− k
+�
+.
+(2) Theorem 4.2 proves that any admissible pinnacle set in SB
+2k is also admissible in SD
+2k; that
+is, APSD
+2k = APSB
+2k.
+(3) In counterpoint to Theorem 4.2, Theorem 4.9 counts the admissible pinnacle sets of type
+B that are not in type D when n = 2k + 1,
+|APSB
+2k+1 \ APSD
+2k+1| =
+�2k − 1
+k
+�
+.
+(4) Theorems 4.11 and 4.12 count the all-positive admissible pinnacle sets of type B that are
+not admissible in type A. Namely, defining APS+
+n := {S ∈ APSB
+n : S ⊂ N}; we prove that
+the sets APS+
+n \ APSn are enumerated by,
+��APS+
+n \ APSn
+�� =
+�
+4k −
+�2k
+k
+�
+if n = 2k + 1, and
+22k−1 −
+�2k
+k
+�
+if n = 2k.
+This article is organized as follows. In Section 2, we introduce all the necessary background and
+notation, defining pinnacles and related notions in type B. In Section 3, we give a characterization
+of admissible signed pinnacle sets and a formula for their enumeration. In Section 4, we provide
+relations between admissible pinnacle sets of type A, B, and D. Lastly, in Section 5, we describe
+some future directions and open conjectures.
+Acknowledgements. The authors thank Patrek K´arason Ragnarsson for the coding and data
+that facilitated the research in this project, and Freyja K´arad´ottir Ragnarsson for the key insight
+to the proof of Theorem 4.9. The authors also thank the American Institute of Mathematics and
+the National Science Foundation for sponsoring the Latinx Mathematicians Research Community,
+which brought together a subset of the authors initially for collaboration.
+2. Background
+Let N = {1, 2, 3, . . .} and for n ∈ N we write [n] := {1, 2, . . . , n}. For any set X, typically of
+positive values, although we make the definition more generally, we define
+−X := {−x : x ∈ X}.
+Finally, we define
+±X = X ∪ −X.
+2
+
+Throughout this paper, we let Sn denote the (type A) symmetric group. That is, Sn is the group
+of bijections from [n] → [n] under function composition. We often write w ∈ Sn using one-line
+notation, as w = w(1)w(2) · · · w(n).
+The type B symmetric group (that is, the hyperoctahedral group) is the group of signed
+permutations SB
+n . These are bijections ±[n] → ±[n] such that
+w(−i) = −w(i) for all i ∈ [n].
+In particular, any w ∈ SB
+n satisfies the property that {|w(1)|, . . . , |w(n)|} = [n].
+The type D symmetric group is the subgroup SD
+n of SB
+n consisting of signed permutations with
+an even number of signs. Namely, these are the signed permutations w for which
+|{i ∈ [n] : w(i) < 0}| is even.
+As in type A, we use one-line notation to denote signed permutations w ∈ SB
+n , where we may
+write only w = w(1)w(2) · · · w(n) since this uniquely determines w(−i) for all positive i. Following
+convention, we write −i = ¯i to ease notation. For example, w = ¯12¯3 is the signed permutation with
+w(1) = −1, w(2) = 2, and w(3) = −3.
+Recall that a permutation w ∈ Sn has a peak at index i ∈ {2, . . . , n − 1} if
+w(i − 1) < w(i) > w(i + 1),
+and the value w(i) is a pinnacle of w. We denote by Peak(w) the set of all peaks of w ∈ Sn. The
+pinnacle set of w ∈ Sn is
+Pin(w) = {w(i) : i ∈ Peak(w)}.
+Definition 2.1. A set P ⊆ [n] is an n-admissible pinnacle set in type A if there exists a permutation
+w ∈ Sn such that Pin(w) = P, and we call the permutation w a witness for the set P.
+For example, the identity permutation is a witness for the admissible pinnacle set ∅ (as is any
+peak-less permutation). Denote by APSn the set of all n-admissible pinnacle sets in type A.
+In order to facilitate our discussions about pinnacles, we introduce terminology about their
+minimal counterparts: a permutation w ∈ Sn has a valley at index i ∈ {2, . . . , n − 1} if
+w(i − 1) > w(i) < w(i + 1),
+and the value w(i) is a vale of w.
+1
+2
+3
+4
+5
+6
+7
+8
+1
+2
+3
+4
+5
+6
+7
+8
+•
+•
+•
+•
+•
+•
+•
+•
+Figure 1. The graph of the permutation 23715648 ∈ S8 with the pinnacles/peaks
+circled in red and the vales/valleys in blue.
+Example 2.2. Consider the permutation w = 23715648 ∈ S8 shown in Figure 1.
+We have
+Peak(w) = {3, 6} and Pin(w) = {6, 7}, and valleys and vales {4, 7} and {1, 4}, respectively.
+3
+
+2.1. Pinnacles in types B and D. Pinnacles were defined in [3] for unsigned permutations, but
+they could just as easily have been defined for signed permutations—or, in fact, for arbitrary strings
+of distinct real numbers. We now expand the type A definitions to type B, and note that since
+SD
+n ⊂ SB
+n , these definitions also hold for type D.
+Definition 2.3. Let w be a signed permutation. A pinnacle of w is a value w(i) that is larger than
+both w(i − 1) and w(i + 1). The pinnacle set of w is the set of its pinnacles.
+In order to define admissible pinnacle sets, it is important to establish which subsets could even
+appear among the one-line notation of a signed permutation.
+Definition 2.4. A signed set (or signed subset, depending on context) is a set I such that x ∈ I
+implies −x ̸∈ I.
+Throughout this paper, we assume that all subsets of ±[n] are signed subsets.
+Definition 2.5. A signed subset S ⊂ ±[n] is an admissible pinnacle set if S is the pinnacle set of
+some signed permutation. That permutation is a witness for S.
+Note that when we study sets that are admissible as pinnacle sets in type D, any witness
+permutation will be required to be in SD
+n for some n.
+As before, we denote by APSB
+n (resp.,
+APSD
+n ) the set of all n-admissible pinnacle sets in type B (resp., type D). Once again, we have
+∅ ∈ APSD
+n ⊆ APSB
+n . For example, 123 · · · n and ¯2¯134 · · · n are both witnesses for ∅.
+While there can be multiple witness permutations for a given admissible pinnacle set, we will
+often refer to a particular witness permutation that we call “canonical.”
+Definition 2.6. For S ∈ APSB
+n , write S = {s1 < s2 < · · · < sk}, and set
+S′ := −[n] \ {−|s| : s ∈ S} = {s′
+1 < s′
+2 < · · · < s′
+n−k}.
+Then the canonical witness permutation is
+w := s′
+1 s1 s′
+2 s2 · · · s′
+k sk s′
+k+1 · · · s′
+n−k ∈ SB
+n .
+If S ∈ APSD
+n , then its canonical (type D) witness permutation is w as defined above if w is in SD
+n ,
+and otherwise its canonical witness is obtained from w by replacing s′
+n−k with |s′
+n−k|.
+Next we establish that the “canonical witness permutations” are, in fact, witnesses and follow
+this by providing canonical witness permutations in Example 2.8.
+Lemma 2.7. The canonical witness permutation for an admissible set S is a witness for S.
+Proof. The set S is admissible, so there is some permutation whose pinnacle set is S. The canonical
+witness has been constructed to have minimal possible non-pinnacle values, and to position the
+smallest non-pinnacle values beside the smallest pinnacle values. Therefore, if any permutations
+were to have S as a pinnacle set (and we know that some permutation does), the permutation given
+in Definition 2.6 would be among them.
+□
+Although SB
+n contains both Sn and SD
+n as subgroups, there are interesting subtleties to the
+pinnacle sets that become admissible when witness permutations can be signed. First, some sets
+will be admissible with type B permutations, but not with type D permutations. And second,
+some sets of all-positive values will be admissible with type B permutations, but not with type A
+(unsigned) permutations. We demonstrate each of these scenarios below.
+4
+
+(a)
+1
+2
+3
+4
+5
+6
+7
+1
+2
+3
+4
+5
+6
+7
+0
+−1
+−2
+−3
+−4
+−5
+−6
+−7
+•
+•
+•
+•
+•
+•
+•
+(b)
+(b)
+1
+2
+3
+4
+5
+6
+7
+1
+2
+3
+4
+5
+6
+7
+0
+−1
+−2
+−3
+−4
+−5
+−6
+−7
+•
+•
+•
+•
+•
+•
+•
+Figure 2. (a) The graph of the permutation ¯7¯4¯61¯52¯3 ∈ SB
+7
+with the pinna-
+cles/peaks circled in red and the vales/valleys in blue. (b) The graph of the permu-
+tation ¯63¯54¯17¯2 ∈ SB
+7 with the pinnacles/peaks circled in red and the vales/valleys
+in blue.
+Example 2.8. The set {¯4, 1, 2} is admissible in SB
+7 , with canonical witness permutation ¯7¯4¯61¯52¯3
+as shown in Figure 2(a).
+However, there is no element of SD
+7 having this pinnacle set.
+That
+is, {¯4, 1, 2} ̸∈ APSD
+7 . The set {3, 4, 7} is admissible in SB
+7 , with canonical witness permutation
+¯63¯54¯17¯2, as shown in Figure 2(b). However, despite its pinnacle set having all positive values, there
+is no type A permutation having this pinnacle set. That is, {3, 4, 7} ̸∈ APSn for any n.
+3. Admissible signed pinnacle sets in type B
+In this section, we characterize and enumerate the admissible pinnacle sets among signed
+permutations. This expands on the work begun in [3] for unsigned permutations, but, as we show,
+the results for signed permutations are subtly different from those in type A.
+3.1. Characterization of admissible pinnacle sets. For the remainder of the article, we will
+often use the fact that given an admissible pinnacle set S ∈ APSB
+n , we can always write
+S = P(S) ⊔ N(S)
+with
+P(S) := S ∩ [n] and N(S) := S ∩ −[n].
+When no confusion will arise, we simply write P := P(S) and N := N(S).
+To give a first inkling of how admissible pinnacle sets in type B are fundamentally different
+from those in type A, we note that there are some sets of positive integers that are never in APSn
+for any n. For example, any set containing 1 or 2 will never be the pinnacle set of any permutation
+in Sn. On the other hand, such a statement is not true in type B.
+Lemma 3.1. Every finite signed subset S is admissible in SB
+n , for some n ∈ N. That is, there
+exists w ∈ SB
+n such that S = PinB(w).
+5
+
+Proof. Write S = {s1 < · · · < sk}. Let m = max{|s| : s ∈ S} (that is, m = max{|s1|, |sk|}). Define
+the set S′ := −[2m + 1] \ {−|s| : s ∈ S}, which we write as S′ = {s′
+1 < · · · < s′
+2m+1−k}. Then
+w = s′
+1 s1 s′
+2 s2 · · · s′
+k sk s′
+k+1 s′
+k+2 · · · s′
+2m+1−k ∈ SB
+2m+1,
+and PinB(w) = S.
+□
+Using a similar argument as the one proving Lemma 3.1, it follows that any finite set of all
+positive values is admissible in some SB
+n .
+Corollary 3.2. Any subset P ⊂ [n] with |P| ≤ n−1
+2
+is admissible in SB
+n .
+Proof. Let P = {p1 < · · · < pk}, and set P ′ := −([n] \ P) = {p′
+1 < · · · < p′
+n−k}. Then
+w = p′
+1 p1 p′
+2 p2 · · · p′
+k pk p′
+k+1 p′
+k+2 · · · p′
+n−k ∈ SB
+n
+and PinB(w) = P.
+□
+This can be particularly interesting when the set P was not admissible in Sn.
+Example 3.3. Consider P = {1, 2} with n = 5.
+The permutation ¯51¯42¯3 ∈ SB
+5 is a witness
+permutation for P, so P ∈ APSB
+5 , while P ̸∈ APSn for any n.
+Our goal is to establish a characterization and formula for the number of admissible pinnacle
+sets in SB
+n . We begin with some preliminary steps, from which those results will follow. The first
+of these is a bijection between admissible pinnacle sets in Sn and those admissible pinnacle sets in
+SB
+n that have no positive values.
+Lemma 3.4. There exists a bijection between APSn and {S ∈ APSB
+n : S ⊂ −N}.
+Proof. Given T ∈ APSn, define T ′ := {t − (n + 1) : t ∈ T}. The set T ′ has no positive elements.
+Let w ∈ Sn be the canonical witness for T. Then w′ := (w(1) − (n + 1)) · · · (w(n) − (n + 1)) ∈ SB
+n
+has pinnacle set T ′, and so T ′ ∈ APSB
+n .
+This process can be inverted: given S ∈ APSB
+n with P(S) = ∅, map this S to S′ := {s + n + 1 :
+s ∈ S}. It follows that S′ ∈ APSn, as before.
+□
+We illustrate Lemma 3.4 with an example.
+Example 3.5. The set {3, 6, 7, 10} ∈ APS10 is in correspondence with {¯8, ¯5, ¯4, ¯1} ∈ APSB
+10. The
+permutations described in the proof of Lemma 3.4, which exhibit these sets as pinnacle sets, are
+shown in Figure 3.
+We have defined admissible pinnacle sets in types A, B, and D, referring to permutations in
+Sn, SB
+n , or SD
+n .
+However, as suggested earlier, there is a natural generalization of admissible
+pinnacle sets to permutations of any totally ordered set.
+Definition 3.6. For any totally ordered set X, let APS(X) be the set of admissible pinnacle sets
+of X. The definitions of witness and canonical witness permutations in this general setting are
+analogous to their definitions in the symmetric groups.
+Because they arise so often, we have been easing notation by writing APS(Sn) as APSn,
+APS(SB
+n ) as APSB
+n , and APS(SD
+n ) as APSD
+n .
+Example 3.7. The set X = {−2, π, 4, 5, 100} has six admissible pinnacle sets:
+∅, {4}, {5}, {100}, {4, 100}, and {5, 100}.
+6
+
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+−1
+−2
+−3
+−4
+−5
+−6
+−7
+−8
+−9
+−10
+0
+...
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+Figure 3. The left-hand figure shows the canonical witness for {3, 6, 7, 10} in S10.
+The right-hand figure shows the corresponding witness permutation for {¯8, ¯5, ¯4, ¯1},
+as defined in the proof of Lemma 3.4.
+Note that if we are only interested in how many admissible pinnacle sets X has, as opposed to
+the sets themselves, then the size of X is what matters.
+Lemma 3.8. For any totally ordered finite set X, |APS(X)| = |APS ([|X|]) | = |APS|X||.
+This calculation will be useful in the enumeration appearing in the next subsection.
+We are now are able to characterize admissible pinnacle sets for signed permutations.
+Theorem 3.9. The sets in APSB
+n are exactly the sets S = P(S) ⊔ N(S) for which
+• |P(S)| + |N(S)| ≤ (n − 1)/2,
+• P(S) ⊂ [n],
+• N(S) ⊂ −([n] \ P(S)), and
+• N(S) ∈ APS(−([n] \ P(S))).
+Proof. First of all, it is clear that any admissible pinnacle set in SB
+n must satisfy the listed require-
+ments.
+Now suppose that a set S satisfies the listed requirements, with P := P(S) = {p1 < · · · < pk}
+and N := N(S) = {n1 < · · · < nr}. In light of the last requirement, let w be the canonical witness
+permutation of the set (−([n] \ P)), having pinnacle set N. That is,
+w = i1 n1 i2 n2 · · · ir nr ir+1 ir+2 ir+3 · · · in−k−r
+where ij < ij+1 and {i1, . . . , in−k−r} = −([n] \ P) \ N. Then
+i1 n1 i2 n2 · · · ir nr ir+1 p1 ir+2 p2 ir+3 · · · pk ir+k+1 ir+k+2 · · · in−r−k
+is a canonical witness for S = P ⊔ N in SB
+n . Hence S ∈ APSB
+n .
+□
+3.2. Enumeration of admissible pinnacle sets. The conditions listed in Theorem 3.9 inform
+our enumeration of the admissible pinnacle sets in SB
+n . In particular, we will construct these sets by
+7
+
+first fixing a collection P of positive pinnacles and then determining how many sets N of negative
+pinnacles exist for which P ∪ N is admissible in SB
+n .
+In order not to have too many pinnacles (that is, not more than ⌊(n − 1)/2⌋), we need to
+understand the following value.
+Definition 3.10. Let pn(d) be the number of admissible pinnacle sets in Sn having cardinality at
+most d. That is,
+pn(d) := |{S ∈ APSn : |S| ≤ d}|.
+This statistic has a particularly nice formula.
+Proposition 3.11. For all integers d ∈ [0, ⌊(n − 1)/2⌋],
+pn(d) =
+�n − 1
+d
+�
+.
+Proof. The admissible pinnacle sets in Sn having cardinality at most d can be partitioned into
+two sets: those that contain n, and those that do not. We claim that the first set is counted by
+pn−1(d − 1), and the second set is counted by pn−1(d).
+Suppose, first, that S ∈ APSn such that n ∈ S and |S| = k ≤ d. Let w ∈ Sn be the canonical
+witness for S. Deleting n from the one-line notation of w will produce a permutation v ∈ Sn−1
+with Pin(v) = S \ {n}. Conversely, given T ∈ APSn−1 with |T| = k − 1, let u ∈ Sn−1 be the
+canonical witness for T. Inserting n between the non-pinnacles u(2k − 1) and u(2k) will produce a
+permutation in Sn whose pinnacle set is T ∪ {n}. This establishes the first part of the claim.
+For the second part of the claim, suppose that S ∈ APSn with n ̸∈ S and |S| = k ≤ d. Let
+w ∈ Sn be the canonical witness for S. Because n ̸∈ S, we have w(n) = n. Thus the permutation
+w(1) · · · w(n − 1) ∈ Sn−1 has pinnacle set S. Conversely, if v ∈ Sn−1 has pinnacle set S, then
+appending n to the end of v will produce a permutation in Sn that also has pinnacle set S.
+This gives the binomial recurrence
+pn(d) = pn−1(d − 1) + pn−1(d).
+To complete the argument, notice that pn(0) = 1 and pn(1) = 1 + (n − 2) = n − 1, for all positive
+integers n.
+□
+Combining Theorem 3.9, which characterizes admissible pinnacle sets for signed permutations,
+with the enumeration in Proposition 3.11, we now count the admissible pinnacle sets for signed
+permutations.
+Theorem 3.12. If n ≥ 2, then
+��APSB
+n
+�� =
+⌊ n−1
+2 ⌋
+�
+k=0
+�n
+k
+�� n − 1 − k
+� n−1
+2
+�
+− k
+�
+.
+Proof. The main idea of the proof will be to construct admissible pinnacle sets in SB
+n following the
+requirements of Theorem 3.9. First, we will select a set P of positive pinnacles. In other words,
+P ⊂ [n] and |P| ≤ (n − 1)/2. Then we add to it any set N ⊂ −([n] \ P) that is in APS(−([n] \ P)),
+so long as |P| + |N| ≤ (n − 1)/2. We are interested in the number of such sets, and Lemma 3.8
+says that we only need to care about the size of P in this process. This and Lemma 3.4 mean that
+such sets N can be counted in terms of admissible pinnacle sets of Sn−|P |.
+8
+
+Fix an integer k ∈ [0, (n − 1)/2], and choose a k-element subset P ⊂ [n]. There are
+�n
+k
+�
+ways
+to do this. We can supplement P with any r-element admissible pinnacle set N ⊂ −([n] \ P), as
+long as k + r ≤ ⌊(n − 1)/2⌋. The number of ways to do this is
+pn−k
+��n − 1
+2
+�
+− k
+�
+.
+Therefore, by Proposition 3.11, the number of admissible pinnacle sets in SB
+n is
+⌊ n−1
+2 ⌋
+�
+k=0
+�n
+k
+�� n − 1 − k
+� n−1
+2
+�
+− k
+�
+,
+as desired.
+□
+In Table 1, we give the number of signed admissible pinnacle sets in type B for 3 ≤ n ≤ 15,
+while permutations in SB
+1 and SB
+2 have no pinnacles. This appears in the OEIS as sequence [12,
+A359066]. The even-indexed terms in the table appear in [12, A240721] and the odd-indexed terms
+appear in [12, A178792].
+n
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+��APSB
+n
+��
+5
+7
+31
+49
+209
+351
+1471
+2561
+10625
+18943
+78079
+141569
+580865
+Table 1. The number of admissible pinnacle sets in SB
+n , for 3 ≤ n ≤ 15.
+In the next Section, we will be able to answer the analogous enumerative question in type D
+(see Corollary 4.10).
+4. Relating admissible pinnacle sets in types A, B, and D
+There is a natural embedding of Sn in SD
+n , and of SD
+n in SB
+n . Having spent Section 3 analyzing
+pinnacle sets that are admissible in SB
+n , it is natural to wonder how these sets are related to those
+that are admissible in SD
+n or, for those elements of APSB
+n without negative values, to those that
+are admissible in Sn. We now give complete characterization of each of these relationships.
+4.1. Comparing admissible pinnacle sets in types B and D. As mentioned before, SD
+n ⊂ SB
+n ,
+thus it is natural to investigate the relationship between those sets that are admissible as pinnacle
+sets in type B and those that are in type D. It is, perhaps, not surprising that this relationship
+depends on the parity of n.
+As a first step in this analysis, we identify a technique that will be handy in proving that a set
+is admissible for type D.
+Lemma 4.1. Suppose that w ∈ SB
+n is a witness for a pinnacle set S. If w(n − 1) > ±w(n) or if
+w(n − 1) < ±w(n), then the permutation w′, defined by
+w′(i) =
+�
+w(i)
+i < n and
+−w(i)
+i = n,
+is also a witness for S. Moreover, S ∈ APSD
+n .
+9
+
+Proof. First note that w′ is an element of SB
+n because changing the sign of the last letter does not
+alter the fact that this is a signed permutation on ±[n]. Next observe that the pinnacle set has
+not changed from w to w′ because none of the inequalities between consecutive letters has been
+altered. Finally, note that the numbers of negative values in w and in w′ differ by 1, meaning that
+one of these permutations is in SD
+n while the other is in SB
+n \ SD
+n .
+□
+We will call on the previous result often throughout our arguments in this section, beginning
+with a description of the simple relationship between APSB
+n and APSD
+n .
+Theorem 4.2. For k ≥ 1, APSB
+2k = APSD
+2k.
+Proof. Certainly anything admissible in type D is also admissible in type B, because signed per-
+mutations include the signed permutations in type D. It remains to show that any pinnacle set
+that is admissible in SB
+2k is also admissible in SD
+2k. Fix S := {s1 < · · · < sl} ∈ APSB
+2k. Because
+l ≤ ⌊(2k − 1)/2⌋, we have l ≤ k − 1. Then the canonical witness w for S satisfies the hypotheses of
+Lemma 4.1, and so in fact S ∈ APSD
+2k.
+□
+The equality shown in Theorem 4.2 relies on the fact that there are always at least two more
+non-pinnacles than there are pinnacles in signed permutations on 2k letters. This not necessarily
+true for signed permutations of an odd number of letters, and hence it is not surprising that the
+relationship between APSB
+2k+1 and APSD
+2k+1 has more nuance than the relationship presented in
+Theorem 4.2. Indeed, we will show that APSD
+2k+1 is a strict subset of APSB
+2k+1, and we will describe
+the elements of the latter that are not elements of the former.
+Lemma 4.3. If S ∈ APSB
+2k+1 \ APSD
+2k+1, then |S| = k.
+Proof. Fix S ∈ APSB
+2k+1 and let w ∈ SB
+2k+1 be the canonical witness for S. If |S| < k, then both
+w(2k) and w(2k + 1) are non-pinnacles and w(2k) < w(2k + 1) < 0. In particular, the hypotheses
+of Lemma 4.1 are satisfied by w, and so S ∈ APSD
+2k+1. Hence, if S ∈ APSB
+2k+1 \ APSD
+2k+1, then
+|S| = k.
+□
+One implication of Lemma 4.3 is that if w ∈ SB
+2k+1 is a witness for S ∈ APSB
+2k+1\APSD
+2k+1, then
+w(3), w(5), . . . , w(2k−1) are all vales. With Lemma 4.3 providing a first step toward understanding
+elements of APSB
+2k+1 \ APSD
+2k+1, we now proceed to describe these sets more clearly.
+Lemma 4.4. Fix S ∈ APSB
+2k+1 \ APSD
+2k+1. In every witness permutation for S, the non-pinnacle
+values are all negative.
+Proof. Fix S ∈ APSB
+2k+1 \ APSD
+2k+1 and w ∈ SB
+2k+1 a witness for S. Following Lemma 4.3, the non-
+pinnacles of w are precisely w(1), w(3), . . . , w(2k + 1). In particular, each w(2i + 1) is less than its
+immediate neighbors. Suppose, for the purpose of obtaining a contradiction, that w(2j +1) > 0 for
+some j. Let w′ ∈ SB
+2k+1 be the permutation obtained from w by replacing w(2j +1) by −w(2j +1).
+Then w′ is still a witness for S. Either w or w′ is in SD
+2k+1, meaning that S must be an element of
+APSD
+2k+1. This is a contradiction, so there is no such j.
+□
+In fact, the negative values of S ∈ APSB
+2k+1 \ APSD
+2k+1 are enough to determine all of S.
+Lemma 4.5. Suppose that S ∈ APSB
+2k+1 \ APSD
+2k+1, with P := S ∩ N and N := S ∩ −N. Then the
+elements of P are the smallest k − |N| values in the set [2k + 1] \ −N. In particular, N determines
+P, and hence all of S.
+10
+
+Proof. Fix S ∈ APSB
+2k+1 \ APSD
+2k+1, with P and N as defined. By Lemma 4.3, we have |S| = k,
+so let S = {s1 < s2 < · · · < sk}. If |N| = k, then there is nothing to check, so assume that
+|N| < k and hence sk > 0. Suppose, for the purpose of obtaining a contradiction, that there exists
+q ∈ ([2k + 1] \ −N) \ P with q < sk. Let w be the canonical witness permutation for S. By
+definition, w(2k) = sk and w(2k + 1) = −q. But then w′, which agrees with w everywhere except
+w′(2k + 1) = q, is also a witness for S, contradicting Lemma 4.4. Therefore P consists precisely of
+the smallest k − |N| values in the set [2k + 1] \ −N.
+□
+Lemma 4.5 gives a necessary condition for elements of APSB
+2k+1 \ APSD
+2k+1. The next result
+establishes that the set N ⊔ P constructed in Lemma 4.5 is, in fact, an admissible signed pinnacle
+set.
+Corollary 4.6. Suppose that N ⊂ −N and N ∈ APSB
+2k+1. Let P be the smallest k − |N| values in
+[2k + 1] \ −N. Then N ⊔ P ∈ APSB
+2k+1.
+Proof. This follows from Theorem 3.9.
+□
+Maintaining the terminology of Corollary 4.6, note that for any set N ⊂ −N, all witness
+permutations for N ⊔ P are forced by construction of P to have the same number of negative
+values: k + 1 + |N|. This yields the following corollary.
+Corollary 4.7. Suppose S ∈ APSB
+2k+1 \ APSD
+2k+1, with N := S ∩ −N. The sets |N| and |S| have
+the same parity.
+Proof. To have S ∈ APSB
+2k+1 \ APSD
+2k+1, we need |S| = k, by Lemma 4.3. Moreover, as discussed
+above, the number of negative values is k + 1 + |N|, and this must be odd because S /∈ APSD
+2k+1.
+Thus k + |N| = |S| + |N| is even, completing the proof.
+□
+The consequence of this collection of results is that if we have a set N ⊂ −N that is, itself,
+admissible in SB
+2k+1, and for which |N| has the same parity as k, then there is a unique ((k − |N|)-
+element) set P ⊂ N for which
+N ⊔ P ∈ APSB
+2k+1 \ APSD
+2k+1.
+Therefore, to enumerate APSB
+2k+1 \ APSD
+2k+1, it suffices to count the elements of APSB
+2k+1 that have
+no positive values and that have size of the form k − 2i.
+Because we want to look at the elements of APSB
+2k+1 having no positive values, we can take
+advantage of Lemma 3.4 to look, instead, at APS2k+1. That is, it will suffice to count
+�
+i≥0
+����{S ∈ APS2k+1 : |S| = k − 2i}
+����.
+The last step of this enumeration requires a parity result.
+Lemma 4.8. For k ≥ 0,
+����{S ∈ APS2k+1 : |S| is even}
+���� =
+����{S ∈ APS2k+1 : |S| is odd}
+����.
+Proof. Fix S ⊂ [2k + 1]. If 2k + 1 ∈ S, then set S′ := S \ {2k + 1}. Clearly if S ∈ APS2k+1 then
+also S′ ∈ APS2k+1, and the sets |S| and |S′| have different parities.
+Now consider S ∈ APS2k+1 with 2k + 1 ̸∈ S. By [3, Theorem 1.8], max(S) > 2|S|. We have
+max(S) < 2k + 1, so |S| < k. Consequently, S has a witness permutation w using at most k
+vales, so there are at least (2k + 1) − (k − 1 + k) = 2 non-pinnacle/non-vale values in this witness
+11
+
+permutation, and one of these is 2k + 1. We can create a new permutation w′ by inserting 2k + 1
+immediately to the right of the largest vale in w. Thus the pinnacle set of w′ is S ∪ {2k + 1}.
+Therefore there is a bijection between even-sized elements of APS2k+1 and odd-sized ones,
+obtained by adding/removing the element 2k + 1. This partitions APS2k+1 into two evenly sized
+parts.
+□
+We have now completed all of the steps necessary to give the desired enumeration.
+Theorem 4.9. For k ≥ 1,
+��APSB
+2k+1 \ APSD
+2k+1
+�� =
+�2k − 1
+k
+�
+.
+Proof. Following Lemmas 4.3 and 4.5 and Corollary 4.7, we can enumerate APSB
+2k+1 \ APSD
+2k+1 by
+counting elements of APS2k+1 that have size {k − 2i : i = 0, 1, . . .}. These are either all of the
+odd-sized sets in APS2k+1 or all of the even-sized ones. By Lemma 4.8, then,
+��APSB
+2k+1 \ APSD
+2k+1
+�� = 1
+2 |APS2k+1| .
+It was shown in [3, Theorem 1.8] that |APS2k+1| =
+�2k
+k
+�
+. Finally, it is straightforward to check that
+1
+2
+�2k
+k
+�
+=
+�2k−1
+k
+�
+.
+□
+We can now use Theorem 3.12, which enumerated APSB
+n , and Theorems 4.2 and 4.9 to enu-
+merate APSD
+n for all n.
+Corollary 4.10. For k ≥ 1,
+��APSD
+2k
+�� =
+��APSB
+2k
+�� and
+��APSD
+2k+1
+�� =
+� k
+�
+i=0
+�2k + 1
+i
+��2k − i
+k − i
+��
+−
+�2k − 1
+k
+�
+.
+In Table 2, we give the number of signed admissible pinnacle sets in type D for 3 ≤ n ≤ 15,
+while permutations in SD
+1 and SD
+2 have no pinnacles.
+This appears in the OEIS as sequence
+A359067.
+The even-indexed terms are identical to even terms in Table 1 and the odd-indexed
+terms are
+�2k−1
+k
+�
+less than the corresponding odd-indexed terms in Table 1.
+n
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+��APSD
+n
+��
+4
+7
+28
+49
+199
+351
+1436
+2561
+10499
+18943
+77617
+141569
+579149
+Table 2. The number of admissible pinnacle sets in SD
+n , for 3 ≤ n ≤ 15.
+4.2. Comparing admissible pinnacle sets in types B and A. Some elements of APSB
+n have
+no negative values, and so one could ask if those sets might also be admissible in Sn. In this section
+we consider how those elements of APSB
+n are related to the admissible pinnacle sets in APSn. To
+make this discussion precise, we introduce:
+APS+
+n := {S ∈ APSB
+n : S ⊂ N};
+in other word, APS+
+n consists of the pinnacle sets that are admissible in SB
+n and that contain no
+negative values.
+12
+
+For example, {1, 3} ∈ APS+
+5 , with canonical witness 51432 ∈ SB
+5 . In fact, by Corollary 3.2,
+any subset of [n] having at most (n − 1)/2 elements is admissible in SB
+n . Contrast this with APSn;
+for example,
+APS+
+5 \ APS5 =
+�
+{1}, {2}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}
+�
+.
+Our goal in this section is to understand APS+
+n \ APSn. As with the comparison of APSB
+n and
+APSD
+n , this will depend on the parity of n.
+Theorem 4.11. For k ≥ 0, |APS+
+2k+1 \ APS2k+1| = 4k −
+�2k
+k
+�
+.
+Proof. Because APS2k+1 ⊂ APS+
+2k+1, the desired value is equal to
+��APS+
+2k+1
+�� − |APS2k+1| .
+Following Corollary 3.2, we can compute
+��APS+
+2k+1
+�� by counting i-element subsets of [2k +1] for all
+i ≤ k. The result follows by recognizing that this yields a sum that is half of a row-sum of Pascal’s
+triangle, and combining this with the enumeration of APS2k+1 from [3]:
+|APS+
+2k+1 \ APS2k+1| =
+��APS+
+2k+1
+�� − |APS2k+1|
+=
+�2k + 1
+0
+�
++
+�2k + 1
+1
+�
++ · · · +
+�2k + 1
+k
+�
+−
+�2k
+k
+�
+= 1
+222k+1 −
+�2k
+k
+�
+= 4k −
+�2k
+k
+�
+.
+□
+We now complete this analysis by considering the even-indexed case.
+Theorem 4.12. For k ≥ 1,
+��APS+
+2k \ APS2k
+�� = 22k−1 −
+�2k
+k
+�
+.
+Proof. This calculation is almost identical to that from the proof of Theorem 4.11, except that we
+will also have to subtract the central term from a row of Pascal’s triangle:
+|APS+
+2k \ APS2k| =
+��APS+
+2k
+�� − |APS2k|
+=
+�2k
+0
+�
++
+�2k
+1
+�
++ · · · +
+� 2k
+k − 1
+�
+−
+�2k − 1
+k − 1
+�
+= 1
+2
+�
+22k −
+�2k
+k
+��
+−
+�2k − 1
+k − 1
+�
+= 22k−1 −
+�1
+2
+�2k
+k
+�
++
+�2k − 1
+k − 1
+��
+= 22k−1 −
+�2k
+k
+�
+.
+□
+We combine the enumerations of Theorems 4.11 and 4.12 in Table 3.
+Specifically, we list
+��APS+
+n \ APSn
+�� for 3 ≤ n �� 15, while permutations in SB
+1 and SB
+2 have no pinnacles. The nth
+term of this appears in the OEIS as double the (n − 1)st term of [12, A294175]. Moreover, the
+odd-indexed terms, enumerated in Theorem 4.11, appear in [12, A068551] and the even-indexed
+terms are double the terms of [12, A008549].
+13
+
+n
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+��APS+
+n \ APSn
+��
+2
+2
+10
+12
+44
+58
+186
+260
+772
+1124
+3172
+4760
+12952
+Table 3. The number of all-positive pinnacle sets that are admissible in SB
+n but
+not in Sn, for 3 ≤ n ≤ 15.
+5. Future directions
+As demonstrated by the results in this paper, admissible pinnacle sets have rich structure and
+properties even beyond the symmetric group. There are many directions for further research on
+this topic, including broad questions about pinnacle sets for families of permutations with certain
+properties, and enumerative specializations.
+As a complement to those large questions, we conclude this work by pointing out that we
+uncovered a possible connection between
+��APSB
+n
+�� and sequence [12, A119258]. In particular, we
+have the following conjecture.
+Conjecture 5.1. Consider the sequence [12, A119258], given by T(n, 0) = T(n, n) = 1 and
+T(n, k) = 2T(n − 1, k − 1) + T(n − 1, k) for 0 < k < n. Then
+��APSB
+n
+�� = T
+�
+n,
+�n − 1
+2
+��
+.
+Appendix A. Data
+Patrek Ragnarsson’s code for computing the data in Tables 1, 2, and 3 can be found at
+https://github.com/PatrekR/Signed-pinnacle-sets.
+Note that the data in Table 2 is the
+difference between the enumerations given in two of the files posted at this GitHub link.
+References
+[1] Sara Billey, Krzysztof Burdzy, and Bruce E. Sagan. Permutations with given peak set. J. Integer Seq., 6(16),
+2013.
+[2] F. Castro-Velez, A. Diaz-Lopez, R. Orellana, J. Pastrana, and R. Zevallos. Number of permutations with same
+peak set for signed permutations. Journal of Combinatorics, 8(4):631–652, 2017.
+[3] Robert Davis, Sarah A. Nelson, T. Kyle Petersen, and Bridget E. Tenner. The pinnacle set of a permutation.
+Discrete Math., 341(11):3249–3270, 2018.
+[4] Alexander Diaz-Lopez, Lucas Everham, Pamela E. Harris, Erik Insko, Vincent Marcantonio, and Mohamed
+Omar. Counting peaks on graphs. Australas. J Comb., 75:174–189, 2019.
+[5] Alexander Diaz-Lopez, Pamela E. Harris, Isabella Huang, Erik Insko, and Lars Nilsen. A formula for enumerating
+permutations with a fixed pinnacle set. Discret. Math., 344:112375, 2021.
+[6] Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko, Mohamed Omar, and Bruce E. Sagan. Descent polynomials.
+Discrete Mathematics, 342(6):1674–1686, 2019.
+[7] Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko, and Darleen Perez-Lavin. Peak sets of classical coxeter
+groups. Involve, 10(2):263–290, 2017.
+[8] Rachel Domagalski, Jinting Liang, Quinn Minnich, Bruce E. Sagan, Jamie Schmidt, and Alexander Sietsema.
+Pinnacle set properties. Discrete Mathematics, 345(7):112882, 2022.
+[9] Justine
+Falque,
+Jean-Christophe
+Novelli,
+and
+Jean-Yves
+Thibon.
+Pinnacle
+sets
+revisited.
+Preprint
+arXiv:2106.05248, 2021.
+[10] Wenjie Fang. Efficient recurrence for the enumeration of permutations with fixed pinnacle set. Disc. Math. The-
+oret. Comp. Sci., 24:#8, 2022.
+[11] Quinn Minnich. Further results on pinnacle sets. Discrete Math., 346(4):Paper No. 113296, 2023.
+[12] OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences, 2022. Published electronically at
+http://oeis.org.
+14
+
+[13] Irena Rusu. Sorting permutations with fixed pinnacle set. Electron. J. Comb., 27:P3.23, 2020.
+[14] Irena Rusu and Bridget Eileen Tenner. Admissible pinnacle orderings. Graphs and Comb., 37:1205–1214, 2021.
+[15] Richard P. Stanley. Enumerative combinatorics, volume 49 of Cambridge Studies in Advanced Mathematics.
+Cambridge University Press, Cambridge, second edition edition, 2012.
+[16] John R. Stembridge. Enriched p-partitions. Transactions of the American Mathematical Society, 349:763–788,
+1997.
+(N. Gonz´alez) Department of Mathematics, University of California, Berkeley, CA, 94720
+Email address: [email protected]
+(P. E. Harris) Department of Mathematical Sciences, University of Wisconsin, Milwaukee, WI 53211
+Email address: [email protected]
+(G. Rojas Kirby) Department of Mathematics and Statistics, San Diego State University, CA 92182
+Email address: [email protected]
+(M. Smit Vega Garcia) Department of Mathematics, Western Washington University, Bellingham,
+WA 98225
+Email address: [email protected]
+(B. E. Tenner) Department of Mathematical Sciences, DePaul University, Chicago, IL 60614
+Email address: [email protected]
+15
+

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+Dynamic Response of Wigner Crystals
+Lili Zhao, Wenlu Lin, and Yang Liu∗
+International Center for Quantum Materials, Peking University, Haidian, Beijing 100871, China
+Yoon Jang Chung, Adbhut Gupta, Kirk W. Baldwin, and Loren N. Pfeiffer
+Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
+The Wigner crystal, an ordered array of electrons, is one of the very first proposed many-body
+phases stabilized by the electron-electron interaction. This electron solid phase has been reported
+in ultra-clean two-dimensional electron systems at extremely low temperatures, where the Coulomb
+interaction dominants over the kinetic energy, disorder potential and thermal fluctuation. We closely
+examine this quantum phase with capacitance measurements where the device length-scale is com-
+parable with the crystal’s correlation length. The extraordinarily high performance of our technique
+makes it possible to quantitatively study the dynamic response of the Wigner crystal within the
+single crystal regime. Our result will greatly boost the study of this inscrutable electron solid.
+Interacting two-dimensional electron system (2DES)
+subjected to high perpendicular magnetic fields (B) and
+cooled to low temperatures exhibits a plethora of exotic
+states [1]. The Wigner crystal (WC) [2] terminates the
+sequence of fractional quantum Hall states at very small
+landau level filling factor [3–24]. This electron solid is
+pinned by the ubiquitous residual disorder, manifests as
+an insulating phase in DC transport [3–11], and the elec-
+trons’ collective motion is evidenced by a resonance in
+AC transport [12–19]. A series of experiments have been
+applied to investigate this correlated solid, such as the
+nonlinear I − V response [4, 16], the noise spectrum [5],
+the huge dielectric constant [20], the weak screening effi-
+ciency [21], the melting process [21–23], the nuclear mag-
+netic resonance [24] and the optics [25, 26].
+Capacitance measurements have revealed a series of
+quantum phenomena [21, 27–38]. In this work, we ex-
+amine the WC formed in an ultra-high mobility 2DES
+at ν <∼ 1/5 using high-precision capacitance measure-
+ment [39, 40]. We find an exceedingly large capacitance
+at low measurement frequency f while the conductance
+is almost zero.
+This phenomenon is inconsistent with
+transporting electrons, but rather an evidence that the
+synchronous vibration of electrons induces a polarization
+current. When we increase f, our high-precision mea-
+surement captures the fine structure of the resonance re-
+sponse with a puzzling ”half-dome” structure. Our sys-
+tematic, quantitative results provide an in-depth insight
+of this murky quantum phase.
+Our sample consists an ultra-clean low-density 2DES
+confined in a 70-nm-wide GaAs quantum well with elec-
+tron density n ≃ 4.4 × 1010 cm−2 and mobility µ ≃ 17
+×106 cm2/(V·s). Each device has a pair of front concen-
+tric gates G1 and G2, whose outer and inner radius are
+r1 and r2, respectively; see the inset of Fig. 1(a) [41]. We
+study four devices with r1 =60 µm and r2 = 60, 80, 100
+and 140 µm, respectively. We measure the capacitance C
+and conductance G between the two gates using a cryo-
+genic bridge and analyze its output with a custom-made
+radio-frequency lock-in amplifier [39–41].
+Fig.
+1(a) shows the C and G measured from the
+r1 = r2 = 60 µm sample. Both C and G decrease as
+we increase the magnetic field B, owing to the mag-
+netic localization where the 2DES conductance σ ∝
+(ne2τ)/m⋆(1+ω2
+cτ 2), m⋆, ωc and τ are the effective mass,
+cyclotron frequency and transport scattering time of the
+electrons, respectively [40]. The C and G are finite at
+ν = 1/2 and 1/4 where the 2DES forms compressible
+composite Fermion Fermi sea. When ν is an integer or
+a certain fraction such as 1/3 and 1/5, the 2DES forms
+incompressible quantum Hall liquids so that both C and
+G vanish [42].
+In all the above cases, the current is carried by trans-
+porting electrons, so that C has a positive dependence
+on G, i.e. C ∝ G3/2, as shown in Fig. 1(b) [40]. Such
+a correlation discontinues when the WC forms at very
+low filling factors ν <∼ 1/5, see the blue shaded regions
+of Fig. 1(a). The vanishing conductance G suggests that
+the electrons are immovable, however, the surprisingly
+large capacitance C evidences that the WC hosts a cur-
+rent even surpassing the conducting Fermi sea at ν = 1/2
+and 1/4 at much lower magnetic field! The phase tran-
+sition between the WC and the liquid states are clearly
+evidenced by spikes in G (marked by solid circles in Fig.
+1(a)) and sharp raises in C. A developing minimum is
+seen in G at 1/5 < ν < 2/9 (marked by the up-arrow)
+when C has a peak. This G minimum develops towards
+zero and the C peak saturates when the solid phase is
+stronger (see black traces in Fig. 3(a)). This is consistent
+with the reentrant insulating phase [3–5, 16, 19, 43, 44].
+It is important to mention that the 2DES in our de-
+vices is effectively “isolated” and we are merely transfer-
+ring charges between different regions within one quan-
+tum phase. Similar to the dielectric materials which also
+have no transporting electrons, the collective motion of
+all electrons, i.e. the k → 0 phonon mode of WC, can
+generate polarization charges and corresponding polar-
+ization current in response to the in-plane component
+of applied electric field.
+An infinitesimally small but
+ubiquitous disorder pins the WC so that electrons can
+arXiv:2301.01475v1  [cond-mat.mes-hall]  4 Jan 2023
+
+2
+0
+2
+4
+6
+8
+10
+12
+14
+B (T)
+0
+0.4
+C (pF)
+1
+0
+G (µS)
+ν=1
+1
+3
+1
+4
+1
+2
+1
+5
+2
+9
+0.0
+0.2
+C (pF)
+0
+1
+G3/2 (µS3/2)
+WC
+Liquid
+(b)
+0.0
+0.2
+0.2
+0.1
+0.0
+ν(∝1/B)
+C (pF)
+(d)
+f=7 MHz
+T=30 mK
+r2=100 µm
+0.0
+0.2
+0.4
+ln(r2/r1)
+8
+4
+0
+1/CWC (1/pF)
+13.5 T
+(e)
+ 
+ 
+12.0 T
+lB
+(c)
+a0
+x
+CWC
+2DES
+E
+G2
+G1
+<<
+<<
+Q ∝ e-d/ζ
+d
+0
+h
+r1=r2=60 µm
+f=7 MHz
+T=30 mK
+(a)
+G2
+G1
+Al2O3
+2DES
+r1
+r2
+FIG. 1. (color online) (a) C and G measured from the r1 = r2 = 60 µm sample with 7 MHz excitation at 30 mK. The horizontal
+dashed lines represent the zeros of C or G. The blue shaded regions mark the presence of WC. Inset is the cartoon of our
+device. (b) The correlation between C and G in panel (a) data. Transporting current dominates at B < 8 T where C ∝ G3/2,
+indicated the red solid line. When the WC polarization current dominates, C ≃ 0.2 pF and G is about zero (the blue box).
+(c) The schematic model describing the collective motion of electrons in the pinned WC. h is the depth of 2DES. The equally
+spaced (by the lattice constant a0) vertical bars represent the equilibrium position of electrons. The gray-scaled solid circles
+represent the electron position at finite external electric field E. The darker gray corresponds to larger electron displacement
+x. The radius of individual electron is about the magnetic length lB. The accumulated charge Q is proportional to ∇ · x,
+and decays exponentially as a function of the distance d from the gate boundary. ζ is the decay length. CWC is the effective
+capacitance of WC in the un-gated region between the two gates. (d) C v.s. ν of the r2=100 µm sample. The black dashed
+line is the zero of C. The red dashed line is the linear extension of data, showing that C = 0 at the extreme quantum limit
+ν = 0. (e) 1/CWC v.s. ln(r2/r1) at two different magnetic field.
+only be driven out of their equilibrium lattice site by
+a small displacement x, as shown in Fig.
+1(c).
+Dur-
+ing the experiments, we use excitation Vin ≃ 0.1 mVrms
+and the measured WC capacitance is ∼ 0.15 pF at 13.5
+T. The polarization charge accumulated under the inner
+gate is Q = CVin ∼ 100 e.
+The corresponding elec-
+tron displacement at the boundary of the inner gate,
+|x(r1)| ≃ Q/(2πr1ne) ∼ 0.6 nm, is much smaller than
+the magnetic length lB =
+�
+¯h/eB ∼ 8 nm, substanti-
+ating our assumption that the electrons vibrate diminu-
+tively around their equilibrium lattice sites.
+An ideal, disorder-free WC is effectively a perfect di-
+electric with infinite permittivity, so that the device ca-
+pacitance should be close to its zero-field value C0 ∼ 1 pF
+when 2DES is an excellent conductor. We note that C0 is
+consistent with the device geometry, ϵ0ϵGaAsπr2
+1/h ≃ 1.3
+pF, where ϵGaAs = 12.8 is the relative dielectric constant
+of GaAs and h ≃ 960 nm is the depth of 2DES. How-
+ever, the measured C ∼ 0.15 pF in the WC regime is
+much smaller than C0. This discrepancy is likely caused
+by the friction-like disorder which poses a pinning force
+≃ −βx on the electrons. When the crystal’s inversion
+symmetry is broken, i.e. x is non-uniform and J (x) is
+finite, the electron-electron interaction generates a restor-
+ing force ≃ −a0µijJ (x), where µij, a0 and J (x) are the
+elastic tensor, WC lattice constant and the Jacobi ma-
+trix of x, respectively. At the low frequency limit, the
+WC is always at equilibrium and all forces are balanced,
+eE − a0µijJ (x) − βx = 0, E is the total parallel electric
+field on the WC.
+E is approximately zero under the metal gates, since
+the gate-to-2DES distance h is small. Therefore, x de-
+creases exponentially when the distance from the gate
+boundary d increases, x ∝ exp(−d/ζ), where ζ = µa0/β
+is the decay length. Deeply inside the gates, electrons
+feel neither parallel electric field nor net pressure from
+nearby electrons, so that their displacement x remains
+approximately zero. This region does not contribute to
+the capacitive response, and the effective gate area re-
+duces to about 2πr1ζ and 2πr2ζ at the inner and outer
+gate, respectively. Because r1 = r2 = 60 µm in Fig. 1(a),
+the experimentally measured C ≈ ϵ0ϵGaAs/h · 2πr1ζ/2 ≃
+0.15 pF at 13.5 T corresponds to a decay length ζ ≃ 6.7
+µm. Interestingly, our result shows a linear dependence
+C ∝ 1/B in Fig. 1(d), suggesting that β ∝ l−2
+B
+if we
+assume µij is independent on B. Especially, the pinning
+becomes infinitely strong, i.e. β → ∞, at the extreme
+quantum limit lB → 0.
+
+3
+The permittivity of a disorder-pinned WC is no longer
+infinitely large, since a non-zero electric field E is neces-
+sary to sustain a finite x. If we assume x is a constant
+in the ring area between the two gates, so that eE = βx.
+The residual E can be modeled as a serial capacitance
+CWC ≈ 2πne2/β · [ln(r2/r1)]−1 in our device. We then
+measure different devices with r1= 60 µm and r2 = 60,
+80, 100 and 140 µm, and calculate the corresponding
+CWC through C−1
+WC = C−1 −(r1 +r2)/r2 ·C−1
+r1=r2, see Fig.
+1(e). By fitting the linear dependence C−1
+WC ∝ ln(r2/r1),
+we estimate the pinning strength β to be about 1.3 ×10−9
+and 1.1 ×10−9 N/m at B = 13.5 and 12 T, respectively
+[45].
+Finally, assuming µij ≈ µ · δij, we can estimate
+the WC elastic modulus µ ≈ β · ζ/a0. For example, µ is
+about 1.6 × 10−7 N/m at 13.5 T.
+0.14
+0.16
+0.18
+0.20
+0.22
+0
+0.2
+C (pF)
+2
+0
+G (μS)
+ν
+30 mK
+95 mK
+110 mK
+125 mK
+145 mK
+200 mK
+1/5
+2/11
+1/7
+(a)
+0
+100
+200
+T (mK)
+0
+0.2
+C (pF)
+ν=0.18
+ν=0.14
+2
+0
+G (μS)
+0.14
+0.20
+ν
+0
+200
+TC (mK)
+FQH liquid
+WC
+Compressible
+liquid
+(b)
+TC
+TC
+r2=80 μm f=17 MHz
+FIG. 2. (color online) (a) C and G vs. ν measured at vari-
+ous temperatures from the r2 = 80 µm sample with 17 MHz
+excitation. (b) Summarized C and G vs. T at ν = 0.14 and
+0.18 from the panel (a) data. A critical temperature Tc at
+certain ν is defined either as the temperature when G has
+a peak at ν in panel (a) or as the temperature when G vs.
+T trace reaches maximum in panel (b); marked by the black
+and red arrows. The panel (b) inset summarizes the Tc us-
+ing the two equivalent definitions using black and red circles,
+respectively. The diagram can be separated into three differ-
+ent regions corresponding to the WC, the fractional quantum
+Hall (FQH) liquid and the compressible liquid.
+Fig. 2 reveals an intriguing temperature-induced solid-
+liquid phase transition when the WC melts. Fig. 2(a)
+shows C and G taken from the r2 = 80 µm sample at
+various temperatures. At a certain temperature, e.g. at
+T ≈ 110 mK, C ∼ 0.2 pF when the 2DES forms WC
+at ν <∼ 0.16 and vanishes when it is a liquid phase at
+ν >∼ 0.18. G has a peak at ν ≃ 0.175 when C vs. ν
+has the maximal negative slope, and it is small when the
+2DES is either a WC at ν < 0.17 or a liquid at ν > 0.19
+[46]. At very high temperature T >∼ 200 mK, both C and
+G are close to zero. In Fig. 2(b), we summarized C and
+G as a function of T at two different filling factors to bet-
+ter illustrate this solid-liquid transition. At ν ≃ 0.14, for
+example, C is large and G is small at T <∼ 100 mK when
+the WC is stable [47], while both of them become small
+at T >∼ 200 mK when the 2DES is a liquid. The G has
+a peak at a critical temperature TC, marked by the red
+arrows, around which the precipitous decrease of C hap-
+pens. Alternatively, TC at a certain filling factor ν can be
+defined as the temperature when the G has a peak (black
+arrow in Fig. 2(a)) at ν. We summarize TC obtained us-
+ing these two equivalent procedures in the Fig. 2(b) inset
+with corresponding red and black symbols. TC has a lin-
+ear dependence on ν whose two intercepts are TC ≃ 340
+mK at the extreme quantum limit ν = 0, and ν ≃ 1/4 at
+TC = 0 mK.
+The Fig.
+2(b) evolution can be qualitatively under-
+stood by the coexistence of transport and polarization
+currents at the solid-liquid transition. The large C re-
+duces to almost zero when the transport current domi-
+nates over the polarization current. G is a measure of
+the 2DES’s capacity to absorb and dissipate power. It is
+negligible if either of these two currents dominates, since
+the polarization current is dissipation-less and the dissi-
+pating transport current is difficult to excite. G becomes
+large when these two currents coexist nip and tuck at
+intermediate T when the excited polarization charge can
+be just dissipated by the transport current.
+The WC exhibits a resonance when we increase the
+excitation frequency. In Fig. 3(a), the C and G measured
+from the r2 = 100 µm sample using different excitation
+frequencies change enormously when the WC presents
+(blue shaded region). G is almost zero and C is large
+at f ≃ 7 MHz, and G becomes finite and C becomes
+even larger at f ≃ 23 MHz. At slightly higher frequency
+27 MHz, G reaches its maximum and C drops to about
+zero. Further increasing f, G gradually declines while
+C first becomes negative at 35 MHz and then gradually
+approaches zero.
+The summarized C and G vs.
+f at
+two certain fillings in Fig. 3(b), resembles qualitatively
+a resonant behavior with resonance frequency fr ≃ 26
+MHz (when C = 0). Fig. 3(c) studies this resonance
+at different temperatures. The data is taken from the
+r2 ≃ 80 µm sample whose resonance frequency is about
+35 MHz [48]. The abrupt change of C near fr becomes
+gradual and the G peak flattens at higher temperatures.
+
+4
+10
+0
+G (μS)
+0.14
+0.16
+0.18
+0.20
+0.22
+0.24
+ν
+0.4
+0
+C (pF)
+  7
+23
+27
+35
+77
+r2=100 μm
+0
+0.2
+-0.2
+ν=0.14
+30
+60
+140
+280
+0
+0.2
+10
+0
+10
+100
+f (MHz)
+r2=80 μm
+10
+0
+10
+100
+f (MHz)
+ν=0.14
+ν=0.213
+r2=100 μm
+T=30 mK
+T=30 mK
+(b)
+(a)
+(c)
+fr=35 MHz
+fr=26 MHz
+ν=0.213
+ f (MHz)
+ T (mK)
+FIG. 3. (color online) (a) C and G vs. ν taken from the r2=100 µm sample using different excitation frequencies f. We see
+a violent change of C and G at different f in the blue region where the WC appears. (b) The C and G vs. f extracted from
+the panel (a) trace at ν = 0.14 and 0.213. The resonance frequency fr, defined as the frequency when C changes its sign, is
+about 26 MHz. (c) The C and G vs. f at ν = 0.14 and different temperatures, data taken from the r2=80 µm sample. The
+resonance disappears at T ≃ 280 mK when C and G remain nearly zero.
+Both C and G become flat zero at T >∼ 280 mK. It is
+noteworthy that, as long as a resonance is seen, fr is
+nearly independent on the filling factor (Fig. 3(b)) and
+temperatures (Fig. 3(c)). This is consistent with another
+experimental study using surface acoustic wave [23].
+The resonance of WC is usually explained by the
+pinning mode [18, 49].
+The resonance frequency is
+related to the mean free path LT of the transverse
+phonon through LT = (2πµt,cl/neBfr)1/2, where µt,cl =
+0.245e2n3/2/4πϵ0ϵGaAs is the classical shear modulus of
+WC. fr = 26 MHz corresponds to LT ≃ 3.2 µm, very
+similar to ζ ≃ 6.7 µm in our Fig.
+1(c) discussion.
+This is justifiable because both LT and ζ describe the
+length-scale within which the collective motion of WC is
+damped/scattered by the random pinning potential.
+Before ending the discussion, we would like to highlight
+the puzzling ”half-dome” structure of the resonance. G
+has a regular-shaped resonance peak, i.e.
+G decreases
+gradually on both sides of fr, when either the WC is weak
+( ν ≃ 0.213 in Fig. 3(b)) or the temperature is high (T ≃
+140 mK in Fig. 3(c)). Surprisingly, the resonance peak
+becomes quite peculiar when the WC is strong at ν ≃
+0.14 and T ≃ 30 mK. G gradually decreases from its peak
+at fr on the high frequency side f > fr, while it vanishes
+instantly when the frequency is lower than fr, resulting in
+a ”half-dome” G vs. f trace. Meanwhile, the C increases
+by ∼ 2 times and then abruptly changes to negative at
+fr. This anomalous ”half-dome” feature is seen in all of
+our devices as long as the WC is strong and temperature
+is sufficiently low, suggesting a threshold frequency for
+the power dissipation.
+In conclusion, using the extraordinarily high-precision
+capacitance measurement technique, we investigate the
+dynamic response of WC systematically. From the quan-
+titative results and using a simple model, we can study
+several physical properties of the WC such as elastic mod-
+ulus, dielectric constant, pinning strength, etc., and dis-
+cover a puzzling ”half-dome” feature in the resonance
+peak. Our results certainly shine light on the study of
+WC and provides new insight on its dynamics.
+We acknowledge support by the National Nature Sci-
+ence Foundation of China (Grant No.
+92065104 and
+12074010) and the National Basic Research Program of
+China (Grant No. 2019YFA0308403) for sample fabrica-
+tion and measurement. This research is funded in part by
+the Gordon and Betty Moore Foundation’s EPiQS Initia-
+tive, Grant GBMF9615 to L. N. Pfeiffer, and by the Na-
+tional Science Foundation MRSEC grant DMR 2011750
+to Princeton University. We thank L. W. Engel, Bo Yang
+and Xin Lin for valuable discussion.
+∗ [email protected]
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+[41] See Supplemental Material for detailed description of our
+sample information and measurement techniques.
+[42] The zero of C and G can be defined either by extrapolat-
+ing their field dependence to B = ∞, or by their values
+at strong quantum hall states such as ν = 1. These two
+approaches are consistent with each other and the dash
+lines in Fig. 1(a) represent the deduced zero.
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+[45] Alternatively, CWC can be modeled as a cylinder ca-
+pacitor whose height equals the effective thickness of
+the 2DES, Z0 ≈ 45 nm. The WC dielectric constant is
+ϵWC = (2πϵ0Z0∂(C−1
+WC)/∂ ln(r2/r1))−1 ≈ 2 × 104 at 13.5
+T, consistent with previous reported value in ref. [20].
+[46] We observe developing minimum at ν = 1/7, 2/11 dur-
+ing the solid-liquid phase transition, signaling that the
+fractional quantum Hall state emerges [8, 11].
+[47] C vs. T has a slightly positive slope in the WC region,
+possibly due to the softening of disorder pinning.
+[48] fr has no obvious dependence with sample geometry,
+which is about 35, 35, 26 and 29 MHz for samples with
+r2 = 60, 80, 100, 140 µm, respectively.
+[49] M. M. Fogler and D. A. Huse, Phys. Rev. B 62, 7553
+(2000).
+
+6
+SUPPLEMENTARY MATERIALS
+Samples
+The sample we studied is made from a GaAs/AlGaAs
+heterostructure wafer grown by molecular beam epitaxy.
+A 70 nm-wide GaAs quantum well is bound by AlGaAs
+spacer-layers and δ-doped layers on each side, and locates
+h ≃ 960 nm below the sample surface. The as-grown den-
+sity of the 2DES is n ≃ 4.4×1010 cm−2, and its mobility
+at 300 mK is µ ≃ 17 ×106 cm2/(V·s). Our sample is a
+2 mm × 2 mm square piece with four In/Sn contacts at
+each corner. The contacts are grounded through a re-
+sistor to avoid signal leaking. We evaporate concentric,
+Au/Ti front gate pair G1 and G2 using standard lift-
+off process, whose outer and inner radius is r1 and r2,
+respectively. We deposit a 20 nm thick Al2O3 layer be-
+tween the two gates to prevent them from shorting with
+each other.
+The four outer-gates are merged into one
+piece so that the area of the outer gate G2 is much larger
+than the inner gate G1.
+Capacitance Measurement Setup
+The capacitance and conductance response is mea-
+sured with a cryogenic bridge similar to refs. [39, 40].
+The kernel of the bridge consists four devices, Rh, Rr,
+Cr and C, as shown in Fig. S1(a). C is the capacitance of
+sample. We change the value of Rh to reach the balance
+condition
+C
+Cr
+= Rh
+Rr
+.
+(1)
+The bridge output Vout is minimum at the balance con-
+dition, from which we calculate the C. This is the so-call
+“V-curve” procedure, see refs. [39, 40] for more informa-
+tion.
+In order to expand the allowed bandwidth of the ex-
+citation frequency, we add an impedance match network
+to the input of the bridge, shown as the Fig. S1(a). Vext
+is the signal source with 50 Ω output impedance. Vext
+drives a signal splitter box (the red dashed box) located
+at the top of the dilution refrigerator through a ∼2 m-
+long semi-rigid coaxial cable.
+The box input is a 1:5
+transformer in series with a 50 Ω resistor. The trans-
+former output drives two serial connected 50 Ω resistors
+differentially. The differential signals are transmitted to
+the cryogenic sample holder (the blue dotted box) by
+two rigid coaxial cables of ∼2 m length. Another pair
+of impedance matching 50 Ω resistors are added at the
+input of the cryogenic bridge, and the 360 Ω resistors are
+chosen by balancing the competition between the perfor-
+mance and heating. The characteristic impedance of all
+coaxial cables in the work is 50 Ω.
+The low-frequency signals Vquasi-DC1 and Vquasi-DC2
+used to measure the value of Rh and Rr, respectively. The
+0.1 µF capacitors are used to separate the high-frequency
+excitation signals and the quasi-DC signal.
+The output Vout is approximately
+Vout ∝ S · (
+Rh
+360 + Rh
+− C
+Cr
+·
+Rr
+360 + Rr
+) · Vext.
+(2)
+S can be obtain from the “V-curve” procedure by linear
+fitting the VX vs. Rh/(360+Rh), as shown in Fig. S1(b).
+VX and VY are the orthogonal component of Vout,
+� VX = |Vout| · cos(θ),
+(3)
+VY = |Vout| · sin(θ),
+(4)
+where θ is the phase of Vout. We can derive the value of
+C using Eq. (2) and (3). The new balance condition of
+the revised bridge is
+C
+Cr
+= Rh
+Rr
+· 360 + Rr
+360 + Rh
+,
+(5)
+where the VX = 0.
+Note that the capacitance C and the conductance G
+of sample lead to the orthogonal component VX and VY,
+respectively. Therefore, the G can be obtained from Eq.
+(2) and (4) by replacing C/Cr with G/2πfCr, where f is
+the excitation frequency.
+Fig. S1(c) shows our calibration measurement using
+different excitation frequencies. The data is almost flat
+from 7 to ∼100 MHz. The measured capacitance begins
+to decline slowly above ∼100 MHz, possibly due to the
+parasitic inductance of bonding wires.
+
+7
+Rh
+Rr
+Cr
+C
+Vin
++
+Vin
+-
+Vout
+360 Ω
+360 Ω
+50 Ω
+50 Ω
+1:5
+50 Ω
+50 Ω
+50 Ω
+0.1 μF
+0.1 μF
+0.1 μF
+0.1 μF
+Vext
+Vquasi-DC1
+Vquasi-DC2
+COAX
+COAX
+(a)
+ 
+ 
+40
+-40
+0
+V (μV)
+0.0
+0.2
+0.4
+0.6
+Rh/(Rh+360)
+Vx
+Vy
+(b)
+Cr= 0.1 pF
+f= 7 MHz
+Rr= 50 Ω
+0.6
+0.0
+0.4
+0.2
+C (pF)
+10
+100
+f (MHz)
+0.5 pF
+0.3 pF
+0.1 pF
+(c)
+COAX
+FIG. S1. (color online) (a) Circuit diagram of measurement bridge with 50 Ω impedance match networks. (b) The VX and VY
+from a typical “V-curve” procedure. C is about 0.25 pF from the balance condition Eq. (5). (c) The calibration results, by
+measuring commercial capacitors with different frequencies.
+

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+Towards AI-controlled FES-restoration of arm movements:
+neuromechanics-based reinforcement learning for 3-D reaching
+Nat Wannawas1 & A. Aldo Faisal1,2
+Abstract— Reaching disabilities affect the quality of life.
+Functional Electrical Stimulation (FES) can restore lost motor
+functions. Yet, there remain challenges in controlling FES
+to induce desired movements. Neuromechanical models are
+valuable tools for developing FES control methods. However,
+focusing on the upper extremity areas, several existing models
+are either overly simplified or too computationally demanding
+for control purposes. Besides the model-related issues, finding
+a general method for governing the control rules for different
+tasks and subjects remains an engineering challenge.
+Here, we present our approach toward FES-based restoration
+of arm movements to address those fundamental issues in
+controlling FES. Firstly, we present our surface-FES-oriented
+neuromechanical models of human arms built using well-
+accepted, open-source software. The models are designed to
+capture significant dynamics in FES controls with minimal
+computational cost. Our models are customisable and can be
+used for testing different control methods. Secondly, we present
+the application of reinforcement learning (RL) as a general
+method for governing the control rules. In combination, our
+customisable models and RL-based control method open the
+possibility of delivering customised FES controls for different
+subjects and settings with minimal engineering intervention.
+We demonstrate our approach in planar and 3D settings.
+Functional Electrical Stimulation, FES, Neuromechanical
+Model, Reinforcement Learning, Arm Movements
+I. INTRODUCTION
+Every year, stroke and spinal cord injury cause the loss of
+motor functions in individuals worldwide through paralysis.
+In these cases the limbs are technically functional but fail to
+receive motor commands from the brain. Arm movement are
+one of the commonly lost motor functions and cause severe
+limitations in performing daily tasks. Functional Electrical
+Stimulation (FES) or neuromuscular stimulation uses low-
+energy electrical signals to stimulate the muscle and induce
+its contraction and eventually movement of the limb. FES can
+be used to animate paralysed muscles and restore lost motor
+functions and may help to restore nature motor functions
+in incomplete paralysis [1]. Early successes included work
+on low limb paralysis, where the ability to cycling through
+rhythmic pedalling motions of the legs was restored and has
+now become an internally recognised bionic sports discipline.
+These success beg the question how to extend this work to
+upper body restoration of movement. While FES induced
+cycling entails the periodic stimulation of muscles without
+1Brain & Behaviour Lab, Imperial College London, London SW7
+2AZ, United Kingdom. (email:[email protected]). 2 Chair
+in Digital Health & Data Science, University of Bayreuth, Bayreuth,
+([email protected]). NW acknowledges his support by the Royal
+Thai Government Scholarship. AAF acknowledges his support by UKRI
+Turing AI Fellowship (EP/V025449/1).
+the need for gravity compensation or end-point precision
+in their control (the feet are strapped to the pedals so all
+motions are on sagittal plane), so movements are relatively
+constrained and requirements on control strategies can be
+relatively, low precision (often involving simple periodic
+stimulation). This is very different in arm movements, where
+reaches towards an end-point require non-linear muscle co-
+ordination in the plane, and gravity compensating activity in
+3D movements in a volume. Early work in this nascent field
+therefore focus on single-joint control, e.g., of the elbow joint
+[2], [3]. The literature on multiple-joint control of arm cases,
+however, is quite limited. This is partly because controlling
+FES in single-joint cases can be achieved through simple,
+model-free, error-based control such as PID controllers ,
+while the multiple-joint cases require significantly more
+complex controls that have to include dynamical models in
+the systems.
+Contrary to the fact that dynamic models play important
+roles in the controls and that the neuromechanics of the
+human arm is complex, many models used in FES control
+studies are relatively simple. In planar arm motion, for
+example, two-joint linkages with six muscles represented by
+straight lines models (Fig.1a) are one of the most commonly
+used [4]–[7]. These models offer fast computation, but may
+not well capture the effects of muscle routes and their
+deformation during the movements. In addition, the muscles’
+properties themselves vary across the studies, thereby lacking
+standardization and making the results difficult to reproduce
+or compare. On the other end, there exist commercial neu-
+romechanical simulation software such as LifeModeler with
+highly detailed models. These models, however, are suitable
+for detailed analyses of particular situations such as er-
+gonomic designs. Besides, the closed-source and commercial
+nature of the software could limit its usage among research
+communities, thereby not addressing the standardization and
+reproducibility issues.
+Besides the model-related issues, governing the control
+rules itself is a major challenge in inducing movements
+using FES. Regards specifically to inducing multiple-joint
+arm movements, successes in real-world settings are limited
+and, oftentimes, require assistive devices [2], [8], [9]. For
+example, the PID controller with inverse dynamics [2] can
+induce a narrow range of movements, while iterative learning
+control [8] can induce a longer range but is limited to
+repetitive trajectories. These limited successes are partly
+attributed to the difficulties of conventional methods in
+dealing with complexities and variations of human arms’
+neuromechanics. To our knowledge, an FES control method
+arXiv:2301.04004v1  [eess.SY]  10 Jan 2023
+
+Fig. 1.
+(a) An example of simple planar arm models. (b) Our neuromechanics Arm-Planar model. (c) An illustration of muscle wrapping at the elbow.
+(d) Our neuromechanics Arm-3D model and (e) its shoulder muscles.
+that can induce arbitrary movements across different subjects
+without intensive parameter tuning has not yet been reported.
+Keeping both model-related and control governing issues
+in focus, we here present our approach toward FES-based
+restoration of arm movements that comprises two elements
+which, as separate entities, can address those issues. The first
+element is the neuromechanics models of the human arm
+built using OpenSim [10], a freely-available, open-source
+neuromechanical simulation software that is well-accepted in
+the communities. This allows us to build the models using
+established biomechanical components, e.g., muscle and joint
+models, thereby addressing the standardisation issue and
+providing state-of-the-art performances [11]. Additionally,
+the open source nature of OpenSim facilitate its uses for
+designing and testing different control methods which help
+promote reproducibility. In this work, we present our two
+arm models designed for surface FES control usages, i.e.,
+they are designed for fast computation while maintaining
+important details. These models could be used as standards
+for comparing different control methods.
+The second element of our approach is to govern the
+control policy using Reinforcement Learning (RL), a ma-
+chine learning algorithm with a learning agent (RL agent)
+that learns to control an environment by interacting with it.
+RL can learn to control complex environments, for which
+hand-crafted control policies are difficult to govern. In FES
+control applications, RL is a promising method for governing
+control policies for any FES control settings. In addition,
+the fact that RL can provide customised stimulation for
+different subjects without intensive manual configuration can
+be an important factor that drives FES-based restoration of
+movements outside the laboratory and toward at-home usage.
+In this work, we present a generic RL setup to learn control
+policies for arbitrary arm-reaching tasks. We demonstrate the
+usage in planar and 3D arbitrary reaching tasks using our
+OpenSim models.
+II. RELATED WORKS
+It is worth mentioning some related works to highlight
+their limitations and the gaps that this work fulfils. Regarding
+neuromechanical arm models, there exist several OpenSim
+models built by the communities. Closely related models are
+the OpenSim core Arm26 and MoBL-ARMS Dynamic Upper
+Limb models [12]. These models have a few critical and
+minor issues as follows. The first critical issue is that they
+produce singularity computation in OpenSim4.4, the latest
+version, at some postures, causing crashes. Secondly, there
+is no mechanism such as joint limits to prevent unnatural
+postures. The minor issues are that there is no joint damping
+that prevents unnatural joint speed and, in some postures,
+the muscle paths are in the wrong positions, e.g., they wrap
+around the wrong side of the joint or go through the bone.
+Regarding the applications of RL in FES control, the early
+studies were based on old RL algorithms, simple planar arm
+models (Fig.1a), and a single, fixed target [4]–[6]. A recent
+study has extended these settings to multiple targets [7]. Our
+previous works investigate cycling motions in simulation [13]
+and single-joint arm movements [3] in the real world. A
+simulation study on 3D arm motions was conducted in [14]
+using a model that does not have muscle, i.e., the RL agent
+directly controls joint torque rather than muscle stimulation.
+III. METHODS
+a) Neuromechanical models: We use two human arm
+models; one is for planar motions (hereafter referred to as
+Arm-Planar) which can be viewed as the detailed version
+of Fig.1a-like model, and the other one is for 3D motions
+(hereafter referred to as Arm-3D). Both models are designed
+at a suitable detail level for surface FES control applications,
+e.g., the muscles that are impossible to be stimulated sepa-
+rately via surface FES are bundled together to minimise the
+computation. The common properties and designs of both
+models are as follows. Both models have the right arms
+connected to the upper bodies located at fixed points in 3D
+space. The elbow and shoulder joints are modelled as pin and
+ball joints, respectively. Both joints have damping and joint
+limit mechanisms that prevent unnatural joint speed and pos-
+tures. The muscles are built using a variant of Hill-type mus-
+cle model DeGrooteFregly2016Muscle. Both models have 4
+muscles crossing elbows: Triceps Medial, Triceps long head
+(biarticular), Brachialis, and Biceps short head (biarticular).
+Note that Triceps lateral head is bundled with Triceps long
+
+Pectoralis
+Arm-Planar
+Arm-3D
+Deltoid Anterior
+major C
+Deltoid Lateral
+Deltoid Posterior
+Deltoid
+Posterior
+ Pectoralis major C
+N
+Biceps short head
+Triceps long head
+Brachialis
+Triceps
+Table
+Medial
+Arm Support
+a
+b
+dhead, and Biceps long head is bundled with Biceps short
+head. These muscles wrap around a cylindrical object at the
+elbows (Fig.1c). The muscles’ excitation-activation delay is
+changed from the default setting of 40 ms to 100 ms to
+capture a longer delay of FES-induced muscle activation.
+The other muscle parameters such as maximum isometric
+force follow those in the Arm26 and MoBL-ARMS Dynamic
+Upper Limb models. The tendon slack length parameters are
+optimised using a genetic algorithm called CMAES [15] to
+equilibrate the passive forces. The other parts of both models
+have slightly different designs described as follows.
+The Arm-Planar model has 6 muscles in total. The
+other muscles besides the aforementioned 4 muscles are the
+Pectoralis Major Clavicular head (Pectoralis Major C) and
+Deltoid posterior. These muscles wrap around a cylindrical
+object at the shoulder. The shoulder joint is only allowed
+to rotate around the vertical axis. The arm is supported at
+the wrist by an arm supporter (Fig.1b) that moves on a
+table with low friction and provides gravity compensation
+to the arm. The Arm-3D model has 8 muscles in total which
+are those 6 muscles of the Arm-Planar model plus Deltoid
+lateral and Deltoid anterior. At the shoulder, there are three
+half ellipsoids functioning as muscle wrapper objects. These
+ellipsoids are carefully placed to support the full range of
+movements and prevent the wrong muscle path issues of the
+existing OpenSim models. The shoulder joint can rotate in all
+directions except the direction that causes the arm to twist.
+b) Reinforcement Learning controllers: The overview
+of RL algorithms is briefly described as follows. RL learns
+a task through reward signals collected from the interaction
+with an environment. The interactions occur in a discrete-
+time fashion, starting with the agent observing the envi-
+ronment’s state st and selecting an action at based on its
+policy π. The action causes the environment to be in a new
+state st+1. The agent then receives an immediate reward
+rt and observes the new state. This interaction experience
+is collected as a tuple (st, at, rt, st+1) which is stored in
+a replay buffer D. This tuple is used to learn an optimal
+policy π∗ that maximises a return R–the sum of discounted
+immediate rewards.
+The RL task here is to apply the muscle stimulation to
+move the arm to the desired pose which is specified by target
+joint angles–shoulder and elbow (θtar,t). The state vector st
+is [θt, ˙θt, θtar,t]T , where θt and ˙θt are the joint angles and
+angular velocities measured at time t, respectively. Note that
+appending the targets into the state vector allows the agents
+to learn goal-directed policies that can perform arbitrary
+reaching tasks. The action vector at comprises normalised
+stimulation intensities (i ∈ [0, 1]). The immediate reward rt
+is simply computed using the square error and action penalty
+as rt = −(θt+1 − θtar,t)2 − Σn
+i=0ai
+n
+, where n is the number
+of stimulated muscles. With this setting, the optimal policy
+π∗ is simply the policy that causes the angles to be close to
+the targets with minimal stimulation.
+The mechanism of finding the optimal policy varies across
+different RL algorithms. In this work, we choose the soft
+actor-critic (SAC) algorithm [16] because of its state-of-the-
+art performance in terms of both sample efficiency and stabil-
+ity across different environments. SAC has two components:
+an actor and a critic. In simple terms, the critic learns to
+estimate the expected return of a state-action pair, known as
+the Q value. The Q value is used to adjust the actor’s policy
+π by increasing the probability of choosing an action with
+a high Q value. Both actor and critic are parameterised by
+neural networks; we, based on empirical experiments and our
+previous works [3], [13], use fully-connected neural networks
+that have two hidden layers. The output layer of the actor
+has a sigmoid activation function to squash the outputs.
+The setups for the planar and 3D cases are slightly
+different. In the planar case, the involved angles are the elbow
+and shoulder angles which rotate about the vertical axes. The
+state vector is therefore s ∈ R6. The action vector a has 4
+elements (ai ∈ [0, 1]) which are the normalised stimulation
+intensities of the Brachialis and Biceps short head, Triceps
+Medial and Triceps long head, Pectoralis Major C, and
+Deltoid posterior. Note that we set the Biceps stimulation
+to affect two muscles because, normally in a real situation,
+only a single pair of electrodes are placed above Biceps (and
+similarly for Triceps). In the 3D case, the shoulder joint can
+rotate in 2 directions, and the Deltoid lateral and Deltoid
+anterior are stimulated via the same pair of electrodes. Hence,
+the state vector becomes s ∈ R9, and the action vector has
+5 elements.
+The RL training is episodic. Each episode starts with a
+random arm pose and target. Each episode has 100 time steps
+with 100 ms time-step size. The target changes to a new
+random value at the 50th time step. Every 5 training episodes,
+the agents’ performances are evaluated on 50 test episodes.
+IV. RESULTS
+RL agents are trained for 250 and 500 episodes on the
+Arm-Planar and Arm-3D models, respectively. The training
+is repeated 10 times to evaluate the robustness. The perfor-
+mance evaluations along the training are shown in Fig.2a.
+In both cases, the best RL’s performances in rmse measure
+are approximately 10◦. The performance development in the
+planar case is significantly quicker than in the 3D case. The
+standard deviations in both cases are in low, confined ranges
+which suggests the robustness.
+Fig.2b and d show examples of control performances
+in planar and 3D cases, respectively. In both cases, the
+RL agents can track arbitrary trajectories that have never
+been assigned during the training. The performance in the
+planar case is slightly better than that in the 3D case as
+the planar movements are less complex. Fig.2c and e show
+the stimulation applied during the tracking tasks. In both
+cases, brief bursts of stimulation appear when the targets
+change, followed by steady stimulation that co-contraction
+the muscles to stabilise the arms. The bursts do not appear
+when the targets change in a ramping manner.
+V. DISCUSSION & CONCLUSION
+We present our approach toward FES-based restoration
+of arm movements. Our approach has two elements. The
+
+Fig. 2.
+(a) Performance evaluation along the training in (red) Arm-3D and (blue) Arm-Planar cases. The solid lines and the shades show the mean and
+standard deviation of 10 runs. The examples of trajectory tracking in (b) Arm-Planar and (d) Arm-3D cases. The dash and solid lines are the targets and
+actual angles that the RL agents achieve, respectively. (c) and (e) show the stimulation along the tracking.
+first element is to use OpenSim to build neuromechanical
+models of the arm. This strategy can help facilitate the build-
+ing process and standardise the models on which different
+control methods are tested and compared. Furthermore, we
+present our two OpenSim models: Arm-Planar and Arm-
+3D. The second element is to govern the control rules by
+using reinforcement learning which can provide customised
+stimulation for different subjects and settings with minimal
+technical intervention. We present a generic RL training
+setup, demonstrate its applications on our OpenSim models
+and show our RL’s performances in performing arbitrary
+reaching tasks.
+Although this approach has promising simulation results,
+several further steps have to be taken to translate it into
+real-world usages. One step is to optimise the models to
+accurately represent the dynamics of a certain subject’s arm.
+This is yet a process that can be done using OpenSimMoCo
+[17]. The customised model can be used for pre-training the
+RL before transferring it to the real subject. Another step is
+to take muscle fatigue into account. The fatigue behaviour
+can be included in OpenSim models without touching the
+source code by using the method presented in our previ-
+ous work [13]. The fatigue will cause the environment’s
+state to become partially observable. Based on [5], [7] and
+our empirical investigation, the fatigue does not cause RL
+to completely fail, but the control performance decreases.
+Lastly, in the early period of the training, the RL-controlled
+stimulation is unpredictable and random. This raises an issue
+about safety. This issue can be mitigated by using offline RL
+in the early period.
+To summarise, the combination of neuromechanical mod-
+els and RL can address existing challenges in FES control.
+Although the translation into real-world usages involves
+several further steps, its potential is emerging.
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+
+Case: Arm-Planar
+RMSE: 5.21 °
+Case: Arm-3D
+RMSE: 6.82
+100
+100
+a
+b
+--
+Elbow
+Arm-3D
+q
+Shoulder-x
+60 
+Arm-Planar
+80
+80 -
+ Shoulder-Z
+60
+60 -
+Angle
+50 
+40
+40
+20 
+20 
+ Shoulder
+Elbow
+0
+100
+100-
+Biceps
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AdE2T4oBgHgl3EQfnAiB/content/tmp_files/load_file.txt

@@ -0,0 +1,314 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf,len=313
+page_content='Towards AI-controlled FES-restoration of arm movements:  neuromechanics-based reinforcement learning for 3-D reaching  Nat Wannawas1 & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Aldo Faisal1,2  Abstract— Reaching disabilities affect the quality of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Functional Electrical Stimulation (FES) can restore lost motor  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Yet, there remain challenges in controlling FES  to induce desired movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Neuromechanical models are  valuable tools for developing FES control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' However,  focusing on the upper extremity areas, several existing models  are either overly simplified or too computationally demanding  for control purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Besides the model-related issues, finding  a general method for governing the control rules for different  tasks and subjects remains an engineering challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Here, we present our approach toward FES-based restoration  of arm movements to address those fundamental issues in  controlling FES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Firstly, we present our surface-FES-oriented  neuromechanical models of human arms built using well-  accepted, open-source software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The models are designed to  capture significant dynamics in FES controls with minimal  computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Our models are customisable and can be  used for testing different control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Secondly, we present  the application of reinforcement learning (RL) as a general  method for governing the control rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' In combination, our  customisable models and RL-based control method open the  possibility of delivering customised FES controls for different  subjects and settings with minimal engineering intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  We demonstrate our approach in planar and 3D settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Functional Electrical Stimulation, FES, Neuromechanical  Model, Reinforcement Learning, Arm Movements  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' INTRODUCTION  Every year, stroke and spinal cord injury cause the loss of  motor functions in individuals worldwide through paralysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  In these cases the limbs are technically functional but fail to  receive motor commands from the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Arm movement are  one of the commonly lost motor functions and cause severe  limitations in performing daily tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Functional Electrical  Stimulation (FES) or neuromuscular stimulation uses low-  energy electrical signals to stimulate the muscle and induce  its contraction and eventually movement of the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' FES can  be used to animate paralysed muscles and restore lost motor  functions and may help to restore nature motor functions  in incomplete paralysis [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Early successes included work  on low limb paralysis, where the ability to cycling through  rhythmic pedalling motions of the legs was restored and has  now become an internally recognised bionic sports discipline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  These success beg the question how to extend this work to  upper body restoration of movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' While FES induced  cycling entails the periodic stimulation of muscles without  1Brain & Behaviour Lab, Imperial College London, London SW7  2AZ, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' (email:nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='wannawas18@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' 2 Chair  in Digital Health & Data Science, University of Bayreuth, Bayreuth,  (aldo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='faisal@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' NW acknowledges his support by the Royal  Thai Government Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' AAF acknowledges his support by UKRI  Turing AI Fellowship (EP/V025449/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  the need for gravity compensation or end-point precision  in their control (the feet are strapped to the pedals so all  motions are on sagittal plane), so movements are relatively  constrained and requirements on control strategies can be  relatively, low precision (often involving simple periodic  stimulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' This is very different in arm movements, where  reaches towards an end-point require non-linear muscle co-  ordination in the plane, and gravity compensating activity in  3D movements in a volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Early work in this nascent field  therefore focus on single-joint control, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=', of the elbow joint  [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The literature on multiple-joint control of arm cases,  however, is quite limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' This is partly because controlling  FES in single-joint cases can be achieved through simple,  model-free, error-based control such as PID controllers ,  while the multiple-joint cases require significantly more  complex controls that have to include dynamical models in  the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Contrary to the fact that dynamic models play important  roles in the controls and that the neuromechanics of the  human arm is complex, many models used in FES control  studies are relatively simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' In planar arm motion, for  example, two-joint linkages with six muscles represented by  straight lines models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='1a) are one of the most commonly  used [4]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' These models offer fast computation, but may  not well capture the effects of muscle routes and their  deformation during the movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' In addition, the muscles’  properties themselves vary across the studies, thereby lacking  standardization and making the results difficult to reproduce  or compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' On the other end, there exist commercial neu-  romechanical simulation software such as LifeModeler with  highly detailed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' These models, however, are suitable  for detailed analyses of particular situations such as er-  gonomic designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Besides, the closed-source and commercial  nature of the software could limit its usage among research  communities, thereby not addressing the standardization and  reproducibility issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Besides the model-related issues, governing the control  rules itself is a major challenge in inducing movements  using FES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Regards specifically to inducing multiple-joint  arm movements, successes in real-world settings are limited  and, oftentimes, require assistive devices [2], [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' For  example, the PID controller with inverse dynamics [2] can  induce a narrow range of movements, while iterative learning  control [8] can induce a longer range but is limited to  repetitive trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' These limited successes are partly  attributed to the difficulties of conventional methods in  dealing with complexities and variations of human arms’  neuromechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' To our knowledge, an FES control method  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='04004v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='SY]  10 Jan 2023  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  (a) An example of simple planar arm models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' (b) Our neuromechanics Arm-Planar model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' (c) An illustration of muscle wrapping at the elbow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  (d) Our neuromechanics Arm-3D model and (e) its shoulder muscles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  that can induce arbitrary movements across different subjects  without intensive parameter tuning has not yet been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Keeping both model-related and control governing issues  in focus, we here present our approach toward FES-based  restoration of arm movements that comprises two elements  which, as separate entities, can address those issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The first  element is the neuromechanics models of the human arm  built using OpenSim [10], a freely-available, open-source  neuromechanical simulation software that is well-accepted in  the communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' This allows us to build the models using  established biomechanical components, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=', muscle and joint  models, thereby addressing the standardisation issue and  providing state-of-the-art performances [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Additionally,  the open source nature of OpenSim facilitate its uses for  designing and testing different control methods which help  promote reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' In this work, we present our two  arm models designed for surface FES control usages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=',  they are designed for fast computation while maintaining  important details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' These models could be used as standards  for comparing different control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  The second element of our approach is to govern the  control policy using Reinforcement Learning (RL), a ma-  chine learning algorithm with a learning agent (RL agent)  that learns to control an environment by interacting with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  RL can learn to control complex environments, for which  hand-crafted control policies are difficult to govern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' In FES  control applications, RL is a promising method for governing  control policies for any FES control settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' In addition,  the fact that RL can provide customised stimulation for  different subjects without intensive manual configuration can  be an important factor that drives FES-based restoration of  movements outside the laboratory and toward at-home usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  In this work, we present a generic RL setup to learn control  policies for arbitrary arm-reaching tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' We demonstrate the  usage in planar and 3D arbitrary reaching tasks using our  OpenSim models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' RELATED WORKS  It is worth mentioning some related works to highlight  their limitations and the gaps that this work fulfils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Regarding  neuromechanical arm models, there exist several OpenSim  models built by the communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Closely related models are  the OpenSim core Arm26 and MoBL-ARMS Dynamic Upper  Limb models [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' These models have a few critical and  minor issues as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The first critical issue is that they  produce singularity computation in OpenSim4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='4, the latest  version, at some postures, causing crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Secondly, there  is no mechanism such as joint limits to prevent unnatural  postures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The minor issues are that there is no joint damping  that prevents unnatural joint speed and, in some postures,  the muscle paths are in the wrong positions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=', they wrap  around the wrong side of the joint or go through the bone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Regarding the applications of RL in FES control, the early  studies were based on old RL algorithms, simple planar arm  models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='1a), and a single, fixed target [4]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' A recent  study has extended these settings to multiple targets [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Our  previous works investigate cycling motions in simulation [13]  and single-joint arm movements [3] in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' A  simulation study on 3D arm motions was conducted in [14]  using a model that does not have muscle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=', the RL agent  directly controls joint torque rather than muscle stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' METHODS  a) Neuromechanical models: We use two human arm  models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' one is for planar motions (hereafter referred to as  Arm-Planar) which can be viewed as the detailed version  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='1a-like model, and the other one is for 3D motions  (hereafter referred to as Arm-3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Both models are designed  at a suitable detail level for surface FES control applications,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=', the muscles that are impossible to be stimulated sepa-  rately via surface FES are bundled together to minimise the  computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The common properties and designs of both  models are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Both models have the right arms  connected to the upper bodies located at fixed points in 3D  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The elbow and shoulder joints are modelled as pin and  ball joints, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Both joints have damping and joint  limit mechanisms that prevent unnatural joint speed and pos-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The muscles are built using a variant of Hill-type mus-  cle model DeGrooteFregly2016Muscle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Both models have 4  muscles crossing elbows: Triceps Medial, Triceps long head  (biarticular), Brachialis, and Biceps short head (biarticular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Note that Triceps lateral head is bundled with Triceps long  Pectoralis  Arm-Planar  Arm-3D  Deltoid Anterior  major C  Deltoid Lateral  Deltoid Posterior  Deltoid  Posterior  Pectoralis major C  N  Biceps short head  Triceps long head  Brachialis  Triceps  Table  Medial  Arm Support  a  b  dhead, and Biceps long head is bundled with Biceps short  head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' These muscles wrap around a cylindrical object at the  elbows (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The muscles’ excitation-activation delay is  changed from the default setting of 40 ms to 100 ms to  capture a longer delay of FES-induced muscle activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  The other muscle parameters such as maximum isometric  force follow those in the Arm26 and MoBL-ARMS Dynamic  Upper Limb models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The tendon slack length parameters are  optimised using a genetic algorithm called CMAES [15] to  equilibrate the passive forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The other parts of both models  have slightly different designs described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  The Arm-Planar model has 6 muscles in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The  other muscles besides the aforementioned 4 muscles are the  Pectoralis Major Clavicular head (Pectoralis Major C) and  Deltoid posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' These muscles wrap around a cylindrical  object at the shoulder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The shoulder joint is only allowed  to rotate around the vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The arm is supported at  the wrist by an arm supporter (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='1b) that moves on a  table with low friction and provides gravity compensation  to the arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The Arm-3D model has 8 muscles in total which  are those 6 muscles of the Arm-Planar model plus Deltoid  lateral and Deltoid anterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' At the shoulder, there are three  half ellipsoids functioning as muscle wrapper objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' These  ellipsoids are carefully placed to support the full range of  movements and prevent the wrong muscle path issues of the  existing OpenSim models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The shoulder joint can rotate in all  directions except the direction that causes the arm to twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  b) Reinforcement Learning controllers: The overview  of RL algorithms is briefly described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' RL learns  a task through reward signals collected from the interaction  with an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The interactions occur in a discrete-  time fashion, starting with the agent observing the envi-  ronment’s state st and selecting an action at based on its  policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The action causes the environment to be in a new  state st+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The agent then receives an immediate reward  rt and observes the new state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' This interaction experience  is collected as a tuple (st, at, rt, st+1) which is stored in  a replay buffer D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' This tuple is used to learn an optimal  policy π∗ that maximises a return R–the sum of discounted  immediate rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  The RL task here is to apply the muscle stimulation to  move the arm to the desired pose which is specified by target  joint angles–shoulder and elbow (θtar,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The state vector st  is [θt, ˙θt, θtar,t]T , where θt and ˙θt are the joint angles and  angular velocities measured at time t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Note that  appending the targets into the state vector allows the agents  to learn goal-directed policies that can perform arbitrary  reaching tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The action vector at comprises normalised  stimulation intensities (i ∈ [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The immediate reward rt  is simply computed using the square error and action penalty  as rt = −(θt+1 − θtar,t)2 − Σn  i=0ai  n  , where n is the number  of stimulated muscles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' With this setting, the optimal policy  π∗ is simply the policy that causes the angles to be close to  the targets with minimal stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  The mechanism of finding the optimal policy varies across  different RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' In this work, we choose the soft  actor-critic (SAC) algorithm [16] because of its state-of-the-  art performance in terms of both sample efficiency and stabil-  ity across different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' SAC has two components:  an actor and a critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' In simple terms, the critic learns to  estimate the expected return of a state-action pair, known as  the Q value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The Q value is used to adjust the actor’s policy  π by increasing the probability of choosing an action with  a high Q value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Both actor and critic are parameterised by  neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' we, based on empirical experiments and our  previous works [3], [13], use fully-connected neural networks  that have two hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The output layer of the actor  has a sigmoid activation function to squash the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  The setups for the planar and 3D cases are slightly  different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' In the planar case, the involved angles are the elbow  and shoulder angles which rotate about the vertical axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The  state vector is therefore s ∈ R6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The action vector a has 4  elements (ai ∈ [0, 1]) which are the normalised stimulation  intensities of the Brachialis and Biceps short head, Triceps  Medial and Triceps long head, Pectoralis Major C, and  Deltoid posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Note that we set the Biceps stimulation  to affect two muscles because, normally in a real situation,  only a single pair of electrodes are placed above Biceps (and  similarly for Triceps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' In the 3D case, the shoulder joint can  rotate in 2 directions, and the Deltoid lateral and Deltoid  anterior are stimulated via the same pair of electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Hence,  the state vector becomes s ∈ R9, and the action vector has  5 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  The RL training is episodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Each episode starts with a  random arm pose and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Each episode has 100 time steps  with 100 ms time-step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The target changes to a new  random value at the 50th time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Every 5 training episodes,  the agents’ performances are evaluated on 50 test episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' RESULTS  RL agents are trained for 250 and 500 episodes on the  Arm-Planar and Arm-3D models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The training  is repeated 10 times to evaluate the robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The perfor-  mance evaluations along the training are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  In both cases, the best RL’s performances in rmse measure  are approximately 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The performance development in the  planar case is significantly quicker than in the 3D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The  standard deviations in both cases are in low, confined ranges  which suggests the robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='2b and d show examples of control performances  in planar and 3D cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' In both cases, the  RL agents can track arbitrary trajectories that have never  been assigned during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The performance in the  planar case is slightly better than that in the 3D case as  the planar movements are less complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='2c and e show  the stimulation applied during the tracking tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' In both  cases, brief bursts of stimulation appear when the targets  change, followed by steady stimulation that co-contraction  the muscles to stabilise the arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The bursts do not appear  when the targets change in a ramping manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' DISCUSSION & CONCLUSION  We present our approach toward FES-based restoration  of arm movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Our approach has two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  (a) Performance evaluation along the training in (red) Arm-3D and (blue) Arm-Planar cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The solid lines and the shades show the mean and  standard deviation of 10 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The examples of trajectory tracking in (b) Arm-Planar and (d) Arm-3D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The dash and solid lines are the targets and  actual angles that the RL agents achieve, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' (c) and (e) show the stimulation along the tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  first element is to use OpenSim to build neuromechanical  models of the arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' This strategy can help facilitate the build-  ing process and standardise the models on which different  control methods are tested and compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Furthermore, we  present our two OpenSim models: Arm-Planar and Arm-  3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The second element is to govern the control rules by  using reinforcement learning which can provide customised  stimulation for different subjects and settings with minimal  technical intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' We present a generic RL training  setup, demonstrate its applications on our OpenSim models  and show our RL’s performances in performing arbitrary  reaching tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Although this approach has promising simulation results,  several further steps have to be taken to translate it into  real-world usages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' One step is to optimise the models to  accurately represent the dynamics of a certain subject’s arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  This is yet a process that can be done using OpenSimMoCo  [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The customised model can be used for pre-training the  RL before transferring it to the real subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Another step is  to take muscle fatigue into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The fatigue behaviour  can be included in OpenSim models without touching the  source code by using the method presented in our previ-  ous work [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' The fatigue will cause the environment’s  state to become partially observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Based on [5], [7] and  our empirical investigation, the fatigue does not cause RL  to completely fail, but the control performance decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Lastly, in the early period of the training, the RL-controlled  stimulation is unpredictable and random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' This raises an issue  about safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' This issue can be mitigated by using offline RL  in the early period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  To summarise, the combination of neuromechanical mod-  els and RL can address existing challenges in FES control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Although the translation into real-world usages involves  several further steps, its potential is emerging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
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+page_content='  Case: Arm-Planar  RMSE: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='21 °  Case: Arm-3D  RMSE: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='82  100  100  a  b  --  Elbow  Arm-3D  q  Shoulder-x  60  Arm-Planar  80  80 -  Shoulder-Z  60  60 -  Angle  50  40  40  20  20  Shoulder  Elbow  0  100  100-  Biceps  Delt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Post;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Biceps  Pect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='Maj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  c  e  30  Triceps  Delt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content=' Lat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Triceps  Deltoid Post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  Stimulation (%)  80  Pect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='Maj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}
+page_content='  80  60  60 -  20 +  40  40  W  000000  20  20 -  10:  0 :  0  50  100  150  200  250  300  0  1015  2025  30  35  40  45  50  55  60  0  5  10  15  20  25  30  Episode  time [s]  time [s]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfnAiB/content/2301.04004v1.pdf'}

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+Complex dynamics of knowledgeable monopoly models with gradient
+mechanisms
+Xiaoliang Lia, Jiacheng Fub, and Wei Niu∗b,c
+aSchool of Digital Economics, Dongguan City University, Dongguan, China
+bSino-French Engineer School, Beihang University, Beijing, China
+cBeihang Hangzhou Innovation Institute Yuhang, Hangzhou, China
+Abstract
+In this paper, we explore the dynamics of two monopoly models with knowledgeable players.
+The first model was initially introduced by Naimzada and Ricchiuti, while the second one is sim-
+plified from a famous monopoly introduced by Puu. We employ several tools based on symbolic
+computations to analyze the local stability and bifurcations of the two models. To the best of our
+knowledge, the complete stability conditions of the second model are obtained for the first time. We
+also investigate periodic solutions as well as their stability. Most importantly, we discover that the
+topological structure of the parameter space of the second model is much more complex than that
+of the first one. Specifically, in the first model, the parameter region for the stability of any periodic
+orbit with a fixed order constitutes a connected set. In the second model, however, the stability
+regions for the 3-cycle, 4-cycle, and 5-cycle orbits are disconnected sets formed by many disjoint
+portions. Furthermore, we find that the basins of the two stable equilibria in the second model are
+disconnected and also have complicated topological structures. In addition, the existence of chaos
+in the sense of Li-Yorke is rigorously proved by finding snapback repellers and 3-cycle orbits in the
+two models, respectively.
+Keywords: monopoly; gradient mechanism; stability; periodic orbit; chaos
+1
+Introduction
+Unlike a competitive market with a large number of relatively small companies producing homogeneous
+products and competing with each other, an oligopoly is a market supplied only by a few firms. It
+is well known that Cournot developed the first formal theory of oligopoly in [7], where players are
+supposed to have the naive expectations that their rivals produce the same quantity of output as in
+the immediately previous period. Cournot introduced a gradient mechanism of adjusting the quantity
+of output and proved that his model has one unique equilibrium, which is globally stable provided
+that only two firms exist in the market.
+A monopoly is the simplest oligopoly, which is a market served by one unique firm. In the existing
+literature, a market supplied by two, three, or even four companies is called a duopoly [17], a triopoly
+[19], or a quadropoly [21], respectively.
+However, a monopoly may also exhibit complex dynamic
+behaviors such as periodic orbits and chaos if the involved firm is supposed to be boundedly rational.
+As distinguished by Matsumoto and Szidarovszky [22], a boundedly rational monopolist is said to be
+knowledgeable if it has full information regarding the inverse demand function, and limited if it does
+not know the form of the inverse demand function but possesses the values of output and price only
+in the past two periods. Knowledgeable and limited players have been considered in several monopoly
+models.
+For example, Puu [26] introduced a monopoly where the inverse demand function is a cubic func-
+tion with an inflection point, and the marginal cost is quadratic. In this model, the monopolist is
+∗Corresponding author: [email protected]
+1
+arXiv:2301.01497v1  [econ.TH]  4 Jan 2023
+
+supposed to be a limited player. Puu indicated that there exist multiple (at most three) equilibria, and
+complex dynamics such as chaos may appear if the reactivity of the monopolist becomes sufficiently
+large. Moreover, Puu’s model was reconsidered by Al-Hdaibat and others in [1], where a numerical
+continuation method is used to compute solutions with different periods and determine their stability
+regions. In particular, they analytically investigated general formulae for solutions with period four.
+It should be mentioned that the equilibrium multiplicity and complex dynamics of Puu’s model
+might depend strictly on the inverse demand function that has an inflection point. In this regard,
+Naimzada and Ricchiuti [25] introduced a simpler monopoly with a knowledgeable player, where the
+inverse demand function is still cubic but has no inflection points. It was discovered that complex
+dynamics can also arise, especially when the reaction coefficient to variation in profits is high. Askar [2]
+and Sarafopoulos [27] generalized the inverse demand function of Naimzada and Ricchiuti to a function
+of a similar form, but the degree of their function could be any positive integer. The difference is that
+the cost function in Askar’s model is linear but quadratic in Sarafopoulos’s.
+Cavalli and Naimzada [4] studied a monopoly model characterized by a constant elasticity demand
+function, in which the firm is also assumed to be knowledgeable with a linear cost. They focused on
+the equilibrium stability as the variation of the price elasticity of demand and proved that there are
+two possible different cases, where elasticity has either a stabilizing or a mixed stabilizing/destabilizing
+effect. Moreover, Elsadany and Awad [8] explored a monopoly game with delays where the inverse
+demand is a log-concave function. Caravaggio and Sodini [3] considered a nonlinear model, where
+a knowledgeable monopolist provides a fixed amount of an intermediate good and then uses this
+good to produce two vertically differentiated final commodities. They found that there are chaotic
+and multiple attractors. Furthermore, continuous dynamical systems have also been applied in the
+study of monopolistic markets. In [23], Matsumoto and Szidarovszky proposed a monopoly model
+formulated in continuous time and investigated the effect of delays in obtaining and implementing the
+output information. Motivated by the aforementioned work, other remarkable contributions including
+[9, 10] were done in this strand of research.
+In our study, we consider two monopoly models formulated with discrete dynamical systems, where
+the players are supposed to be knowledgeable. The two models are distinct mainly in their inverse
+demand functions. The first model uses the inverse demand of Naimzada and Ricchiuti [25], while the
+second one employs that of Puu [26]. For both models, we analyze the existence and local stability of
+equilibria and periodic solutions by using tools based on symbolic computations such as the method
+of triangular decomposition and the method of partial cylindrical algebraic decomposition. It should
+be mentioned that different from numerical computations, symbolic computations are exact, thus the
+results can be used to rigorously prove economic theorems in some sense.
+The main contributions of this paper are as follows. To the best of our knowledge, the complete
+stability conditions of the second model are obtained for the first time. We also investigate the periodic
+solutions in the two models as well as their stability. Most importantly, we find different topological
+structures of the parameter spaces of the two considered models. Specifically, in the first model, the
+parameter region for the stability of any periodic solution with a fixed order constitutes a connected
+set. In the second model, however, the stability regions for the 3-cycle, 4-cycle, and 5-cycle orbits are
+disconnected sets formed by many disjoint portions. In other words, the topological structures of the
+regions for stable periodic orbits in Model 2 are much more complex than those in Model 1. This may
+be because the inverse demand function of Model 2 has an inflection point. Furthermore, according
+to our numerical simulations of Model 2, it is discovered that the basins of the two stable equilibria
+are disconnected and also have complex topological structures. In addition, the existence of chaos in
+the sense of Li-Yorke is rigorously proved by finding snapback repellers and 3-cycle orbits in the two
+models, respectively.
+The rest of this paper is organized as follows. In Section 2, we revisit the construction of the two
+models. In Section 3, the local stability of the equilibrium is thoroughly studied, and bifurcations
+through which the equilibrium loses its stability are also investigated. In Section 4, the existence and
+stability of periodic orbits with relatively lower orders are explored for the two models. In Section 5,
+we rigorously derive the existence of chaotic dynamics in the sense of Li-Yorke. The paper is concluded
+with some remarks in Section 6.
+2
+
+2
+Basic Models
+Suppose a monopolist exists in the market, and the quantity of its output is denoted as x. We use P(x)
+to denote the price function (also called inverse demand function), which is assumed to be downward
+sloping, i.e.,
+dP(x)
+dx
+< 0,
+for any x > 0.
+(1)
+It follows that P(x) is invertible.
+The demand function (the inverse of P(x)) exists and is also
+downward sloping. Furthermore, the cost function is denoted as C(x). Then the profit is
+Π(x) = P(x)x − C(x).
+The monopolist is assumed to adopt a gradient mechanism of adjusting its output to achieve
+increased profits.
+Suppose that the firm is a knowledgeable player, which means that it has full
+information regarding the inverse demand function P(x) and has the capability of computing the
+marginal profit dΠ/dx. The firm adjusts its output by focusing on how the variation of x affects the
+variation of Π(x). Specifically, the adjustment process is formulated as
+x(t + 1) = x(t) + K dΠ(x(t))
+dx(t)
+,
+K > 0.
+Since K > 0, a positive marginal profit induces the monopolist to adjust the quantity of its output in
+a positive direction and vice versa.
+The first model considered in this paper was initially proposed by Naimzada and Ricchiuti [25],
+where a cubic price function without the inflection point is employed. We restate the formulation of
+this model in the sequel.
+Model 1. The price function is cubic and the cost function is linear as follows.
+P(x) = a − bx3,
+C(x) = cx,
+where a, b, c are parameters. The downward sloping condition (1) is guaranteed if dP/dx = −3bx2 < 0,
+that is if b > 0. Moreover, assume that the marginal cost dC/dx = c > 0. We adopt the general
+principle of setting price above marginal cost, i.e., P(x) − c > 0 for any x ≥ 0. Therefore, we must
+have that a > c. One knows the profit function is
+Π(x) = P(x)x − C(x) = (a − bx3)x − cx = (a − c)x − bx4.
+Thus, the gradient adjustment mechanism can be described as
+x(t + 1) = x(t) + K(a − c − 4bx3(t)),
+K > 0.
+Without loss of generality, we denote f = 4bK and e = (a − c)/4b. Then, the model is simplified into
+a map with only two parameters:
+x(t + 1) = x(t) + f(e − x3(t)),
+e, f > 0.
+(2)
+The second model considered in this paper is simplified from a famous monopoly model introduced
+by Puu [26]. We retain the same inverse demand function and cost function. The only difference is
+that the monopolist in our model is knowledgeable, whereas the monopolist in Puu’s original model
+is limited.
+Model 2. The price function is cubic of a more general form
+P(x) = a1 − b1x + c1x2 − d1x3,
+where a1, b1, c1, d1 > 0 are parameters. The cost function is also cubic and has no fixed costs, i.e.,
+C(x) = a2x − b2x2 + c2x3,
+3
+
+where a2, b2, c2 > 0. Hence, the profit function becomes
+Π(x) = P(x)x − C(x) = (a1 − a2)x − (b1 − b2)x2 + (c1 − c2)x3 − d1x4,
+which can be denoted as
+Π(x) = ax − bx2 + cx3 − dx4
+with
+a = a1 − a2, b = b1 − b2, c = c1 − c2, and d = d1.
+For the sake of simplicity, we assume that a, b, c, d > 0. The marginal profit dΠ/dx is directly obtained
+and the gradient adjustment mechanism can be formulated as
+x(t + 1) = x(t) + K(a − 2bx(t) + 3cx2(t) − 4dx3(t)),
+a, b, c, d > 0.
+(3)
+3
+Local Stability and Bifurcations
+Firstly, we explain the main idea of the symbolic approach used in this paper by analyzing stepwise
+the local stability of Model 1. Then the theoretical results of Model 2 are reported without giving all
+the calculation details.
+3.1
+Model 1
+Proposition 1. Model 1 always has a unique equilibrium, which is stable if
+4b(a − c)2K3 < 8
+27
+Moreover, there is a period-doubling bifurcation if
+4b(a − c)2K3 = 8
+27.
+The above proposition is a known result, which was first derived by Naimzada and Ricchiuti
+[25].
+Indeed, this proposition can be easily proved since the analytical expression of the unique
+equilibrium can be obtained, i.e., x∗ = ( a−c
+4b )1/3. However, we would like to provide another proof in
+a computational style to demonstrate in detail how our symbolic approach works.
+In what follows, the model formulation (2) is taken. By setting x(t + 1) = x(t) = x, we acquire
+the equilibrium equation x = x + f(e − x3). An equilibrium x of the one-dimensional iteration map is
+locally stable if
+�����
+dx(t + 1)
+dx(t)
+����
+x(t)=x
+����� =
+��1 − 3fx2�� < 1.
+Moreover, we say the equilibrium x to be feasible if x > 0. Thus, a stable and feasible equilibrium can
+be characterized as a real solution of
+�
+�
+�
+�
+�
+x = x + f(e − x3),
+��1 − 3fx2�� < 1,
+x > 0, e > 0, f > 0.
+(4)
+Although system (4) is so simple that one can solve the closed-form expression of x from the
+equality part, the problem is how we handle a general polynomial that may have no closed-form
+solutions.
+Furthermore, it is also a nontrivial task to identify the conditions on the parameters
+whether a system with inequalities has real solutions. In [18], the first author of this paper and his
+coworker proposed an algebraic approach to systematically tackle these problems. The main idea of
+this approach is as follows.
+The parametric system (4) is univariate in x. For a univariate system, we introduce a key concept
+called border polynomial in the sequel. One useful property of a border polynomial is that its real
+zeros divide the parameter space into separated regions and the solution number of the original system
+is invariant for all parameter points in each region.
+4
+
+Definition 1 (Border Polynomial). Consider a univariate system
+�
+P(u, x) = �m
+i=0 ai(u) xi = 0,
+Q1(u, x) > 0, . . . , Qs(u, x) > 0,
+(5)
+where P and Q1, . . . , Qs are univariate polynomials in x, and u stands for all parameters. The product
+am(u) · discr(P) ·
+s
+�
+i=1
+res(P, Qi)
+is called the border polynomial of system (5). Here, res(F, G) stands for the resultant of two polyno-
+mials F and G, while discr(F) denotes the discriminant of F.
+More specifically, the formal definitions of the resultant and the discriminant in the above definition
+are given as follows. Let
+F =
+m
+�
+i=0
+ai xi,
+G =
+l
+�
+j=0
+bj xj
+be two univariate polynomials in x with coefficients ai, bj in the field of complex numbers, and am, bl ̸=
+0. The determinant
+���������������
+am
+am−1
+· · ·
+a0
+...
+...
+...
+...
+am
+am−1
+· · ·
+a0
+bl
+bl−1
+· · ·
+b0
+...
+...
+...
+...
+bl
+bl−1
+· · ·
+b0
+���������������
+�
+�
+� l
+�
+�
+� m
+is called the Sylvester resultant (or simply resultant) of F and G, and denoted by res(F, G). The
+resultant of F and its derivative dF/dx, i.e., res(F, dF/dx), is called the discriminant of F and
+denoted by discr(F). The following lemma is one of the well-known properties of resultants, which
+could be found in [24].
+Lemma 1. Two univariate polynomials F and G have common zeros in the field of complex numbers
+if and only if res(F, G) = 0. Moreover, a univariate polynomial F has a multiple zero in the field of
+complex numbers if and only if discr(F) = 0.
+It is worth noticing that the number of real zeros of P may change when the leading coefficient
+am(u) or the discriminant discr(P) goes from non-zero to zero and vice versa. In addition, if res(P, Qi)
+goes across zero, then the zeros of P will pass through the boundaries of Qi > 0, which means that
+the number of real roots of (5) may change. Therefore, the following lemma is derived.
+Lemma 2. Consider a univariate system as (5). Let A and B be two points in the space of parameters
+u. Suppose that any of A, B does not annihilate the border polynomial of system (5). If there exists
+a real path C from A to B such that any point on C is not a root of the border polynomial, then the
+number of real solutions of system (5) evaluated at A is the same as that at B.
+Since 1 − 3fx2 < 1, we know that system (4) is equivalent to
+�
+�
+�
+�
+�
+x3 − e = 0,
+2 − 3fx2 > 0,
+x > 0, e > 0, f > 0.
+(6)
+We have am = 1 and discr(x3 − e) = 27e2.
+Moreover, res(x3 − e, 2 − 3fx2) = −27e2f3 + 8 and
+res(x3−e, x) = e. According to Definition 1, the border polynomial of system (6) is 27e3(−27e2f3+8),
+the zeros of which are marked in blue as shown in Figure 1. This blue curve divides the parameter
+set {(e, f) | e > 0, f > 0} into two (the northeast and the southwest) regions.
+5
+
+S2
+S1
+A
+Real path C
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+e
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+f
+Figure 1: Partitions of the parameter space of Model 1 and sample points
+Notice the two points S2 and A in Figure 1. One can find a real path C from A to S2 such that
+it does not pass through the blue curve. According to Lemma 2, system (6) has the same number of
+real roots with the parameters evaluated at S2 and A. This means that the number of real solutions
+of system (6) is invariant in the northeast region. Therefore, we can choose a sample point from each
+region to determine the root number. For this simple system, sample points might be selected directly
+by eyes, e.g., S1 = (1, 1/2), S2 = (1, 1). However, the choosing process might be extremely complex in
+general, which could be done automatically by using, e.g., the method of partial cylindrical algebraic
+decomposition or called the PCAD method [5].
+For each region, one can determine the root number by counting roots of the non-parametric system
+of (6) evaluated at the corresponding sample point. Take S1 as an example, where (6) becomes
+�
+x3 − 1 = 0, 2 − 3
+2x2 > 0, x > 0
+�
+.
+(7)
+In order to count the number of its real roots, an obvious way is directly solving x3 −1 = 0, i.e., x = 1,
+and then checking whether 2 − 3
+2x2 > 0 and x > 0 are satisfied. The result is true, which means that
+there exists one unique real solution of (7). However, it is difficult to precisely obtain all real zeros
+of a general univariate system since root formulae do not exist for polynomials with degrees greater
+than 4. Therefore, a more systematic method called real root counting [31] is generally needed here,
+and we demonstrate how this method works by using (7) as an example.
+It is noted that x3 −1, 2− 3
+2x2 and x have no common zeros, i.e., they have no factors in common.
+Otherwise, one needs to reduce the common factors from the inequalities first. After that, we isolate
+all real zeros of 2 − 3
+2x2 and x by rational intervals, e.g.,
+�
+−12
+10, −11
+10
+�
+,
+�
+− 1
+10, 1
+10
+�
+,
+�11
+10, 12
+10
+�
+.
+(8)
+Although it is trivial for this simple example, the isolation process could be particularly tough for
+general polynomials, which may be handled by using, e.g., the modified Uspensky algorithm [6].
+Moreover, the intervals can be made as small as possible to guarantee no zeros of x3 − 1 lie in these
+intervals, which could be checked by using, e.g., Sturm’s theorem [28]. Thus, the real zeros of x3 − 1
+must be in the complement of (8):
+�
+−∞, −12
+10
+�
+,
+�
+−11
+10, − 1
+10
+�
+,
+� 1
+10, 11
+10
+�
+,
+�12
+10, +∞
+�
+.
+(9)
+In each of these open intervals, the signs of 2 − 3
+2x2 and x are invariant and can be determined
+by checking them at selected sample points.
+For instance, to determine the sign of 2 − 3
+2x2 on
+6
+
+(12/10, +∞), we check the sign at a sample point, e.g., x = 2. We have that 2 − 3
+2x2|x=2 = −4 < 0,
+thus 2 − 3
+2x2 < 0 on (12/10, +∞). Similarly, it is obtained that the signs of 2 − 3
+2x2 and x at (9) are
+−, +, +, − and −, −, +, +, respectively. Hence, (1/10, 11/10) is the only interval such that the two
+inequalities 2 − 3
+2x2 > 0 and x > 0 of system (7) are simultaneously satisfied.
+We focus on (1/10, 11/10). Using Sturm’s theorem, we can count the number of the real zeros
+of x3 − 1 at (1/10, 11/10), which is one. Therefore, system (6) has one real root at S1 = (1, 1/2).
+The above approach works well for a system formulated with univariate polynomial equations and
+inequalities although some steps seem silly and not necessary for this simple example. Similarly, we
+know that system (6) has no real roots at S2 = (1, 1).
+In conclusion, system (6) has one real root if the parameters take values from the southwest region
+where S1 lies, and has no real roots if the parameters take values from the northeast region where S2
+lies. Furthermore, the inequalities of some factors of the border polynomial may be used to explicitly
+describe a given region. It is evident that 27e2f3 −8 < 0 describes the region where S1 lies. Therefore,
+Model 1 has one unique stable equilibrium provided that
+e2f3 =
+�a − c
+4b
+�2
+(4bK)3 = 4b(a − c)2K3 < 8
+27,
+which is consistent with Proposition 1.
+According to the classical bifurcation theory, for a one-dimensional iteration map x(t+1) = F(x(t)),
+we know that bifurcations may occur if
+�����
+dx(t + 1)
+dx(t)
+����
+x(t)=x
+����� =
+����
+dF
+dx
+���� = 1.
+More specifically, if dF/dx = −1, then the system may undergo a period-doubling bifurcation (also
+called flip bifurcation), where the dynamics switch to a new behavior with twice the period of the
+original system. On the other hand, if dF/dx = 1, then the system may undergo a saddle-node (fold),
+transcritical, or pitchfork bifurcation. One might determine the type of bifurcation from the change
+in the number of the (stable) equilibria. In the case of saddle-node bifurcation, one stable equilibrium
+(a node) annihilates with another unstable one (a saddle). Before and after a transcritical bifurcation,
+there is one unstable and one stable equilibrium, and the unstable equilibrium becomes stable and
+vice versa. In the case of pitchfork bifurcation, the number of equilibria changes from one to three or
+from three to one, while the number of stable equilibria changes from one to two or from one to zero.
+Accordingly, it is concluded that Model 1 may undergo a period-doubling bifurcation if
+e2f3 = 4b(a − c)2K3 = 8
+27,
+and there are no other bifurcations.
+3.2
+Model 2
+According to (3), by setting x(t + 1) = x(t) = x, we know that Model 2 has at most three equilibria.
+The analytical expressions of the equilibria exist, but are complex, i.e.,
+x1 =
+3√
+M
+12d − 8bd − 3c2
+4d
+3√
+M
++ c
+4d,
+x2,3 = −
+3√
+M
+24d + 8bd − 3c2
+8d
+3√
+M
++ c
+4d ± i
+√
+3
+2
+�
+3√
+M
+12d + 8bd − 3c2
+4d
+3√
+M
+�
+,
+(10)
+where
+M = 12d
+√
+3
+�
+108a2d2 − 108abcd + 27 ac3 + 32b3d − 9b2c2 + 216ad2 − 108bcd + 27c3.
+7
+
+Furthermore, an equilibrium x is locally stable provided that
+�����
+dx(t + 1)
+dx(t)
+����
+x(t)=x
+����� =
+��1 + K(−2b + 6cx − 12dx2)
+�� < 1.
+Hence, a stable equilibrium of map (3) is a real solution of
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+x = x + K(a − 2bx + 3cx2 − 4dx3),
+K(−2b + 6cx − 12dx2) < 0,
+2 + K(−2b + 6cx − 12dx2) > 0,
+x > 0, a > 0, b > 0, c > 0, d > 0.
+(11)
+Obviously, analyzing the stable equilibrium by substituting the closed-form solutions (10) into (11)
+is complicated and impractical. In comparison, the approach applied in the analysis of Model 1 does
+not require explicitly solving any closed-form equilibrium. If the analytical solution has a complicated
+expression or even if there are no closed-form solutions, our approach still works in theory.
+Concerning the border polynomial of system (11), we compute
+discr(K(a − 2bx + 3cx2 − 4dx3)) = −16K5dR1,
+res(K(a − 2bx + 3cx2 − 4dx3), K(−2b + 6cx − 12dx2)) = −16K5dR1,
+res(K(a − 2bx + 3cx2 − 4dx3), 2 + K(−2b + 6cx − 12dx2)) = −16K2dR2,
+res(K(a − 2bx + 3cx2 − 4dx3), x) = −Ka,
+where
+R1 = 108a2d2 − 108abcd + 27ac3 + 32b3d − 9b2c2,
+R2 = 108K3a2d2 − 108K3abcd + 27K3ac3 + 32K3b3d − 9K3b2c2 − 24Kbd + 9Kc2 − 8d.
+Therefore, the border polynomial is −16384 d4K14aR2
+1R2, the zeros of which divide the parameter set
+{(a, b, c, d, K) | a, b, c, d, K > 0} into separated regions. The PCAD method [5] permits us to select at
+least one sample point from each region. In Table 1, we list the 30 selected sample points and the
+corresponding numbers of distinct real solutions of system (11).
+Table 1: Selected Sample Points in {(a, b, c, d, K) | a, b, c, d, K > 0}
+(a, b, c, d, K)
+num
+R1
+R2
+(a, b, c, d, K)
+num
+R1
+R2
+(1, 1, 1/4, 1/64, 1/2)
+2
+−
+−
+(1, 1, 1/4, 1/64, 1)
+1
+−
++
+(1, 1, 1/4, 1/64, 2)
+0
+−
+−
+(1, 1, 1/4, 19/1024, 1)
+2
+−
+−
+(1, 1, 1/4, 19/1024, 2)
+1
+−
++
+(1, 1, 1/4, 19/1024, 3)
+0
+−
+−
+(1, 1, 1/4, 1/16, 1)
+1
++
+−
+(1, 1, 1/4, 1/16, 2)
+0
++
++
+(1, 1, 1/4, 1, 1/2)
+1
++
+��
+(1, 1, 1/4, 1, 1)
+0
++
++
+(1, 1, 3/8, 1/64, 1/8)
+1
++
+−
+(1, 1, 3/8, 1/64, 1)
+0
++
++
+(1, 1, 3/8, 1/32, 1/4)
+2
+−
+−
+(1, 1, 3/8, 1/32, 1)
+1
+−
++
+(1, 1, 3/8, 1/32, 17)
+0
+−
+−
+(1, 1, 3/8, 49/1024, 1)
+2
+−
+−
+(1, 1, 3/8, 49/1024, 4)
+1
+−
++
+(1, 1, 3/8, 49/1024, 8)
+0
+−
+−
+(1, 1, 3/8, 1/16, 1)
+1
+−
++
+(1, 1, 3/8, 1/16, 3)
+0
+−
+−
+(1, 1, 3/8, 1, 1/2)
+1
++
+−
+(1, 1, 3/8, 1, 1)
+0
++
++
+(1, 1, 15/32, 1/16, 1/2)
+1
++
+−
+(1, 1, 15/32, 1/16, 1)
+0
++
++
+(1, 1, 15/32, 3/32, 1)
+1
++
+−
+(1, 1, 15/32, 3/32, 8)
+0
++
++
+(1, 1, 15/32, 1, 1/2)
+1
++
+−
+(1, 1, 15/32, 1, 1)
+0
++
++
+(1, 1, 1, 1, 1/2)
+1
++
+−
+(1, 1, 1, 1, 1)
+0
++
++
+According to Table 1, one can see that system (11) has one real solution if and only if R1 < 0, R2 > 0
+or R1 > 0, R2 < 0. Moreover, a necessary condition that system (11) has two real solutions is that
+8
+
+R1 < 0 and R2 < 0, which is not a sufficient condition, however. For example, at (a, b, c, d, K) =
+(1, 1, 1/4, 1/64, 2), system (11) has no real solutions but R1 < 0 and R2 < 0 are fulfilled. To acquire
+the necessary and sufficient condition, additional polynomials (R3 and R4) are needed, which can be
+found in the so-called generalized discriminant list and can be picked out by repeated trials. Regarding
+the generalized discriminant list, readers may refer to [32] for more details. Due to space limitations,
+we directly report below the necessary and sufficient condition that system (11) has two real solutions
+without giving the calculation details:
+R1 < 0, R2 < 0, R3 > 0, R4 < 0,
+where
+R3 = 8Kbd − 3Kc2 + 8d,
+R4 = 432K2a2d3 − 432K2abcd2 + 108K2ac3d + 128K2b3dt2 − 36K2b2c2d + 192Kb2d2
+− 144Kbc2d + 27Kc4 + 64bd2 − 24c2d.
+We continue to analyze the bifurcations of this model. An equilibrium x of map (3) may undergo
+a period-doubling bifurcation if
+dx(t + 1)
+dx(t)
+����
+x(t)=x
+= 1 + K(−2b + 6cx − 12dx2) = −1.
+Hence, a period-doubling bifurcation may occur if the following system has at least one real solution.
+�
+�
+�
+�
+�
+x = x + K(a − 2bx + 3cx2 − 4dx3),
+K(−2b + 6cx − 12dx2) + 2 = 0,
+x > 0, a > 0, b > 0, c > 0, d > 0.
+(12)
+By using the method of triangular decomposition1, we transform the solutions of the first two equations
+of system (12) into zeros of the triangular set
+T = [(8Kbd − 3Kc2 + 4d)x − 6adK + bcK − c, R2].
+Obviously, the system {T = 0, x > 0, a > 0, b > 0, c > 0, d > 0} has at least one real positive
+solution if R2 = 0 and x = (6adK − bcK + c)/(8Kbd − 3Kc2 + 4d) > 0, i.e.,
+R2 = 0, R5 > 0,
+where
+R5 = (6adK − bcK + c)(8Kbd − 3Kc2 + 4d)
+= 48K2abd2 − 18K2ac2d − 8K2b2cd + 3K2bc3 + 24Kad2 + 4Kbcd − 3Kc3 + 4cd.
+Similarly, concerning the occurrence of a pitchfork bifurcation, we consider
+�
+�
+�
+�
+�
+x = x + K(a − 2bx + 3cx2 − 4dx3),
+K(−2b + 6cx − 12dx2) = 0,
+x > 0, a > 0, b > 0, c > 0, d > 0,
+(13)
+and count the number of stable equilibria. More details are not reported here due to space limitations.
+We summarize all the obtained results in the following theorem.
+1The method of triangular decomposition can be viewed as an extension of the method of Gaussian elimination. The
+main idea of both methods is to transform a system into a triangular form. However, the triangular decomposition
+method is available for polynomial systems, while the Gaussian elimination method is just for linear systems. Refer to
+[30, 16, 12, 29] for more details.
+9
+
+Theorem 1. Model 2 has at most two stable equilibria.
+Specifically, there exists just one stable
+equilibrium if
+R1 < 0, R2 > 0 or R1 > 0, R2 < 0,
+and there exist two stable equilibria if
+R1 < 0, R2 < 0, R3 > 0, R4 < 0.
+Moreover, there is a period-doubling bifurcation if
+R2 = 0, R5 > 0,
+and there is a pitchfork bifurcation if
+R1 = 0, R2 > 0, R6 > 0 or R1 = 0, R2 > 0, R4 < 0, R6 > 0,
+where
+R6 = 48abd2 − 18ac2d − 8b2cd + 3bc3.
+Remark 1. To the best of our knowledge, the stability results regarding the parameters a, b, c, d, K
+reported in Theorem 1 are new although the special case of a = 3.6, b = 2.4, c = 0.6, d = 0.05 has
+been discussed in [22]. The two parameters K, a play more ambitious roles than others in practice for
+K controls the speed of adjusting the monopolist’s output and a is the difference between the initial
+product price of the market without any supply and the initial marginal cost of the firm without any
+production. By fixing b = 2.4, c = 0.6 and d = 0.05, we depict the (a, K) parameter plane in Figure
+2, where the region for the existence of one stable equilibrium is colored in yellow, while the region for
+the existence of two stable equilibria is colored in blue-gray. Model 2 behaves differently from typical
+oligopolies with gradient mechanisms. As shown by Figure 2, for instance, even if the adjustment
+speed K is quite large, there always exist some values of a such that Model 2 is stable. Moreover, for
+a fixed value of K greater than around 1.7, Model 2 undergoes from instability to stability and then
+back to instability twice as the parameter a changes from low to high.
+Figure 2: The two-dimensional (a, K) parameter plane of Model 2 with the other parameters fixed:
+b = 2.4, c = 0.6, and d = 0.05. The region for the existence of one stable equilibrium is colored in
+yellow, while that of two stable equilibria is colored in blue-gray.
+10
+
+6-
+pitchfork bifurcation
+5
+curves (R1=0)
+4-
+K 3
+period-doubling bifurcation
+curves (R2=0)
+2 
+1
+0
+0
+1
+2
+3
+4
+5
+6
+a
+R=0
+R=04
+Periodic Solutions
+From an economic point of view, it is realistic to assume that a boundedly rational firm can not learn
+the pattern behind output and profits if periodic dynamics take place. In this regard, we investigate
+the existence and stability of periodic solutions with relatively lower orders in this section.
+Let I be an interval of real numbers, and let F : I → R be a function. If x ∈ I, suppose that
+F 0(x) represents x and F n+1(x) denotes F(F n(x)) for n ∈ {0, 1, . . .}. A point p ∈ I is said to be a
+periodic point with period n or order n if p = F n(p), and p ̸= F k(p) for any 1 ≤ k < n. If p is a point
+with period n, we call p �→ F 1(p) �→ · · · �→ F n(p) = p a n-cycle orbit. Furthermore, a point y ∈ I
+with period k is said to be asymptotically stable if there exists δ such that |F k(x) − y| < |x − y| for all
+x ∈ (y − δ, y + δ).
+The following lemma can be found in [15], which provides an algebraic criterion to verify the
+stability of a periodic point.
+Lemma 3. Assume that y ∈ I is a periodic point of F with period k. If F is differentiable at the points
+y, F(y), . . . , F k−1(y), then y is asymptotically stable if
+�����
+k−1
+�
+i=0
+d
+dxF(yi)
+����� < 1,
+where yi = F i(y).
+4.1
+Model 1
+We start by considering the existence of periodic orbits with order two. Assume that there is a 2-cycle
+orbit x �→ y �→ x, where �→ stands for the iteration map (2). Thus, we have
+y = x + f(e − x3),
+x = y + f(e − y3).
+(14)
+Obviously, x ̸= y should be guaranteed. Otherwise, x �→ y �→ x will degenerate into an equilib-
+rium. Then, the problem of determining the existence of 2-cycles is transformed into determining the
+existence of real solutions of
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+y = x + f(e − x3),
+x = y + f(e − y3),
+x ̸= y,
+x > 0, y > 0, e > 0, f > 0.
+(15)
+Since the above system involves two variables x and y, the approach used in Section 3 (feasible
+only for univariate systems) might not be directly employed herein.
+Remark 2. However, we can transform system (15) equivalently into univariate systems based on its
+triangular decomposition. Specifically, the triangular decomposition method permits us to decompose
+the equation part (14) into the following two triangular sets.
+T11 = [y − x, x3 − e],
+T12 = [y + fx3 − x − ef, f3x6 − 3f2x4 − 2ef3x3 + 3fx2 + 3ef2x + e2f3 − 2].
+Since the first polynomial in T11 is y − x, which implies that x = y. Thus, the zeros of T11 are not
+of our concern. We only focus on T12, where the first polynomial y + fx3 − x − ef has degree one
+with respect to y. Therefore, one can directly solve y = −fx3 + x + ef and substitute it into relative
+inequalities of system (15). In short, system (15) can be equivalently transformed into the following
+univariate system.
+�
+�
+�
+�
+�
+f3x6 − 3f2x4 − 2ef3x3 + 3fx2 + 3ef2x + e2f3 − 2 = 0,
+− fx3 + x + ef > 0,
+x > 0, e > 0, f > 0.
+After that, the approach in Section 3 can be applied. The results show that the above system has two
+real solutions if and only if 8/27 < e2f3 < 2. It is evident that these two real solutions belong to the
+11
+
+same 2-cycle orbit because x, y are symmetric and can be replaced with each other. Therefore, there
+exists at most one 2-cycle orbit in Model 1.
+According to Lemma 3, to determine whether the discovered 2-cycle is stable, we consider (15)
+together with the condition
+����
+d(x + f(e − x3))
+dx
+× d(y + f(e − y3))
+dy
+���� < 1,
+i.e.,
+��(1 − 3fx2)(1 − 3fy2)
+�� < 1.
+The technique introduced in Remark 2 is needed to transform the system into a univariate one.
+According to our calculations, the unique 2-cycle orbit is stable if and only if 729e4f6 − 3294e2f3 +
+1664 > 0 or equivalently 8/27 < e2f3 < (61 − 11
+√
+17)/27. We collect the aforementioned results in
+the following theorem.
+Theorem 2. Model 1 has at most one 2-cycle orbit, which exists if
+8/27 < e2f3 < 2.
+Furthermore, this unique 2-cycle is stable if
+8/27 < e2f3 < 61 − 11
+√
+17
+27
+,
+or approximately
+0.2962962963 < e2f3 < 0.5794754859.
+The measurement of the magnitude of periodic orbits is economically interesting for it characterizes
+the size of fluctuations in dynamic economies. For a n-cycle orbit p1 �→ p2 �→ · · · pn �→ p1, a direct
+definition of the magnitude measure is
+d = |p1 − p2| + |p2 − p3| + · · · + |pn−1 − pn| + |pn − p1|.
+However, to obtain better mathematical properties, we square each item and define the magnitude
+measure to be
+d = (p1 − p2)2 + (p2 − p3)2 + · · · + (pn−1 − pn)2 + (pn − p1)2.
+For a 2-cycle orbit x �→ y �→ x in Model 1, the magnitude measure becomes d = (x − y)2 + (y − x)2.
+Thus, we have
+�
+�
+�
+�
+�
+d − (x − y)2 − (y − x)2 = 0,
+− y + x + f(e − x3) = 0,
+− x + y + f(e − y3) = 0.
+Using the method of triangular decomposition, we decompose the solutions of the above system into
+zeros of the following two triangular sets.
+T21 = [ y − x, x3 − e, d ],
+T22 = [ y + x3f − ef − x,
+(d2f3 + 4df2 + 4f)x2 + (−6def3 − 12ef2)x + 36e2f3 − 2f2d2 − 8fd − 8,
+f3d3 − 12f2d2 − 60fd + 216e2f3 − 64 ].
+The first polynomial y − x in T21 implies that x = y. Thus, T21 is not of concern since it corresponds
+to equilibria rather than 2-cycle orbits. We focus on the last polynomial f3d3 − 12f2d2 − 60fd +
+216e2f3 − 64 in T22. By solving d from this polynomial, we obtain three solutions:
+d1 = 2
+f
+�3H
+2 + 6
+H + 2
+�
+,
+d2, d3 = 2
+f
+�
+−3H
+4 − 3
+H + 2 ± i
+√
+3
+2
+�3H
+2 − 6
+H
+��
+,
+12
+
+where
+H =
+3�
+8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2.
+Here, only the real solution d1 is meaningful. Therefore, the magnitude measure of the unique 2-cycle
+orbit in Model 1 can be expressed as
+d = 2
+f
+�
+3 3�
+8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2
+2
++
+6
+3�
+8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2 + 2
+�
+.
+In the rest of this section, similar calculations as above are repeated. We omit these computation
+details due to space limitations. Concerning 3-cycle orbits in Model 1, we need to count real solutions
+of
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+y = x + f(e − x3),
+z = y + f(e − y3),
+x = z + f(e − z3),
+x ̸= y, x ̸= z,
+x > 0, y > 0, z > 0, e > 0, f > 0.
+Based on a series of computations, we derive the following theorem.
+Theorem 3. Model 1 has no 3-cycle orbits for all possible parameter values such that e, f > 0.
+For a 4-cycle orbit x �→ y �→ z �→ w �→ x, we have the system
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+y = x + f(e − x3),
+z = y + f(e − y3),
+w = z + f(e − z3),
+x = w + f(e − w3),
+x ̸= y, x ̸= z, x ̸= w,
+x > 0, y > 0, z > 0, e > 0, f > 0.
+Furthermore, the following condition is required to guarantee that the considered 4-cycle is stable.
+����
+d(x + f(e − x3))
+dx
+× d(y + f(e − y3))
+dy
+× d(z + f(e − z3))
+dz
+× d(w + f(e − w3))
+dw
+���� < 1,
+i.e.,
+��(1 − 3fx2)(1 − 3fy2)(1 − 3fz2)(1 − 3fw2)
+�� < 1.
+As the polynomials involved in the conditions of the existence and stability of 4-cycle orbits are
+extremely complicated, we report below the obtained results in an approximate style.
+Theorem 4. Model 1 has at most one 4-cycle orbit, which exists if
+0.5794754859 < e2f3 < 1.237575627.
+Furthermore, this unique 4-cycle is stable if
+0.5794754859 < e2f3 < 0.6673871142.
+Figure 3 (a) depicts the phase diagram of the unique 4-cycle in Model 1 with e = 0.6 and f = 1.2.
+Since e2f3 = 0.62208 ∈ (0.5794754859, 0.6673871142), this unique 4-cycle in Model 1 is asymptotically
+stable according to Theorem 4. Actually, the horizontal coordinates of A, B, C, D, i.e., x, y, z, w, are
+the four points in the 4-cycle orbit. For the sake of simplicity, we connect A, B, C, D with lines and
+use the simplified phase diagram as Figure 3 (b) to demonstrate periodic solutions in the rest of this
+paper.
+13
+
+(a) phase diagram.
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+x(t-1)
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+x(t)
+A
+B
+D
+C
+(b) simplified phase diagram.
+Figure 3: The unique stable 4-cycle in Model 1 with e = 0.6 and f = 1.2.
+Furthermore, by using the same approach as we computed the magnitude of the 2-cycle orbit, we
+conclude that if a 4-cycle x �→ y �→ z �→ w �→ x exists in Model 1, its magnitude measure equals to
+d = 4
+f
+�
+3 3�
+8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2
+2
++
+6
+3�
+8 − 4e2f3 + 4(e4f6 − 4e2f3)1/2 + 2
+�
+,
+which is twice as large as that of the 2-cycle orbit.
+The parameter plane of Model 1 is shown in Figure 4. One can see that the parameter region for
+the stability of the unique equilibrium (2-cycle or 4-cycle orbit) constitutes a connected set. Moreover,
+the three regions for the stability of the equilibrium, 2-cycle, and 4-cycle adjoin without any gap. In
+the next subsection, one will find that the topological structure of the parameter space of Model 2 is
+much more complex than that of Model 1.
+Figure 4: The parameter plane of Model 1.
+The light blue, yellow, and light orange regions are
+the parameter regions for the stability of the 4-cycle orbit, the 2-cycle orbit, and the equilibrium,
+respectively.
+Figure 5 depicts the two-dimensional bifurcation diagram of Model 1 for (e, f) ∈ [0.6, 1.6]×[0.6, 1.6].
+For additional information regarding two-dimensional bifurcation diagrams, readers can refer to [13].
+14
+
+c
+A
+1
+0.8
+0.6
+B
+0.4
+D
+0.2
+-
+-
+x
+Z
+y
+0
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.
+x(t-1)21.2In the numerical simulations of Figure 5, we set the initial state to be x(0) = 1.0. Parameter points
+corresponding to periodic orbits with different orders are marked in different colors. For example,
+parameter points are colored in dark red if the order is just one (equilibria) and are marked in black
+if the order is greater than or equal to 24 (complex trajectories). In the case that the order is greater
+than 24, the black points may be viewed as the parameter values where complex dynamics such as
+chaos take place. Moreover, we also use black to mark those parameter points where the trajectories
+diverge to ∞. One can see that Figure 5 confirms the theoretical results presented in Figure 4.
+In Figure 5, the transitions between different types of periodic orbits can also be observed. One can
+see that the equilibrium loses its stability through a series of period-doubling bifurcations as the value
+of e or f increases. For example, along the line of e = 1.0, the unique stable equilibrium bifurcates
+into a stable 2-cycle orbit at f = 0.6665, which further bifurcates into a 4-cycle orbit at f = 0.8339.
+There is a stable 8-cycle orbit when f ∈ (0.8744, 0.8826). Finally, chaotic dynamics take place if the
+value of f is large enough. Additional details can be found in the one-dimensional bifurcation diagram
+presented in Figure 6, where we fix e = 1.0 and choose x(0) = 1.1 to be the initial state of iterations.
+Figure 5: The two-dimensional bifurcation diagram of Model 1 for (e, f) ∈ [0.6, 1.6] × [0.6, 1.6]. We
+choose x(0) = 1.0 to be the initial state of the iterations.
+4.2
+Model 2
+The formulation (3) of Model 2 involves five parameters, which might be particularly complex for
+symbolic computations of searching periodic solutions. In what follows, we keep K as the only pa-
+rameter and assume that a = 3.6, b = 2.4, c = 0.6, and d = 0.05. This setting is meaningful and
+has been discussed by several economists, e.g., Puu [26], Al-Hdaibat and others [1], Matsumoto and
+Szidarovszky [22].
+15
+
+1.6
+1.5
+20
+1.4
+1.3
+15
+1.2
+f 1.1
+1.0
+10
+0.9
+0.8
+5
+0.7
+0.6 -
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+eFigure 6: The one-dimensional bifurcation diagram of Model 1 with respect to f by fixing e = 1.0.
+We choose x(0) = 1.1 to be the initial state of the iterations.
+Let x �→ y �→ x be a 2-cycle orbit. Hence, we have
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+y = x + K(3.6 − 4.8x + 1.8x2 − 0.2x3),
+x = y + K(3.6 − 4.8y + 1.8y2 − 0.2y3),
+x ̸= y,
+x > 0, y > 0, K > 0.
+(16)
+Furthermore, the following condition is required if the stability of the 2-cycle is considered.
+|S(x) · S(y)| < 1,
+where
+S(x) = d(x + K(3.6 − 4.8x + 1.8x2 − 0.2x3)
+dx
+= 1 − K(4.8 − 3.6x + 0.6x2).
+(17)
+According to our computations, the following theorem is obtained.
+Theorem 5. In Model 2, the possible number of 2-cycle orbits is zero (no real solutions in system (16))
+or three (six real solutions in system (16)). There exist three 2-cycle orbits if K > 5/3. Moreover,
+two of them are stable if
+5/3 < K < (5
+√
+5 − 5)/3,
+or approximately
+1.666666667 < K < 2.060113296.
+To measure the magnitude of a 2-cycle orbit x �→ y �→ x, we also use d = (x − y)2 + (y − x)2. The
+method of triangular decomposition permits us to decompose the solutions of
+�
+�
+�
+�
+�
+d = (x − y)2 + (y − x)2,
+y = x + K(3.6 − 4.8x + 1.8x2 − 0.2x3),
+x = y + K(3.6 − 4.8y + 1.8y2 − 0.2y3)
+16
+
+1.5
+1.0
+0.5
+X
+0.0
+-0.5
+-1.0
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+finto zeros of the following triangular systems.
+T31 = [ y − 3, x − 3, d ],
+T32 = [ y − x, x2 − 6x + 6, d ],
+T33 = [ y + x − 6, Kx2 − 6Kx + 6K − 10, Kd − 24K − 80 ],
+T34 = [ 5y + x3K − 9Kx2 + (24K − 5)x − 18K,
+K2x4 − 12K2x3 + (51K2 − 5K)x2 + (−90K2 + 30K)x + 54K2 − 45K + 25,
+Kd − 6K + 10 ],
+where the last two polynomials Kd − 24K − 80 and Kd − 6K + 10 in T33 and T34 are of our concern.
+We conclude that d = (24K + 80)/K or d = (6K − 10)/K. One can see that two of the three 2-cycle
+orbits possess the same magnitude.
+For a 3-cycle orbit x �→ y �→ z �→ x, we consider the system
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+y = x + K(3.6 − 4.8x + 1.8x2 − 0.2x3),
+z = y + K(3.6 − 4.8y + 1.8y2 − 0.2y3),
+x = z + K(3.6 − 4.8z + 1.8z2 − 0.2z3),
+x ̸= y, x ̸= z,
+x > 0, y > 0, z > 0, K > 0,
+(18)
+as well as the stability condition
+|S(x) · S(y) · S(z)| < 1,
+(19)
+where S(x) is given in (17). Based on a series of calculations, we have the following theorem.
+Theorem 6. In Model 2, all possible cases for the number of (stable) 3-cycle orbits are listed in Table
+2, where
+m1 ≈ 2.417401607, m2 ≈ 2.434714456, m3 ≈ 3.302953127, m4 ≈ 3.303122765.
+Readers can refer to Remark 3 to understand how these mi are obtained.
+Table 2: Numbers of (stable) 3-cycle orbits in Model 2
+K ∈
+(0, m1)
+(m1, m2)
+(m2, m3)
+(m3, m4)
+(m4, +∞)
+3-cycles
+0
+4
+4
+8
+8
+Stable 3-cycles
+0
+2
+0
+2
+0
+Remark 3. As aforementioned, the border polynomial plays an important role. However, one can derive
+that the properties of the border polynomial reported in Lemma 2 will retain if we use the squarefree
+part of the border polynomial. The squarefree part SP of the border polynomial of (18)+(19) is
+simpler, which is given in Appendix. In Theorem 6, m1, . . . , m4 are the real roots of SP. A rigorous
+style of writing Theorem 6 is to express the conditions using factors in SP. However, this would be
+quite tedious. Since only one parameter, i.e., K, is involved in SP, the regions divided by zeros of
+SP are indeed intervals and can be approximately described as in Theorem 6. However, it should be
+noticed that the values of m1, . . . , m4 can be made arbitrarily accurate if we want because the exact
+expression of SP has already been obtained.
+Figure 7 depicts all the 3-cycle orbits in Model 2 with K = 3.303 ∈ (m3, m4), where the 6 unstable
+cycles are marked in red and the 2 stable cycles are marked in blue. It is worth noting that two of
+the unstable 3-cycle orbits in red almost coincide with the stable ones in blue, but they are different.
+We should underline that this dynamic phenomenon, derived by symbolic computations, may be too
+subtle to observe through numerical simulations.
+17
+
+0
+1
+2
+3
+4
+5
+6
+x(t-1)
+0
+1
+2
+3
+4
+5
+6
+x(t)
+Figure 7: The 3-cycle orbits in Model 2 with K = 3.303. The 6 unstable cycles are marked in red,
+while the 2 stable ones are marked in blue.
+Moreover, if measuring the magnitude of the 3-cycle orbit x �→ y �→ z �→ x with d = (x − y)2 +
+(y − z)2 + (z − x)2, then we have
+K4d4 + (−54K4 − 90K3)d3 + (972K4 + 2700K3 + 1800K2)d2
++ (−6696K4 − 19440K3 − 5400K2 + 27000K)d
++ 15552K4 + 38880K3 − 32400K2 − 162000K + 270000 = 0.
+The above condition on K and d is plotted in Figure 8.
+Figure 8: The magnitude d of the possible 3-cycle orbits in Model 2 as the variation of K.
+Similarly, we analyze the 4-cycle and 5-cycle orbits in Model 2, and report the obtained results in
+the sequel.
+Theorem 7. In Model 2, all possible cases for the number of (stable) 4-cycle orbits are given in Table
+3, where
+m1 ≈ 2.060113296, m2 ≈ 2.146719591, m3 ≈ 2.579725065, m4 ≈ 2.581385365, m5 ≈ 3.062775154,
+m6 ≈ 3.070194019, m7 ≈ 3.279225134, m8 ≈ 3.279260335, m9 ≈ 3.319881360, m10 ≈ 3.319889702.
+18
+
+Readers can refer to Remark 3 to understand how these mi are obtained.
+Table 3: Numbers of (stable) 4-cycle orbits in Model 2
+K ∈
+(0, m1)
+(m1, m2)
+(m2, m3)
+(m3, m4)
+(m4, m5)
+(m5, m6)
+4-cycles
+0
+2
+2
+6
+6
+10
+Stable 4-cycles
+0
+2
+0
+2
+0
+2
+K ∈
+(m6, m7)
+(m7, m8)
+(m8, m9)
+(m9, m10)
+(m10, +∞)
+4-cycles
+10
+14
+14
+18
+18
+Stable 4-cycles
+0
+2
+0
+2
+0
+In Figure 9, we show all the 4-cycle orbits in Model 2 with K = 3.319885 ∈ (m9, m10), where the
+16 unstable cycles are marked in red and the 2 stable ones are marked in blue. If we measure the
+magnitude of the 4-cycle orbit x �→ y �→ z �→ w �→ x with d = (x−y)2 +(y −z)2 +(z −w)2 +(w −x)2,
+then d must satisfy one of the following equations.
+Kd − 12K + 20 = 0,
+Kd − 48K − 160 = 0,
+K2d2 + (−36K2 − 60K)d + 288K2 + 960K + 1600 = 0,
+C4(K, d) = 0,
+where C4(K, d) is a complex polynomial given in Appendix.
+Figure 9: The 4-cycle orbits in Model 2 with K = 3.319885. The 16 unstable cycles are marked in
+red, while the 2 stable ones are marked in blue.
+Theorem 8. In Model 2, all possible cases for the number of (stable) 5-cycle orbits are listed in Table
+4, where
+m1 ≈ 2.323208379, m2 ≈ 2.326320457, m3 ≈ 2.509741151, m4 ≈ 2.510528490,
+m5 ≈ 2.632885028, m6 ≈ 2.633089005, m7 ≈ 2.997641294, m8 ≈ 2.997736262,
+m9 ≈ 3.113029799, m10 ≈ 3.113069634, m11 ≈ 3.197332995, m12 ≈ 3.197354147,
+m13 ≈ 3.219425160, m14 ≈ 3.219440784, m15 ≈ 3.269613400, m16 ≈ 3.269618202,
+m17 ≈ 3.288059620, m18 ≈ 3.288062995, m19 ≈ 3.314977518, m20 ≈ 3.314978815,
+m21 ≈ 3.324008184, m22 ≈ 3.324008826, m23 ≈ 3.332961824, m24 ≈ 3.332961850.
+19
+
+5
+4
+x(t)
+2
+1
+2
+3
+4
+5
+x(t-1)9Readers can refer to Remark 3 to understand how these mi are obtained.
+Table 4: Numbers of (stable) 5-cycle orbits in Model 2
+K ∈
+(0, m1)
+(m1, m2)
+(m2, m3)
+(m3, m4)
+(m4, m5)
+5-cycles
+0
+4
+4
+8
+8
+Stable 5-cycles
+0
+2
+0
+2
+0
+K ∈
+(m5, m6)
+(m6, m7)
+(m7, m8)
+(m8, m9)
+(m9, m10)
+5-cycles
+12
+12
+16
+16
+20
+Stable 5-cycles
+2
+0
+2
+0
+2
+K ∈
+(m10, m11)
+(m11, m12)
+(m12, m13)
+(m13, m14)
+(m14, m15)
+5-cycles
+20
+24
+24
+28
+28
+Stable 5-cycles
+0
+2
+0
+2
+0
+K ∈
+(m15, m16)
+(m16, m17)
+(m17, m18)
+(m18, m19)
+(m19, m20)
+5-cycles
+32
+32
+36
+36
+40
+Stable 5-cycles
+2
+0
+2
+0
+2
+K ∈
+(m20, m21)
+(m21, m22)
+(m22, m23)
+(m23, m24)
+(m24, +∞)
+5-cycles
+40
+44
+44
+48
+48
+Stable 5-cycles
+0
+2
+0
+2
+0
+In Figure 10, we plot all possible 5-cycle orbits in Model 2 with K = 3.33296183 ∈ (m23, m24),
+where the 46 unstable cycles are marked in red and the 2 stable ones are marked in blue.
+Figure 10: The 5-cycle orbits in Model 2 with K = 3.33296183. The 46 unstable cycles are marked in
+red, while the 2 stable ones are marked in blue.
+By Theorems 6, 7 and 8, one can see the parameter space of Model 2 is quite different from
+that of Model 1 in the sense that the stability regions for the 3-cycle, 4-cycle and 5-cycle orbits
+are disconnected sets formed by many disjoint portions. Therefore, the topological structures of the
+regions for stable periodic orbits in Model 2 are much more complex than those in Model 1. This may
+be because the inverse demand function of Model 2 has an inflection point. However, the following
+observations of Model 2 are similar to Model 1. Theorem 5 shows that the stability region for the
+2-cycles is a connected interval. In Model 2, the right boundary of the stability region for the 2-cycles
+is the same as the left boundary of the stability region for the 4-cycles. When a = 3.6, b = 2.4, c = 0.6,
+and d = 0.05, by Theorem 1 we know that Model 2 has stable equilibria if K ∈ (0, 5/3), which adjoins
+the stability region for the 2-cycles. Moreover, in Model 2, the stability regions for cycles with distinct
+periods may not intersect with each other, which means that multistability might only arise among
+20
+
+5
+4
+x(t) 3
+2
+0
+1
+2
+3
+4
+5
+x(t-1)96periodic orbits with the same period.
+Figure 11 depicts the two-dimensional bifurcation diagram of Model 2 for (a, K) ∈ [2.5, 5.0] ×
+[0.0, 3.0]. We fix the parameters b = 2.4, c = 0.6, d = 0.05, and set the initial state to be x(0) = 1.0.
+Similarly, we use different colors to mark parameter points corresponding to trajectories with different
+periods. Parameter points are marked in black if the corresponding orbits have orders greater than
+24.
+Furthermore, we also use black to mark the parameter points where the trajectories diverge.
+One can see that Figure 11 confirms the theoretical results reported in Theorem 1. However, Fig-
+ure 11 generated by numerical simulations is not accurate compared to Figure 2 based on symbolic
+computations.
+Figure 11: The two-dimensional bifurcation diagram of Model 2 for (a, K) ∈ [2.5, 5.0] × [0.0, 3.0]. We
+fix the parameters b = 2.4, c = 0.6, d = 0.05, and choose x(0) = 1.0 to be the initial state of the
+iterations.
+Figure 12 depicts the one-dimensional bifurcation diagrams of Model 2 with respect to K by fixing
+a = 3.3, b = 2.4, c = 0.6, and d = 0.05. The bifurcation diagrams are different if the selected initial
+states of the iterations are distinct.
+For example, in Figure 12 (a) and (b), the initial states are
+selected to be x(0) = 1.0 and x(0) = 4.0, respectively. The difference may be because two stable
+equilibria exist when K is relatively small and distinct initial states approach distinct equilibria. As
+shown by Figure 12 (a), the trajectory converges to 1.058 when K < 1.1996 and converges to 4.384
+when K > 1.9874. In Figure 12, the occurrence of period-doubling bifurcations can also be observed.
+Figure 13 depicts the one-dimensional bifurcation diagrams of Model 2 with respect to a by fixing
+K = 2.2, b = 2.4, c = 0.6, and d = 0.05. In Figure 13 (a) and (b), the initial states of the iterations
+are selected to be x(0) = 1.0 and x(0) = 4.0, respectively. Similarly, the two bifurcation diagrams are
+different because of the selection of distinct initial states. Furthermore, pitchfork bifurcations can be
+observed in Figure 13, where the number of stable equilibria changes from one to zero.
+In Model 2, two stable equilibria may coexist (see the blue-gray region in Figure 2). The equilibrium
+selection problem is interesting. The final outcome of the iterations depends not only on the values of
+21
+
+3.0
+2.7
+20
+2.4
+2.1 -
+15
+1.8
+K 1.5
+1.2
+10
+0.9
+0.6
+5
+0.3
+0.0 -
+2.5
+2.75
+3.0
+3.25
+3.5
+3.75
+4.0
+4.25
+4.5
+4.75
+5.0
+a(a) x(0) = 1.0.
+(b) x(0) = 4.0.
+Figure 12: The one-dimensional bifurcation diagrams of Model 2 with respect to K by fixing a = 3.3,
+b = 2.4, c = 0.6, and d = 0.05.
+(a) x(0) = 1.0.
+(b) x(0) = 4.0.
+Figure 13: The one-dimensional bifurcation diagrams of Model 2 with respect to a by fixing K = 2.2,
+b = 2.4, c = 0.6, and d = 0.05.
+22
+
+4.5
+4.0
+3.5
+3.0
+x2.5
+2.0
+1.5-
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+K4.6
+4.5
+4.4
+4.3
+X
+4.2
+4.1
+4.0
+3.9
+0
+2
+3
+1
+4
+5
+K6
+5
+4
+X
+3 -
+2 
+1
+3.2
+3.3
+3.4
+3.5
+3.6
+3.7
+3.8
+3.9
+4.0
+a5.5
+5.0
+4.5
+4.0
+X3.5
+3.0
+2.5
+2.0
+1.5
+3.2
+3.3
+3.4
+3.5
+3.6
+3.7
+3.8
+3.9
+4.0
+athe parameters but also on the starting conditions of the game. According to our numerical simulations
+of Model 2, the basins of attraction of coexisting equilibria have complicated structures. For example,
+by fixing K = 0.5, a = 3.5, b = 2.4, c = 0.6, d = 0.05, we have two stable equilibria E1 = 1.19 and
+E2 = 4.64. The basin of E1 is
+B(E1) = (0, 3.168) ∪ (6.518, 7.577) ∪ (7.745, 7.781) ∪ (7.786, 7.789),
+while that of E2 is
+B(E2) = (3.168, 6.518) ∪ (7.577, 7.745) ∪ (7.781, 7.786).
+Furthermore, when the initial state x(0) > 7.786, the trajectory will not converge to any of the two
+stable equilibria but diverge to +∞. Take K = 1 and a = 4 as the other example. If the other
+parameters keep unchanged, i.e., b = 2.4, c = 0.6, and d = 0.05, there are two stable equilibria
+E1 = 4.99 and E2 = 1.99. Our simulations show that the basins of these two equilibria are
+B(E1) = (0, 0.807) ∪ (2.0, 6.192) ∪ (6.431, 6.647) ∪ (6.653, 6.659),
+and
+B(E2) = (0.807, 2.0) ∪ (6.192, 6.431) ∪ (6.647, 6.653),
+respectively. The escape set is (6.659, +∞), where the trajectory diverges. In short, in Model 2, the
+basins of the two stable equilibria are disconnected sets and have complex topological structures.
+5
+Chaotic Dynamics
+In the bifurcation diagrams (Figures 6 and 12), one can observe that the dynamics of the two considered
+models transition to chaos through period-doubling bifurcations as the adjustment speed increases.
+From an economic point of view, if chaos appears, the pattern behind output and profits is nearly
+impossible to learn even for completely rational players. Therefore, it is extremely hard for a firm to
+handle a chaotic economy, where no market rules could be discovered and followed.
+In this section, we rigorously prove the existence of chaos for the two models.
+The following
+famous lemma was first derived by Li and Yorke [15], which is mathematically deep and facilitates the
+exploration of complicated dynamics arising in one-dimensional discrete dynamical systems.
+Lemma 4. Let I be an interval of real numbers, and let F : I → R be a continuous function. Assume
+that there exists a point x ∈ I such that
+F 3(x) ≤ x < F(x) < F 2(x)
+or
+F 3(x) ≥ x > F(x) > F 2(x),
+(20)
+then the following two statements are true.
+1. For each k ∈ {1, 2, . . .}, there is a point pk ∈ I with period k, i.e., F k(pk) = pk, and F i(pk) ̸= pk
+for 1 ≤ i < k.
+2. There is an uncountable set S ⊂ I (containing no periodic points), which satisfies the following
+conditions:
+(a) for any p, q ∈ S with p ̸= q,
+lim sup
+n→∞ |F n(p) − F n(q)| > 0,
+(21)
+and
+lim inf
+n→∞ |F n(p) − F n(q)| = 0;
+(22)
+(b) for every point p ∈ S and every periodic point q ∈ I,
+lim sup
+n→∞ |F n(p) − F n(q)| > 0.
+(23)
+23
+
+Remark 4. Eq. (22) means that every trajectory in S can wander arbitrarily close to every other.
+However, by (21) we know that no matter how close two distinct trajectories in S may come to each
+other, they must eventually wander away. Furthermore, by (23) it is clear that every trajectory in S
+goes away from any periodic orbit in I. If the two statements in the above lemma are both satisfied,
+we say that there exist chaotic dynamics or chaos in the sense of Li-Yorke.
+Therefore, we can conclude that “period three implies chaos” for one-dimensional discrete dynam-
+ical systems. In Section 4, we have rigorously derived the existence of 3-cycle orbits in Model 2 if
+K > 2.417401607, which proves that chaos would arise for an uncountable set of initial states in the
+sense of Li-Yorke.
+But in Model 1, we have proved that there are no solutions with period three. However, it can
+not be concluded that there exist no chaotic trajectories since the existence of period three is not
+a necessary but only a sufficient condition of chaos. In [20], Marotto indicated that the existence
+of snapback repellers also implies chaos for general n-dimensional systems. However, Li and Chen
+[14] pointed out that Marotto’s original definition of snapback repeller may result in an insufficiency,
+and proposed the Marotto-Li-Chen Theorem. Thus, we give the following lemma for one-dimensional
+systems by simplifying the Marotto-Li-Chen Theorem. Readers can refer to [11] for additional details.
+Lemma 5. Let I be an interval of real numbers, and let F : I → R be a differentiable function. Assume
+that
+1. x ∈ I is an equilibrium, i.e., F(x) = x;
+2. there exists a close interval S ⊂ I such that x is an inner point of S, and the derivative of F
+has the absolute value greater than 1 at every point p ∈ S, i.e., |F ′(p)| > 1;
+3. for some integer m > 1, there exists a point y ∈ S such that y ̸= x, F m(y) = x, and F ′(F k(y)) ̸=
+0 for all 1 ≤ k ≤ m.
+Then the system x(t + 1) = F(x(t)) is chaotic in the sense of Li-Yorke.
+For Model 1, we have F(x) = x + f(e − x3) and F ′(x) = 1 − 3fx2. Then |F ′(x)| > 1 and x > 0
+imply that x >
+�
+2
+3f . Thus, if we can find x, y with x ̸= y such that both |F ′(x)| > 1 and |F ′(y)| > 1
+are satisfied, then there must exist one closed interval S containing x, y as inner points. In such a
+case, it is obvious that |F ′(p)| > 1 for every point p ∈ S. Naturally, we start from m = 2 to verify the
+conditions of Lemma 5 by counting real solutions of the following system.
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+x = x + f(e − x3),
+x = F 2(y) = y + f(e − y3) + f(e − (y + f(e − y3))3),
+|1 − 3fx2| > 1,
+|1 − 3fy2| > 1,
+|1 − 3f(y + f(e − y3))2| ̸= 0,
+x ̸= y,
+x > 0, y > 0, e > 0, f > 0.
+The technique introduced in Remark 2 should be conducted first to transform the above system into
+a univariate one. According to our calculations, the above system has at least one real solution if and
+only if 8/27 < e2f3 < 64/27. Therefore, we conclude that Model 1 is chaotic in the sense of Li-Yorke
+provided that 8/27 < e2f3 < 64/27.
+6
+Concluding Remarks
+It is known that a monopoly may exhibit complex dynamics such as periodic orbits and chaos al-
+though it is the simplest oligopoly. In this study, we investigated two monopoly models with gradient
+mechanisms, where the monopolists are knowledgeable firms. The two models are distinct mainly in
+24
+
+their inverse demand functions. Model 1 uses the inverse demand function of Naimzada and Ricchiuti
+[25], while Model 2 employs that of Puu [26]. Different from widely applied numerical methods such
+as numerical simulations and bifurcation continuation approaches, symbolic methods were applied in
+this paper to analyze the local stability, periodic solutions, and even chaotic dynamics. Numerical
+methods have some shortcomings, e.g., the computations may encounter the problem of instability,
+which makes the results completely useless. In comparison, symbolic computations are exact, thus the
+obtained results can be used to rigorously prove economic theorems in some sense.
+By reproving the already-known results (Proposition 1) of the local stability and bifurcations
+of Model 1, we explained in detail how our symbolic approach works. Afterward, the analysis of the
+stability and bifurcations of Model 2 was conducted based on this approach. We acquired the complete
+conditions of the local stability and bifurcations of Model 2 for the first time (see Theorem 1). In
+Figure 2, it was observed that Model 2 behaves quite differently from typical oligopoly models with
+gradient mechanisms. For example, even if the adjustment speed K is quite large, there always exist
+some values of a (the difference between the initial commodity price and the initial marginal cost)
+such that Model 2 has a stable equilibrium. Moreover, Model 2 may go from instability to stability
+and then back to instability twice as the value of a increases.
+From an economic point of view, the study of periodic solutions is of practical importance. Under
+the assumption of bounded rationality, firms can not learn the pattern behind output and profits if
+periodic dynamics take place. For the two models, we explored the periodic solutions with lower orders
+as well as their local stability. Differences between the two models were found, e.g., 3-cycle orbits exist
+in Model 2 but not in Model 1. In Model 1, the parameter region for the stability of the periodic
+solution with a fixed order constitutes a connected set. In Model 2, however, the stability regions for
+the 3-cycle, 4-cycle, and 5-cycle orbits are disconnected sets formed by many disjoint portions. In other
+words, the topological structures of the regions for stable periodic orbits in Model 2 are much more
+complex than those in Model 1. The above differences may be because the inverse demand function of
+Model 2 has an inflection point. According to the numerical simulations of Model 2, we found that the
+basins of the two stable equilibria are disconnected sets and also have complex topological structures.
+For a n-cycle orbit p1 �→ p2 �→ · · · pn �→ p1, we defined the magnitude measure to be
+d = (p1 − p2)2 + (p2 − p3)2 + · · · + (pn−1 − pn)2 + (pn − p1)2.
+For the two considered models, we analytically investigated the formulae for the magnitude of periodic
+orbits with lower orders.
+Furthermore, it is extremely hard for a firm to handle an economy when chaos appears. In such
+a case, no market rules can be discovered and followed, and the pattern behind output and profits
+is nearly impossible to learn even for completely rational players. In the bifurcation diagrams of the
+two models, it seems that chaos occurs when the adjustment speed is large enough. We clarified this
+observation analytically. By virtue of the fact “period three implies chaos”, we derived that Model 2
+is chaotic in the sense of Li-Yorke by proving the existence of 3-cycle orbits. However, there are no
+3-cycles in Model 1, but the Marotto-Li-Chen Theorem permitted us to prove the existence of chaos
+by finding snapback repellers.
+In this paper, we take the assumption of knowledgeable players, which means the enterprise has
+full information regarding the inverse demand function and can compute its marginal profit at any
+time. In the real world, however, it is more reasonable to assume players to be limited rather than
+knowledgeable. In this case, the enterprise does not know the form of the inverse demand function,
+but possesses the values of output and price only in the past periods and estimates its marginal profit
+with a simple difference formula. The investigation of the dynamics of limited firms might be an
+important direction for our future study.
+Acknowledgments
+The authors wish to thank Dr. Bo Huang for the beneficial discussions and are grateful to the anony-
+mous referees for their helpful comments.
+25
+
+This work has been supported by Philosophy and Social Science Foundation of Guangdong under
+Grant No. GD21CLJ01, Major Research and Cultivation Project of Dongguan City University under
+Grant Nos. 2021YZDYB04Z and 2022YZD05R, National Natural Science Foundation of China under
+Grant No. 11601023, and Beijing Natural Science Foundation under Grant No. 1212005.
+Declaration of competing interest
+The authors declare no conflict of interest.
+Appendix
+SP = (972K8 + 19440K7 + 127575K6 + 162000K5 − 1552500K4 − 6412500K3 − 5062500K2
++ 23437500K + 67187500)(8503056K12 + 191318760K11 + 1523464200K10
++ 3754532250K9 − 14134854375K8 − 101982543750K7 − 146939062500K6
++ 399469218750K5 + 1522072265625K4 + 261457031250K3 − 4576816406250K2
+− 1938867187500K + 13981445312500),
+C4(K, d) = K8d8 + (−126K8 − 210K7)d7 + (6660K8 + 21300K7 + 17800K6)d6 + (−192024K8
+− 874800K7 − 1382400K6 − 731000K5)d5 + (3285360K8 + 18688320K7 + 41115600K6
++ 39438000K5 + 13350000K4)d4 + (−33957792K8 − 221940000K7 − 588016800K6
+− 728172000K5 − 379740000K4 − 45500000K3)d3 + (206172864K8 + 1453101120K7
++ 4191652800K6 + 5433912000K5 + 2183760000K4 − 1105200000K3 − 478000000K2)d2
++ (−672686208K8 − 4870886400K7 − 14246409600K6 − 16185744000K5
++ 2054160000K4 + 13262400000K3 − 7632000000K2 − 11520000000K)d + 906992640K8
++ 6500113920K7 + 18223833600K6 + 13351392000K5 − 25284960000K4
+− 27302400000K3 + 65376000000K2 + 30720000000K − 102400000000.
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@@ -0,0 +1,2587 @@
+Draft version January 13, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+Cosmological-Scale HI Distribution Around Galaxies and AGN
+Probed with the HETDEX and SDSS Spectroscopic Data
+Dongsheng Sun,1, 2 Ken Mawatari,3, 1 Masami Ouchi,3, 1, 4 Yoshiaki Ono,1 Hidenobu Yajima,5 Yechi Zhang,1, 2, 4
+Makito Abe,5 William P. Bowman,6 Erin Mentuch Cooper,7, 8 Dustin Davis,7 Daniel J. Farrow,9, 10
+Karl Gebhardt,7 Gary J. Hill,8, 7 Chenxu Liu,11, 7 and Donald P. Schneider12, 13
+1Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan
+2Department of Astronomy, Graduate School of Science, the University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
+3National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
+4Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), The University of Tokyo, 5-1-5 Kashiwanoha,
+Kashiwa, Chiba, 277-8583, Japan
+5Center for Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1 Tsukuba, Ibaraki 305-8577, Japan
+6Department of Astronomy, Yale University, New Haven, CT 06520
+7Department of Astronomy, The University of Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712, USA
+8McDonald Observatory, The University of Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712, USA
+9University Observatory, Fakult¨at f¨ur Physik, Ludwig-Maximilians University Munich, Scheinerstrasse 1, 81679 Munich, Germany
+10Max-Planck Institut f¨ur extraterrestrische Physik, Giessenbachstrasse 1, 85748 Garching, Germany
+11South-Western Institute for Astronomy Research, Yunnan University, Kunming, Yunnan, 650500, People’s Republic of China
+12Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA
+13Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
+ABSTRACT
+We present cosmological-scale 3-dimensional (3D) neutral hydrogen (Hi) tomographic maps at z =
+2−3 over a total of 837 deg2 in two blank fields that are developed with Lyα forest absorptions of 14,736
+background Sloan Digital Sky Survey (SDSS) quasars at z=2.08-3.67. Using the tomographic maps,
+we investigate the large-scale (≳ 10 h−1cMpc) average Hi radial profiles and two-direction profiles of
+the line-of-sight (LoS) and transverse (TS) directions around galaxies and AGN at z = 2 − 3 identified
+by the Hobby-Eberly Telescope Dark Energy eXperiment (HETDEX) and SDSS surveys, respectively.
+The peak of the Hi radial profile around galaxies is lower than the one around AGN, suggesting that
+the dark-matter halos of galaxies are less massive on average than those of AGN. The LoS profile of
+AGN is narrower than the TS profile, indicating the Kaiser effect. There exist ionized outskirts at
+≳ 30 h−1cMpc beyond Hi rich structures of galaxies and AGN found in the LoS profiles that can
+be explained by the influence of high energy photons propagating over a long distance. Our findings
+indicate that the Hi radial profile of AGN has transitions from proximity zones (≲ a few h−1cMpc)
+to the Hi rich structures (∼ 1 − 30 h−1cMpc) and the ionized outskirts (≳ 30 h−1cMpc). Although
+there is no significant dependence of AGN types (type-1 vs. type-2) on the Hi profiles, the peaks of
+the radial profiles anti-correlate with AGN luminosities, suggesting that AGN’s ionization effects are
+stronger than the gas mass differences.
+Keywords: galaxies: formation — galaxies: evolution — galaxies: high-redshift — intergalactic medium
+1. INTRODUCTION
+Galaxy formation in the Universe is closely related
+to the neutral hydrogen (Hi) gas in the intergalactic
+Corresponding author: Dongsheng Sun
+[email protected]
+medium (IGM). Within the modern paradigm of galaxy
+formation, galaxies form and evolve in the filament
+structure of Hi gas (e.g., Meiksin 2009; Mo et al. 2010).
+Cosmological hydrodynamics simulations suggest that
+the picture of galaxy formation and evolution is asso-
+ciated with large-scale baryonic gas exchange between
+the galaxy and the IGM (fox 2017; van de Voort 2017).
+arXiv:2301.05100v1  [astro-ph.GA]  12 Jan 2023
+
+2
+Sun et al.
+Enormous rivers of cold gas (∼ 104 K) flow into the
+galaxy and trigger the star formation. (e.g., Dekel et al.
+2009; Kereˇs et al. 2005) The cold gas is heated by star
+formation and then ejected by the powerful galactic-
+scale outflows due to feedback caused by stellar winds
+and supernovae.
+The circulation of gas is one of the keys to under-
+standing galaxy formation and evolution. The interplay
+of gravitational and feedback-driven processes can have
+surprisingly large effects on the large scale behavior of
+the IGM. Some of the radiation produced by massive
+stars and black hole accretion disks can escape from
+the dense gaseous environments and propagate out of
+galaxies and photoionize the Hi gas in the circumgalac-
+tic medium (CGM) and even in the IGM (National
+Academies of Sciences, Engineering 2021; Mukae et al.
+2020).
+Great progress has been achieved in exploring the Hi
+distribution around galaxies and active galactic nuclei
+(AGN). The cross-correlation of the Hi in the IGM and
+galaxies has been detected by Lyα absorption features
+in the spectra of background quasars (e.g., Rauch 1998;
+Faucher-Gigu`ere et al. 2008a; Prochaska et al. 2013)
+and bright star-forming galaxies (Steidel et al. 2010;
+Mawatari et al. 2016; Thomas et al. 2017). The Keck
+Baryon Structure Survey (KBSS: Rudie et al. 2012; Ra-
+kic et al. 2012; Turner et al. 2014), the Very Large
+Telescope LBG Redshift Survey (VLRS: Crighton et al.
+2011; Tummuangpak et al. 2014), and other spectro-
+scopic programs (e.g., Adelberger et al. 2003, 2005) have
+investigated the detailed properties of the Hi distri-
+bution around galaxies. These observations target Hi
+gas around galaxies on the scale of the circumgalactic
+medium (CGM). Recently, 3-dimensional (3D) Hi to-
+mography mapping, a powerful technique to reconstruct
+the large scale structure of Hi gas, has been developed
+by Lee et al. (2014, 2016, 2018). Hi tomography map-
+ping is originally proposed by Pichon et al. (2001) and
+Caucci et al. (2008) with the aim of reconstructing the
+3D matter distribution from the Hi absorption of mul-
+tiple sightlines. By this technique, the COSMOS Lyα
+Mapping and Tomography Observations (CLAMATO)
+survey (Lee et al. 2014, 2018) has revealed Hi large
+scale structures with spatial resolutions of 2.5 h−1 co-
+moving Megaparsec (cMpc). This survey demonstrates
+the power of 3D Hi tomography mapping in a number
+of applications, including the study of a protocluster at
+z = 2.44 (Lee et al. 2016) and the identification of cos-
+mic voids (Krolewski et al. 2018). Due to an interpola-
+tion algorithm (Section 4.3) used in the reconstruction
+of the 3D Hi tomography map, we are able to estimate
+the Hi distribution along lines-of-sight where there are
+no available background sources. Based on the 3D Hi to-
+mography map of the CLAMATO survey, Momose et al.
+(2021) have reported measurements the IGM Hi–galaxy
+cross-correlation function (CCF) for several galaxy pop-
+ulations. Due to the limited volume of the CLAMATO
+3D IGM tomography data, Momose et al. (2021) can-
+not construct the CCFs at scales over 24 h−1cMpc in
+the direction of transverse to the line-of-sight. Mukae
+et al. (2020) have investigated a larger field than the
+one of Momose et al. (2021) using 3D Hi tomography
+mapping and report that a huge ionized structure of
+Hi gas associated with an extreme QSO overdensity re-
+gion in the EGS field. Mukae et al. (2020) interpret the
+large ionized structure as the overlap of multiple prox-
+imity zones which are photoionized regions created by
+the enhanced ultraviolet background (UVB) of quasars.
+However, Mukae et al. (2020) found only one example of
+a huge ionized bubble, and no others have been reported
+in the literature.
+Dispite the great effort made by previous studies,
+the limited volume of previous work prevents us from
+understanding how ubiquitous or rare these large ion-
+ized structures are.
+In order to answer this ques-
+tion, we must investigate the statistical Hi distribu-
+tions around galaxies and AGN at much larger spatial
+scales (≳ 10 h−1cMpc). Although Momose et al. (2021)
+derived CCFs for different populations: Lyα emitters
+(LAEs), Hα emitters (HAEs), [Oiii] emitters (O3Es),
+active galactic nuclei (AGN), and submillimeter galaxies
+(SMGs), on a scale of more than 20 h−1cMpc, the lim-
+ited sample size results in large uncertainties in the CCF
+at large scales and prevents definitive conclusions to be
+made regarding the statistical Hi distributions around
+galaxies and AGN.
+Another open question is the luminosity and AGN
+type dependence of the large scale Hi distribution
+around AGN. Font-Ribera et al. (2013) have estimated
+the Hi distribution around AGN using the Sloan Dig-
+ital Sky Survey (SDSS; York et al. 2000) data release
+9 quasar catalog (DR9Q; (Pˆaris et al. 2011)) and find
+no dependence of the Hi distribution on AGN luminos-
+ity. In this study, we investigate the luminosity depen-
+dence using the SDSS data release 14 quasar (DR14Q;
+Pˆaris et al. 2018) catalog, which includes sources ∼ 2
+magnitude fainter than those used by Font-Ribera et al.
+(2013).
+In the AGN unification model (Antonucci &
+Miller 1985; see also Spinoglio & Fern´andez-Ontiveros
+2021), which provides a physical picture that a hot ac-
+cretion disk of super-massive blackhole is obscured by a
+dusty torus, the type-1 and type-2 classes are produced
+by different accretion disk viewing angles. In this pic-
+ture, the type-1 (type-2) AGN is biased to AGN with
+
+Cosmological-Scale Hi Distribution Around Galaxies and AGN
+3
+a wide (narrow) opening angle. In the case of type-1
+AGN, one can directly observe the accretion disks and
+the broad line region, while for type-2 AGN, only the
+narrow line region is observable. Previous studies have
+identified the proximity effect that the IGM of type-1
+AGN is statistically more ionized due to the local en-
+hancement of the UV background on the line-of-sight
+passing near the AGN (Faucher-Gigu`ere et al. 2008b).
+Based on the unification model, the type-2 AGN ob-
+scured on the line of sight statistically radiates in trans-
+verse direction. The investigation of the AGN type de-
+pendence on the surrounding Hi can reveal the large
+scale Hi distribution influenced by the direction of radi-
+ation from the AGN.
+To investigate the Hi distributions around galaxies
+and AGN on large scales, over tens of h−1cMpc, we
+need conduct a new study in a field with length of any
+side larger than 100 h−1cMpc.
+We reconstruct a 3D
+Hi tomography maps of Hi distribution at z ∼ 2 − 3
+in a total area of 837 deg2.
+We use ≳ 15, 000 back-
+ground sightlines from SDSS quasars (Pˆaris et al. 2018;
+Lyke et al. 2020) for the Hi tomography map recon-
+struction and have a large number of unbiased galaxies
+and AGN from the Hobby Eberly Telescope Dark En-
+ergy eXperiment (HETDEX; Gebhardt et al. 2021) and
+SDSS surveys for the investigations of the large scale Hi
+distributions around galaxies and AGN.
+This paper is organized as follows. Section 2 describes
+the details of the HETDEX survey and our spectroscopic
+data. Our foreground and background samples of galax-
+ies and AGN are presented in Section 3. The technique
+of creating the Hi tomography mapping and the recon-
+structed Hi tomography map are described in Section 4,
+and the observational results of Hi distributions around
+galaxies and AGN are given in Section 5. In this section,
+we also interpret our results in the context of previous
+studies, and investigate the dependence of out tomog-
+raphy maps on AGN type and luminosity. We adopt
+a cosmological parameter set of (Ωm, ΩΛ, h) = (0.29,
+0.71, 0.7) in this study.
+2. DATA
+2.1. HETDEX Spectra
+HETDEX provides an un-targeted, wide-area, integral
+field spectroscopic survey, and aims to determine the
+evolution of dark energy in the redshift range 1.88 −
+3.52 using ∼ 1 million Lyman-α emitters (LAEs) over
+540 deg2 in the northern and equatorial fields that are
+referred to as “Spring” and “Fall” fields, respectively.
+The total survey volume is ∼ 10.9 comoving Gpc3.
+The HETDEX spectroscopic data are gathered us-
+ing the 10 m Hobby-Eberly Telescope (HET; Ramsey
+et al. 1994; Hill et al. 2021) to collect light for the Visi-
+ble Integral-field Replicable Unit Spectrograph (VIRUS;
+Hill et al. 2018, 2021) with 78 integral field unit (IFUs;
+Kelz et al. 2014) fiber arrays. VIRUS covers a wave-
+length, with resolving power ranging from 750 − 950.
+Each IFU has 448 fibers with a 1′′.5 diameter. The 78
+IFUs are spread over the 22 arcmin field of view, with
+a 1/4.6 fill factor.
+Here we make use of the data re-
+lease 2 of the HETDEX (HDR2; Cooper et al. 2023)
+over the Fall and Spring fields. In this study, we inves-
+tigate the fields where HETDEX survey data are taken
+between 2017 January and 2020 June. The effective area
+is 11542 arcmin2. The estimated depth of an emission
+line at S/N= 5 reaches 3 − 4 × 10−17 erg cm−2 s−1.
+2.2. Subaru HSC Imaging
+The HETDEX-HSC imaging survey was carried out
+in a total time allocation of 3 nights in 2015 − 2018
+(semesters S15A, S17A, and S18A; PI: A. Schulze) and
+2019 − 2020 (semester S19B; PI: S. Mukae) over a ∼250
+deg2 area in the Spring field, accomplishing a 5σ limit-
+ing magnitude of r = 25.1 mag. The SSP-HSC program
+has obtained deep multi-color imaging data on the 300
+deg2 sky, half of which overlaps with the HETDEX foot-
+prints. In this study, we use the r-band imaging data
+from the public data release 2 (PDR2) of SSP-HSC. The
+5σ depth of the SSP-HSC PDR2 r-band imaging data
+is typically 27.7 mag for the 3′′.0 diameter aperture.
+The data reduction of HETDEX-HSC survey and SSP-
+HSC program are processed with HSC pipeline software,
+hscPipe (Bosch et al. 2018) version 6.7.
+Because the spectral coverage width of the HETDEX
+survey is narrow, only 2000 ˚A, most sources appear as
+single-line emitters. Furthermore, since the Oii doublet
+is not resolved, we rely on the equivalent width (EW) to
+distinguish Lyα from Oii. The high-z Lyα emission is
+typically stronger than low-z [Oii] lines, due to the in-
+trinsic line strengths and the cosmological effects. The
+continuum estimate from the HETDEX spectra reach
+about g= 25.5 (Davis et al. 2021; Cooper et al. 2023)
+and we improve on this using the deep HSC imaging.
+We estimate EW using continua measured from two sets
+of images taken by HSC r-band imaging survey for HET-
+DEX (HETDEX-HSC survey) and the Subaru Strategic
+Program (SSP-HSC; Aihara et al. 2018). Davis et al.
+and Cooper et al. find that our contamination of Oii
+emitters in the LAE sample to be below 2%.
+2.3. SDSS-IV eBOSS Spectra
+We use quasar data from eBOSS (Dawson et al. 2016),
+which is publically available in the SDSS Data Release
+14 and 16 quasar catalog (DR14Q, DR16Q; Pˆaris et al.
+
+4
+Sun et al.
+2018; Lyke et al. 2020). The cosmology survey, eBOSS,
+is part of SDSS-IV. The eBOSS quasar targets are se-
+lected by the XDQSOz method (Bovy et al. 2012) and
+the color cut
+mopt − mW ISE ≥ (g − i) + 3,
+(1)
+where mopt is a weighted stacked magnitude in the g, r
+and i bands and mW ISE is a weighted stacked magni-
+tude in the W1 and W2 bands of the Wide-Field In-
+frared Survey (WISE; Wright et al. 2010). The aim of
+the eBOSS is to accomplish precision angular-diameter
+distance measurements and the Hubble parameter deter-
+mination at z ∼ 0.6 − 3.5 using different tracers of the
+underlying density fields over 7500 deg2. Its final goal is
+to obtain spectra of ∼ 2.5 million luminous red galaxies,
+∼ 1.95 million emission line galaxies, ∼ 450,000 QSOs at
+0.9 ≤ z ≤ 2.2, and the Lyman-α forest of 60,000 QSOs
+at z > 2 over four years of operation.
+The eBOSS program is conducted with twin SDSS
+spectrographs (Smee et al. 2013), which are fed by 1,000
+fibers connected from the focal plane of the 2.5m Sloan
+telescope (Gunn et al. 2006) at Apache Point Observa-
+tory. SDSS spectrographs have a fixed spectral band-
+pass of 3600 − 10000 ˚A over the 7 deg2 field of view.
+The spectral resolution varies from 1300 at the blue end
+to 2600 at the red end, where one pixel corresponds to
+1.8 − 5.2 ˚A.
+3. SAMPLES
+Our study aims to map the statistical distribution of
+Hi gas on a cosmological scale around foreground galax-
+ies and AGN by the 3D Hi tomography mapping tech-
+nique with background sources at z = 2−3. We use the
+foreground galaxies, foreground AGN, and background
+sources presented in Sections 3.1, 3.2, and 3.3, respec-
+tively.
+Two of the goals of this study are to explore the de-
+pendence of luminosity and AGN type on the Hi distri-
+bution. To examine statistical results, we need a large
+number of bright AGN and type-2 AGN. Compared to
+moderately bright AGN and type-1 AGN, bright AGN
+and type-2 AGN are relatively rare. To obtain a suffi-
+ciently large samples of bright AGN and type-2 AGN, we
+expand the Spring and Fall fields of the HETDEX sur-
+vey, from which we are able to investigate the statistical
+luminosity and AGN type dependence of the HI distribu-
+tion around AGN (Section 3.2). The northern extended
+Spring field flanking the HETDEX survey fields, referred
+to as the “ExSpring field”, covers over 738 deg2, while
+the equatorial extended Fall field flanking the HETDEX
+survey fields, here after “ExFall field”, covers 99 deg2.
+The total area of our 3D Hi tomography mapping field
+is 837 deg2 in the ExSpring and ExFall fields that is re-
+ferred to as “our study field”. Our analysis is conducted
+in our study field where the foreground galaxies+AGN
+and the background sources overlap on the sky. As an
+example, we present the foreground galaxies+AGN in
+the ExFall field at z = 2.0 − 2.2 in Figure 1. We also
+present the sky distribution of the background sources
+within the ExFall field in Figure 2. The rest of the fore-
+ground and background sources are shown in the Ap-
+pendix.
+3.1. Foreground Galaxy Sample
+We make a sample of foreground galaxies from the
+data of the HETDEX spectra (Section 2.1) and the Sub-
+aru HSC images (Section 2.2). With these data, Zhang
+et al. (2021) have build a catalog of LAEs that have
+the rest-frame equivalent widths (EW0) of EW0 > 20
+˚A and the HETDEXs Emission Line eXplorer (ELiXer)
+probabilities (Davis et al. 2021; Davis et al. 2023) larger
+than 1. This EW0 cut is similar to previous LAE studies
+(e.g., Gronwall et al. 2007; Konno et al. 2016). This cat-
+alog of LAEs is composed of 15959 objects. Because the
+LAE catalog of Zhang et al. (2021) consists of galaxies,
+type-1 AGN, and type-2 AGN, we isolate galaxies from
+the sources of the LAE catalog with the limited observa-
+tional quantities, Lyα and UV magnitude (MUV), that
+can be obtained from the HETDEX and Subaru/HSC
+data. Because type-1 AGN have broad-line Lyα emis-
+sion, we remove sources with broad-line Lyα whose full
+width half maximum (FWHM) of the Lyα emission lines
+are greater than 1000 km s−1. To remove clear type-2
+AGN from the LAE catalog, we apply a UV magnitude
+cut of MUV < −22 mag that is the bright end of the UV
+luminosity function dominated by galaxies (Zhang et al.
+2021). We then select sources in our study field, and
+apply the redshift cut of z = 2.0 − 3.0 (as measured by
+the principle component analysis of multiple lines; Pˆaris
+et al. 2018) to match the redshift range over which we
+construct Hi tomography map. These redshifts are mea-
+sured with Lyα emission (Zhang et al. 2021), because
+Lyα is the only emission available for all of the sources.
+By these selections, we obtain 14130 galaxies from the
+LAE catalog. These 14130 galaxies are referred to as the
+“Galaxy” sample.
+3.2. Foreground AGN Samples
+In this subsection, we describe how we select fore-
+ground AGN from two sources, (a) the combination of
+the HETDEX spectra and the HSC imaging data and
+(b) the SDSS DR14Q catalog.
+The type-1 AGN are
+identified with the sources of (a) and (b), while the type-
+2 AGN are drawn from the source of (b).
+
+Cosmological-Scale Hi Distribution Around Galaxies and AGN
+5
+Figure 1.
+Sky distribution of the foreground AGN and galaxies at z = 2.0 − 2.2 in the ExFall field. The squares present
+the positions of All-AGN sample sources. Pink (magenta) squares represent the sources of the T1-AGN (T2-AGN) sample.
+The cyan and blue dots show the positions of the Galaxy and T1-AGN(H) sample sources, respectively. The black dashed line
+indicates the border of the Hi tomography map in the Exfall field.
+Figure 2. Sky distribution of background AGN in the ExFall field. The gray crosses indicate background AGN that are used
+to reconstruct our Hi tomography map. The back dashed line has the same meaning as that in Figure 1.
+Table 1. Sample size of foreground samples at z = 2 − 3
+Name of sample
+ExFall
+ExSpring
+Total
+Survey
+Criteria
+Galaxy
+3431
+11436
+14867
+HETDEX
+EW0 > 20 ˚A, FWHMLyα < 1000 km/s, Muv> −22 mag
+T1-AGN(H)
+438
+1349
+1787
+HETDEX
+EW0 > 20 ˚A, FWHMLyα > 1000 km/s
+T1-AGN
+2393
+12300
+14693
+SDSS
+FWHMLyα > 1000 km/s
+T2-AGN
+436
+1633
+2069
+SDSS
+FWHMLyα < 1000 km/s
+Table 2. Sample size of background sample at z = 2.08 − 3.67
+Name of sample
+ExFall
+ExSpring
+Total
+Survey
+Criteria
+background AGN
+2181
+12555
+14736
+SDSS
+⟨S/N⟩Lyαforest > 1.4
+With the source (a) that is the same as the one stated
+in Section 3.1, Zhang et al. (2021) have constructed
+the LAE catalog. We use the catalog of Zhang et al.
+(2021) to select LAEs at z ∼ 2 − 3 that fall in our study
+field. Applying a Lyα line width criterion of FWHM
+> 1000 km s−1 with the HETDEX spectra, we identify
+broad-line AGN, i.e. type-1 AGN, from the LAEs. We
+thus obtain 1829 type-1 AGN that are referred to as
+T1-AGN(H).
+We use the width of Lyα emission line for the selection
+of type-1 AGN. This is because no other emission lines
+characterising AGN, e.g. Civ, are available for all of the
+LAEs due to the limited wavelength coverage and the
+sensitivity of HETDEX. Similarly, the redshifts of T1-
+
+Dec.[deg]
+2
+0
+2
+35
+30
+25
+20
+15
+10
+5
+R.A.[deg]Dec.[deg]
+1
+35
+30
+25
+20
+15
+10
+5
+R.A.[deg]6
+Sun et al.
+AGN(H) objects are measured with Lyα emission whose
+redshifts may be shifted from the systemic redshifts by
+up to a few 100 km s−1 (See Section 3.1). We do not
+select type-2 AGN from the source of (a), because we
+cannot identify type-2 AGN easily with the given data
+set of source (a).
+From the source (b), we obtain the other samples of
+foreground AGN. We first choose objects with a classi-
+fication of QSOs of the SDSS DR14Q, and remove ob-
+jects outside the redshift range of z = 2.0 − 3.0 in our
+study field. We obtain 23721 AGN. For 16762 out of
+23721 AGN, Lyα FWHM measurements are available
+from Rakshit et al. (2020).
+The other AGN without
+FWHM measurement are removed due to the poor qual-
+ity of the Lyα line. We thus use these 16762 AGN with
+good quality of the Lyα line to compose our AGN sam-
+ple, referred to as All-AGN sample.
+To investigate the type dependence, we classify these
+16762 AGN into type-1 and type-2 AGN. In the same
+manner as the T1-AGN(H) sample construction, we use
+Lyα line width measurements of Rakshit et al. (2020)
+for the type-1 and type-2 AGN classification. For the
+16762 AGN, we apply the criterion of Lyα FWHM >
+1000 km s−1 (Villarroel & Korn 2014; Panessa & Bassani
+2002) to select type-1 AGN, and obtain 14693 type-1
+AGN. Following Villarroel & Korn (2014); Panessa &
+Bassani (2002), we classify type-2 AGN by the criterion
+of Lyα FWHM < 1000 km s−1 and obtain 2069 type-
+2 AGN (c.f. Alexandroff et al. 2013; Zakamska et al.
+2003). These type-1 and type-2 AGN are referred to as
+T1-AGN and T2-AGN, respectively.
+Table 1 presents the summary of foreground samples.
+We obtain 14693 and 1829 type-1 AGN, which referred
+to as T1-AGN and T1-AGN(H), from the SDSS and
+HETDEX surveys, respectively. We select 2069 type-2
+AGN that are referred to as T2-AGN from the SDSS
+survey.
+3.3. Background Source Sample
+In this subsection, we describe how the background
+sources are selected. We select the background sources
+with the SDSS DR16Q catalog, following the three steps
+below.
+In the first step, we extract QSOs in our study field
+from the SDSS DR16Q catalog. We then select QSOs
+falling in the range of redshifts from 2.08 to 3.67. The
+lower and upper limits of the redshift range are deter-
+mined by the Lyα forest. Our goal is to probe Hi ab-
+sorbers at z = 2.0−3.0 with the Lyα forest. Because the
+Lyα forest is observed in the rest-frame 1040 − 1185 ˚A
+of the background sources, we obtain the lower and up-
+per limits of the redshifts, 2.08 and 3.67, by 1216 × (1 +
+2.0)/1185−1 = 2.08 and 1216×(1+3.0)/1040−1 = 3.67,
+respectively. By this step, we have selected 26899 back-
+ground source candidates.
+In the second step, we choose background source can-
+didates with good quality. We calculate the average sig-
+nal to noise ratio, ⟨S/N⟩, in the wavelength range of the
+Lyα forest for the 26899 background source candidates,
+and select 15573 candidates with ⟨S/N⟩ greater than 1.4.
+To maximize the special resolution of the tomography
+map, we set the threshold, ⟨S/N⟩ > 1.4, smaller than
+the value used by Mukae et al. (2020). This threshold
+is more conservative than the value, 1.2, used in Lee
+et al. (2018). In the third step, we remove damped Lyα
+absorbers (DLAs) and broad absorption lines (BALs)
+from the Lyα forest of the 15573 candidates, because the
+DLAs and BALs cause an overestimation of the absorp-
+tion of the Lyα forest. We identify and remove DLAs
+using the catalog of Chabanier et al. (2022), which is
+based on the SDSS DR16Q (Lyke et al. 2020). We mask
+out the wavelength ranges contaminated by the DLAs of
+the Chabanier et al. (2022) catalog (see Section 4.1 for
+the procedures). We conduct visual inspection for the
+15573 candidates to remove 115 BALs. In Figure 3, we
+show the spectrum with BALs identified by visual in-
+spection. In this way, we obtain 15458 (= 15573 − 115)
+sources whose spectra are free from DLAs and BALs,
+which we refer to as the background source sample. Ta-
+ble 2 lists the number of background sources in each
+field.
+Figure 3. Spectrum of background AGN with BALs. The
+black line represents the spectrum of a background source.
+The vertical dashed lines present the central wavelengths
+of the metal absorptions.
+The yellow hatches show the
+wavelength ranges of the BALs.
+The gray hatches indi-
+cate the wavelength ranges not used for the reconstruction
+of Hi tomography maps. The SDSS ID of this spectrum is
+106584616, whose redshift is 3.067837.
+4. HI TOMOGRAPHY AND MAPPING
+
+8
+Flux density
+6
+2
+4500
+000S
+5500
+6000
+6500
+Wavelength [A]Cosmological-Scale Hi Distribution Around Galaxies and AGN
+7
+In this section we describe the process to construct Hi
+tomography maps with the spectra of the background
+sources. For Hi tomography, we need to obtain intrin-
+sic continua of the background sources. Section 4.2 ex-
+plains masking the biasing absorption features in the
+background sources, while Section 4.3 determines the
+intrinsic continua of the background source spectra. In
+Section 4.3, we construct Hi tomography maps with the
+intrinsic continuum spectra.
+4.1. DLA and Intrinsic Absorption Masking
+Because a DLA is an absorption system with a high
+neutral hydrogen column density NHI > 2 × 1020 cm−2,
+the intervening DLA completely absorbs a large por-
+tion of the Lyα forest over ∆v ∼ 103 km s−1, which
+gives bias in the estimates of the intrinsic continua of
+the background sources. For the spectra of the back-
+ground sources, we mask out the DLAs identified in
+Section 3.3. We determine the range of wavelengths for
+masking with the IDL code of Lee et al. (2012). The
+wavelength range corresponds to the equivalent width
+of each DLA (Draine 2011):
+W ∼ λα
+� e2
+mec2 NHIfαλα
+�γαλα
+c
+��1/2
+.
+(2)
+In the formula, λα is the rest-frame wavelength of the
+hydrogen Lyα line (i.e.
+1216 ˚A), while c, e, me, fα,
+NHi, and γα are the speed of light, the electron charge,
+the electron mass, the Lyα oscillator strength, the Hi
+column density of the DLA, and the sum of the Einstein
+A coefficients. We mask out these wavelength ranges of
+the background source spectra. In Figure 5, the masked
+DLA is indicated by yellow hatches.
+We also mask out the intrinsic absorption lines of the
+metal absorption lines, which are the other sources of
+bias. We mask SIv λ1062, Nii λ1084, Ni λ1134, and
+Ciii λ1176 (Lee et al. 2012), which are shown by the
+dashed lines in Figure 5. Because the spectral resolu-
+tions of SDSS DR14Q are ∆λ = 1.8 − 5.2 ˚A, we adopt
+the masking size of 10 ˚A in the observed frame.
+4.2. Intrinsic Continuum Determination
+In order to obtain the intrinsic continuum of the back-
+ground source (Section 3.3) in the Lyα forest wavelength
+range (LAF-WR; 1040−1185 ˚A), we conduct mean-flux
+regulated principle component analysis (MF-PCA) fit-
+ting with the IDL code (Lee et al. 2012) for the back-
+ground sources after the masking (Section 4.1).
+There are two steps in the MF-PCA fitting process.
+The first step is to predict the shape of the intrinsic
+continuum of the background sources in the LAF-WR.
+We conduct least-squares principle component analysis
+(PCA) fitting (Suzuki et al. 2005; Lee et al. 2012) to the
+background source spectrum in the rest frame 1216 −
+1600 ˚A :
+fPCA(λ) = µ(λ) +
+8
+�
+j=1
+cjξj(λ),
+(3)
+where λ is the rest-frame wavelength. The values of cj
+are the free parameters for the weights. The function of
+µ(λ) is the average spectrum calculated from the 50 lo-
+cal QSO spectra in Suzuki et al. (2005). The function of
+ξj(λ) represents the jth principle component (or ‘eigen-
+spectrum’) out of the 8 principle components taken from
+the PCA template shown in Figure 4.
+In the second step, we predict the intrinsic continuum
+of the background source in the LAF-WR. Because the
+PCA template is obtained with the local QSO spectra,
+the best-fit fPCA in the LAF-WR does not include cos-
+mic evolution on the average transmission rate of the
+Lyα forest. On average, the best-fit fPCA in the LAF-
+WR should agree with the cosmic mean-flux evolution
+(Faucher-Gigu`ere et al. 2008c):
+⟨F(z)⟩ = exp[−0.001845(1 + z)3.924],
+(4)
+where z is the redshift of the absorber. We use fPCA and
+a correction function of a + bλ to estimate the intrinsic
+continuum fintrinsic(λ) for large-scale power along the
+line of sight with the equation:
+fintrinsic(λ) = fPCA(λ) × (a + bλ),
+(5)
+where a and b are the free parameters.
+Because the
+ratio of fobs(λ)/fintrinsic(λ) should agree with the cosmic
+average ⟨F(z)⟩ for z = (λ/1216)−1 in the LAF-WR, we
+conduct least-squares-fitting to find the values of a and
+b providing the best fit between the mean ratio and the
+cosmic average. The red line shown by the bottom panel
+of Figure 5 presents a MF-PCA fitted continuum derived
+from the spectrum of one of our background sources.
+By the MF-PCA fitting, we have obtained the esti-
+mates of fintrinsic(λ) for 14736 out of the 15458 back-
+ground sources. We find the other background sources
+show poor fitting results found by visual inspection. We
+do not use these background sources in the following
+analyses.
+Figure 6 shows an example of poor fitting
+result due to the unknown absorption. We adopt con-
+tinuum fitting errors of ∼ 7%, ∼ 6%, and ∼ 4% for Lyα
+forests with mean S/N values of < 4, 4 − 10, and > 10,
+respectively (Lee et al. 2012).
+4.3. HI Tomography Map Reconstruction
+We reconstruct our Hi tomography maps by a proce-
+dure similar to Lee et al. (2018). We define Lyα forest
+
+8
+Sun et al.
+Figure 4. Principle components and mean flux taken from
+Suzuki et al. (2005). The top panel shows the normalized
+mean flux of 50 local QSOs in rest-frame wavelength. The
+bottom 8 panels show the 1st − 8th principle components
+that are used in the PCA fitting in our study. Each principle
+component is normalized to the mean flux.
+fluctuations δF at each pixel on the spectrum by
+δF = fobs/fintrinsic
+⟨F(z)⟩
+− 1
+(6)
+, where fobs and fintrinsic are the observed spectrum
+and estimated intrinsic continuum, respectively. ⟨F(z)⟩
+is the cosmic average transmission. We calculate δF with
+our background source spectra. The top panel of Figure
+5 shows the ‘spectrum’ of δF derived from the fobs and
+fintrinsic in the bottom panel.
+For the pixels in the
+wavelength ranges of masking (Section 4.1), we do not
+use δF in our further analyses. We thus obtain δF in
+876,560 pixels.
+For the the HI tomography map of the Extended Fall
+field, we define the cells of the Hi tomography map in the
+three-dimensional comoving space. We choose a volume
+of 30◦ × 3.3◦ in the longitudinal and latitudinal dimen-
+Figure 5. Example of a background source spectrum that
+was used for the reconstruction of the Hi tomography map.
+Bottom panel: Estimation of intrinsic continuum. The thin
+black line is the spectrum of a background source taken from
+the SDSS survey.
+The red and magenta lines are the re-
+sults of MF-PCA and PCA fitting, respectively. The vertical
+dashed lines present the central wavelengths of the metal ab-
+sorptions. The gray hatches represent the wavelength ranges
+that are not used for the Hi tomography map reconstruc-
+tions. The yellow hatch indicates the wavelength ranges of
+DLA. Top panel: Spectrum of δF extracted from the bottom
+panel in the LAF-WR. The vertical yellow and gray hatches
+are the same as those in the bottom panel. The black and
+pink lines show the spectrum of δF and the error of δF at the
+corresponding wavelength extracted from the bottom panel.
+The horizontal line indicates the cosmic average of Lyα forest
+transmission.
+Figure 6. Same as the bottom panel of Figure 5, but for
+the background spectrum with a poor fitting result.
+The
+red and magenta lines are the results of MF-PCA and PCA
+continuum fitting, respectively. The yellow hatch indicates
+the wavelength range of unknown absorption.
+sions, respectively, in the redshift range of 2.0 < z < 3.0.
+The comoving size of our Hi tomography map is 2257
+h−1cMpc × 233 h−1cMpc × 811 h−1cMpc in the right
+
+5
+Mean flux
+0
+2
+1st Component
+0
+....
+0.2
+2nd Component
+0.0
+0.1
+3rd Component
+0.0
+0.1
+0.05
+0.00
+4th Component
+0.05
+0.1
+0.0
+5th Component
+0.1
+0.1
+0.0
+6th Component
+0.1
+0.2
+0.0
+7th Component
+0.2
+0.1
+0.0
+-0.1
+8th Component
+1000
+1100
+1200
+1300
+1400
+1500
+1600
+Rest-frame wavelength [A]0.6
+0.3
+AF0.0
+-0.3
+-0.6
+12
+Flux density
+080
+4000
+4500
+000S
+5500
+6000
+Wavelength [A]6
+Flux density
+2
+0
+3500
+4000
+4500
+5000
+5500
+Wavelength [A]Cosmological-Scale Hi Distribution Around Galaxies and AGN
+9
+ascension (R.A.), declination (Dec), and z directions,
+respectively in the same manner as Mukae et al. (2020).
+Our Hi tomography map has 451 × 46 × 162 cells, and
+one cell is a cubic with a size of 5.0 h−1cMpc on a side,
+where the line-of-sight distance is estimated under the
+assumption of the Hubble flow.
+We conduct a Wiener filtering scheme for reconstruct-
+ing the sightlines that do not have background sources.
+We use the calculation code developed by Stark et al.
+(2015). The solution for each cell of the reconstructed
+sightline is obtained by
+δrec
+F
+= CMD · (CDD + N)−1 · δF,
+(7)
+where CMD, CDD, and N are the map-data, data-data,
+and noise covariances, respectively. We assume Gaus-
+sian covariances between two points r1 and r2:
+CMD = CDD = C(r1, r2),
+(8)
+C(r1, r2) = σ2
+F exp
+�
+−(∆r∥)2
+2L2
+∥
+�
+exp
+�
+−(∆r⊥)2
+2L2
+⊥
+�
+,
+(9)
+where ∆r∥ and ∆r⊥ are the distances between r1 and
+r2 in the directions of parallel and transverse to the line
+of sight, respectively. The values of L⊥ and L∥ are the
+correlation lengths for vertical and parallel to the line-
+of-sight (LoS) direction, respectively, and defined with
+L⊥ = L∥ = 15 h−1cMpc. The value of σ2
+F is the normal-
+ization factor that is σ2
+F = 0.05. Stark et al. (2015) de-
+velop this Gaussian form to obtain a reasonable estimate
+of the true correlation function of the Lyα forest. We
+perform the Wiener filtering reconstruction with the val-
+ues of δF at the 898390 pixels, using the aforementioned
+parameters of the Stark et al. (2015) algorithm with a
+stopping tolerance of 10−3 for the pre-conditioned con-
+jugation gradient solver. As noted by Lee et al. (2016),
+the boundary effect that leads to an additional error
+on δF occurs at the positions that are near the bound-
+aries of an Hi tomography map. The boundary effect
+is caused by the background sightlines not covering the
+region that contribute to the calculation of the δF values
+for cells near the Hi tomography map boundaries. To
+avoid the boundary effect, we extend a distance of 40
+h−1cMpc for each side of the Hi tomography map of the
+ExFall field. The resulting map is shown in Figure 7.
+For the HI tomography map reconstruction of the Ex-
+tended Spring field (hereafter ExSpring field), we per-
+form almost the same procedure as the one of the Ex-
+Fall field. The area of the ExSpring field is more than
+6 times larger than that of the ExFall field. We sep-
+arate the ExSpring field into 8 × 3 = 24 footprints to
+save calculation time. Each footprint covers an area of
+10◦ × 5◦ in the R.A. and Dec directions, respectively.
+We reconstruct the Hi tomography map one by one for
+the footprints of the ExSpring field.
+To weaken the boundary effect, we extend a distance
+of 40 h−1cMpc for each side of the footprints. The ex-
+tensions mean that every two adjacent footprints has an
+overlapping region of 80 h−1cMpc width. The width of
+the overlapping regions is a conservative value to weaken
+the boundary effect since it is much larger than the res-
+olution, 15 h−1cMpc, of our Hi tomography maps. By
+the 40 h−1cMpc extension, we reduce the uncertainty
+in the δF value for the edge of each footprint caused by
+boundary effect to ±0.01. This value corresponds to the
+1/10 of the typical error for each cell of the Hi tomogra-
+phy map (Mukae et al. 2020) The remaining additional
+error caused by boundary effect is negligible compared
+to the statistical uncertainties in the HI distributions ob-
+tained in Section 5. Then we follow the reconstruction
+procedure for the ExFall field to reconstruct HI tomog-
+raphy maps of the footprints and cut off all the cells
+within 40 h−1cMpc to the borders that are affected by
+the boundary effect. Finally we obtain the Hi tomogra-
+phy map of the ExSpring field with a special volume of
+3475 h−1cMpc × 1058 h−1cMpc × 811 h−1cMpc in the
+R.A., Dec, and z directions, respectively (Figure 8).
+5. RESULTS AND DISCUSSIONS
+5.1. Average HI Profiles around AGN: Validations of
+our AGN Samples
+In this section we present the Hi profile, δF as a func-
+tion of distance, with the All-AGN sample sources, us-
+ing the reconstructed Hi tomography maps. We com-
+pare the Hi profile of the All-AGN sample to the one of
+the previous study (Font-Ribera et al. 2013). We also
+present the comparison of the Hi profiles between T1-
+AGN(H) and T1-AGN samples that are made with the
+HETDEX and SDSS data. In this study, we only discuss
+the structures having size ≳ 15 h−1cMpc corresponding
+to the resolution of our 3D Hi tomography maps.
+For the Hi profiles with the All-AGN sample, we
+extract δF values around the 16978 All-AGN sample
+sources in the Hi tomography map.
+We cut the Hi
+tomography map centered at the positions of the All-
+AGN sample sources, and stack the δF values to make a
+two dimensional (2D) map of the average δF distribution
+around the sources that is referred to as a 2D Hi profile
+of the All-AGN sample sources. The two dimensions of
+the 2D Hi profile correspond to the transverse distance
+DTrans and the LoS Hubble distance. The velocity corre-
+sponding to the LoS Hubble distance is referred to as the
+LoS velocity. Here we define the Lyα forest absorption
+fluctuation
+AF ≡ −δF
+(10)
+
+10
+Sun et al.
+Figure 7. 3D Hi tomography map of the ExFall field. The color contours represent the values of δF from negative (red) to
+positive (blue). The spatial volume of the Hi tomography map is 2257×233×811 h−3cMpc3. The redshift range is z = 2.0−3.0.
+that is an indicator of the amount of the Hi absorption.
+Figure 9 shows the 2D Hi profile with values of AF
+(δF) for All-AGN sample. The solid black lines denote
+the contours of AF. In each cell of the 2D Hi profile,
+we define the 1σ error with the standard deviation of
+AF values of the 100 mock 2D Hi profiles. Each mock
+2D Hi profile is obtained in the same manner as the
+real 2D Hi profile, but with random positions of sources
+whose number is the same as the one of All-AGN sample
+sources.
+In Figure 9, the dotted black lines indicate
+the contours of the 6σ, 9σ and 12σ confidence levels,
+respectively. We find the 19.5σ level detection of AF at
+the source position (0,0). The AF value at the source
+position indicates the averaging value over the ranges of
+(−7.5 h−1cMpc, +7.5 h−1cMpc) in both the LoS and
+transverse directions. The 19.5σ level detection at the
+source position is suggestive that rich Hi gas exists near
+the All-AGN sources on average The 2D Hi profile is
+more extended in the transverse direction than along
+the line of sight. We discuss this difference in Section
+5.2.
+We then define a 3D distance, D, under the assump-
+tion of the Hubble flow in the LoS direction. We derive
+AF as a function of D that is referred to as ”Hi radial
+profile”, averaging AF values of the 2D Hi profile over
+the 3D distance. Figure 10 shows the Hi radial profile
+of the All-AGN sample. We find that the AF values de-
+crease towards a large distance. This trend is consistent
+with the one found by Ravoux et al. (2020) with the
+SDSS quasars.
+Ravoux et al. (2020) have obtained the average Hi
+absorption distribution around the AGN taken from the
+SDSS data release 16 quasar (SDSS DR16Q) catalog in
+the field of Strip 82. The criteria of the target selection
+for the SDSS DR16Q and SDSS DR14Q sources are the
+same. The luminosity distribution of AGN for Ravoux
+et al. (2020) is almost the same as that of our All-AGN
+sample sources that are taken from the SDSS DR14Q
+catalog. We derive the average radial Hi profile of the
+Ravoux et al. (2020) AGN sources by the same method
+as for our All-AGN sample, using the 3D Hi tomogra-
+phy map reconstructed by Ravoux et al. (2020).
+We
+compare the radial Hi profile of the All-AGN sample
+with the one derived from the 3D Hi tomography map
+of Ravoux et al. (2020). The comparison is shown in
+Figure 10. Our result agrees with that of Ravoux et al.
+(2020) within the error range at scale D > 10 h−1 cMpc.
+The peak values of AF are comparable, AF ≃ 0.02. The
+
+0.300
+0.214
+0.129
+Dec[cMpc]
+0.0429
+2250
+300
+2000
+-0.0429
+750
+1500
+200
+1250
+-0.129
+1000
+750
+100
+750
+500
+RA
+[cMpc
+500
+-0.214
+250
+250
+0
+z[cMpc]
+-0.300Cosmological-Scale Hi Distribution Around Galaxies and AGN
+11
+Figure 8. Same as Figure 7, but for the ExSpring field. The spatial volume of the Hi tomography map is 3475 × 1058 × 811
+h−3cMpc3.
+slight difference between the peak values of our and
+Ravoux et al.’s results can be explained by the differ-
+ent approaches of the estimation for the intrinsic con-
+tinuum adopted by Ravoux et al. and us. Ravoux et al.
+conduct power law fitting, which is different from the
+MF-PCA fitting that we used, for the intrinsic contin-
+uum in the wavelength range of the Lyα forest. Given
+the low (∼ 15 h−1) spatial resolution of both our Hi to-
+mography map and that of Ravoux et al. (2020), neither
+studies are able to search for the proximity effect mak-
+ing a photoionization region around AGN (D’Odorico
+et al. 2008). From the comparison shown by Figure 10,
+we conclude that the Hi distribution derived from our
+Hi tomography map is reliable.
+To check the reliability of the HETDEX survey results,
+we use the reliable result of the SDSS AGN to compare
+with the result derived by the HETDEX AGN.
+We select type-1 AGN from the HETDEX’s T1-
+AGN(H) and SDSS’s T1-AGN samples to make sub-
+samples of T1-AGN(H) and T1-AGN whose rest-frame
+1350 ˚A luminosity (L1350) distributions are the same.
+For T1-AGN, the measurements directly from the SDSS
+spectra (Lspec
+1350) are available (Rakshit et al. 2020). For
+T1-AGN(H), we do not have Lspec
+1350 measurements from
+the HETDEX spectra, we estimate it using HSC r-band
+imaging.
+Since the central wavelength of the r-band
+imaging is rest-frame ∼ 1700˚A, we calibrate the conver-
+sion between r-band luminosity, Lphot
+UV , and Lspec
+1350. We
+examine the 283 type-1 AGN sources that appear in
+both the SDSS and HETDEX surveys (and, thus, have
+both Lspec
+1350 measurements from SDSS and r-band lumi-
+nosities from HSC) to calibrate the relationship. The
+results are displayed in Figure 11. The Lphot
+UV are always
+smaller than those of Lspec
+1350 (Rakshit et al. 2020). Due
+to the blue UV slope of the spectra for the AGN both
+categorized in the T1-AGN(H) and T1-AGN samples,
+the luminosity of the rest-frame 1350 ˚A always shows
+a larger value than the one of rest-frame 1700 ˚A. We
+conduct linear fitting to the data points of Figure 11,
+and obtain the best-fit linear function. With the best-fit
+linear function, we estimate Lspec
+1350 values for the HET-
+DEX’s T1-AGN(H) sample sources.
+We show the Lspec
+1350 distributions of all the T1-AGN(H)
+and T1-AGN sample sources in the upper panel of Fig-
+ure 12. We make the sub-samples of T1-AGN and T1-
+AGN(H) that consist of the sources in the overlapping
+area of Lspec
+1350 distributions. We present the Lspec
+1350 distri-
+butions of the T1-AGN and T1-AGN(H) sub-samples in
+
+0.300
+0.214
+Dec [cMpc]
+0.129
+11000
+0.0429
+900
+800
+-0.0429
+700
+35Q0
+600
+3250
+3000
+-0.129
+2750
+500
+2500
+2250
+400
+2000
+1750
+-0.214
+1500
+300
+RA[cMpcl
+1250
+200
+1000
+750
+750
+-0.300
+10Q
+500
+500
+250
+250
+0
+z[cMpc]12
+Sun et al.
+Figure 9.
+2D Hi profile of the All-AGN sample sources. The color map indicates the AF (δF) values of each cell of the 2D
+Hi profile. The solid lines denote constant AF (δF) values in steps of 0.01 (−0.01) starting at 0.01 (−0.01). The dotted lines
+correspond to multiples of 3σ starting at 6σ.
+Figure 10.
+Hi radial profile of the All-AGN and Ravoux
+et al. (2020) AGN samples. The black and gray data points
+and error bars show the Hi radial profiles of our All-AGN
+sample sources and the AGN of Ravoux et al. 2020, respec-
+tively. The horizontal dashed line shows the cosmic average
+Hi absorption, AF = 0 (δF = 0).
+the bottom panel of Figure 12. We obtain 540 and 4338
+type-1 AGN for the sub-samples of T1-AGN(H) and T1-
+AGN, respectively, whose Lspec
+1350 distributions are shown
+in the bottom panel of Figure 11.
+We derive the Hi radial profiles for the sub-samples
+of T1-AGN(H) and T1-AGN sample sources, as shown
+in Figure 13. The Hi radial profiles of T1-AGN(H) and
+T1-AGN sub-sample sources are in good agreement.
+Figure 11. Relations of Lphot
+UV
+against Lspec
+1350 for the sources
+both categorized in the T1-AGN(H) and T1-AGN samples.
+The Lphot
+UV
+and Lspec
+1350 are measured from the HSC r-band
+imaging and SDSS spectra (Rakshit et al. 2020), respectively.
+The gray points show the distribution of Lspec
+1350 − Lphot
+UV
+re-
+lations for the sources both categorized in the T1-AGN(H)
+and T1-AGN samples. The black dashed line indicates the
+relation where Lspec
+1350 = Lphot
+UV . The red dashed line represents
+the linear best fit of the blue points.
+5.2. AGN Average Line-of-Sight and Transverse Hi
+Profiles
+
+0.04
+0.04
+Ravoux+20 AGN
+0.03
+0.03
+All-AGN
+AF
+0.02
+-0.02
+0.010F
+0.01
+0.00
+0.00
+0.01
+0.01
+0
+10 20 3040506070
+D [h-1cMpc]best fit
+46
+[erg s
+.
+45
+logL
+44
+45
+46
+logL
+spec
+[erg s-1]
+1350LoS Velocity [km s-1]
+LoS Hubble distance
+7500
+75
+0.01
+0.01
+[h-1cMpc]
+5000
+50
+0.010.01
+AF
+2500
+25
+0.030.03
+0
+0
+204060
+DTrans [h-1cMpc]Cosmological-Scale Hi Distribution Around Galaxies and AGN
+13
+Figure 12. Top panel: Lspec
+1350 distributions of the T1-AGN
+and T1-AGN(H) samples with blue and red histograms, re-
+spectively. Bottom panel: Same as the top panel, but for the
+T1-AGN and T1-AGN(H) sub-sample sources.
+Figure 13.
+Hi radial profiles of the T1-AGN and T1-
+AGN(H) sub-samples. The blue and red triangles show the
+values of AF as a function of distance, D, for the T1-AGN
+and T1-AGN(H) sample sources, respectively.
+The hori-
+zontal dashed line shows the cosmic average Hi absorption,
+AF = 0. The right y-axis shows the corresponding δF values.
+Based on the 2D Hi profile of the All-AGN sample
+(Figure 9), we find that the Hi distributions of the All-
+AGN sample sources are more extended in the trans-
+verse direction. In this section, we present the Hi radial
+profiles of All-AGN sample in the LoS and transverse
+directions and compare these two Hi radial profiles.
+To derive the Hi radial profile of the All-AGN sample
+with the absolute LoS distance, which is referred to as
+the LoS Hi radial profile (Figure 15), we average AF val-
+ues of the 2D Hi profiles of All-AGN over DTrans < 7.5
+h−1cMpc (from −7.5 h−1cMpc to +7.5 h−1cMpc in the
+transverse direction) that corresponds to the spatial res-
+olution of the 2D Hi profile map, 15 h−1cMpc. Among
+the 16,978 All-AGN sample sources, 10,884 sources are
+used as both background and foreground sources. In this
+case, the Hi absorption (AF) of these 10,884 sources at
+the LoS velocity ≲ −5250 km s−1 is estimated mainly
+from their own spectrum. As the discussion in Youles
+et al. (2022), the redshift uncertainty of the SDSS AGN
+causes the overestimation of intrinsic continuum and the
+underestimation of AF around the metal emission lines
+such as Ciii λ1176. This leads to a systemics toward
+positive AF in the Hi radial profile of LoS velocity (LoS
+distance) at the LoS velocity ≲ 5250 km s−1 (Figure 14).
+The Hi radial profile of LoS velocity (LoS distance) is de-
+rived by averaging AF values over DTrans < 7.5 h−1cMpc
+as a function of the negative and positive LoS velocity
+(LoS distance). In this study, we only use the values of
+AF at the LoS distance > −52.5h−1cMpc (LoS velocity
+> −5250 km s−1) to derive the LoS Hi radial profile
+of the All-AGN sample (Figure 15). The scale, LoS dis-
+tance > −52.5h−1cMpc (LoS velocity > −5250 km s−1),
+is determined by the maximum wavelength of the Lyα
+forest we used, the smoothing scale of the Wiener filter-
+ing scheme, and the AGN redshift uncertainty, assumed
+by Youles et al. (2022). After removing the AF values af-
+fected the systemics in the 2D Hi profile, we present the
+LoS Hi radial profile of the All-AGN sample in Figure
+15.
+We estimate the Hi radial profiles of DTrans, which is
+referred to as the Transverse Hi radial profile,by averag-
+ing the AF values over the LoS velocity of (−750, +750)
+km s−1 whose velocity width corresponds to 15 h−1
+cMpc in the Hubble-flow distance. The Hi radial profile
+of DTrans is also shown in Figure 15.
+We compare the LoS and Transverse Hi radial profile.
+The AF value decrease more rapidly in the LoS direc-
+tion than those in the Transverse direction (Figure 15).
+This difference may be explained by an effect similar to
+the Kaiser effect (Kaiser 1987), doppler shifts in AGN
+redshifts caused by the large-scale coherent motions of
+the gas towards the AGN. The LoS Hi radial profile is
+negative, AF ∼ −0.002±0.0008, at the large scale, ≳ 30
+h−1cMpc. In Section 5.5, we discuss the negative AF
+values of LoS Hi radial profiles at large scale and com-
+pare our observational result to the models of a previous
+study, Font-Ribera et al. (2013).
+5.3. Source Dependences of the AGN Average HI
+Profiles
+In this section, we present 2D and Hi radial profiles
+of the AGN sub-samples to investigate how the average
+Hi density depends on luminosity and AGN type.
+5.3.1. AGN Luminosity Dependence
+
+Fraction
+0.15
+T1-AGN
+0.10
+T1-AGN(H)
+0.05
+0.00
+Fraction
+0.10
+0.05
+0.00
+42 43 4445 46 470.04
+0.04
+T1-AGN
+0.03
+0.03
+T1-AGN(H)
+AF
+0.02
+0.02
+TT
+0.018F
+0.01
+0.00
+0.00
+-0.01
+0.01
+0
+10 20 30 40 506070
+D [h-1cMpc]14
+Sun et al.
+Figure 14. Hi radial profiles of LoS velocity (LoS distance)
+for the All-AGN sample. The black solid line shows the AF
+values as a function of LoS velocity (LoS distance) for the
+All-AGN sample. The vertical dashed line presents the posi-
+tion of LoS velosity = 0 km s−1 (LoS distance = 0 h−1cMpc).
+The horizontal dashed indicates the cosmic average Hi ab-
+sorption, AF = 0. The gray shaded area shows the range of
+the AF not used to derive LoS Hi radial profile.
+Figure 15. LoS and Transverse Hi radial profiles the All-
+AGN sample. The black and gray lines show the AF (δF)
+values as a function of LoS distance and DT rans, respectively.
+The horizontal dashed line indicates AF (δF) = 0 (= 0).
+We study the AGN-luminosity dependence of the
+average Hi profiles.
+Figure 16 presents the Lspec
+1350
+distribution of All-AGN. We make 3 sub-samples
+of All-AGN that are All-AGN-L3, All-AGN-L2 and
+All-AGN-L1.
+The luminosity ranges of the sub-
+samples are 43.70
+<
+log(Lspec
+1350/[erg s−1])
+<
+45.41,
+45.41 < log(Lspec
+1350/[erg s−1]) < 45.75, and 45.75 <
+log(Lspec
+1350/[erg s−1]) < 47.35, respectively. The luminos-
+ity ranges of the 3 sub-samples are defined in a way that
+the numbers of the AGN are same 5695 in each subsam-
+ples. We derive the 2D Hi profiles of the sub-samples
+in the same manner as Section 5.1, and present the pro-
+files in Figures 17. In these 2D Hi profiles, The bright-
+est sub-sample of All-AGN-L1 (the faintest sub-sample
+of All-AGN-L3) shows the weakest (the strongest) Hi
+absorptions around the source position, D = 0.
+We then extract the Hi radial profiles from the 2D
+Hi profiles of the All-AGN sub-samples, and present
+the Hi radial profiles in Figure 18. In this figure, we
+find that the peak values of AF for the All-AGN sub-
+samples is anti-correlates with AGN luminosities. The
+peak AF values near the source position drops from
+the faintest All-AGN-L3 subsample to the brightest All-
+AGN-L1 subsample.
+The gas densities around bright
+AGN are higher than (or comparable to) those around
+faint AGN, this result would suggest that the ioniza-
+tion fraction of the hydrogen gas around bright AGN is
+higher than the one around faint AGN on average.
+We also present the LoS and Transverse Hi radial pro-
+files of the All-AGN sub-samples derived by the same
+method as that for the All-AGN sample in Figure 19.
+Similar to what we found in the comparison of the Hi
+radial profiles for the All-AGN sub-samples, the peak
+values of the LoS and Transverse Hi profiles also de-
+crease from the faintest sub-sample, All-AGN L3, to the
+brightest sub-sample, All-AGN L1. For the LoS (Trans-
+verse) Hi radial profiles at the scales beyond 25 h−1
+cMpc, we do not find any significant differences in the
+comparison of the LoS (Transverse) Hi radial profiles for
+the All-AGN sub-samples.
+Figure 16. logLspec
+1350 distribution of the bright and All-AGN
+sample sources. The vertical dashed lines indicate the board-
+ers of Lspec
+1350 where log(Lspec
+1350/[erg s−1]) = 45.41 and 45.75, re-
+spectively. These three borders separate the All-AGN sam-
+ple into 3 sub-samples of All-AGN-L3, All-AGN-L2, and All-
+AGN-L1, respectively.
+5.3.2. AGN Type Dependence
+
+LoS distance [h-1cMpc]
+75
+-50-25
+0
+25 1 50
+75
+0.03
+0.03
+All-AGN LoS
+0.02
+0.02
+A
+F0.01
+0.01
+H
+0.00
+0.00
+-7500-5000-2500
+0
+2500 5000 7500
+LoS velocity [km s-1]Los Velocity [km s-1]
+0
+2500
+5000
+7500
+0.03
+0.03
+All-AGN LoS
+All-AGN Trans
+0.02
+0.02
+AF
+0.01
+0.01
+0.00
+0.00
+0.01
+25
+50
+75
+[
+D [h-1cMpc]0.10
+All-AGN-L3
+Fraction
+All-AGN-L2
+All-AGN-L1
+--
+0.05
+0.00
+-
+44
+45
+46
+47Cosmological-Scale Hi Distribution Around Galaxies and AGN
+15
+Figure 17. Same as Figure 9, but for the All-AGN-L3 (top),
+All-AGN-L2 (middle) and All-AGN-L1 (bottom) samples.
+Figure 18.
+Same as Figure 13, but for the All-AGN-L3
+(red), All-AGN-L2 (gray) and All-AGN-L1 (black) samples.
+We investigate the dependence of Hi profiles on type-1
+and type-2 AGN. To remove the effects of the AGN lumi-
+Figure 19. LoS and Transverse Hi radial profiles of the All-
+AGN-L3, All-AGN-L2, and All-AGN-L1 sub-samples. The
+top figure (bottom figure) presents the LoS (Transverse) Hi
+radial profiles of the All-AGN-L3, All-AGN-L2, and All-
+AGN-L1 sub-samples, shown by the red, gray, and black
+lines, respectively.
+The meaning of the horizontal dashed
+lines both in the top and bottom figures are the same as the
+one in Figure 10.
+nosity dependence (Section 5.3.1), we make sub-samples
+of T1-AGN and T2-AGN with the same Lspec
+1350 distribu-
+tion by the same manner as the one we conduct for the
+selection of T1-AGN and T1-AGN(H) sub-samples in
+Section 5.1.
+The top panel of Figure 20 presents the
+Lspec
+1350 distributions of T1-AGN and T2-AGN samples,
+while the bottom panel of Figure 20 shows those of the
+T1-AGN and T2-AGN sub-samples. The sub-samples
+of T1-AGN and T2-AGN are composed of 10329 type-
+1 AGN and 1462 type-2 AGN, respectively. We derive
+the 2D Hi profiles from the T1-AGN and T2-AGN sub-
+samples. The profiles are presented in Figure 21. We
+
+LoS Velocity [km s-1]
+LoS Hubble distance
+7500
+75
+0.01
+0.01
+[h-1cMpc]
+5000
+0.010.01
+AF
+2500
+25
+0.030.03
+0
+0
+204060
+DTrans [h-1cMpc]LoS Velocity [km s-1]
+LoS Hubble distance
+7500
+75
+0.01
+0.01
+[h-1cMpc]
+5000
+50
+-0.010.01
+AF
+2500
+25
+0.030.03
+0
+0
+204060
+DTrans [h-1cMpc]Los Velocity [km s-1]
+LoS Hubble distance
+7500
+75
+0.01
+0.01
+[h-1cMpc]
+5000
+50
+-0.010.01
+AF
+2500
+25
+0.030.03
+0
+0
+204060
+DTrans [h-1cMpc]0.04
+0.04
+A1l-AGN-L3
+0.03
+0.03
+All-AGN-L2
+AF
+0.02
+All-AGN-L1
+0.02
+0.010F
+0.01
+0.00
+0.00
+-0.01
+0.01
+0
+10 20 3040506070
+D [h-1cMpc]Los Velocity [km s-1]
+0
+2500
+5000
+7500
+0.03
+0.03
+All-AGN-L3 LoS
+All-AGN-L2 LoS
+0.02
+All-AGN-L1 LoS
+-0.02
+AF 0.01
+0.01
+0.00
+0.00
+0.01
+25
+50
+75
+D [h-1cMpc]0.03
+0.03
+All-AGN-L3 Trans
+All-AGN-L2 Trans
+0.02
+All-AGN-L1 Trans
+0.02
+AF 0.01
+0.01
+OF
+0.00
+0.00
+-0.01
+25
+50
+75
+D [h-1cMpc]16
+Sun et al.
+find 17.7 and 7.9 σ detections at the source center po-
+sition (0,0) of the T1-AGN and T2-AGN sub-samples,
+respectively.
+We calculate the Hi radial profiles from
+the 2D Hi profiles of the T1-AGN and T2-AGN sub-
+samples. In Figure 22, we compare the Hi radial profiles
+of the T1-AGN and T2-AGN sub-samples. No notable
+difference is found within 1σ error. The peak value of
+AF of the T2-AGN subsample is within 1σ error of the
+peak value of the T1-AGN subsample near the source
+position.
+To compare the Hi distributions of type-1 and type-2
+AGN in the LoS and transverse directions, we derive the
+LoS and Transverse Hi radial profiles of the T1-AGN
+and T2-AGN sub-samples and present the profiles in
+Figure 23. Similar to the trend of the Hi radial profiles,
+the peak values of the LoS and Transverse Hi radial
+profiles for T1-AGN and T2-AGN sub-samples are not
+significantly different. The comparable peak values of
+the LoS and Transverse Hi radial profiles suggest that
+the selectively different orientation and opening angles
+of the dusty tori of the type-1 and type-2 AGN do not
+significantly affect the Hi distribution at the scale ≲ 15
+h−1cMpc.
+For the Hi radial profiles at the scale > 15 h−1cMpc,
+we find that the AF value for the LoS Hi radial pro-
+file of the T1-AGN sub-sample is greater than those of
+the T2-AGN sub-sample over the 1σ error bar at the
+scale around 25 h−1cMpc.
+This result may hint that
+the type-2 AGN have a stronger power of ionization at
+25 h−1cMpc than the type-1 AGN. The interpretation
+of ionization at large-scales is in Section 5.5.
+Figure 20. Same as Figure 12, but for the T1-AGN (blue)
+and T2-AGN (red) samples.
+Figure 21.
+Same as Figure 9, but for the T1-AGN (top
+figure) and T2-AGN (bottom figure) sub-samples.
+Figure 22. Same as Figure 13, but for the T1-AGN (blue)
+and T2-AGN (red) sub-samples and the Galaxy (gray) sam-
+ple.
+5.4. Average HI Profiles around Galaxy
+We derive the 2D Hi profile at the positions of the
+Galaxy sample sources in the same manner as the one
+of the All-AGN sample sources. Figure 24 presents the
+2D Hi profile of the Galaxy sample sources. There is
+a clear 10.5σ detection at the source position of (0,0).
+Similarly, we calculate the Hi radial profile from the
+2D Hi profile of the Galaxy sample (Figure 25). The Hi
+radial profile of the Galaxy sample shows a trend similar
+to those of the All-AGN sample. Both for the Galaxy
+
+raction
+0.15
+T1-AGN
+0.10
+T2-AGN
+0.05
+0.00
+Fraction
+0.10
+0.05
+0.00
+424344454647LoS Velocity [km s-1]
+LoS Hubble distance
+7500
+75
+0.01
+0.01
+[h-1cMpc]
+5000
+50
+0.010.01
+AF
+2500
+25
+0.030.03
+0
+0
+204060
+DTrans [h-1cMpc]Los Velocity [km s-1]
+LoS Hubble distance
+7500
+75
+0.01
+0.01
+[h-1cMpc]
+5000
+0.010.01
+AF
+2500
+25
+0.030.03
+0
+0
+204060
+DTrans [h-1cMpc]0.04
+0.04
+T1-AGN
+0.03
+0.03
+T2-AGN
+AF
+0.02
+Galaxy
+-0.02
+0.010F
+0.01
+0.00
+0.00
+-0.01
+0.01
+0
+10 203040506070
+D [h-1cMpc]Cosmological-Scale Hi Distribution Around Galaxies and AGN
+17
+Figure 23. Same as Figure 19, but for the T1-AGN and
+T2-AGN sub-samples.
+and All-AGN samples, the Hi radial profile decreases
+towards the large scales, reaching AF ∼ 0.
+In Figure 24, we find that the Hi distributions in the
+LoS and transverse directions are different. A similar
+difference between the values of AF in LoS and trans-
+verse directions of 2D Hi profiles is claimed by Mukae
+et al. (2020). To investigate the difference between the
+Hi distributions in LoS and transverse directions for the
+Galaxy sample, we present the LoS and Transverse Hi
+radial profiles of the Galaxy sample in Figure 27. We
+find that the LoS and Transverse Hi radial profiles of the
+Galaxy sample show different gradient of the decreasing
+AF at the scale D ∼ 3.75−50 h−1cMpc. This difference
+can be explained by the gas version of the Kaiser effect
+that we discussed in Section 5.2. In the LoS Hi radial
+profile of the Galaxy sample, we find that the AF val-
+ues are negative on the scale of D = 25 − 70 h−1cMpc,
+which is similar to the negative AF values we found on
+the large scale of the LoS Hi radial profile for the All-
+AGN sample. We discuss these negative AF values on
+the LoS Hi radial profile of the Galaxy sample in Section
+5.5.
+Figure 24. 2D Hi profile of the Galaxy sample sources. The
+color map indicates the AF (δF) values of each cell of the 2D
+Hi profile. The dotted lines show confidence level contours
+of 3σ and 6σ. The solid line presents the contour where AF
+= 0.01 (δF = −0.01).
+5.4.1. Galaxy-AGN Dependence
+We derive 2D Hi profiles for the T1-AGN(H) sample
+constructed from the HETDEX data. Figure 24 and 25
+show the 2D Hi profiles of the Galaxy and T1-AGN(H)
+samples. We find 7.6σ detection around the source posi-
+tion for the T1-AGN(H) sample. Figure 26 presents the
+Hi radial profiles of the Galaxy and T1-AGN(H) samples
+derived from the 2D Hi profiles. We also compare the
+Hi radial profiles of the Galaxy sample with those of T1-
+AGN and T2-AGN in Figure 22. In the Hi radial profiles
+of the Galaxy and T1-AGN(H) samples, the AF values
+increase toward the source position D = 0. In Figure
+26 (22), we find that the AF values of T1-AGN(H) (T1-
+AGN and T2-AGN) are larger than those of the galaxies
+at ≲ 20 h−1 cMpc. These AF excesses of the AGN may
+be explained by the hosting dark matter halos of the
+AGN being more massive than those of the galaxies.
+Momose et al. (2021) also investigate the Hi radial pro-
+file around AGN, and find Hi absorption decrement at
+the source center (≲ 5 h−1Mpc). They argure that this
+trend can be explained by the proximity effect. On the
+other hand, their result is different from ours that the
+AF values monotonically increase with decreasing dis-
+tance. This difference between our and Momose et al.’s
+results is produced by the fact that our results for ≲ 10
+h−1 cMpc are largely affected by the Hi absorption at
+∼ 10 h−1 cMpc due to the coarse resolution of our Hi
+tomography map, 15 h−1 cMpc, in contrast with 2.5 h−1
+cMpc for the resolution of Momose et al. (2021).
+We then derive the LoS and Transverse radial Hi pro-
+file of the T1-AGN(H) sample. The results of the profiles
+
+0
+2500
+5000
+7500
+0.03
+0.03
+T1-AGN LoS
+T2-AGN LoS
+0.02
+0.02
+AF 0.01
+0.01
+0.00
+0.00
+-0.01
+0
+25
+50
+75
+D [h-1cMpc]0.03
+0.03
+T1-AGN Trans
+T2-AGN Trans
+0.02
+0.02
+AF
+0.01
+0.01
+0.00
+0.00
+0.01
+25
+50
+0
+75
+D [h-1cMpc]Los Velocity [km s-1]
+LoS Hubble distance
+7500
+75
+0.01
+0.01
+[h-1cMpc]
+5000
+-0.010.01
+AF
+2500
+25
+0.030.03
+0
+0
+204060
+DTrans [h-1cMpc]18
+Sun et al.
+are shown in Figure 27. Similar to the LoS and Trans-
+verse Hi radial profiles of the All-AGN and Galaxy sam-
+ples, the gas version of the Kaiser effect and the nega-
+tive AF in the LoS direction on the scale beyond D = 25
+h−1cMpc are also found in those of the T1-AGN(H) sam-
+ple.
+Figure 25.
+Same as Figure 17, but for the T1-AGN(H)
+sample.
+Figure 26. Same as Figure 18, but for Galaxy (gray) and
+T1-AGN(H) (black) samples.
+5.5. Comparison with Theoretical Models
+There are theoretical models of Hi radial profiles
+around AGN that are made by Font-Ribera et al. (2013).
+Font-Ribera et al. (2013) present their Hi radial profiles
+with the LoS distance in the form of cross-correlation
+function (CCF).
+We first calculate theoretical CCFs of All-AGN, fol-
+lowing the definition of the CCF presented in Font-
+Ribera et al. (2013). Font-Ribera et al. (2013) assume
+the linear cross-power spectrum of the QSOs and Lyα
+forest,
+PqF(k, z) = bq(z)[1+βq(z)µ2
+k]bF(z)[1+βF(z)µ2
+k]PL(k, z),
+(11)
+Figure 27.
+Same as Figure 15, but for the Galaxy and
+T1-AGN(H) samples.
+where PL(k, z) is the linear matter power spectrum.
+Here µk is the cosine of the angle between the Fourier
+mode and the LoS (Kaiser 1987). The values of bq and bF
+(βq and βF) are the bias factors (redshift space distortion
+parameters) of the QSO and Lyα density, respectively.
+The redshift distortion parameter of QSO obeys the
+relation βq = f(Ω)/bq, where f(Ω) is the logarithmic
+derivative of the linear growth factor (Kaiser 1987),
+bq = 3.8±0.3 (White et al. 2012). We use the condition
+of Lyα forest, bF(1 + βF) = −0.336 for bF ∝ (1 + z)2.9,
+that is determined by observations of Lyα forest at
+z ≃ 2.25 (Slosar et al. 2011). Font-Ribera et al. (2013)
+estimate the CCF of QSOs by the Fourier transform of
+PqF (Hamilton 1992):
+ξ(r) = ξ0(r)P0(µ) + ξ2(r)P2(µ) + ξ4(r)P4(µ),
+(12)
+where µ is the cosine of angle between the position r
+and the LoS in the redshift space. The values of P0,
+P2, and P4 are the Legendre polynomials, P0 = 1, P2 =
+(3µ2 − 1), and P4 = (35µ4 − 30µ2 + 3)/8, respectively.
+The functions of ξ0, ξ2, and ξ4 are:
+ξ0(r) = bqbF[1 + (βq + βF)/3 + βqβF/5]ζ(r),
+(13)
+ξ2(r) = bqbF[2/3(βq+βF)+4/7βqβF][ζ(r)− ¯ζ(r)], (14)
+ξ4(r) = 8/35bqbFβqβF[ζ(r) − 5/2¯ζ(r) − 7/2¯¯ζ(r)]. (15)
+The function ζ(r) is the standard CDM linear correla-
+tion function in real space (Bardeen et al. 1986; Hamil-
+ton et al. 1991). The functions ¯ζ(r) and ¯¯ζ(r) are given
+by:
+¯ζ(r) ≡ 3r−3
+� r
+0
+ζ(s)s2ds,
+(16)
+
+LoS Velocity [km s-1]
+LoS Hubble distance
+7500
+75
+0.01
+0.01
+[h-1cMpc]
+5000
+0.0110.01
+AF
+2500
+25
+0.030.03
+0
+204060
+DTrans [h-1cMpc]0.04
+0.04
+Galaxy
+0.03
+0.03
+T1-AGN(H)
+AF
+0.02
+0.02
+0.018F
+0.01
+0.00
+0.00
+0.01
+0.01
+0
+10 203040506070
+D [h-1cMpc]0
+2500
+5000
+7500
+0.03
+0.03
+Galaxy LoS
+Galaxy Trans
+0.02
+T1-AGN(H) LoS
+-0.02
+T1-AGN(H) Trans
+AF
+0.01
+0.01
+OF
+0.00
+0.00
+0.01
+25
+50
+75
+D [h-1cMpc]Cosmological-Scale Hi Distribution Around Galaxies and AGN
+19
+¯¯ζ(r) ≡ 5r−5
+� r
+0
+ζ(s)s4ds.
+(17)
+Here we define
+ξ′(r) ≡ −ξ(r).
+(18)
+In Figure 28, we present Dξ′ as a function of the LoS
+distance for the model of Font-Ribera et al. (2013) that
+is calculated under the assumption of the mean over-
+density of the 15 h−1cMpc corresponding to the spatial
+resolution of our observational results.
+To compare our observational measurements with the
+model CCF of Font-Ribera et al. (2013), we calculate
+the value of ξ′ for our All-AGN sample. The value of ξ′
+in each cell ξ′cell is calculated by
+ξ′
+cell =
+�
+i∈cell ωiAFi
+�
+i∈cell ωi
+,
+(19)
+where ωi is the weight determined by the observational
+errors and the intrinsic variance of the Lyα forest. The
+value of ωi is obtained by
+ωi =
+�
+σ2
+F(zi) +
+1
+⟨S/N⟩2 × ⟨F(zi)⟩2
+�−1
+,
+(20)
+where σF(zi) is the intrinsic variance of the Lyα forest.
+The value of ⟨F(zi)⟩ is the cosmic average Lyα transmis-
+sion (Eq.4). We adopt ⟨S/N⟩ = 1.4 that is the criterion
+of the background source selection (Section 3.3). The
+intrinsic variance, σF(zi), of the Lyα forest taken from
+Font-Ribera et al. (2013) is:
+σ2
+F(zi) = 0.065[(1 + zi)/3.25]3.8.
+(21)
+We calculate ξ′ with our All-AGN sample via the
+Equations 19, 20, and 21, using the binning sizes same
+as those in Font-Ribera et al. (2013).
+We present ξ′
+multiplied by D with the black squares in Figure 28.
+(explanation of Momose+21) For reference, we also de-
+rive the ξ′ for our Galaxy sample shown by the blue
+triangles.
+In Figure 28, we find that the Dξ′ profile of our All-
+AGN sample show a trend similar to the one of the
+model predicted by Font-Ribera et al. (2013). The ob-
+servational Dξ′ profile of our All-AGN sample shows
+a good agreement with the model Dξ′ profile of Font-
+Ribera et al. (2013) at the scale of D > 30 h−1cMpc. Al-
+though the model Dξ′ profile of Font-Ribera et al. (2013)
+is slightly higher than the Dξ′ profiles of the observa-
+tions at ≳ 60 h−1cMpc, the general trend of the negative
+Dξ′ profiles at ≳ 30 h−1cMpc are the same. Font-Ribera
+et al. (2013) suggests that the negative Dξ
+′ values at the
+large scale of ≳ 30 h−1cMpc are explained by the ion-
+ization. In the model of ionization, Font-Ribera et al.
+(2013) assume the spectrum of the AGN at D = 0 with
+Lν ∝ ν−α, where α = 1.5 (1.0) for the frequency ν over
+(below) the Lyman limit. The luminosity of λ = 1420
+˚A is normalized as Lν = 3.1 × 1030 erg/s/Hz, which
+is taken from the mean luminosity of the SDSS data re-
+lease 9 quasars. No assumptions of AGN type have been
+made in the models of Font-Ribera+13. Based on the
+model of ionization, Font-Ribera et al. (2013) calculate
+ξ for the homogeneous gas radiated by AGN, and obtain
+the function
+ξ = 0.0065(20 h−1cMpc/D)2.
+(22)
+With the ξ function, we calculate Dξ
+′ that is presented
+with the cyan dashed curve in Figure 28.
+The cyan
+dashed curve shows the plateau at D ≥ 40 h−1cMpc
+with negative Dξ
+′ values that is comparable with the
+model Dξ′ profile of Font-Ribera et al. (2013). It indi-
+cates that the negative Dξ
+′ values are originated from
+the ionization of radiation including the hard radiation.
+Similarly, the negative Dξ
+′ values of our All-AGN at the
+large scale towards ≳ 40 h−1cMpc may be explained by
+the ionization of radiation.
+To distinguish the large-
+scale negative Dξ
+′ values, which are referred to as the
+‘ionized outskirts’, from the proximity zone created by
+the proximity effect, we plot the observational CCF of
+AGN obtained by Momose et al. (2021) in Figure 28.
+The AGN CCF obtained by Momose et al. shows a de-
+creasing Hi absorption toward source position (D = 0
+h−1cMpc) caused by the proximity effect. Our findings
+indicate that the Hi radial profile of AGN has transi-
+tions from proximity zones (≲ a few h−1cMpc) to the
+Hi structures (∼ 1 − 30 h−1cMpc) and the ionized out-
+skirts (≳ 30 h−1cMpc). The hard radiation may pass
+through the Hi structure due to the small cross-section
+and ionizes the Hi gas in the regions of ionized out-
+skirts. Because of the low recombination rate, the Hi
+gas remains ionized in the ionized outskirt.
+Interestingly, the Dξ′ profile of our Galaxy sample also
+shows negative Dξ
+′ values towards ≳ 30 h−1cMpc which
+is similar to those of the model and our All-AGN sam-
+ple. This result may suggest that the Hi gas at large
+scale (≳ 20 h−1cMpc) around galaxies has been ion-
+ized. The ionizing source causing the structure of neg-
+ative Dξ
+′ values at the large scale may not be a single
+galaxy, but a group of galaxies within a radius of a few
+cMpc. Regions around galaxies are special as galaxies
+are clustered together. Galaxies in this work are bright
+with MUV < −22 mag. The galaxies can be hosted by
+massive haloes, and are likely to distribute at overden-
+sity regions. The overdensity region suggests that each
+galaxy can be surrounded by several satellite galaxies.
+Although it is difficult for a galaxy to ionize the Hi gas
+
+20
+Sun et al.
+on a scale of ≳ 20 h−1cMpc, a group galaxies may have
+enough ionizing photons to ionize the Hi on this scale.
+Figure 28. Comparison between our All-AGN and Galaxy
+results and the models of Font-Ribera et al. (2013) in the LoS
+CCF (ξ
+′) multiplied by distance (D). The black and blue
+points are the results derived from the All-AGN and Galaxy
+samples sources, respectively. The orange curve is the LoS
+CCF of QSOs with the Lyα forest derived by Font-Ribera
+et al. (2013). The cyan dashed curve shows the ionization
+of radiation effect taken from Font-Ribera et al. (2013). The
+pink line presents the CCF of AGN obtained by Momose
+et al. (2021). The gray shade presents the range of the Hi
+structure. Two white areas show the regions of proximity
+zone and ionized outskirt. The horizontal gray line indicates
+the cosmic average where Dξ
+′ = 0.
+6. SUMMARY
+We reconstruct two 3D Hi tomography maps based
+on the Lyα forests in the spectra of 14763 background
+QSOs from the SDSS survey with no signatures of
+damped Lyα system or broad absorption lines.
+The
+maps cover the extended Fall and Spring fields defined
+by the HETDEX survey. The spatial volume of the re-
+constructed 3D Hi tomography maps are 2257×233×811
+h−3cMpc3 and 3475 × 1058 × 811 h−3cMpc3. We inves-
+tigate Hi distribution around galaxies and AGN with
+samples made from HETDEX and SDSS survey results
+in our study field. Our results are summarized below.
+• We derive the 2D Hi and Hi radial profiles of the
+All-AGN sample consisted of SDSS AGN. We find
+that the 2D Hi profile is more extended in the
+transverse direction than along the line of sight. In
+the Hi radial profile All-AGN sample, the values
+of Hi absorption, AF, decrease toward the large
+scale, touching to AF ∼ 0.
+• We compare the Hi radial profiles derived from
+the T1-AGN and T1-AGN(H) sub-samples, whose
+Lspec
+1350 distributions are the same.
+We find that
+the Hi radial profile of the T1-AGN sub-sample
+agrees with that of the T1-AGN(H) sub-sample.
+This agreement suggests that the systematic un-
+certainty between the SDSS and the HETDEX
+survey results is negligible.
+• We examine the dependence of the Hi profile on
+AGN luminosity by deriving the 2D Hi, Hi ra-
+dial, LoS Hi radial, and Transverse Hi radial pro-
+files of the All-AGN-L3 (the faintest), All-AGN-
+L2, and All-AGN-L1 (the brightest) sub-samples.
+We find that the Hi absorption is the greatest in
+the lowest-luminosity AGN sub-sample, and that
+the Hi absorption becomes weaker with increasing
+AGN luminosity This result suggests that, on av-
+erage, if the density of Hi gas around the bright
+AGN is greater than (or comparable to) those of
+the faint AGN, the ionization fraction of Hi gas
+around bright AGN is higher than that around
+faint AGN.
+• We investigate the AGN type dependence of Hi
+distribution around type-1 and type-2 AGN by the
+2D Hi, Hi radial, LoS Hi radial, and Transverse
+Hi radial profiles extracted from the T1-AGN and
+T2-AGN sub-samples with the same Lspec
+1350 distri-
+butions. The comparison between the Hi radial
+profiles of T1-AGN and T2-AGN sub-samples in-
+dicates that the Hi absorption around the T2-
+AGN sub-sample is comparable to the one of the
+T1-AGN sub-sample on average. This trend sug-
+gests that, the selectively different opening angle
+and orientation of the dusty torus for type-1 and
+type-2 AGN do not have a significant impact on
+the Mpc-scale Hi distribution.
+• We compare the Hi distributions around galax-
+ies and type-1 AGN with the 2D Hi, Hi radial,
+LoS Hi radial, and Transverse Hi radial profiles
+derived from the Galaxy and T1-AGN(H) sample
+sources.
+The Hi absorption values, AF, around
+the T1-AGN(H) sample are larger than those of
+the Galaxy sample on average. This result may
+be caused by the dark matter halos of type-1 AGN
+having a larger mass than the one of galaxies on
+average.
+• We find that the Hi radial profiles of the LoS dis-
+tance for the Galaxy and All-AGN samples show
+negative AF values, which means weak Hi absorp-
+tion, at the scale over ∼ 30 h−1cMpc. We extract
+
+LoS velocity [km s-1]
+0
+2000
+4000
+6000
+8000
+0.50
+HI structure
+Ionized Outskirt
+0.25
+DS
+0.00
+Zone
+-0.25
+Momose+21
+CCF LoS model
+Ionization model
+-0.50
+Galaxy Los
+All-AGN LoS
+0
+102030
+4050
+60
+70
+80
+D [h-1cMpc]Cosmological-Scale Hi Distribution Around Galaxies and AGN
+21
+the Dξ′ profile of our Galaxy and All-AGN sam-
+ples to compare with the model CCF of AGN from
+Font-Ribera et al. (2013). The general trend of the
+negative Dξ′ at ≳ 30 h−1cMpc is the same as the
+model CCF. This results suggest that the Hi ra-
+dial profile of AGN has transitions from proximity
+zones (≲ a few h−1cMpc) to the Hi rich struc-
+tures (∼ 1−30 h−1cMpc) and the ionized outskirts
+(≳ 30 h−1cMpc).
+ACKNOWLEDGEMENTS
+We thank Nobunari Kashikawa, Khee-Gan Lee, Akio
+Inoue, Rikako Ishimoto, Shengli Tang, Yongming Liang,
+Rieko Momose, and Koki Kakiichi for giving us helpful
+comments.
+HETDEX is led by the University of Texas at Austin
+McDonald Observatory and Department of Astron-
+omy with participation from the Ludwig-Maximilians-
+Universit¨at
+M¨unchen,
+Max-Planck-Institut
+f¨ur
+Ex-
+traterrestrische Physik (MPE), Leibniz-Institut f¨ur As-
+trophysik Potsdam (AIP), Texas A&M University,
+Pennsylvania State University, Institut f¨ur Astrophysik
+G¨ottingen, The University of Oxford, Max-Planck-
+Institut f¨ur Astrophysik (MPA), The University of
+Tokyo and Missouri University of Science and Tech-
+nology.
+In addition to Institutional support, HET-
+DEX is funded by the National Science Foundation
+(grant AST-0926815), the State of Texas, the US Air
+Force (AFRL FA9451-04-2- 0355), and generous sup-
+port from private individuals and foundations. The ob-
+servations were obtained with the Hobby-Eberly Tele-
+scope (HET), which is a joint project of the University
+of Texas at Austin, the Pennsylvania State University,
+Ludwig-Maximilians-Universit¨at M¨unchen, and Georg-
+August-Universit¨at G¨ottingen. The HET is named in
+honor of its principal benefactors, William P. Hobby and
+Robert E. Eberly. The authors acknowledge the Texas
+Advanced Computing Center (TACC) at The University
+of Texas at Austin for providing high performance com-
+puting, visualization, and storage resources that have
+contributed to the research results reported within this
+paper. URL: http://www.tacc.utexas.edu
+VIRUS is a joint project of the University of Texas
+at Austin, Leibniz-Institut f¨ur Astrophysik Potsdam
+(AIP), Texas A&M University (TAMU), Max-Planck-
+Institut f¨ur Extraterrestrische Physik (MPE), Ludwig-
+Maximilians-Universit¨at Muenchen, Pennsylvania State
+University, Institut fur Astrophysik G¨ottingen, Univer-
+sity of Oxford, and the Max-Planck-Institut f¨ur As-
+trophysik (MPA). In addition to Institutional support,
+VIRUS was partially funded by the National Science
+Foundation, the State of Texas, and generous support
+from private individuals and foundations.
+This work is supported in part by MEXT/JSPS KAK-
+ENHI Grant Number 21H04489 (HY), JST FOREST
+Program, Grant Number JP-MJFR202Z (HY).
+K. M. acknowledges financial support from the Japan
+Society for the Promotion of Science (JSPS) through
+KAKENHI grant No. 20K14516.
+This paper is supported by World Premier Inter-
+national Research Center Initiative (WPI Initiative),
+MEXT, Japan, the joint research program of the In-
+stitute of Cosmic Ray Research (ICRR), the Univer-
+sity of Tokyo, and KAKENHI (19H00697, 20H00180,
+and 21H04467) Grant-in-Aid for Scientific Research (A)
+through the Japan Society for the Promotion of Science.
+
+22
+Sun et al.
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+
+24
+Sun et al.
+APPENDIX
+Figure 29. Continued from Figure 1. The different panels denote the coverages over different redshift ranges shown at the top
+left of each panel.
+
+z = 2.2 - 2.4
+2
+-
+2
+35
+30
+25
+20
+15
+10
+nz = 2.4 - 2.6
+Dec.[deg]
+2
+-
+2
+35
+30
+25
+20
+15
+10
+5z = 2.6 - 2.8
+2
+-
+2
+35
+30
+25
+20
+15
+10
+5z = 2.8 - 3.0
+Dec.[deg]
+2
+品
+0
+品
++
+-
++.品
+中
+-
+2
+35
+30
+25
+20
+15
+10
+5
+R.A.[deg]Cosmological-Scale Hi Distribution Around Galaxies and AGN
+25
+Figure 30.
+Same as Figure 1, but for the foreground sources in the ExSpring field.
+
+Z=2
+2.0 - 2.2
+60
+口
+55
+电
+50
+45
+160
+170
+180
+190
+200
+210
+220
+230
+240Z=2
+2.2 - 2.4
+60
+55
+50
+45
+160
+170
+180
+190
+200
+210
+220
+230
+240Z = 2.4 - 2.6
+60
+50
+45
+160
+170
+180
+190
+200
+210
+220
+230
+240
+R.A.[deg]26
+Sun et al.
+Figure 31. Continued from Figure 30.
+Figure 32.
+Same as Figure 2, but for the background sources in the ExSpring field.
+
+Z=2
+2.6 - 2.8
+60
+品
+-
+日
+.
+55
+50
+45
+160
+170
+180
+190
+200
+210
+220
+230
+240Z=
+2.8 -3.0
+60
+品
+..
+-
+55
++
+T
+50
+45
+160
+170
+180
+190
+200
+210
+220
+230
+240
+R.A.[deg]60
+Dec.[deg]
+55
+50
+45
+160
+170
+180
+190
+200
+210
+220
+230
+240
+R.A.[deg]

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+arXiv:2301.11800v1  [math.CV]  25 Jan 2023
+COMMUTING TOEPLITZ OPERATORS AND MOMENT MAPS
+ON CARTAN DOMAINS OF TYPE III.
+DAVID CUEVAS-ESTRADA AND RAUL QUIROGA-BARRANCO
+Abstract. Let DIII
+n
+and Sn be the Cartan domains of type III that con-
+sist of the symmetric n × n complex matrices Z that satisfy ZZ < In and
+Im(Z) > 0, respectively.
+For these domains, we study weighted Bergman
+spaces and Toeplitz operators acting on them. We consider the Abelian groups
+T, R+ and Symm(n, R) (symmetric n × n real matrices), and their actions on
+the Cartan domains of type III. We call the corresponding actions Abelian
+Elliptic, Abelian Hyperbolic and Parabolic. The moment maps of these three
+actions are computed and functions of them (moment map symbols) are used
+to construct commutative C∗-algebras generated by Toeplitz operators. This
+leads to a natural generalization of known results for the unit disk. We also
+compute spectral integral formulas for the Toeplitz operators corresponding to
+the Abelian Elliptic and Parabolic cases.
+1. Introduction
+Bounded symmetric domains, weighted Bergman spaces on such domains and
+Toeplitz operators acting on Bergman spaces constitute three fundamental objects
+in operator theory. The reason is that they are specific enough to make explicit
+computations that lead to interesting results, and at the same time they are com-
+plicated enough so that such results are non-trivial and enlightening.
+For some years now, operator theory analysts have found plenty of examples of
+commutative C∗-algebras generated by Toeplitz operators when the corresponding
+set of symbols is suitably restricted. The first such example was considered in [11],
+where it was proved that Toeplitz operators on the unit disk D with radial symbols
+are diagonal with respect to the orthogonal monomial basis. Clearly, a symbol on
+D is radial if it is invariant under the natural T-action. We note that the T-action
+on the unit disk D realizes, up to conjugacy, all the elliptic M¨obius transformations.
+The introduction in [11] of Toeplitz operators with radial symbols was followed
+by a series of developments found in [3, 4, 5].
+These references considered all
+three fundamental types of M¨obius transformations on the unit disk D: elliptic,
+hyperbolic and parabolic.
+It was proved that symbols that are invariant under
+the corresponding groups of M¨obius transformations yield Toeplitz operators that
+generate commutative C∗-algebras. Then, it was found in [7] that, under suitable
+smoothness conditions, these constructions yield the only commutative C∗-algebras
+generated by Toeplitz operators acting on every weighted Bergman space on the
+unit disk D.
+2020 Mathematics Subject Classification. Primary 47B35 30H20; Secondary 53D20.
+Key words and phrases. Toeplitz operators, Bergman spaces, Cartan domains, Lie groups,
+K¨ahler manifolds, moment maps.
+1
+
+2
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+The next step was to study the behavior in the case of higher dimensional
+bounded symmetric domains, and the unit ball Bn in Cn was the first natural
+example to consider. It was found in [16, 17] that there exists exactly, up to con-
+jugacy, n + 2 maximal Abelian subgroups (MASGs) of biholomorphisms each one
+of which yields invariant symbols whose Toeplitz operators generate commutative
+C∗-algebras. This is a natural generalization of the situation observed for the unit
+disk D, since in this case we have n = 1 from which it follows the existence of three
+MASGs. Nevertheless, some simplicity is lost because the number of MASGs grows
+with the dimension of the unit ball Bn.
+After these works, many other results have been found where a suitable symmetry
+of the symbols yields commuting Toeplitz operators. Such symmetry is in most
+cases a consequence of the invariance with respect to a certain biholomorphism
+group.
+This has been observed for every bounded symmetric domain on every
+weighted Bergman space. We refer to [1] for a very general collection of related
+results.
+In a parallel line of development, symplectic geometry has been found to play
+an special role in the construction of symbols whose Toeplitz operators generate
+commutative C∗-algebras. It was proved in [14] that, for the unit ball Bn and on
+any of its weighted Bergman spaces, every single Abelian connected group of bi-
+holomorphisms provides symbols with mutually commuting Toeplitz operators. For
+such a group H acting on Bn this is achieved by considering the so-called moment
+map symbols for H instead of H-invariant symbols. We refer to Section 3 for the
+details of the definitions and properties involved. However, we mention here that
+the moment map of an action is a mapping defined on the corresponding bounded
+symmetric domain using its symplectic manifold structure, and the moment map
+symbols are functions of such moment maps. Another example of the use of moment
+map symbols is given by the results found in [15], where the bounded symmetric
+domain considered is the Cartan domain of type IV.
+The goal of this work is to apply these ideas to study Toeplitz operators with
+moment map symbols acting on the weighted Bergman spaces of Cartan domains of
+type III. We recall that such domains are realized by the so-called generalized unit
+disk DIII
+n
+and Siegel’s generalized upper half-plane Sn (see Section 2). In fact, as
+we show in Section 4, to these domains we can associate three biholomorphic actions
+that naturally generalize the three actions described above for the unit disk. Hence,
+we call these actions on either DIII
+n
+or Sn the Elliptic, Hyperbolic and Parabolic
+Actions (see subsection 4.1).
+These come from the groups U(n), GL(n, R) and
+Symm(n, R), respectively, of which only the last one is Abelian for every n. Hence,
+we introduce actions that we call Abelian Elliptic and Abelian Hyperbolic (see
+Definition 4.2). As noted in Remark 4.3 all three Abelian actions can be seen as
+coming from the corresponding centers of the original groups involved.
+We present in Section 2 all the theory needed to understand the Riemannian
+and symplectic geometry background used in the rest of the work. In particular,
+we compute in subsection 3.2 the Bergman metric and the K¨ahler form for both
+DIII
+n
+and Sn. We use this to compute in subsection 4.2 the moment maps for our
+three distinguished actions: Abelian Elliptic, Abelian Hyperbolic and Parabolic.
+We introduce in Section 5 Toeplitz operators with special symbols. First, we
+consider invariant symbols in subsection 5.1 and we recall some known commutative
+C∗-algebras generated by Toeplitz operators for our setup. Second, we introduce
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+3
+moment map symbols in Definition 5.3, and with the use of our moment map
+computations we obtain explicit formulas for moment map symbols for our three
+distinguished Abelian actions.
+We obtain the following general description (see
+Proposition 5.4 for the precise statements)
+• Abelian Elliptic symbols: Z �−→ f
+�
+tr(ZZ)
+�
+.
+• Abelian Hyperbolic symbols: Z �−→ f
+�
+tr(Im(Z)−1Re(Z))
+�
+.
+• Parabolic symbols: Z �−→ f(Im(Z)).
+From a quick comparison with the notions considered in the current literature, we
+observe that these three types of symbols are natural, almost canonical, generaliza-
+tions from the unit disk D to the domains DIII
+n
+and Sn of the symbols obtained
+from the elliptic, hyperbolic and parabolic actions on D.
+We prove in Theorem 5.8 that the three types of symbols above yield Toeplitz op-
+erators that generate commutative C∗-algebras on every weighted Bergman space.
+Our method of proof is based on the fact that these moment map symbols have an
+additional invariance: they are invariant under the group from which the Abelian
+group is the center. This allows to use the results from subsection 5.1. On the
+other hand, it is interesting to observe the importance of having only three types
+of symbols in Theorem 5.8 as a generalization of the corresponding result for the
+unit disk. This is explained in Remark 5.9.
+Finally, we obtain in Section 6 integral formulas for the Toeplitz operators with
+moment map symbols that provide simultaneous diagonalization for them. This is
+done for the Abelian Elliptic and Parabolic Actions; we leave the Abelian Hyper-
+bolic case as an important project to develop. The relevant results are Theorems 6.3
+and 6.8. The simplicity of the formulas presented in Theorem 6.3 highlights the
+importance of using symplectic geometry to solve these operator theory problems.
+Likewise, Theorem 6.8 has very natural formulas that involve a Fourier-Laplace
+transform obtained in Theorem 6.7.
+2. The Cartan domains of type III and their analysis
+We recall the basic geometric and analytic properties of the Cartan domains of
+type III.
+2.1. Bounded and unbounded realizations. In the rest of this work, and for F
+either R or C, we will denote by Mat(n, F) the space of n × n matrices over F and
+by Symm(n, F) its subspace of symmetric matrices. As usual, GL(n, F) will denote
+the Lie group of invertible elements of Mat(n, F).
+Definition 2.1. The n-dimensional Cartan domain of type III is the complex
+domain given by DIII
+n
+= {Z ∈ Symm(n, C) | In − ZZ > 0}.
+The domain DIII
+n
+is clearly bounded. On the other hand, there is a natural
+unbounded domain associated to DIII
+n
+.
+Definition 2.2. The n-dimensional Siegel domain is the complex domain given by
+Sn = {Z ∈ Symm(n, C) | Im(Z) > 0}.
+We note that DIII
+1
+and S1 are precisely the unit disk D and the upper half-plane
+H, respectively, in the complex plane C. For this reason, the domains DIII
+n
+and Sn
+are also known as the generalized unit disk and generalized upper-half plane, respec-
+tively. Furthermore, these domains are related in a way similar to the well known
+1-dimensional case. For the next result we refer to [9, Exercise C, Chapter VIII].
+
+4
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+Proposition 2.3. The map ϕ : Sn → DIII
+n
+given by
+Z �→ (In + iZ)(In − iZ)−1,
+is a biholomorphism from Sn onto DIII
+n
+.
+Because of the previous result, the domain Sn is also known as the unbounded
+realization of the n-dimensional Cartan domain of type III.
+2.2. Biholomorphism groups. In this section we describe the groups of biholo-
+morphisms of the domains DIII
+n
+and Sn introduced above. We start by considering
+the matrices
+In,n =
+�
+In
+0
+0
+−In
+�
+,
+Jn =
+�
+0
+−In
+In
+0
+�
+.
+These naturally yield the next Lie groups.
+Sp(n, C) = {M ∈ Mat(2n, C) | M ⊤JnM = Jn},
+Sp(n, R) = {M ∈ Mat(2n, R) | M ⊤JnM = Jn},
+U(n, n) = {M ∈ Mat(2n, C) | M ∗In,nM = In,n}.
+We recall the notion of a bounded symmetric domain.
+Definition 2.4. A domain D ⊂ CN is called symmetric if for every z ∈ D there
+exists a biholomorphism ϕz : D → D such that ϕz(w) = w if and only if w = z. If
+D is also bounded, then D is called a bounded symmetric domain. If D satisfies
+tD = D, for every t ∈ T, then the domain D is called circled.
+Through suitable actions of the groups introduced above, one can prove that
+the domains DIII
+n
+and Sn are symmetric.
+For the next result we refer to [13,
+Paragraph (2.3)] (see also [9]). From now on, for any given matrix M ∈ Mat(2n, C)
+a decomposition of the form
+M =
+�A
+B
+C
+D
+�
+,
+will always be taken so that A, B, C, D have size n × n.
+Proposition 2.5. The action via generalized M¨obius transformations given by
+Sp(n, C) ∩ U(n, n) × DIII
+n
+−→ DIII
+n
+�A
+B
+C
+D
+�
+· Z �−→ (AZ + B)(CZ + D)−1,
+realizes the biholomorphism group of DIII
+n
+. Furthermore, DIII
+n
+is a circled bounded
+symmetric domain and it is given as the quotient
+DIII
+n
+≃ Sp(n, C) ∩ U(n, n)/U(n),
+where U(n) embedded in Sp(n, C) ∩ U(n, n) by
+A �−→
+�
+A
+0
+0
+A
+�
+corresponds to the group of biholomorphisms of DIII
+n
+that fix the origin.
+Similarly, we have the next description of the biholomorphism group of the do-
+main Sn. We now refer to [9, Exercise C, Chapter VIII].
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+5
+Proposition 2.6. The action via generalized M¨obius transformations given by
+Sp(n, R) × Sn −→ Sn
+�
+A
+B
+C
+D
+�
+· Z �−→ (AZ + B)(CZ + D)−1,
+realizes the biholomorphism group of Sn. Furthermore, Sn is a symmetric domain
+and it is given as the quotient
+Sn = Sp(n, R)/U(n),
+where U(n) embedded in Sp(n, R) by
+A �−→
+�
+Re(A)
+Im(A)
+−Im(A)
+Re(A)
+�
+corresponds to the group of biholomorphisms of Sn that fix the matrix iIn.
+Remark 2.7. By Proposition 2.3 it follows that the biholomorphism groups of
+DIII
+n
+and Sn are isomorphic. In fact, it is easy to prove that Sp(n, C) ∩ U(n, n)
+and Sp(n, R) are conjugated (see [13]).
+2.3. Bergman spaces and Toeplitz operators. From now on, D will denote
+either of the domains DIII
+n
+or Sn, and dZ the Lebesgue measure on Symm(n, C).
+A number of invariants can be associated to any symmetric domain. The simplest
+one is the dimension, which for D is n(n + 1)/2. For other invariants we refer to
+[19] for further details on their definitions and here we simply state their known
+values with some remarks.
+• The rank is defined as the dimension of maximal linearly embedded poly-
+disks. For D the rank is n.
+• The multiplicities are defined as the main invariants that describe the Jor-
+dan triple system associated to the symmetric domain. For D the multi-
+plicities are a = 1, b = 0. The vanishing of the latter implies that DIII
+n
+has
+a tubular realization which is in fact given by Sn. For this we observe that
+Sn = Symm(n, R) ⊕ iPos(n, R),
+where Pos(n, R) denotes the cone of positive definite n × n real matrices.
+In the rest of this work we will denote Ωn = Pos(n, R).
+• For a tubular domain, the genus is given as p = 2d/r, where d and r are
+the dimension and the rank of the domain, respectively. Hence, for D we
+have p = n + 1.
+We will make use of the multi-gamma function (see [19, Definition 2.4.2]) that
+we will consider only for Cartan domains of type III. Such function is associated
+to the cone part of a tubular realization of a tube type symmetric domain. In our
+case, it is defined by
+ΓΩn(λ) = (2π)
+n(n−1)
+4
+n
+�
+j=1
+Γ
+�
+λ − j − 1
+2
+�
+,
+for every λ > (n−1)/2. It is well known (see [10, 19]) that the volume of a bounded
+symmetric domain can be expressed in terms of the multi-gamma functions. In this
+
+6
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+case we have (see [10])
+Vol(DIII
+n
+) = π
+n(n+1)
+2
+ΓΩn
+� n+1
+2
+�
+ΓΩn(n + 1)
+.
+Hence, we consider the normalized measure on Symm(n, C)
+dv(Z) =
+ΓΩn(n + 1)
+π
+n(n+1)
+2
+ΓΩn
+� n+1
+2
+� dZ.
+In particular, dv(Z) is a probability measure on DIII
+n
+.
+Definition 2.8. The (weightless) Bergman space A2(D) is the subspace of L2(D, v)
+that consists of holomorphic functions. In other words, we have
+A2(D) = {f ∈ L2(D, v) | f is holomorphic }.
+It is a well known fact that A2(D) is a closed subspace of L2(D, v) (see [9, 19]).
+We will denote by BD : L2(D, v) → A2(D) the corresponding orthogonal projection.
+It is called the (weightless) Bergman projection. Moreover, it is also well known
+that A2(D) is a reproducing kernel Hilbert space (see [9, Chapter VIII]) in the
+sense that the evaluation map
+evZ : A2(D) −→ C
+f �−→ f(Z),
+is continuous for every Z ∈ D.
+This implies the existence of a unique smooth
+function KD : D × D → C, holomorphic in the first variable and anti-holomorphic
+in the second variable, satisfying KD(Z, ·) ∈ A2(D) for every Z ∈ D and for which
+the Bergman projection is given by
+BD(f)(Z) =
+�
+D
+f(W)KD(Z, W) dv(W).
+for every f ∈ L2(D, v) and Z ∈ D. The function KD is called the (weightless)
+Bergman kernel of D.
+The Bergman kernels of symmetric domains have closed known expressions. In
+particular, it follows from Examples 2.4.17 and 2.9.15 in [19] that the Bergman
+kernels of DIII
+n
+and Sn are given by the expressions
+KDIII
+n (Z, W) = det(In − ZW)−(n+1),
+(2.1)
+KSn(Z, W) = det(−i(Z − W))−(n+1),
+(2.2)
+respectively. We note that a linear biholomorphism has to be applied in order to
+obtain the above expression for KSn from the one found in [19]. More precisely, our
+unbounded realization of DIII
+n
+is obtained from the one considered in [19] through
+the map Z �→ −iZ.
+The next standard construction is to use powers of the Bergman kernel to obtain
+weighted measures. The following formula, which holds for every λ > n, is useful
+to normalize such weighted measures (see [19, Lemma 2.9.18])
+�
+DIII
+n
+det(In − ZZ)λ−n−1 dZ = π
+n(n+1)
+2
+ΓΩn
+�
+λ − n+1
+2
+�
+ΓΩn(λ)
+.
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+7
+Hence, we consider for every λ > n the measure
+dvλ(Z) =
+ΓΩn (λ)
+π
+n(n+1)
+2
+ΓΩn
+�
+λ − n+1
+2
+� det(In − ZZ)λ−n−1 dZ
+which is a probability measure on DIII
+n
+, and we also consider the normalized mea-
+sure
+d�vλ(Z) =
+ΓΩn (λ)
+π
+n(n+1)
+2
+ΓΩn
+�
+λ − n+1
+2
+� det(−i(Z − Z))λ−n−1 dZ.
+on the domain Sn.
+Definition 2.9. For λ > n, the weighted Bergman spaces on DIII
+n
+and Sn with
+weight λ are given by
+A2
+λ(DIII
+n
+) = {f ∈ L2(DIII
+n
+, vλ) | f is holomorphic },
+A2
+λ(Sn) = {f ∈ L2(Sn, �vλ) | f is holomorphic },
+respectively. We will denote by A2
+λ(D) the corresponding weighted Bergman space
+when D is DIII
+n
+or Sn.
+Note that for λ = n+1, we obtain A2
+λ(DIII
+n
+) = A2(DIII
+n
+) and A2
+λ(Sn) = A2(Sn),
+which are the weightless Bergman spaces.
+As before, it is well known that every weighted Bergman space is closed in
+the corresponding L2 space in such a way that it is a reproducing kernel Hilbert
+space.
+In particular, for D either DIII
+n
+or Sn there exists a smooth function
+KD,λ : D × D → C, holomorphic and anti-holomorphic in the first and second
+variable (respectively), such that the orthogonal projection onto A2
+λ(D) is given by
+BD,λ(f)(Z) =
+�
+D
+f(W)KD,λ(Z, W) dνλ(W),
+for every f ∈ L2(D, νλ) and Z ∈ D, where νλ denotes either vλ or �vλ according
+to whether D is DIII
+n
+or Sn.
+This projection is called the weighted Bergman
+projection. It follows by Propositions 2.4.22 and 2.9.24 from [19] that the weighted
+Bergman kernels for these domains are given by the following expressions
+KDIII
+n
+,λ(Z, W) = det(In − ZW)−λ,
+KSn,λ(Z, W) = det(−i(Z − W))−λ,
+for every λ > n. In particular, we have KD,λ(Z, W) = KD(Z, W)
+λ
+n+1 for every
+Z, W ∈ D.
+The previous constructions allow us to define our main object of study.
+Definition 2.10. For every weight λ > n and a ∈ L∞(D), the Toeplitz operator
+with symbol a is the bounded operator T (λ)
+a
+acting on A2
+λ(D) that is given by
+T (λ)
+a
+= BD,λ ◦ Ma.
+It is interesting to note that the Bergman spaces A2
+λ(DIII
+n
+) and A2
+λ(Sn) are uni-
+tarily equivalent, thus simplifying some computations. This unitary equivalence is
+stated without proof in the next result. Its proof is a straightforward generalization
+of the arguments provided to obtain Theorem 4.9 in Chapter IV from [18].
+
+8
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+Theorem 2.11. The map ϕ given in Proposition 2.3 induces the unitary operator
+given by
+Uϕ : A2
+λ(DIII
+n
+) −→ A2
+λ(Sn)
+f �−→ JC(ϕ)
+λ
+n+1 f ◦ ϕ,
+where JC(ϕ) = det(dϕC) denotes the complex Jacobian.
+3. Geometry of Cartan domains of type III
+3.1. Symplectic and K¨ahler geometry. We discuss here some basic material
+from symplectic geometry, which will be essential for the main results of this work.
+Definition 3.1. A symplectic manifold is a pair (M, ω), where M is a smooth
+manifold and ω is a closed 2-form which yields a non-degenerate bilinear form at
+every point.
+Some of the most important examples of symplectic manifolds come from com-
+plex differential geometry. We recall that a manifold M is complex if their charts
+map onto open sets of complex vector spaces so that the changes of coordinates are
+holomorphic. For such a manifold M, this yields a complex structure Jz on every
+tangent space TzM, for every z ∈ M. In turn, this provides a tensor field J known
+as the complex structure tensor of M. In particular, we have J2 = −I the identity
+tensor acting on the fibers of the tangent bundle T M. We refer to [13] for further
+details.
+The next definition describes well behaved Riemannian metrics with respect to
+these constructions.
+Definition 3.2. Let M be a complex manifold with complex structure tensor J
+and a given Riemannian metric g. We say that M is a Hermitian manifold if it
+satisfies
+gz(Jzu, Jzv) = gz(u, v)
+for every z ∈ M and u, v ∈ TzM.
+We now proceed to relate Hermitian manifolds to symplectic geometry. We will
+explain the main constructions and refer to [13] for further details. Let us start
+by considering a complex manifold M with complex structure tensor J. Then, the
+tangent bundle can be complexified to a complex tangent bundle denoted by T CM,
+and the action of J on T M can also be complexified to obtain a tensor JC acting
+on T CM. Such complexifications are performed fiberwise.
+Since (JC
+z )2 = −I, for every z ∈ M, if we define the spaces
+T 1,0
+z
+M = {v ∈ T C
+z M | JC
+z v = iv},
+T 0,1
+z
+M = {v ∈ T C
+z M | JC
+z v = −iv},
+then we have T C
+z M = T 1,0
+z
+M ⊕ T 0,1
+z
+M. These spaces are known as the subspaces of
+holomorphic and anti-holomorphic tangent vectors. If (z1, . . . , zn) is a holomorphic
+chart with real components obtained from the decomposition zj = xj + iyj, then
+the usual Wirtinger differential operators are given by
+∂
+∂zj
+= 1
+2
+�
+∂
+∂xj
+− i ∂
+∂yj
+�
+,
+∂
+∂zj
+= 1
+2
+�
+∂
+∂xj
++ i ∂
+∂yj
+�
+,
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+9
+for every j = 1, . . . , n. The first set of operators define at every point in the domain
+of the chart a basis for the corresponding fibers of T 1,0M. Similarly, the second set
+of operators define a basis for the fibers of T 0,1M. The corresponding dual basis
+are given by
+dzj = dxj + i dyj,
+dzj = dxj − i dyj,
+where j = 1, . . . , n.
+Let us now consider a Riemannian metric g on M for which M is a Hermitian
+manifold.
+We can complexify g to a complex bilinear tensor gC defined on the
+complexified tangent bundle T CM. This yields a positive definite Hermitian form
+T 1,0
+z
+M × T 1,0
+z
+M −→ C,
+(u, v) �−→ gC
+z (u, v),
+for every z ∈ M. In local coordinates, this can be written as
+n
+�
+j,k=1
+gjk(z)dzj ⊗ dzk.
+For this reason, we will denote this field of complex Hermitian forms with the same
+symbol g. To more easily distinguish between the two of them, we will refer to
+the original g as the Riemannian metric of M and we will call the previous field of
+Hermitian forms the Hermitian metric of M.
+The previous setup and constructions allow to introduce the next important
+geometric object.
+Definition 3.3. For a Hermitian manifold M with Hermitian metric g as con-
+structed above, the associated 2-form is given by
+ω = g(J(·), ·) = −2Im(g)
+where the first occurrence of g is the Riemannian metric and the second one is the
+corresponding Hermitian metric. The Hermitian manifold M is called K¨ahler if its
+associated 2-form is closed. In this case, ω is called the K¨ahler form of M.
+It is straightforward to check that the associated 2-form of any Hermitian mani-
+fold is non-degenerate. Hence, every K¨ahler manifold is a symplectic manifold, and
+in this case the K¨ahler form is its symplectic form.
+One can alternatively provide a K¨ahler structure on a complex manifold by
+introducing a field of Hermitian bilinear forms. This is the content of the next
+result which is a particular case of Proposition 1 in page 18 from [13].
+Proposition 3.4. Let M be a complex manifold and let g be a tensor field of positive
+definite Hermitian bilinear forms on T 1,0M. Assume that for every holomorphic
+coordinate chart (z1, . . . , zn), in a family of charts covering M, there is some real
+valued function F such that
+g =
+n
+�
+j,k=1
+∂2F
+∂zj∂zk
+dzj ⊗ dzk
+in the domain of the given chart. Then, the tensor 2Re(g) is a Riemannian metric
+that yields a K¨ahler structure on M whose Hermitian metric is given by g.
+
+10
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+3.2. The Bergman metric and its K¨ahler form. We now use the results pre-
+viously obtained to construct a K¨ahler structure on the Cartan domains of type
+III. The next fundamental theorem is a particular case of the discussion in the first
+part of Chapter 4 in [13] (see also [9, 18]). Note that, from now on, we will use the
+canonical complex linear coordinates of Symm(n, C).
+Theorem 3.5. Let D be either of DIII
+n
+or Sn and let KD(Z, W) be the reproducing
+Bergman kernel of D. Then, the tensor given by
+�
+1≤l≤m≤n
+1≤j≤k≤n
+∂2 log KD(Z, Z)
+∂zlm∂zjk
+dzlm ⊗ dzjk,
+is a field of positive definite Hermitian forms that yields a structure of K¨ahler man-
+ifold on D. Furthermore, both the corresponding Riemannian metric and associated
+K¨ahler form are invariant under the group of biholomorphisms.
+We use Theorem 3.5 to introduce K¨ahler structures on DIII
+n
+and Sn by nor-
+malizing the tensor considered in its statement. These normalization will simplify
+some formulas below.
+Definition 3.6. Let D be either of DIII
+n
+or Sn and KD(Z, W) the Bergman kernel
+of D. The Bergman metric of D is the field of Hermitian forms given by
+gD = cD
+�
+1≤l≤m≤n
+1≤j≤k≤n
+∂2 log KD(Z, Z)
+∂zlm∂zjk
+dzlm ⊗ dzjk,
+where cDIII
+n
+=
+1
+n+1 and cSn =
+4
+n+1.
+The next two results are very well known properties of the Wirtinger differ-
+ential operators that will be useful in this work. We state them for the sake of
+completeness.
+Lemma 3.7. For any smooth function f : CN −→ C we have
+df =
+N
+�
+j=1
+� ∂f
+∂zj
+dzj + ∂f
+∂zj
+dzj
+�
+.
+Lemma 3.8 (Chain rule for Wirtinger derivatives). Let g : Cn → Cm and f :
+Cm → C be smooth functions. Then, we have
+∂(f ◦ g)
+∂zj
+=
+m
+�
+k=1
+� ∂f
+∂zk
+◦ g ∂gk
+∂zj
++ ∂f
+∂zk
+◦ g ∂gk
+∂zj
+�
+,
+∂(f ◦ g)
+∂zj
+=
+m
+�
+k=1
+� ∂f
+∂zk
+◦ g ∂gk
+∂zj
++ ∂f
+∂zk
+◦ g ∂gk
+∂zj
+�
+.
+The following elementary computation will be used latter on. We provide its
+proof for the sake of completeness.
+Lemma 3.9. The differential of det : Mat(n, C) → C is given by
+d(det)A = tr
+�
+adj(A)dA
+�
+,
+for every A ∈ Mat(n, C), where adj(A) (adjugate of A) is the transpose of the
+cofactor matrix of A.
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+11
+Proof. If A = (alm) ∈ Mat(n, C) and clm is the cofactor of alm, then the cofactor
+expansion of the determinant along the k-th column is given by
+det A =
+n
+�
+l=1
+clkalk =
+n
+�
+l=1
+�
+adj(A)T �
+lkalk.
+It follows that
+∂ det
+∂ajk
+(A) = cjk =
+�
+adj(A)T �
+jk
+and we obtain the differential
+d(det)A =
+n
+�
+j,k=1
+∂ det
+∂ajk
+(A) dajk =
+n
+�
+j,k=1
+cjk dajk
+=
+n
+�
+j,k=1
+�
+adj(A)T �
+jk dajk =
+n
+�
+j,k=1
+�
+adj(A)
+�
+kj dajk
+=
+n
+�
+k=1
+(adj(A) dA)kk = tr(adj(A) dA).
+□
+We now obtain explicit formulas for the Bergman metrics of the Cartan domains
+of type III. Note that we have provided coordinate free expressions. This will be
+useful for our computations in the rest of this work.
+Theorem 3.10. The Bergman metrics on DIII
+n
+and Sn are respectively given by
+gDIII
+n
+Z
+(U, V ) = tr
+�
+(In − ZZ)−1U(In − ZZ)−1V
+�
+,
+gSn
+Z (U, V ) = tr
+�
+Im(Z)−1UIm(Z)−1V
+�
+,
+for every U, V ∈ Symm(n, C). In particular, the K¨ahler forms of DIII
+n
+and Sn are
+respectively given by
+ωDIII
+n
+Z
+(U, V ) = i tr
+�
+(In − ZZ)−1U(In − ZZ)−1V
+�
+− i tr
+�
+(In − ZZ)−1U(In − ZZ)−1V
+�
+,
+ωSn
+Z (U, V ) = 2 tr
+�
+Im(Z)−1Re(U)Im(Z)−1Im(V )
+�
+− 2 tr
+�
+Im(Z)−1Im(U)Im(Z)−1Re(V )
+�
+,
+for every U, V ∈ Symm(n, C).
+Proof. In this proof we will consider the complex vector spaces Symm(n, C) and
+Mat(n, C) whose coordinates will be denoted in both cases by zjk, even though they
+have different meanings for such spaces. However, from the context where these
+coordinates are used it will be easy to identify the actual meaning.
+We start by computing the Bergman metric on DIII
+n
+. First, we observe that we
+have the following partial derivative
+∂(In − ZZ)
+∂zjk
+(Z) = −Z ∂Z
+∂zjk
+= −ZEjk,
+where Ejk is the n × n symmetric matrix that has 1 in the entries (j, k) and (k, j)
+and 0 elsewhere. Note that these matrices are the basis with respect to which we
+are considering the canonical coordinates in Symm(n, C).
+
+12
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+Next, using the previous computation, applying Lemmas 3.9 and 3.8, Equa-
+tion (2.1) and using the fact that det is holomorphic, we obtain
+1
+n + 1
+∂
+∂zjk
+log KDIII
+n (Z, Z) =
+=
+1
+det(In − ZZ)
+n
+�
+l,m=1
+∂ det
+∂zlm
+(In − ZZ)(ZEjk)lm
+=
+1
+det(In − ZZ)
+n
+�
+l,m=1
+(adj(In − ZZ)T )lm(ZEjk)lm
+=
+1
+det(In − ZZ)tr
+�
+adj(In − ZZ)ZEjk
+�
+= tr
+�
+(In − ZZ)−1ZEjk
+�
+.
+Now, we will use the easy to prove relations
+(In − ZZ)−1Z = Z(In − ZZ)−1,
+Z(In − ZZ)−1 = (In − ZZ)−1Z,
+which hold for every Z ∈ DIII
+n
+. Using the identities obtained so far we compute
+1
+n + 1
+∂2
+∂zlm∂zjk
+log KDIII
+n (Z, Z) =
+∂
+∂zlm
+tr
+�
+(In − ZZ)−1ZEjk
+�
+= tr
+�
+(In − ZZ)−1ElmZ(In − ZZ)−1ZEjk
+�
++ tr
+�
+(In − ZZ)−1ElmEjk
+�
+= tr
+�
+(In − ZZ)−1Elm(In − ZZ)−1ZZEjk
+�
++ tr
+�
+(In − ZZ)−1ElmEjk
+�
+= tr
+�
+(In − ZZ)−1Elm(In − ZZ)−1(ZZ + (In − ZZ))Ejk
+�
+= tr
+�
+(In − ZZ)−1Elm(In − ZZ)−1Ejk
+�
+.
+This implies that the metric gDIII
+n
+Z
+satisfies the required identity on the basic el-
+ements of the vector space Symm(n, C), thus proving the result for the Bergman
+metric of DIII
+n
+. The corresponding computation of the Bergman metric for Sn is
+obtained similarly.
+From Definition 3.3 the K¨ahler form of DIII
+n
+is given by
+ωDIII
+n
+Z
+(U, V ) = −2Im
+�
+gDIII
+n
+Z
+(U, V )
+�
+= i
+�
+gDIII
+n
+Z
+(U, V ) − gDIII
+n
+Z
+(U, V )
+�
+,
+which yields the stated formula from our computation of the Bergman metric of
+DIII
+n
+.
+Finally, for the K¨ahler form of Sn we compute
+ωSn
+Z (U, V ) = −2Im
+�
+gSn
+Z (U, V )
+�
+= −2Im
+�
+tr
+�
+Im(Z)−1(Re(U) + iIm(U))Im(Z)−1(Re(V ) − iIm(V ))
+��
+,
+from which the stated formula is easily obtained.
+□
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+13
+3.3. Moment maps. Now we turn back our attention to symplectic geometry. It
+will provide the main geometric tools and objects that we will apply to the study
+of Toeplitz operators. We refer to [12] for the symplectic geometry facts stated
+without proof.
+In the rest of this subsection (M, ω) will denote a fixed symplectic manifold. A
+diffeomorphism ϕ : M → M is called a symplectomorphism if ϕ∗(ω) = ω. In other
+words, a symplectomorphism is a diffeomorphism preserving the symplectic form.
+If H is a Lie group with a smooth action on M, then we say that the H-action
+is symplectic if the map
+M −→ M
+z �−→ h · z
+is a symplectomorphism for every h ∈ H.
+There are two important types of vector fields on M. From now on, we will
+denote by X(M) the Lie algebra of vector fields over M. A field X ∈ X(M) is
+called a symplectic vector field if and only if the 1-form ω(X, ·) is closed, and it is
+called a Hamiltonian vector field if and only if the form ω(X, ·) is exact. We will
+denote by X(M, ω) the space of symplectic vector fields on M. It is a well known
+fact that X(M, ω) is a Lie subalgebra of X(M).
+For any smooth function f : M → R, the non-degeneracy of ω implies the
+existence of a unique element Xf ∈ X(M) such that
+df = ω(Xf, ·).
+In this case, Xf is called the Hamiltonian vector field associated to f.
+Symplectic vector fields can be characterized by symplectomorphisms.
+More
+precisely, it is well known that an element X ∈ X(M) belongs to X(M, ω) if and
+only if the local flow generated by X acts by (locally defined) symplectomorphisms.
+An important converse to the previous fact relates symplectic actions to sym-
+plectic vector fields as follows. Let us consider a symplectic action of a Lie group H
+on M. Then, for every X ∈ h (the Lie algebra of H), we define the induced vector
+field on M by
+X♯
+z = d
+ds
+���
+s=0 exp(sX) · z.
+for every z ∈ M, where exp : h → H is the exponential map of H. Then, the fact
+that the H-action is symplectic implies that X♯ ∈ X(M, ω) for every X ∈ h.
+In the previous discussion, we have shown two different constructions that map
+into the space X(M, ω) of symplectic vector fields. Hence, a natural problem to
+consider is the existence of a map h → C∞(M) that makes the following diagram
+commute
+C∞(M)
+�
+h
+�①
+①
+①
+①
+①
+①
+①
+①
+①
+� X(M, ω)
+where the vertical arrow is the map f �→ Xf and the horizontal arrow is the map
+X �→ X♯. The existence of such diagonal map yields the notion of a moment map
+for the H-action. The precise definition requires some additional conditions. We
+recall that Ad = AdH : H → GL(h) denotes the adjoint representation of the Lie
+
+14
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+group H, and that Ad∗ denotes the dual representation on h∗. In particular, we
+have Ad∗(h) = Ad(h−1)⊤ for every h ∈ H.
+Definition 3.11. Let (M, ω) be a symplectic manifold and let H be a Lie group
+acting by symplectomorphisms on M. A moment map for the H-action is a smooth
+map µ : M → h∗, where h∗ is the vector space dual of h, that satisfies the following
+properties.
+(1) For every X ∈ h consider the map µX : M → R given by µX(z) = ⟨µ(z), X⟩.
+Then, the Hamiltonian vector field associated to µX is X♯, for every X ∈ h.
+In other words, it holds
+dµX = ω(X#, ·),
+for every X ∈ h.
+(2) The map µ is H-equivariant. In other words, we have
+µ(h · z) = Ad∗(h)(µ(z)),
+for every z ∈ M and h ∈ H.
+Remark 3.12. If H is an Abelian group, then its adjoint representation satisfies
+Ad(h) = Ih for every h ∈ H. Hence, in this case, condition 2. in Definition 3.11
+reduces to
+µ(h · z) = µ(z),
+for every h ∈ H and z ∈ M. In other words, this requires the smooth map to be
+H-invariant.
+4. Three Abelian biholomorphism groups and their moment maps
+In this section we study three special types of subgroups of biholomorphisms
+acting on Cartan domains of type III. For the corresponding Abelian groups, we
+compute the moment maps. We will see later on that these moment maps are a
+powerful tool to find commutative C∗-algebras generated by Toeplitz operators.
+4.1. Elliptic, Hyperbolic, and Parabolic Actions. The Cartan domains DIII
+n
+and their unbounded realizations Sn carry three interesting actions of subgroups of
+biholomorphisms. As we will see, these actions generalize the three different types
+of M¨obius transformations found for the unit disk D and the upper half plane H.
+Proposition 2.5 provides the action
+U(n) × DIII
+n
+−→ DIII
+n
+U · Z = UZU ⊤,
+which yields the subgroup of biholomorphisms that fixes the origin. Up to con-
+jugacy, this characterizes the subgroups that fix some point in the domain DIII
+n
+.
+This is so because of the homogeneity of this domain. We will call this the Elliptic
+Action on DIII
+n
+.
+Next we observe that there is a canonical homomorphism of Lie groups given by
+GL(n, R) −→ Sp(n, R)
+A �−→
+�
+A
+0
+0
+(A−1)⊤
+�
+.
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+15
+A straightforward computation shows that this assignment is indeed a homomor-
+phism whose image lies in Sp(n, R). Hence, this homomorphism and Proposition 2.6
+provide the action
+GL(n, R) × Sn −→ Sn
+A · Z = AZA⊤.
+It is easily seen that this action realizes the subgroup of biholomorphisms that fixes
+the origin, a boundary point of the domain Sn. For this reason, we will call this
+the Hyperbolic Action on Sn.
+Finally, we have a canonical homomorphism of Lie groups given by
+Symm(n, R) −→ Sp(n, R)
+S �−→
+�In
+S
+0
+In
+�
+,
+where Symm(n, R) is endowed with the Lie group structure with operation given
+by the sum of matrices. Again, it is straightforward to show that this map is indeed
+a homomorphism into the group Sp(n, R). We now have that this homomorphism
+together with Proposition 2.6 provide the action
+Symm(n, R) × Sn −→ Sn
+S · Z = Z + S.
+This action realizes the subgroup of biholomorphisms of the tube type domain Sn
+that correspond to translations on the real vector space part. Since this action
+clearly generalizes the translation action on the real part on the upper half-plane
+H, we will call this action on Sn the Parabolic Action.
+In fact, all three actions introduced above generalize the behavior observed in
+the 1-dimensional case.
+This is stated in the following well known result.
+We
+recall that two biholomorphisms are conjugated if they are so under some other
+biholomorphism.This result justifies our choice of notation for the actions considered
+above.
+Corollary 4.1. Let us denote by D either D or H. If ϕ is a biholomorphism of D,
+then the following equivalences hold.
+(1) The M¨obius transformation ϕ is elliptic if and only if it is conjugated to a
+transformation that belongs to the action T × D → D given by z �→ tz.
+(2) The M¨obius transformation ϕ is hyperbolic if and only if it is conjugated to
+a transformation that belongs to the action R+ × H → H given by z �→ rz.
+(3) The M¨obius transformation ϕ is parabolic if and only if it is conjugated to
+a transformation that belongs to the action R × H → H given by z �→ z + s.
+We note that the Elliptic and Hyperbolic Actions are given by actions of Abelian
+groups if and only if n = 1. Nevertheless, the Parabolic Action is given by an
+Abelian Lie group in any dimension. For these reason, we introduce in the next
+definition actions of Abelian groups associated to the Elliptic and Hyperbolic cases.
+Definition 4.2. The Abelian Elliptic Action on DIII
+n
+is defined by
+T × DIII
+n
+−→ DIII
+n
+t · Z = t2Z.
+
+16
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+The Abelian Hyperbolic Action on Sn is defined by
+R+ × Sn −→ Sn
+r · Z = r2Z.
+Remark 4.3. We note that the Abelian Elliptic and Abelian Hyperbolic actions are
+obtained by considering the center of the groups corresponding to the non-Abelian
+actions. More precisely, we have the centers
+Z(U(n)) = TIn,
+Z(GL(n, R)) = R+In ∪ (−R+In),
+and the actions in Definition 4.2 are the restriction of the previously defined actions
+to these center groups. On the other hand, Symm(n, R) is already Abelian so that
+it coincides with its center, in other words we have
+Z(Symm(n, R)) = Symm(n, R).
+Hence, the most obvious definition of “Abelian Parabolic Action” would yield what
+we already have defined as the Parabolic Action. We also observe that these three
+actions of Abelian groups of biholomorphisms, the Abelian Elliptic, Abelian Hyper-
+bolic and Parabolic, are natural generalizations of the actions described in Corol-
+lary 4.1.
+4.2. Moment maps of the Abelian actions. We will now compute moment
+maps for all three Abelian actions introduced in this section. We refer to Defini-
+tion 4.2 and Remark 4.3. It follows from Theorem 3.5 that every biholomorphism
+of either of the domains DIII
+n
+and Sn preserves the corresponding K¨ahler form.
+Hence, all the groups considered above act by symplectomorphisms. In particular,
+the notion of moment map given in Definition 3.11 can be applied to such actions.
+4.2.1. Moment map of the Abelian Elliptic Action. The group in this case is T acting
+on DIII
+n
+. The Lie algebra of this group is R. The latter is canonically isomorphic
+to its dual R∗, so we will compute a moment map as a function DIII
+n
+→ R.
+For every element t ∈ R the corresponding induced vector field on DIII
+n
+is
+given by
+t♯
+Z = d
+ds
+���
+s=0 exp(st) · Z = d
+ds
+���
+s=0 exp(2ist)Z = 2itZ,
+for every Z ∈ DIII
+n
+. Note that we have used the fact that the (Lie group) exponen-
+tial map R → T satisfies t �→ exp(it).
+Proposition 4.4. The function given by
+µT : DIII
+n
+−→ R
+µT(Z) = −2tr
+�
+(In − ZZ)−1�
+,
+is a moment map for the Abelian Elliptic Action on DIII
+n
+.
+Proof. We start by computing ωDIII
+n
+Z
+(t♯
+Z, ·) for every t ∈ R and Z ∈ DIII
+n
+. For this
+first computation we use the above formula for t♯ and the expression for the K¨ahler
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+17
+form of DIII
+n
+obtained in Theorem 3.10. We have
+ωDIII
+n
+Z
+(t♯
+Z, V ) = i tr
+�
+(In − ZZ)−12itZ(In − ZZ)−1V
+�
+− i tr
+�
+(In − ZZ)−12itZ(In − ZZ)−1V
+�
+,
+= − 2t tr
+�
+(In − ZZ)−1Z(In − ZZ)−1V
+�
+− 2t tr
+�
+(In − ZZ)−1Z(In − ZZ)−1V
+�
+,
+for every V ∈ Symm(n, C).
+On the other hand, we consider the function µt : DIII
+n
+→ R defined by
+µt(Z) = ⟨µT(Z), t⟩ = tµT(Z),
+and we compute its differential as follows
+d(µt)Z(V ) =
+= − 2t d
+ds
+���
+s=0 tr
+��
+In − (Z + sV )(Z + sV )
+�−1�
+= − 2t tr
+��
+In − ZZ
+�−1�
+V Z + ZV
+��
+In − ZZ
+�−1�
+= − 2t tr
+��
+In − ZZ
+�−1V
+�
+In − ZZ
+�−1Z
+�
+− 2t tr
+�
+Z
+�
+In − ZZ
+�−1V
+�
+In − ZZ
+�−1�
+where we applied in the last identity the commutation relations between Z, (In −
+ZZ)−1 and their conjugates used in the proof of Theorem 3.10. We conclude that
+d(µt)Z(V ) = ωDIII
+n
+Z
+(t♯
+Z, V )
+for every V ∈ Symm(n, C) and Z ∈ DIII
+n
+. It follows that the first condition in
+Definition 3.11 is satisfied by the map in the statement. It remains to prove the
+T-invariance of this map, but this is established through the identities
+µT(t · Z) = µT(t2Z) = −2 tr
+�
+(In − t2Zt2Z)−1�
+= −2 tr
+�
+(In − ZZ)−1�
+= µT(Z)
+that hold for every t ∈ T and Z ∈ DIII
+n
+.
+□
+Remark 4.5. For the case n = 1, the Abelian Elliptic Action yields the T-action
+on the unit disk D given by t · z = t2z. With this assumption, Proposition 4.4
+provides the moment map
+µT(z) = −2
+1
+1 − |z|2 .
+We observe that for actions of Abelian groups we can add to a given moment map an
+arbitrary, but fixed, constant to obtain another moment map (see Definition 3.11).
+Hence, the map given by
+µ(z) = µT(z) + 2 = −2
+|z|2
+1 − |z|2 ,
+is a moment map as well for our T-action on D. This recovers, up to the multi-
+plicative constant 2, the moment map obtained in [14, Proposition 4.1] for n = 1.
+
+18
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+This referenced result computes the moment map for the natural action of the n-
+dimensional torus on the n-dimensional unit ball. We note that the factor 2 comes
+from the reparameterization involved in using the action t · z = t2z instead of the
+action t · z = tz.
+4.2.2. Moment map of the Abelian Hyperbolic Action. We now have the group R+
+acting on Sn. The Lie algebra of this group is R itself, which is canonically isomor-
+phic to its dual. Hence, the moment map will be computed as a function Sn → R.
+For every t ∈ R the induced vector field on Sn is obtained as follows. This
+computation uses the fact that the (Lie group) exponential map is given in this
+case by t �→ exp(t).
+t♯
+Z = d
+ds
+���
+s=0 exp(st) · Z = d
+ds
+���
+s=0 exp(2st)Z = 2tZ,
+for every Z ∈ Sn.
+Proposition 4.6. The function given by
+µR+ : Sn −→ R
+µR+(Z) = −4tr
+�
+Im(Z)−1Re(Z)
+�
+is a moment map for the Abelian Hyperbolic Action on Sn.
+Proof. We compute ωSn
+Z (t♯
+Z, ·), for every t ∈ R and Z ∈ Sn. For this we use the
+previous computations and the expression of the K¨ahler form of Sn obtained in
+Theorem 3.10. We have in this case
+ωSn
+Z (t♯
+Z, V ) = 2 tr
+�
+Im(Z)−1Re(2tZ)Im(Z)−1Im(V )
+�
+− 2 tr
+�
+Im(Z)−1Im(2tZ)Im(Z)−1Re(V )
+�
+= 4t tr
+�
+Im(Z)−1Re(Z)Im(Z)−1Im(V )
+�
+− 4t tr
+�
+Im(Z)−1Re(V )
+�
+,
+for every V ∈ Symm(n, C).
+On the other hand, we consider the function µt : Sn → R given by
+µt(Z) = ⟨µR+(Z), t⟩ = tµR+(Z),
+for which we compute the differential as follows
+d(µt)Z(V ) =
+= −4t d
+ds
+���
+s=0 tr
+�
+Im(Z + sV )−1Re(Z + sV )
+�
+= −4t d
+ds
+���
+s=0 tr
+��
+Im(Z) + sIm(V )
+�−1�
+Re(Z) + sRe(V )
+��
+= 4t tr
+�
+Im(Z)−1Im(V )Im(Z)−1Re(Z)
+�
+− 4t tr
+�
+Im(Z)−1Re(V )
+�
+,
+for every V ∈ Symm(n, C). From this we conclude that
+d(µt)Z(V ) = ωSn
+Z (t♯
+Z, V ),
+for every V ∈ Symm(n, C) and Z ∈ Sn. Hence, by Definition 3.11 it remains to
+show that µR+ is R+-invariant, and this is verified in the next computation
+µR+(r · Z) = µR+(r2Z) = −4tr
+�
+Im(r2Z)−1Re(r2Z)
+�
+= −4tr
+�
+Im(Z)−1Re(Z)
+�
+= µR+(Z),
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+19
+which holds for every r ∈ R+ and Z ∈ Sn.
+□
+Remark 4.7. For n = 1, the Abelian Hyperbolic Action yields the R+-action
+on the upper half-plane H given by r · z = r2z.
+Under this restriction, from
+Proposition 4.6 we obtain the moment map
+µR+(z) = −4Re(z)
+Im(z).
+This recover, up to a constant factor, the moment map obtained in [14, Proposi-
+tion 4.3] for n = 1. In this case the factor comes from two sources. Firstly, we
+use the action r · z = r2z, instead of the action r · z = rz used in [14]. Secondly,
+our formula for the K¨ahler form for S1 = H differs by a constant factor from the
+corresponding formula found in [14].
+4.2.3. Moment map of the Parabolic Action. Finally, we consider the group Symm(n, R)
+acting on Sn. Since Symm(n, R) is a vector group, it follows that it coincides with
+its Lie algebra and its exponential map is the identity. There is a canonical isomor-
+phism between Symm(n, R) and its dual space given by the positive definite inner
+product
+⟨A, B⟩ = tr(AB),
+defined for A, B ∈ Symm(n, R).
+For every S ∈ Symm(n, R) the corresponding induced vector field on Sn satisfies
+for every Z ∈ Symm(n, C)
+S♯
+Z = d
+ds
+���
+s=0 exp(sS) · Z = d
+ds
+���
+s=0 (Z + sS) = S,
+which is the constant vector with value S.
+Proposition 4.8. The function given by
+µSymm(n,R) : Sn −→ Symm(n, R)
+µSymm(n,R)(Z) = −2Im(Z)−1,
+is a moment map for the Parabolic Action on Sn.
+Proof. For every S ∈ Symm(n, R) and Z ∈ Sn, using the above computations and
+Theorem 3.10 we obtain
+ωSn
+Z (S♯
+Z, V ) = 2 tr
+�
+Im(Z)−1Re(S)Im(Z)−1Im(V )
+�
+− 2 tr
+�
+Im(Z)−1Im(S)Im(Z)−1Re(V )
+�
+= 2 tr
+�
+Im(Z)−1S Im(Z)−1Im(V )
+�
+,
+for every Z ∈ Sn.
+On the other hand, we consider for every S ∈ Symm(n, R) the map µS : Sn →
+Symm(n, R) defined by
+µS(Z) = −2tr
+�
+Im(Z)−1S
+�
+,
+for which we compute
+d(µS)Z(V ) = −2 d
+ds
+���
+s=0 tr
+�
+Im(Z + sV )−1S
+�
+= 2 tr
+�
+Im(Z)−1Im(V )Im(Z)−1S
+�
+,
+
+20
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+for every V ∈ Symm(n, C) and Z ∈ Sn. This immediately yields
+d(µS)Z(V ) = ωSn
+Z (S♯
+Z, V ),
+for every V ∈ Symm(n, C) and Z ∈ Sn. By Definition 3.11 it remains to establish
+the Symm(n, R)-invariance of µSymm(n,R), and this achieved by noting that
+µSymm(n,R)(S · Z) = µSymm(n,R)(Z + S) = −2
+�
+Im(Z + S)−1�
+= −2
+�
+Im(Z)−1�
+= µSymm(n,R)(Z)
+for every Z ∈ Sn and S ∈ Symm(n, R).
+□
+Remark 4.9. For n = 1, the Parabolic Action yields the R-action on the upper
+half-plane H given by s · z = z + s. And in this situation, Proposition 4.8 provides
+the moment map
+µR(z) = −2
+1
+Im(z).
+As in the previous cases, this recovers, up to a constant factor, the moment map
+obtained in [14, Proposition 4.2] for n = 1. As in the case of Remark 4.7 the factor
+comes from a different normalization of the K¨ahler form on this work and [14].
+5. Toeplitz operators with special symbols
+We will now describe Toeplitz operators with special symbols using two related
+alternatives: symbols invariant under biholomorphism groups and symbols depend-
+ing on the moment maps of such groups. Both cases yield, under suitable conditions,
+commutative C∗-algebras generated by Toeplitz operators.
+First we introduce a general notation. As before, in the rest of this work D
+denotes either of the domains DIII
+n
+or Sn. For A ⊂ L∞(D) a set of essentially
+bounded symbols, we denote by T (λ)(A) the C∗-algebra generated by the Toeplitz
+operators T (λ)
+a
+where a ∈ A.
+5.1. Invariant symbols. Let H be a closed subgroup of biholomorphisms of D.
+We will denote by L∞(D)H the subspace of L∞(D) consisting of H-invariant sym-
+bols. In other words, we have
+L∞(D)H = {a ∈ L∞(D) : h · a = a, for all h ∈ H},
+where, for a given a ∈ L∞(D) and h ∈ H, we define
+(h · a)(Z) = a(h−1 · Z),
+for almost every Z ∈ D.
+Symmetric pairs associated to symmetric domains can be used to obtain commu-
+tative C∗-algebras generated by Toeplitz operators by considering invariant sym-
+bols. The definitions and precise statements can be found in [1]. In this work, we
+will use the fact that the pairs (Sp(n, R), GL(n, R)) and (Sp(n, C) ∩ U(n, n), U(n))
+are symmetric in order to obtain the following consequence of Theorem 5.1 from
+[1].
+Theorem 5.1 ([1]). For every λ > n, the C∗-algebras T (λ)(L∞(DIII
+n
+)U(n)) and
+T (λ)(L∞(Sn)GL(n,R)) acting on the weighted Bergman spaces A2
+λ(DIII
+n
+) and A2
+λ(Sn),
+respectively, are commutative.
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+21
+With the notation from subsection 4.1, Theorem 5.1 states that for the Elliptic
+and Hyperbolic actions on DIII
+n
+and Sn, respectively, the symbols invariant under
+such actions yield Toeplitz operators that generate commutative C∗-algebras.
+The Parabolic Action provides the same sort of conclusion. This follows from
+the next consequence of [1, Theorem 5.8]. We note that in this case the group
+Symm(n, R) does not yield a symmetric pair in the group Sp(n, R) of biholomor-
+phisms of Sn.
+Theorem 5.2 ([1]). For every λ > n, the C∗-algebra T (λ)(L∞(Sn)Symm(n,R))
+acting on the weighted Bergman space A2
+λ(Sn) is commutative.
+5.2. Moment map symbols. Following [14, 15] we define the notion of moment
+map symbol for the setup of this work.
+Definition 5.3. Let D be either of the domains DIII
+n
+or Sn and H a closed
+subgroup of the biholomorphism group of D. If µH : D → h∗ is a moment map
+for the action of H on D, then a moment map symbol for H or a µH-symbol is a
+symbol a ∈ L∞(D) that can be written in the form a = f ◦µH for some measurable
+function f. We denote by L∞(D)µH the space of all essentially bounded measurable
+µH-symbols on D.
+We have computed moment maps for the Abelian Elliptic, Abelian Hyperbolic
+and Parabolic actions in subsection 4.2. These computations allow us to provide
+the following explicit description of moment map symbols for these three actions.
+Proposition 5.4. Let a ∈ L∞(DIII
+n
+) and b ∈ L∞(Sn) be given. Then, the follow-
+ing equivalences hold
+(1) The measurable function a is a µT-symbol if and only if there exists a mea-
+surable function f such that a(Z) = f
+�
+tr(ZZ)
+�
+, for almost every Z ∈ DIII
+n
+.
+(2) The measurable function b is a µR+-symbol if and only if there exists a
+measurable function f such that b(Z) = f
+�
+tr(Im(Z)−1Re(Z))
+�
+, for almost
+every Z ∈ Sn.
+(3) The measurable function b is a µSymm(n,R)-symbol if and only if there exists
+a measurable function f such that b(Z) = f(Im(Z)), for almost every Z ∈
+Sn.
+Proof. We note that the claims on the symbols b ∈ L∞(Sn) are immediate conse-
+quences of Definition 5.3 and Propositions 4.6 and 4.8. Hence, we consider the case
+of moment maps for the Abelian Elliptic Action.
+By Proposition 4.4 and Definition 5.3, a symbol a ∈ L∞(DIII
+n
+) is a µT-symbol if
+and only if there is a measurable function g such that
+a(Z) = g
+�
+tr
+�
+(In − ZZ)−1��
+,
+for almost every Z ∈ DIII
+n
+. In the cone Pos(n, C) of positive definite n× n complex
+matrices let us consider the open subsets given by
+(0, In) = {Z ∈ Pos(n, C) | Z < In}
+(In, ∞) = {W ∈ Pos(n, C) | In < W}.
+It is straightforward to verify that the maps
+F : (0, In) −→ (In, ∞)
+G : (In, ∞) −→ (0, In)
+Z �−→ (In − Z)−1
+W �−→ In − W −1
+
+22
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+are well defined smooth maps, such that they are inverses of each other. In partic-
+ular, these maps satisfy
+(In − ZZ)−1 = F(ZZ),
+ZZ = G
+�
+(In − ZZ)−1�
+,
+for every Z ∈ DIII
+n
+. The case of the Abelian Elliptic Action clearly follows from
+these remarks.
+□
+By definition, the moment map symbols for Abelian groups are invariant under
+the corresponding actions. It turns out that the moment maps of the first two
+actions are in fact invariant under larger groups, those considered in Theorem 5.1.
+This is the content of the next two results.
+Proposition 5.5. Let µT : DIII
+n
+→ R be the moment map for the T-action on DIII
+n
+given in Proposition 4.4. Then, µT is a U(n)-invariant function. In particular, we
+have L∞(DIII
+n
+)µT ⊂ L∞(DIII
+n
+)U(n).
+Proof. Recall that U(n) acts on DIII
+n
+by U · Z = UZU T . Using the expression of
+µT obtained in Proposition 4.4, we have for every U ∈ U(n)
+µT(g · Z) = −2tr
+�
+(In − UZU TUZU T))−1�
+= −2tr
+�
+(In − UZZU T )−1�
+= −2tr
+�
+(U(In − ZZ)U T )−1�
+= −2tr
+�
+U(In − ZZ)−1U −1�
+= −2tr
+�
+(In − ZZ)−1�
+= µT(Z),
+for every Z ∈ DIII
+n
+. The last claim is now an immediate consequence of Defini-
+tion 5.3.
+□
+Proposition 5.6. Let µR+ : Sn → R be the moment map of the R+-action on
+Sn given in Proposition 4.6.
+Then, µR+ is a GL(n, R)-invariant function.
+In
+particular, we have L∞(Sn)µR+ ⊂ L∞(Sn)GL(n,R).
+Proof. Recall that GL(n, R) acts on Sn by A · Z = AZAT .
+We now use the
+expression of µR+ obtained in Proposition 4.6, and for every A ∈ GL(n, R) we
+compute
+µR+(A · Z) = −4tr
+�
+(AIm(Z)A⊤)−1ARe(Z)A⊤�
+= −4tr
+�
+(A⊤)−1Im(Z)−1A−1ARe(Z)A⊤�
+= −4tr
+�
+Im(Z)−1Re(Z)
+�
+= µR+(Z),
+for every Z ∈ Sn. Again, the last claim now follows immediately.
+□
+For the Parabolic Action, it turns out that Symm(n, R)-invariance and being a
+µSymm(n,R)-symbol are equivalent. This is the content of the next result.
+Proposition 5.7. For the moment map µSymm(n,R) : Sn → Symm(n, R) of the
+Symm(n, R)-action on Sn given in Proposition 4.8, we have
+L∞(Sn)µSymm(n,R) = L∞(Sn)Symm(n,R).
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+23
+Proof. The Symm(n, R)-action on Sn is given by the expression S · Z = Z + S.
+Hence, for a given a ∈ L∞(Sn) we have the following sequence of equivalences
+a is Symm(n, R)-invariant
+⇐⇒ for every S ∈ Symm(n, R) : a(Z + S) = a(Z) for a.e. Z ∈ Sn
+⇐⇒ for some measurable f : a(Z) = f(Im(Z)) for a.e. Z ∈ Sn,
+and the result follows from the last case in Proposition 5.4.
+□
+5.3. Commuting Toeplitz operators with moment maps symbols. We now
+state one of our main results: for D either of the domains DIII
+n
+or Sn, there are three
+Abelian groups of biholomorphisms of D to which we can associate commutative
+C∗-algebras generated by Toeplitz operators.
+Theorem 5.8. Let D be either of the domains DIII
+n
+or Sn. The Abelian Elliptic
+Action, the Abelian Hyperbolic Action and the Parabolic Action on D yield three
+Abelian groups of biholomorphisms of D which provide, for every λ > n, the fol-
+lowing commutative C∗-algebras generated by Toeplitz operators.
+Abelian Elliptic: The C∗-algebra T (λ)�
+L∞(DIII
+n
+)µT�
+, acting on A2
+λ(DIII
+n
+),
+obtained from the moment map of the T-action on DIII
+n
+.
+Abelian Hyperbolic: The C∗-algebra T (λ)�
+L∞(Sn)µR+�
+, acting on A2
+λ(Sn),
+obtained from the moment map of the R+-action on Sn.
+Parabolic: The C∗-algebra T (λ)�
+L∞(Sn)µSymm(n,R)�
+, acting on A2
+λ(Sn), ob-
+tained from the moment map of the Symm(n, R)-action on Sn.
+Proof. First, we note that Propositions 5.5 and 5.6 imply the inclusions
+T (λ)(L∞(DIII
+n
+)µT) ⊂ T (λ)(L∞(DIII
+n
+)U(n))
+T (λ)(L∞(Sn)µR+) ⊂ T (λ)(L∞(Sn)GL(n,R)),
+and so the cases of the Abelian Elliptic and Abelian Hyperbolic Actions follow from
+Theorem 5.1.
+For the Parabolic Action, we note that Proposition 5.7 yields the identity
+T (λ)(L∞(Sn)µSymm(n,R)) = T (λ)(L∞(Sn)Symm(n,R)),
+and now the result is a consequence of Theorem 5.2.
+□
+Remark 5.9. It follows from the discussion in subsection 4.1 (see Corollary 4.1 and
+Definition 4.2) that for n = 1 the three actions considered in Theorem 5.8 reduce to
+the usual elliptic, hyperbolic and parabolic actions known from complex analysis.
+These three actions have been previously used to obtain commutative C∗-algebras
+generated by Toeplitz operators, notably in the results found in [3, 4, 5, 7] (see also
+[11]). In fact, the commutative C∗-algebras generated by Toeplitz operators from
+Theorem 5.8 reduce to those from these previous works when n = 1.
+One of the main guiding lights in this line of study of Toeplitz operators has
+been to find generalizations to higher dimensions of these commutative C∗-algebras
+observed in the case of the unit disk. This was achieved for the unit ball Bn in Cn
+through the use of maximal Abelian subgroups of the biholomorphism group of Bn
+(see [16, 17]). However, the result for the unit ball Bn from these references lead to
+
+24
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+n + 2 different commutative C∗-algebras, which is in contrast with the simplicity
+of only three Abelian groups for the case of the unit disk.
+On the other hand, our Theorem 5.8 recovers for higher dimensions the simplic-
+ity observed in the case of the unit disk. More precisely, we consider the generalized
+unit disk DIII
+n
+and its unbounded realization Sn, Siegel’s generalized upper half-
+plane. For these domains, Theorem 5.8 yields three commutative C∗-algebras gen-
+erated by Toeplitz operators, acting on the Bergman spaces of DIII
+n
+and Sn, which
+can be seen as natural extensions of the case of the unit disk. And this is achieved
+while using only three Abelian groups for any dimension. This is possible due to
+the fact that we have replaced invariant symbols with moment map symbols. This
+highlights the importance of using symplectic geometry to study Toeplitz operators
+acting on Bergman spaces in higher dimensions.
+6. Spectral integral formulas for Toeplitz operator with moment
+map symbols
+In this final section we present explicit integral formulas that simultaneously
+diagonalize Toeplitz operators.
+This will be done for the Abelian Elliptic and
+Parabolic Actions.
+6.1. Toeplitz operators with Abelian Elliptic symbols. In this case, we are
+dealing with symbols that belong to T (λ)(L∞(DIII
+n
+)µT) ⊂ T (λ)(L∞(DIII
+n
+)U(n)). In
+particular, it is useful to consider the U(n)-action on the Bergman spaces over DIII
+n
+.
+We recall some properties of such action and refer to [19, 2] for further details.
+For every λ > n, there is a unitary representation given by
+πλ : U(n) × A2
+λ(DIII
+n
+) −→ A2
+λ(DIII
+n
+)
+πλ(A)(f) = f ◦ A−1.
+This representation leaves invariant the subspace of (holomorphic) polynomials on
+DIII
+n
+⊂ Symm(n, C), that we will denote by P(Symm(n, C)) = P, for simplicity. In
+particular, for every λ > n, the decomposition of A2
+λ(DIII
+n
+) into irreducible U(n)-
+submodules is the same as the one corresponding to the U(n)-action on P. Let us
+denote by −→
+N n the set of integer n-tuples that satisfy α1 ≥ · · · ≥ αn ≥ 0. Then,
+using the representation πλ, one can show that, for every λ > n, there is a Hilbert
+direct sum decomposition
+(6.1)
+A2
+λ(DIII
+n
+) =
+�
+α∈−
+→
+N n
+Pα,
+where
+�
+Pα�
+α∈−
+→
+N n is family of mutually non-isomorphic U(n)-submodules of P. For
+the proof of this claim we refer to [19, Chapter 2] (see also [2, 8]).
+We consider the polynomials given by
+∆j(Z) = det(Zj)
+where Zj is the upper-left corner j × j submatrix of Z. For every α ∈ −→
+N n we will
+also consider the polynomial
+∆α(Z) =
+n
+�
+j=1
+∆j(Z)αj−αj+1
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+25
+where we agree to define αn+1 = 0. These are known as the conical polynomials
+for the representation of U(n) on P (see [19, Chapter 2]).
+With the previous notation, the following result is an application of Proposi-
+tion 4.7 and Theorem 4.11 from [2] to our current setup.
+Theorem 6.1 ([2]). Let a ∈ L∞(DIII
+n
+)U(n) and λ > n be given. Then, the Toeplitz
+operator T (λ)
+a
+acting on the Bergman space A2
+λ(DIII
+n
+) preserves the Hilbert direct
+sum (6.1). Furthermore, we have
+T (λ)
+a
+|Pα = ca,λ(α)IPα,
+where the complex constant ca,λ(α) is given by
+ca,λ(α) =
+�
+0<X<In
+a(
+√
+X)∆α(X)∆n(In − X)λ−n−1 dX
+�
+0<X<In
+∆α(X)∆n(In − X)λ−n−1 dX
+,
+for every α ∈ −→
+N n. The condition 0 < X < In denotes an open subset of Symm(n, R)
+and dX the Lebesgue measure on the latter.
+Note that the integrals in Theorem 6.1 are taken over an open subset of the
+vector space Symm(n, R), which is n(n + 1)/2-dimensional. We will now simplify
+these expressions, and so the results of [2], to obtain integral formulas over lower
+dimensional spaces for Toeplitz operators with U(n)-invariant symbols. Our goal
+is to reduce the number of variables over which the corresponding symbols have to
+be integrated. More precisely, we will obtain integral formulas involving the set
+−−−→
+(0, 1)n = {x ∈ (0, 1)n | xn > · · · > x1 > 0},
+which is only n-dimensional. In the rest of this work, we will denote by D(x) the
+diagonal n × n matrix with diagonal elements given by x ∈ Rn. Also, for a given
+x ∈ Rn
++ we will write √x = (√x1, . . . , √xn).
+Theorem 6.2. Let a ∈ L∞(DIII
+n
+)U(n) and λ > n be given. Then, the complex
+constants (ca,λ(α))α∈−
+→
+N n such that
+T (λ)
+a
+|Pα = ca,λ(α)IPα,
+for every α ∈ −→
+N n, as obtained in Theorem 6.1, are given by
+ca,λ(α) =
+�
+−−→
+(0,1)n
+a(D(√x))
+�
+n
+�
+j=1
+xj
+�αn�
+n
+�
+j=1
+(1 − xj)
+�λ−n−1� �
+j<k
+(xj − xk)
+�
+hα(x) dx
+�
+−−→
+(0,1)n
+�
+n
+�
+j=1
+xj
+�αn�
+n
+�
+j=1
+(1 − xj)
+�λ−n−1� �
+j<k
+(xj − xk)
+�
+hα(x) dx
+,
+
+26
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+where the functions hα : −−−→
+(0, 1)n → [0, ∞) are defined by
+hα(x) =
+�
+O(n)
+n−1
+�
+j=1
+∆j(AD(x)A⊤)αj−αj+1 dA
+for every α ∈ −→
+N n, and dA is a fixed Haar measure on O(n).
+Proof. Let us denote
+Ia,λ(α) =
+�
+0<X<In
+a(
+√
+X)∆α(X)∆n(In − X)λ−n−1 dX.
+As noted above, this integral is taken over the open subset of Ωn that consists of
+the matrices X satisfying 0 < X < In. The linear maps that leave invariant Ωn is
+realized by the action of the group GL(n, R) given by
+A · X = AXA⊤,
+where A ∈ GL(n, R) and X ∈ Ωn. In particular, the isotropy group of symmetries
+of the cone Ωn that fixes In is realized by the corresponding action of the group
+O(n). We refer to [19, Section 1.3] for the proof of these claims.
+By the previous discussion, it follows from [19, Proposition 1.3.63] that there is
+a constant C > 0 such that
+Ia,λ(α) = C
+�
+−−→
+(0,1)n
+�
+O(n)
+a
+��
+AD(x)A⊤
+�
+∆α
+�
+AD(x)A⊤�
+×
+× ∆n
+�
+In − AD(x)A⊤�λ−n−1 �
+j<k
+(xj − xk) dA dx.
+Note that we have used the fact that the symmetric cone Ωn has rank n and
+characteristic multiplicity a = 1. We now observe that for every A ∈ O(n) and
+x ∈ −−−→
+(0, 1)n we have
+a
+��
+AD(x)A⊤
+�
+= a
+�
+AD(√x)A⊤�
+= a(A · D(√x)) = a(D(√x)),
+because a is U(n)-invariant. On the other hand, we have
+∆α
+�
+AD(x)A⊤�
+=
+n−1
+�
+j=1
+∆j
+�
+AD(x)A⊤�αj−αj+1 det
+�
+AD(x)A⊤�αn
+=
+n−1
+�
+j=1
+∆j
+�
+AD(x)A⊤�αj−αj+1 det(D(x))αn
+=
+n−1
+�
+j=1
+∆j
+�
+AD(x)A⊤�αj−αj+1
+�
+n
+�
+j=1
+xj
+�αn
+and a similar computation yields
+∆n
+�
+In − AD(x)A⊤�λ−n−1 = det(In − D(x))λ−n−1
+=
+�
+n
+�
+j=1
+(1 − xj)
+�λ−n−1
+,
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+27
+where the last two computations hold for every A ∈ O(n) and x ∈ −−−→
+(0, 1)n as well.
+Collecting these identities we obtain
+Ia,λ(α) = C
+�
+−−→
+(0,1)n
+a(D(√x))
+�
+n
+�
+j=1
+xj
+�αn�
+n
+�
+j=1
+(1 − xj)
+�λ−n−1
+×
+×
+� �
+j<k
+(xj − xk)
+�
+hα(x) dx,
+where hα(x) is given as in the statement. The result now follows once we observe
+that
+ca,λ(α) = Ia,λ(α)
+I1,λ(α),
+for every α ∈ −→
+N n.
+□
+The next goal in this subsection is to apply Theorem 6.2 to the case of the
+moment map symbols corresponding to the Abelian Elliptic Action. The description
+of such symbols provided by Proposition 5.4 will greatly simplify our formulas.
+Theorem 6.3. Let a ∈ L∞(DIII
+n
+)µT and λ > n be given. Let f be a measurable
+function such that a(Z) = f
+�
+tr(ZZ)
+�
+, for almost every Z ∈ DIII
+n
+.
+Then, the
+complex constants (ca,λ(α))α∈−
+→
+N n such that
+T (λ)
+a
+|Pα = ca,λ(α)IPα,
+for every α ∈ −→
+N n, as obtained in Theorem 6.1, are given by
+ca,λ(α) =
+�
+−−→
+(0,1)n
+f(∥x∥1)
+�
+n
+�
+j=1
+xj
+�αn�
+n
+�
+j=1
+(1 − xj)
+�λ−n−1� �
+j<k
+(xj − xk)
+�
+hα(x) dx
+�
+−−→
+(0,1)n
+�
+n
+�
+j=1
+xj
+�αn�
+n
+�
+j=1
+(1 − xj)
+�λ−n−1� �
+j<k
+(xj − xk)
+�
+hα(x) dx
+,
+where the functions hα : −−−→
+(0, 1)n → [0, ∞) are those defined in Theorem 6.2.
+Proof. It is an immediate consequence of Theorem 6.2 and the computation
+a(D(√x)) = f
+�
+tr(D(√x)D(√x))
+�
+= f
+�
+tr(D(x))
+�
+= f(∥x∥1),
+which holds for every x ∈ −−−→
+(0, 1)n.
+□
+6.2. Toeplitz operators with Parabolic symbols. We recall from subsection 2.3
+the decomposition
+Sn = Symm(n, R) ⊕ iΩn,
+where Ωn = Pos(n, R). With respect to this decomposition, and for every λ > n,
+the weighted measure �vλ decomposes as
+d�vλ(Z) = Cλ,n det(2Y )λ−n−1 dX dY,
+
+28
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+with the coordinates Z = X + iY , (X ∈ Symm(n, R) and Y ∈ Ωn) and for the
+positive constant
+Cλ,n =
+ΓΩn(λ)
+π
+n(n+1)
+2
+ΓΩn
+�
+λ − n+1
+2
+�.
+This yields the natural isometry
+L2(Sn, �v) ≃ L2(Symm(n, R), dX) ⊗ L2(Ωn, Cλ,n det(2Y )λ−n−1 dY ),
+that we will use in the rest of this work.
+Let us consider the unitary operator U = F ⊗ I defined on L2(Sn, �vλ), where F
+is the Fourier transform on Symm(n, R). More precisely, we have
+(F(f))(X) =
+1
+(2π)
+n(n+1)
+4
+�
+Symm(n,R)
+e−itr(Xξ)f(ξ) dξ
+for every f ∈ L1(Symm(n, R))∩L2(Symm(n, R)). In particular, we use as canonical
+inner product on Symm(n, R) the one induced by the trace. We recall that, with
+respect to such inner product, the cone Ωn is self-dual in the sense that
+Ωn = {ξ ∈ Symm(n, R) | tr(ξX) > 0 for all X ∈ Ωn \ {0}}.
+We will use this fundamental property (see [19]) to apply some well known formulas
+associated to symmetric cones.
+The next two results allow to describe the Bergman spaces after applying the
+unitary map U.
+Lemma 6.4. Let Hλ(Sn) = U(A2
+λ(Sn)) be the image of the Bergman space
+A2
+λ(Sn) under the unitary map U. Then, the operator given by
+Sλ : L2(Ωn) −→ L2(Symm(n, R)) ⊗ L2(Ωn, Cλ,n det(2Y )λ−n−1 dY )
+(Sλ(f))(X, Y ) = (2π)
+n(n+1)
+4
+ΓΩn(λ)
+1
+2 χΩn(X)f(X) det(X)
+λ
+2 − n+1
+4 e−tr(XY ),
+is an isometry onto Hλ(Sn).
+Proof. From the basic properties of the Fourier transform, the Cauchy-Riemann
+equations on Sn are transformed under U to the equations
+�
+Xjk +
+∂
+∂Yjk
+�
+ϕ = 0,
+which must hold for every 1 ≤ j ≤ k ≤ n. The general solution of these equations
+is ϕ(X, Y ) = ψ(X)e−tr(XY ). Next, we need to consider the L2-integrability of these
+solutions, and for this we evaluate
+�
+Sn
+|ϕ(X, Y )|2Cλ,n det(2Y )λ−n−1 dX dY =
+= Cλ,n
+�
+Sn
+|ψ(X)|2e−2tr(XY ) det(2Y )λ−n−1 dX dY
+= Cλ,n
+�
+Symm(n,R)
+|ψ(X)|2
+� �
+Ωn
+e−2tr(XY ) det(2Y )λ−n−1 dY
+�
+dX.
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+29
+For this to be finite it is necessary that supp(ψ) ⊂ Ωn. On the other hand, [19,
+Equation 2.4.30] implies that (after some simple changes of variable) we have
+�
+Ωn
+e−2tr(XY ) det(2Y )λ−n−1 dY = ΓΩn
+�
+λ − n+1
+2
+�
+2
+n(n+1)
+2
+det(X)
+n+1
+2
+−λ
+Hence, in the above solution of the Cauchy-Riemann equations we replace ψ(X) by
+the function
+ψ(X) = (2π)
+n(n+1)
+4
+ΓΩn(λ)
+1
+2 χΩn(X)f(X) det(X)
+λ
+2 − n+1
+4
+for a suitable function f. With these choices and the previous computations we
+obtain
+∥ϕ∥2
+Hλ(Sn) = Cλ,n
+�
+Ωn
+(2π)
+n(n+1)
+2
+ΓΩn(λ)
+|f(X)|2 det(X)λ− n+1
+2 ×
+× ΓΩn
+�
+λ − n+1
+2
+�
+2
+n(n+1)
+2
+det(X)
+n+1
+2
+−λ dX
+= Cλ,n
+π
+n(n+1)
+2
+ΓΩn
+�
+λ − n+1
+2
+�
+ΓΩn(λ)
+∥f∥2
+L2(Ωn) = ∥f∥2
+L2(Ωn),
+where we have used the definition of the constant Cλ,n. The last set of identities
+completes the proof by the definition of Sλ.
+□
+Lemma 6.5. The adjoint operator of Sλ from Lemma 6.4 is a partial isometry
+with initial space Hλ(Sn) and final space L2(Ωn). Furthermore, we have
+(S∗
+λ(ϕ))(X) =
+2
+n(n+1)
+4
+ΓΩn(λ)
+1
+2
+π
+n(n+1)
+4
+ΓΩn
+�
+λ − n+1
+2
+� det(X)
+λ
+2 − n+1
+4 ×
+×
+�
+Ωn
+ϕ(X, Y )e−tr(XY ) det(2Y )λ−n−1 dY,
+for every ϕ ∈ L2(Symm(n, R)) ⊗ L2(Ωn, Cλ,n det(2Y )λ−n−1 dY ).
+Proof. The first claim follows from Lemma 6.4. On the other hand, the expression
+for S∗
+λ is a consequence of the following straightforward computation for f and ϕ
+in the corresponding spaces
+⟨Sλ(f),ϕ⟩ =
+= Cλ,n
+�
+Sn
+(Sλ(f))(X, Y )ϕ(X, Y ) det(2Y )λ−n−1 dX dY
+= Cλ,n
+(2π)
+n(n+1)
+4
+ΓΩn(λ)
+1
+2
+�
+Ωn
+f(X)×
+× det(X)
+λ
+2 − n+1
+4
+� �
+Ωn
+ϕ(X, Y )e−tr(XY ) det(2Y )λ−n−1
+�
+dX,
+where we have used again the value of Cλ,n.
+□
+The next result provides a formula for the Bergman projection after applying
+the unitary map U.
+
+30
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+Lemma 6.6. Let Bλ = UBSn,λU ∗ be the orthogonal projection
+L2(Symm(n, R)) ⊗ L2(Ωn, Cλ,n det(2Y )λ−n−1 dY ) −→ Hλ(Sn).
+Then, we have the identities
+S∗
+λSλ = IL2(Ωn),
+SλS∗
+λ = Bλ.
+In particular, the orthogonal projection Bλ is given by
+(Bλ(ϕ))(X, Y ) =
+2
+n(n+1)
+2
+ΓΩn
+�
+λ − n+1
+2
+�χΩn(X) det(X)λ− n+1
+2 e−tr(XY )×
+×
+�
+Ωn
+e−tr(Xη) det(2η)λ−n−1ϕ(X, η) dη.
+Proof. By Lemma 6.4, the operator Sλ is a partial isometry with initial space
+L2(Ωn) and final space Hλ(Sn). This implies the first two identities in the state-
+ment. Hence, it remains to compute SλS∗
+λ explicitly, and this is done as follows for
+every ϕ ∈ Hλ(Sn)
+((SλS∗
+λ)(ϕ))(X, Y ) =
+= (2π)
+n(n+1)
+4
+ΓΩn(λ)
+1
+2 χΩn(X)(S∗
+λ(ϕ))(X) det(X)
+λ
+2 − n+1
+4 e−tr(XY )
+= (2π)
+n(n+1)
+4
+ΓΩn(λ)
+1
+2 χΩn(X)
+�
+2
+n(n+1)
+4
+ΓΩn(λ)
+1
+2
+π
+n(n+1)
+4
+ΓΩn
+�
+λ − n+1
+2
+� det(X)
+λ
+2 − n+1
+4 ×
+×
+�
+Ωn
+ϕ(X, η)e−tr(Xη) det(2η)λ−n−1 dη
+�
+,
+which clearly simplifies to the required expression.
+□
+The constructions considered so far allow us to introduce in the next result a
+Fourier-Laplace transform from A2
+λ(Sn) onto L2(Ωn). We refer to [19, Proposi-
+tion 2.4.26] for a similar related construction.
+Theorem 6.7. With the current notation and for every λ > n, the operator Rλ =
+S∗
+λU : L2
+λ(Sn, �vλ) → L2(Ωn) is a partial isometry with initial space A2
+λ(Sn) and
+final space L2(Ωn). In particular, its adjoint
+R∗
+λ : L2(Ωn) −→ L2(Sn, �vλ)
+is an isometry onto A2
+λ(Sn). Furthermore, we have
+(R∗
+λ(f))(Z) =
+1
+ΓΛ(λ)
+1
+2
+�
+Ωn
+f(ξ) det(ξ)
+λ
+2 − n+1
+4 eitr(ξZ) dξ,
+for every f ∈ L2(Ωn) and Z ∈ Sn.
+Proof. Since U is a unitary operator mapping A2
+λ(Sn) onto Hλ(Sn), it follows from
+Lemma 6.5 that Rλ is a partial isometry with the indicated initial and final spaces.
+From this we now conclude that R∗
+λ is an isometry from L2(Ωn) onto A2
+λ(Sn).
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+31
+It only remains to find the expression stated for R∗
+λ, which is achieved in the
+following computation. For every f ∈ L2(Ωn) and Z ∈ Sn we have
+(R∗
+λ(f))(Z) = ((U ∗Sλ)(f))(Z) = ((F−1 ⊗ I) ◦ Sλ(f))(Z)
+= (F−1 ⊗ I)
+�(2π)
+n(n+1)
+4
+ΓΩn(λ)
+1
+2 χΩn(X)f(X) det(X)
+λ
+2 − n+1
+4 e−tr(XY )
+�
+=
+1
+ΓΩn(λ)
+1
+2
+�
+Ωn
+f(ξ) det(ξ)
+λ
+2 − n+1
+4 e−tr(ξY )eitr(Xξ) dξ
+=
+1
+ΓΩn(λ)
+1
+2
+�
+Ωn
+f(ξ) det(ξ)
+λ
+2 − n+1
+4 eitr(ξZ)dξ,
+where Z = X + iY , with X, Y real matrices.
+□
+We recall from Proposition 5.7 that
+L∞(Sn)µSymm(n,R) = L∞(Sn)Symm(n,R).
+In other words, the Symm(n, R)-invariant symbols and the moment map symbols
+for the Symm(n, R)-action on Sn are the same. Hence, the next result provides
+integral formulas that simultaneously diagonalizes Toeplitz operators with either
+type of symbols.
+Theorem 6.8. Let a ∈ L∞(Sn) be a Symm(n, R)-invariant symbol and λ > n
+be given.
+Then, for Rλ the operator from Theorem 6.7 we have a commutative
+diagram
+A2
+λ(Sn)
+T (λ)
+a
+�
+Rλ
+� L2(Ωn)
+Mγa,λ
+�
+A2
+λ(Sn)
+Rλ
+� L2(Ωn)
+where γa,λ ∈ L∞(Ωn) is given by
+γa,λ(X) =
+= 2
+n(n+1)
+2
+det(X)λ− n+1
+2
+ΓΩn
+�
+λ − n+1
+2
+�
+�
+Ωn
+a(Y )e−2tr(XY ) det(2Y )λ−n−1 dY,
+for every X ∈ Ωn.
+Proof. For our given Symm(n, R)-invariant symbol a ∈ L∞(Sn) we have T (λ)
+a
+=
+BSn,λ ◦ Ma. Then, Theorem 6.7 implies that
+RλT (λ)
+a
+R∗
+λ = RλBSn,λMaBSn,λR∗
+λ
+= Rλ(R∗
+λRλ)Ma(R∗
+λRλ)R∗
+λ
+= RλMaR∗
+λ = S∗
+λUMaU ∗Sλ
+= S∗
+λMaSλ,
+where we have used that UMaU ∗ = Ma, since U = F ⊗ I and a depends only on
+Y = Im(Z).
+
+32
+CUEVAS-ESTRADA AND QUIROGA-BARRANCO
+We now evaluate the last composition as follows for every f ∈ L2(Ωn) and
+X ∈ Ωn
+(S∗
+λMaSλ(f))(X) =
+=
+2
+n(n+1)
+4
+ΓΩn(λ)
+1
+2
+π
+n(n+1)
+4
+ΓΩn
+�
+λ − n+1
+2
+� det(X)
+λ
+2 − n+1
+4 ×
+�
+Ωn
+�
+a(Y )(2π)
+n(n+1)
+4
+ΓΩn(λ)
+1
+2 f(X) det(X)
+λ
+2 − n+1
+4 e−tr(XY )
+�
+×
+e−tr(XY ) det(2Y )λ−n−1 dY
+= 2
+n(n+1)
+2
+det(X)λ− n+1
+2
+ΓΩn
+�
+λ − n+1
+2
+�
+f(X)
+�
+Ωn
+a(Y )e−2tr(XY ) det(2Y )λ−n−1 dY,
+which yields the required conclusion.
+□
+Remark 6.9. Theorem 6.8 can be seen as a generalization of some of the results
+found in [20]. More precisely, [20, Theorem 4.1] provides the diagonalization of
+Toeplitz operators with the so-called cone component symbols, and such results
+holds for every tubular domain. However, [20] considers only the weightless case.
+On the other hand, we have considered only the tubular domain Sn, but our result
+holds for arbitrarily weighted Bergman spaces and Toeplitz operators with symbols
+that depend only on the cone coordinates.
+Acknowledgment. This research was supported by a Conacyt scholarship, SNI-
+Conacyt and Conacyt grants 280732 and 61517.
+References
+[1] Dawson, Matthew; ´Olafsson, Gestur and Quiroga-Barranco, Raul:
+Commuting Toeplitz
+operators on bounded symmetric domains and multiplicity-free restrictions of holomorphic
+discrete series. J. Funct. Anal. 268 (2015), no. 7, 1711–1732.
+[2] Dawson, Matthew and Quiroga-Barranco, Raul: Radial Toeplitz operators on the weighted
+Bergman spaces of Cartan domains. Representation theory and harmonic analysis on sym-
+metric spaces, 97–114, Contemp. Math., 714, Amer. Math. Soc., Providence, RI, 2018.
+[3] Grudsky, S., Karapetyants, A. and Vasilevski, N.: Toeplitz operators on the unit ball in Cn
+with radial symbols. J. Operator Theory 49 (2003), no. 2, 325–346.
+[4] Grudsky, S., Karapetyants, A. and Vasilevski, N.: Dynamics of properties of Toeplitz oper-
+ators on the upper half-plane: hyperbolic case. Bol. Soc. Mat. Mexicana (3) 10 (2004), no.
+1, 119–138.
+[5] Grudsky, S., Karapetyants, A. and Vasilevski, N.: Dynamics of properties of Toeplitz oper-
+ators on the upper half-plane: parabolic case. J. Operator Theory 52 (2004), no. 1, 185–214.
+[6] Grudsky, S., Karapetyants, A. and Vasilevski, N.: Dynamics of properties of Toeplitz oper-
+ators with radial symbols. Integral Equations Operator Theory 50 (2004), no. 2, 217–253.
+[7] Grudsky, S., Quiroga-Barranco, R. and Vasilevski N.: Commutative C∗-algebras of Toeplitz
+operators and quantization on the unit disk. J. Funct. Anal. 234 (2006), no. 1, 1–44.
+[8] Johnson, Kenneth D.: On a ring of invariant polynomials on a Hermitian symmetric space.
+J. Algebra 67 (1980), no. 1, 72–81.
+[9] Helgason, Sigurdur: Differential geometry, Lie groups, and symmetric spaces. Corrected
+reprint of the 1978 original. Graduate Studies in Mathematics, 34. American Mathematical
+Society, Providence, RI, 2001.
+[10] Hua, L. K.: Harmonic analysis of functions of several complex variables in the classical
+domains. Translated from the Russian by Leo Ebner and Adam Kor´anyi American Mathe-
+matical Society, Providence, R.I. 1963.
+
+TOEPLITZ AND MOMENT MAPS ON DOMAINS OF TYPE III
+33
+[11] Korenblum, Boris and Zhu, Ke He: An application of Tauberian theorems to Toeplitz oper-
+ators. J. Operator Theory 33 (1995), no. 2, 353–361.
+[12] McDuff, Dusa and Salamon, Dietmar: Introduction to symplectic topology. Third edition.
+Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2017.
+[13] Mok, Ngaiming: Metric rigidity theorems on Hermitian locally symmetric manifolds. Series
+in Pure Mathematics, 6. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.
+[14] Quiroga-Barranco, Raul and Sanchez-Nungaray, Armando: Moment maps of Abelian groups
+and commuting Toeplitz operators acting on the unit ball, Journal of Functional Analysis
+281 (2021), no. 3, article 109039.
+[15] Quiroga-Barranco, Raul and Seng, Monyrattanak: Commuting Toeplitz operators on Cartan
+domains of type IV and moment maps. Complex Anal. Oper. Theory 16 (2022), no. 7, Paper
+No. 102, 41 pp.
+[16] Quiroga-Barranco, Raul and Vasilevski, Nikolai: Commutative C∗-algebras of Toeplitz oper-
+ators on the unit ball. I. Bargmann-type transforms and spectral representations of Toeplitz
+operators. Integral Equations Operator Theory 59 (2007), no. 3, 379–419.
+[17] Quiroga-Barranco, Raul and Vasilevski, Nikolai: Commutative C∗-algebras of Toeplitz oper-
+ators on the unit ball. II. Geometry of the level sets of symbols. Integral Equations Operator
+Theory 60 (2008), no. 1, 89–132.
+[18] Range, R. Michael: Holomorphic functions and integral representations in several complex
+variables. Graduate Texts in Mathematics, 108. Springer-Verlag, New York, 1986.
+[19] Upmeier, Harald: Toeplitz operators and index theory in several complex variables. Operator
+Theory: Advances and Applications, 81. Birkh¨auser Verlag, Basel, 1996.
+[20] Vasilevski, N. L.: Bergman space on tube domains and commuting Toeplitz operators. Pro-
+ceedings of the Second ISAAC Congress, Vol. 2 (Fukuoka, 1999), 1523–1537, Int. Soc. Anal.
+Appl. Comput., 8, Kluwer Acad. Publ., Dordrecht, 2000.
+Centro de Investigaci´on en Matem´aticas, Guanajuato, Guanajuato, M´exico
+Email address: [email protected]
+Centro de Investigaci´on en Matem´aticas, Guanajuato, Guanajuato, M´exico
+Email address: [email protected]
+

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+arXiv:2301.05554v1  [quant-ph]  11 Jan 2023
+Synergies Between Operations Research and Quantum Information
+Science
+Ojas Parekh
+Quantum Algorithms and Applications Collaboratory (QuAAC)
+Sandia National Laboratories
+[email protected]
+Abstract
+This article highlights synergies between quantum information science (QIS) and operations
+research for QIS-curious operations researchers (and vice-versa).
+1
+Introduction
+Operations researchers are no strangers to transferring their expertise to new domains to realize
+an impact. Such endeavors often entail understanding enough domain specifics to effectively
+build models and solve problems. This can be an iterative and challenging process, as sometimes
+idiosyncrasies that have been internalized by domain experts need to be sussed out. However,
+overcoming such obstacles may become particular points of pride, in addition to overall success.
+Even though quantum information science (QIS) is a broad, vibrant, and intensely growing
+field, I advocate approaching QIS the same way we might a more specialized domain. Instead of
+being daunted by, for example, never having taken a quantum physics course, we might try to
+stick to mathematical descriptions or other abstractions, with the understanding of likely being
+oblivious to a body of underlying intuition that has been well earned by physicists. Time and
+experience may help remedy the latter if desired.
+I highlight seminal or recent advances in QIS attained through the lens of optimization. I
+also offer suggestions on how engaging with QIS might lead to advances in more traditional
+operations research (OR). Finally, I present strategies for operations researchers to engage QIS.
+I encourage operations researchers to cultivate new synergies with QIS.
+2
+Quantum Information Science applications of Opera-
+tions Research
+Properties of quantum states and channels.
+Without worrying about additional de-
+tails or quantum-mechanical interpretations, we may think of a quantum state, ρ, on n quantum
+bits (qubits) as a 2n ×2n matrix with complex entries. Many basic properties of quantum states
+and the quantum channels describing operations on them may be readily cast as semidefinite
+programs. In fact the defining properties of a state ρ are that it has trace equal to one and is
+positive semidefinite. For more details, a self-contained account of five accessible applications
+of semidefinite programming to properties of quantum states and channels appears in [ST21].
+The high-level perspective of the remainder of the article will not expect an understanding
+of qubits or quantum states, beyond the fact that quantum states are exponentially large in the
+number of qubits. The latter opens the door to potential exponential advantages over classical
+computation, as clever physical manipulations of n qubits may enable nature to implicitly process
+exponentially large quantum states in meaningful and useful ways. However, the same kind of
+1
+
+statement could be made about randomized classical algorithms, where manipulating n random
+bits yields an implicit distribution over an exponentially large set of outcomes. Thus we seek to
+identify features of quantum physics that are not accessible classically.
+2.1
+Differentiating classical and quantum physics
+What does a universe endowed with quantum physics offer that is simply impossible under the
+laws of classical physics (i.e., conventional non-quantum physics)? Let me highlight such an
+example. In a nonlocal game Alice and Bob are to be posed questions x and y and must agree on
+a strategy that generates answers a and b. The value of a nonlocal game, V (a, b | x, y) ∈ {0, 1}
+determines correctness of the answers, and Alice and Bob’s goal is to maximize the expectation
+of V over a given distribution on x, y (and potential randomness in selecting a, b). It turns
+out that random strategies do not offer any advantage over deterministic strategies; however,
+strategies where Alice and Bob each have one of a pair of entangled1 quantum bits (qubits) are
+able to outperform classical strategies (see the survey, [BCP+14]).
+Advantages offered by quantum strategies to nonlocal games are intimately related to John
+Bell’s seminal tests for nonlocality (or “quantumness”) in physical systems. Quantum violations
+of Bell inequalities, that classical physical systems must satisfy, have been demonstrated experi-
+mentally under a variety of settings ([BCP+14], Section VII). The values of nonlocal games may
+in turn be approximated by hierarchies of semidefinite programs (SDPs) ([BCP+14], Section
+II.C); however, the size of the resulting programs grows exponentially (or worse) in the size of
+the game. Imagine the gratification in finally successfully solving or analyzing a thorny SDP
+and subsequently receiving a message from your experimental-physicist friend that nature agrees
+with your findings – a nice pat on the back from the universe. Nonlocal games have fundamental
+connections to models of (quantum) computation [MNY21], and a nonlocal games perspective
+has recently enabled a landmark result resolving two longstanding problems: Tsirelson’s problem
+in quantum mechanics and Connes’ embedding problem in operator algebras [JNV+21, PV16].
+2.2
+Computational quantum advantages
+A foremost direction in QIS is leveraging non-classical features of quantum physics to realize
+forms of quantum computation that are able to enjoy advantages over conventional classical
+computation. Shor’s seminal quantum algorithm for factoring integers runs exponentially faster
+than any known classical algorithm (see e.g., [NC10], Section 5.3); however, as far as we currently
+know, exponentially faster classical factoring algorithms might well exist. Yet, if we consider
+computational resources beyond execution time, exponential quantum advantages over best-
+possible classical algorithms are known.
+One such setting is query complexity, where we are only concerned with the number of
+queries to a problem’s input data rather than overall execution time. Exponential quantum
+advantages in query complexity were among the first quantum algorithms discovered (see e.g.,
+[NC10], Section 1.4). The scope for exponential advantages in query complexity for graph prob-
+lems is generally well understood ([BDCG+20]; see the related survey, [MdW16]). Remarkably,
+finding the best quantum algorithm for a problem in the query model can be captured, within
+constant factors, as SDPs [Rei11, BSS03], and this perspective has helped design quantum
+graph algorithms (e.g., [DKW19]). Recent work has characterized the precise query complexity
+in terms of the completely bounded norm of a tensor [ABP19], which in turn is expressible as
+an SDP [GL19]. A relaxed notion of query complexity in expectation turns out to be intimately
+related to the Sherali-Adams hierarchy in the classical case and the Lasserre/Sum-of-Squares
+(SoS) hierarchy in the quantum case [KLW15]. For open problems in quantum query complexity
+see [Aar21], and for a broader survey of exponential quantum speedups see [Aar22].
+1Rather than offering a concise but potentially misleading description of entanglement here, I suggest the popular
+article, [Wil16].
+2
+
+2.3
+Building better quantum computers
+The biggest open question in quantum computing is perhaps whether we can indeed design and
+engineer scalable fault-tolerant quantum computers to realize theoretically supported quantum
+advantages. The world is a particularly hostile place for a quantum computer, with magnetic
+fields, variations in temperature, and a host of other sources of noise that are disruptive to com-
+putation. Noise induces errors that are amplified the longer a computation executes. Schemes
+to correct such errors are known; however, they demand considerable overhead in terms of ex-
+tra error-correction qubits. Designing efficient quantum error correction schemes with desirable
+resiliency properties may be cast as an optimization problem, and semidefinite programming
+techniques have been used to design and analyze such schemes [FSW07, KL09, BBFS21].
+Quantum processors may be built upon different physical substrates, though the selection is
+drastically constrained compared to classical processors. Each brings unique design and engi-
+neering challenges, as well as associated optimization problems. Quantum processors are gen-
+erally put through quantum characterization, validation and verification hoops to ensure they
+behave as expected (see the tutorial, [KR21] and review, [EHW+20]). Well-behaved quantum
+processors sit under software stacks that offer further opportunities for optimization, includ-
+ing compiling high-level quantum algorithms into quantum circuits consisting of native gates.
+Interdisciplinary teams including operations researchers are increasingly addressing optimiza-
+tion problems at many levels of quantum computer system design [NBGJ22, NLC21, TTS+21,
+FBL+22, MCL+22, BPLP20].
+2.4
+Relaxations for problems in quantum physics
+The above applications suggest a recurring theme: convex programs naturally model quantum-
+mechanical phenomena, but faithful models require exponential-size programs in the number
+of qubits. A technique OR may bring to the table is finding smaller but reasonably strong
+relaxations of such exponential-sized convex programs. This could help accelerate approaches for
+exactly or approximately computing quantities of physical interest. I will illustrate this concept
+by drawing connections between discrete optimization problems and quantum counterparts,
+using the well-known classical Max Cut problem as an example.
+Max Cut as an eigenvalue problem.
+For a graph G = (V, E) on n vertices, Max Cut
+seeks to find a set S ⊆ V that maximizes the number of edges between S and V \ S. We
+will encode Max Cut as finding the largest eigenvalue of a matrix exponentially large in n, for
+more direct comparison with problems from quantum physics. Imagine the columns and rows
+of a diagonal matrix H ∈ R2n×2n are labeled with the 2n possible vertex sets S ⊆ V , and set
+HS,S to the number of edges in the cut (S, V \ S). Now since H is diagonal, λmax(H) (the
+maximum eigenvalue of H) is the maximum cut value, achieved by a set S∗. The corresponding
+eigenvector v∗ ∈ R2n corresponds to a basis vector with a one in the position labeled S∗ and zeros
+elsewhere. Even though v∗ lives in an exponentially large space, it has a succinct description
+that only depends on S∗. The matrix H itself also has a succinct description that only depends
+on the edges in G. In summary, Max Cut (or discrete optimization problems in general2) may
+be cast as finding λmax(H) for a diagonal matrix H that is exponentially large in n with a
+description of size polynomial in n.
+Sampling-based problem models.
+What happens if a symmetric H is not required to
+be diagonal? In this case a solution eigenvector v∗ may have exponentially many nonzeros,
+which is naturally a major obstacle for efficient algorithms or heuristics. We can circumvent
+this by asking for statistics about v∗ instead. One option is to instead request samples from a
+distribution over the labels of the elements of v∗, such that the label l is obtained with probability
+proportional to (v∗
+l )2 (or |v∗
+l |2 if H is Hermitian and v∗ is complex). This output model still
+2H could be labeled with subsets of any discrete domain, with infeasibility modeled by large-magnitude values.
+3
+
+captures Max Cut, since we would obtain some optimal set S∗ as a label with probability 1.
+Such models are perhaps as not as foreign as they might first appear; for example, Markov
+Chain Monte Carlo methods are tailored for similar settings.
+Polynomial-time quantum algorithms for linear algebra (see the primer, [DHM+18]) and
+machine learning (see the survey, [BWP+17]) are known in the above kind of model, where the
+implicitly defined matrices involved are exponentially larger than the number qubits necessary to
+describe them, and output vectors are only accessible through samples. However, there are some
+critical caveats for obtaining exponential quantum advantages in this context [Aar15, Aar22]. In
+breakthrough work, quantum-inspired polynomial-time classical algorithms of the same flavor
+have been recently discovered (e.g., [Tan19, CGL+20]); however, they rely on a particular kind of
+classical data access model that may be impractical [CHM21]. Recent empirical demonstrations
+of quantum advantages are also based on sampling problems [AAB+19, MLA+22].
+The Local Hamiltonian problem.
+Returning to the problem of computing λmax(H)
+as described above, replacing “diagonal” with “Hermitian” in the requirements on H takes us
+from an NP-complete discrete optimization problem (e.g., Max Cut) to a fundamental quantum
+optimization problem: the Local Hamiltonian problem. Here H is called a Hamiltonian, and
+“local” refers to a kind of succinct implicit description of H, in the vein of our Max Cut example
+above. Physical systems may be described by local Hamiltonians that dictate how they evolve
+over time, where the eigenvectors of the Hamiltonian correspond to stable states of the system.
+In fact nature is constantly trying to solve optimization problems all around us! Indeed nature
+strives to heuristically put physical systems in their ground states, corresponding to minimum-
+eigenvalue eigenvectors of the corresponding Hamiltonian. Consequently, studying ground states
+of physical systems is a fundamental problem that aids in better understanding and exploiting
+exotic properties of materials, for example, [Min09].
+From a computational perspective, Local Hamiltonian is a cornerstone in understanding the
+power and limitations of different models of quantum computing, serving a role akin to that of
+the Boolean Satisfiability problem (SAT) in classical complexity theory. Local Hamiltonian is
+complete for the complexity class Quantum Merlin Arthur (QMA), which contains and is the
+natural quantum analogue of NP. We do not expect polynomial-time quantum algorithms to
+solve QMA-hard problems (quantum advantages are more subtle than solving NP-hard problems
+[AC16, Aar22]. Yet, as with NP-hard problems in the classical regime, aspiring to solve QMA-
+hard problems may spark new approaches for heuristic solutions, rigorous approximations, or
+exact solutions in special cases.
+The Quantum Max Cut problem.
+A desire to help shape the nascent field of quantum
+approximation algorithms underlaid my foray into QIS. Sevag Gharibian and I introduced Quan-
+tum Max Cut, an instance of Local Hamiltonian that is closely related to both the Heisenberg
+model, a physical model of quantum magnetism, as well as classical Max Cut [GP19]. It turns
+out that the celebrated Goemans-Williamson SDP-based approximation algorithm for Max Cut
+[GW95] can be generalized to give approximation algorithms for Quantum Max Cut [GP19].
+The SDP relaxation employed for Max Cut is an instance of the Lasserre/SoS hierarchy, and
+relaxations for Quantum Max Cut [PT21a, PT22] may be obtained from a non-commutative3
+counterpart of the Lasserre/SoS hierarchy [PNA10].
+As as a canonical constraint-satisfaction and discrete-optimization problem, studying Max
+Cut has had far-reaching consequences in both computer science [KKMO07] and OR [DL97],
+including exponential lower bounds on polyhedral formulations of the Traveling Salesperson
+problem (via the related correlation polytope) [FMP+15]. The goal is for Quantum Max Cut
+to serve as a testbed for designing approaches to better solve more general Local Hamiltonian
+3Max Cut may be cast as maxzi
+�
+ij∈E(1 − zizj)/2 for commutative variables z2
+i = 1, while Quantum Max Cut
+is maxxi,yi,zi λmax(�
+ij∈E(1 − xixj − yiyj − zizj)/4), for non-commutative variables (i.e., matrices) x2
+i = 1, y2
+i = 1,
+z2
+i = 1 with the additional constraints that variables with different indices commute, while different variables with
+the same index anti-commute.
+4
+
+problems [PT21b, PT22]. See Section 7 of [HNP+21] for an introduction to Quantum Max Cut
+and Section 3 of [PT22] for additional parallels between Max Cut and Quantum Max Cut.
+Challenges.
+Although analogies between Max Cut and Quantum Max Cut have helped
+direct research into the latter, it largely remains enigmatic. The Goemans-Williamson 0.878-
+approximation for Max Cut [GW95] is the best possible under the Unique Games Conjecture
+[KKMO07]. Although an optimal approximation for Quantum Max Cut is known in a special
+setting [PT22], the currently best-known approximations for the general problem seem far from
+optimal [HNP+21]. Thus a primary challenge is better approximation algorithms for Quantum
+Max Cut, as well as more general local Hamiltonians problems.
+Another direction is designing effective and practical heuristics.
+I expect OR-influenced
+heuristics will likely be different and complementary to those currently employed by physi-
+cists.
+More sophisticated OR-style relaxations, based on bespoke valid inequalities or new
+types of mathematical-programming hierarchies, are unexplored. Better understanding the non-
+commutative Lasserre/SoS hierarchy for Local Hamiltonian is a promising direction, since this
+may also yield new types of quantum entanglement constraints [PT22]. Finally, Max Cut and
+Quantum Max Cut are natural unconstrained optimization problems; models, relaxations, and
+approximations for constrained Local Hamiltonian problems are virtually nonexistent.
+2.5
+Additional applications
+I mention a few more applications in passing. QIP is a model of computation based on quan-
+tum interactive proofs, while PSPACE is the model in which a classical computer is granted
+polynomial space (but no explicit limit on execution time). Proofs of the celebrated result that
+QIP = PSPACE rely on the multiplicative weights method for solving SDPs [JJUW10]. A re-
+cent demonstration of self-concordant barrier functions for quantum relative entropy programs
+implies more efficient interior-point approaches for such problems [FS22]. Non-commutative
+SDP hierarchies can be used to better understand mutually unbiased bases, which have many
+applications in quantum information and beyond [GP21]. There are a wide variety of further
+such applications in QIS, and there are likely many more waiting to be discovered.
+2.6
+Discerning a quantum advantage
+How will we know if we have witnessed a true and significant quantum advantage? On the
+theoretical side, worst-case and asymptotic analysis suggests provable exponential quantum
+advantages are possible on fault-tolerant quantum computers; however, as previously discussed,
+problems admitting provable quantum advantages may not have direct classical counterparts,
+rendering an apples-to-apples comparison difficult. Even when such comparisons are possible,
+the big question is whether nature will allow us to engineer scalable quantum computers beyond
+the near-term noisy intermediate-scale quantum (NISQ) regime [Pre18].
+On the practical side, promising empirical benchmarks of current early stage NISQ computers
+may not be indicative of sustainable advantages into the future. In addition, recent work points
+to theoretical limitations of NISQ computing [CCHL22, AGL+22, Kal20]. Empirical benchmarks
+are typically focused on a relatively small or otherwise limited set of instances, and even when
+quantum computers appear superior, better classical algorithms may be right around the corner.
+To mitigate such factors, identifying problems appearing to admit empirical quantum advantages
+and issuing open challenges in the vein of the DIMACS Implementation Challenges [DIM22] may
+be a fruitful practice. While this is not a direct QIS application of OR, it is something the latter
+may share of its culture to benefit the former.
+3
+Quantum-inspired Operations Research Advances
+Diversifying instance libraries.
+OR should be proud of its well-defined problem models,
+curated libraries of problem instances, and systematic benchmarks across ranges of solvers (e.g.,
+5
+
+[MS21]). QIS-inspired instances of optimization problems are likely to have a different charac-
+ter than traditional OR instances and would make nice additions to existing instance libraries.
+Moreover, solving such instances effectively may drive improvements or new techniques or fea-
+tures in solvers. Systematic benchmarks and instance collections of quantum-inspired problems
+may also be of benefit to the QIS community.
+3.1
+Quantum algorithms for classical optimization problems
+Quantum advantages are known for a variety of bread-and-butter optimization problems within
+OR. Pursuing polynomial-factor quantum speedups for optimization problems including gra-
+dient descent [GAW19, KP20], linear programming [Nan22], second-order cone programming
+[KPS21], semidefinite programming [HJS+22, ANTZ21], and convex programming [CCLW20,
+vAGGdW20] has been a fruitful endeavor. Yet such quantum algorithms do not always improve
+upon classical counterparts for all the problem parameters, and lower bounds suggest expo-
+nential quantum speedups are not possible [GKNS21, CCLW20, vAGGdW20]. A “natural” or
+“practical” optimization problem admitting a rigorous exponential quantum advantage remains
+elusive.
+In place of a speedup, we may consider how well a quantum algorithm might approximate a
+problem relative to classical algorithms. The Quantum Approximate Optimization Algorithm
+(QAOA) [FGG14] is a quantum-algorithmic framework for formulating and solving discrete op-
+timization problems (see the thesis, [Had18]). QAOA resembles mathematical programming in
+that discrete optimization problems may be relatively easily expressed in the framework, and
+the overall efficacy of the algorithm is often highly dependent on the particular formulation em-
+ployed. QAOA has garnered considerable QIS community interest, and it is a natural candidate
+for implementation on NISQ systems. Moreover, QAOA is a parameterized algorithm that lends
+itself to a natural hybrid quantum-classical loop, wherein a classical optimization routine is used
+to obtain parameter values by leveraging a quantum computer to execute QAOA. The perfor-
+mance of QAOA is assessed on the current parameter values, informing subsequent parameter
+updates. It is not currently known if there is a problem where a polynomial-time execution of
+QAOA on a quantum computer is able to yield a provably better approximation than possible
+classically. Some theoretical limitations of QAOA are known [AM22, CLSS22, MH22, BKKT20].
+Even if QAOA is unable to provide an advantage on classical problems, QAOA might be able
+to achieve better approximations than possible classically for quantum optimization problems,
+such as Quantum Max Cut. Such directions are understudied [AGM20, AGKS21]. Better un-
+derstanding the power and limitations of QAOA, applying it to broader problem classes, and
+devising QAOA-inspired classical algorithms remain challenges.
+3.2
+New types of problems inspired by quantum physics
+Beyond helping to solve quantum optimization problems, I urge operations researchers to allow
+quantum problems to inspire new types of classical ones. Let me supply an example. Opera-
+tions researchers are well aware that our models frequently fail to capture the nuances of the
+underlying practical problems we endeavor to solve. For this reason and others, a diverse set of
+near-optimal solutions may be preferable to a single optimal solution.
+Consider the following version of Max Cut that seeks to promote diverse solutions:
+max
+x
+E[c(x)] +
+�
+x,y∈{0,1}n
+�
+Pr[x = x]Pr[x = y]h(x, y),
+where x ∈ {0, 1}n is a random variable, c(x) is the cost of the cut induced by x, and {0, 1} ∋
+h(x, y) = 1 if and only if xi ̸= yi for exactly one position i. The above problem then seeks
+to find a distribution over cuts, represented by x, that is incentivized for both having a large
+expected cut value as well as having support on pairs of cuts that differ in exactly one vertex.
+Here h(x, y) is meant to measure diversity between x and y, and other options are possible.
+6
+
+The notion of diversity above arises naturally as what is known as a transverse-field term
+in the quantum Ising model studied in statistical mechanics.
+We could think of the above
+problem as a transverse-field Max Cut problem, and it is actually equivalent to the well-studied
+transverse-field Ising model, which is a Local Hamiltonian problem.
+Fresh insights into the
+problem may have an impact on the physical side of things, as I already argued for more
+general Local Hamiltonian problems. However, here I would like to emphasize a complementary
+story.
+We may readily adapt x and c(x) above to derive transverse-field versions of other
+familiar optimization problems, where we likewise seek distributions over diverse solutions. In
+this case insights gleaned from a physical perspective might suggest new avenues for these and
+related optimization problems. The challenge is then to more broadly frame physical models
+and solution techniques in a way that might impact OR.
+4
+Suggestions for Engaging QIS
+QIS is a diverse and multi-disciplinary field collecting physicists, chemists, engineers, computer
+scientists, mathematicians and increasingly, operations researchers. Here the notion of an “out-
+sider” is perhaps diminished relative to more homogeneous fields. Reach out to your friendly
+neighborhood quantum information scientist with questions or for guidance. Seek out OR col-
+leagues who have started dabbling in QIS. Members of the INFORMS ICS Working Group on
+Quantum Computing and others have been active in organizing workshops bridging OR and
+QIS – be on the lookout for such opportunities.
+Suggested reading.
+I have explicitly called out references to surveys, primers, tutorials,
+reviews, and theses above as hints for further reading. As far as textbooks go, Nielsen and
+Chuang is the standard introduction to quantum computing and quantum information for good
+reason [NC10]. Several quantum information scientists maintain web pages with pointers for
+learning QIS topics (e.g., [Har22]).
+I want to stress that you do not have to necessarily understand quantum physics to have an
+impact. Working in interdisciplinary teams or settling into a comfortable mathematical formal-
+ism are some ways to mitigate a proper physics background. If your regular time commitments
+are at odds with cover-to-cover reading of new material, of course you should not feel ashamed
+to focus on the bits and pieces that interest you most, with the hope that you will eventually
+be able to fill in gaps as necessary or time permits. If you would like to roll up your sleeves and
+learn interactively, quantum programming tutorials such as [M+21] might be a good place to
+start.
+Concluding Remarks
+Improved capabilities for solving optimization problems arising in quantum physics can help
+us better understand how nature works at a fundamental level. Both practical approaches for
+solving specific kinds of problems as well as furthering theoretical foundations are valuable.
+Historically, a stronger command of nature has fueled transformative impacts on civilization
+and society.
+Acknowledgements
+This material is based upon work supported by the U.S. Department of Energy, Office of Sci-
+ence, Office of Advanced Scientific Computing Research, Accelerated Research in Quantum
+Computing, Fundamental Algorithmic Research for Quantum Computing (FAR-QC).
+This article has been authored by an employee of National Technology & Engineering Solu-
+tions of Sandia, LLC under Contract No. DE-NA0003525 with the U.S. Department of Energy
+7
+
+(DOE). The employee owns all right, title and interest in and to the article and is solely respon-
+sible for its contents. The United States Government retains and the publisher, by accepting the
+article for publication, acknowledges that the United States Government retains a non-exclusive,
+paid-up, irrevocable, world-wide license to publish or reproduce the published form of this ar-
+ticle or allow others to do so, for United States Government purposes. The DOE will provide
+public access to these results of federally sponsored research in accordance with the DOE Public
+Access Plan https://www.energy.gov/downloads/doe-public-access-plan.
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+

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+arXiv:2301.12924v1  [math.CO]  30 Jan 2023
+STRONG EDGE-COLORING OF 2-DEGENERATE GRAPHS
+GEXIN YU1 AND RACHEL YU2
+1Department of Mathematics, William & Mary, Williamsburg, VA 23185, USA.
+2Jamestown High School, Williamsburg, VA 23185, USA.
+Abstract. A strong edge-coloring of a graph G is an edge-coloring in which every color class is an
+induced matching, and the strong chromatic index χ′
+s(G) is the minimum number of colors needed
+in strong edge-colorings of G. A graph is 2-degenerate if every subgraph has minimum degree at
+most 2. Choi, Kim, Kostochka, and Raspaud (2016) showed χ′
+s(G) ≤ 5∆ + 1 if G is a 2-degenerate
+graph with maximum degree ∆. In this article, we improve it to χ′
+s(G) ≤ 5∆ − ∆1/2−ǫ + 2 when
+∆ ≥ 41/(2ǫ) for any 0 < ǫ ≤ 1/2.
+1. Introduction
+A strong edge-coloring of a graph G is an edge-coloring in which every color class is an induced
+matching; that is, there are no 2-edge-colored triangles or paths of three edges. The strong chromatic
+index χ′
+s(G) is the minimum number of colors in a strong edge-coloring of G. This notion was
+introduced by Fouquet and Jolivet [9] and one of the main open problems was proposed by Erd˝os
+and Neˇsetˇril [8] during a seminar in Prague:
+Conjecture 1 (Erd˝os and Neˇsetˇril, 1985). If G is a simple graph with maximum degree ∆, then
+χ′
+s(G) ≤ 5∆2/4 if ∆ is even, and χ′
+s(G) ≤ (5∆2 − 2∆ + 1)/4 if ∆ is odd.
+This conjecture is true for ∆ ≤ 3, see [1, 10]. For ∆ = 4, Huang, Santana and Yu [11] showed
+that χ′
+s(G) ≤ 21, one more than the conjectured upper bound 20. Chung, Gy´arf´as, Trotter, and
+Tuza (1990, [6]) confirmed the conjecture for 2K2-free graphs. Using probabilistic methods, Molloy
+and Reed [14], Bruhn and Joos [3], Bonamy, Perrett, and Postle [2], and recently Hurley, Verclos,
+and Kang [12] showed that χ′
+s(G) ≤ 1.772∆2 for sufficiently large ∆.
+Sparse graphs have also attracted a lot of attention. Interested readers may see the survey paper
+[7] for more details. In this article, we are interested in k-degenerate graphs when k = 2. A graph
+is k-degenerate if every subgraph has minimum degree at most k.
+Let G be a 2-degenerate graph with maximum degree ∆ ≥ 2. Chang and Narayanan [4] proved
+that χ′
+s(G) ≤ 10∆ − 10. Luo and Yu [13] improved it to χ′
+s(G) ≤ 8∆ − 4. For arbitrary values of
+k, Wang [15] improved the result of Yu [16] and showed the following result.
+Theorem 2. If G is a k-degenerate graph with maximum degree ∆ ≥ k, then χ′
+s(G) ≤ (4k −2)∆−
+2k2 + 1.
+This implies that χ′
+s(G) ≤ 6∆ − 7 for 2-degenerate graph G.
+Choi, Kim, Kostochka, and
+Raspaud [5] further improved it to χ′
+s(G) ≤ 5∆ + 1 in 2016. Many believe that the optimal bound
+should be 4∆ + C for some constant C, but no progress has yet been made.
+E-mail address: [email protected].
+1
+
+In this article, we show that for any 0 < ǫ ≤ 1/2, and ∆ ≥ 41/(2ǫ), χ′
+s(G) ≤ 5∆ − ∆1/2−ǫ + 2 for
+2-degenerate graph G with maximum degree ∆.
+2. Main result and its proof
+A special vertex is a vertex with at most two neighbors of degree more than two.
+Every 2-
+degenerate graph contains special vertices, which are the new 2−-vertices after we remove all vertices
+of degree at most two. Let G be a 2-degenerate graph and S be the set of special vertices of G.
+For each u ∈ S, there exists a set Wu of vertices such that each w ∈ Wu shares some 2-neighbors
+with u. The capacity of special vertices of G is the maximum number of common 2-neighbors that
+are shared by a vertex in S and other vertices in G. A pendant edge is an edge incident with a leaf
+(a vertex of degree one). Below is the main result of this article.
+Theorem 3. For any 0 < ǫ ≤ 1/2, let D be a positive integer when D ≥ 41/(2ǫ), for any 2-
+degenerate graph G with maximum degree ∆, if a vertex u is adjacent to at least d(u) − D leaves
+when it has d(u) > D, and
+• ∆ ≤ D + 2 when the capacity of special vertices is at least D1/2−ǫ, and
+• ∆ ≤ D + D1/2−ǫ when the capacity of special vertices is less than D1/2−ǫ,
+then χ′
+s(G) ≤ 5D − D1/2−ǫ + 2.
+Proof. Let G be a counterexample with the fewest number of vertices of degree at least two.
+Since G is 2-degenerate, G must have a set S of special vertices. Let u ∈ S and Wu be the set
+whose vertices share 2-neighbors with u such that the maximum number of common 2-neighbors
+of u and vertices of Wu is the capacity of special vertices of G. Let Wu = {w1, . . . , ws} and u1, u2
+be the two neighbors of u with degree more than 2. For each wi ∈ Wu, let Wi = {vi,1, . . . , vi,ti}
+be the common 2-neighbors of wi and u. Then N(u) = {u1, u2} ∪ �s
+i=1{vi,1, . . . , vi,ti}. We assume
+that t1 ≥ t2 ≥ . . . ≥ ts ≥ 1. Then t1 is the capacity of special vertices of G. It is not hard to see
+that u has no neighbors of degree one and wi has degree at least two for each i.
+For edge uv, let N2(uv) be the set of edges xy such that x or y is adjacent to u or v. By definition,
+uv should have a color different from the colors on edges in N2(uv) in a valid strong edge-coloring.
+Case 1. t1 < D1/2−ǫ. In this case, we have ∆ ≤ D + D1/2−ǫ.
+Let G′ be the graph obtained from G−{uv1,1, . . . , uvs,ts} by adding up to D
+1
+2 −ǫ pendant neighbors
+to each of {w1, . . . , ws} so that wi has degree at most D+D1/2−ǫ and wi has at least D
+1
+2 −ǫ pendant
+neighbors.
+Then the graph G′ has fewer vertices of degree at least 2 and can be colored with
+5D − D1/2−ǫ + 2 colors. We modify the coloring of G′ to obtain a coloring of G according to the
+following algorithm.
+(1) Keep the colors of edges that appear in both G and G′, but if wivi,j for some i, j in G has
+the same color as uu1 or uu2, then we switch color of wivi,j with a color on other pendant
+edges incident to wi in G′.
+(2) For each i, if a color c appears on both a pendant edge incident to wi in G′ and an edge
+incident to u1 or u2 (not including uu1 and uu2), then we switch the color of wivi,j for some
+j with the color c.
+(3) After (2), if a color c appears on pendant edges of two or more vertices in Wu in G′, then
+we switch the color of wivi,j for some j with c for each such vertex wi ∈ Wu.
+(4) After (2) and (3), we color the edges uv1,1, . . . , uvs,ts in reverse order with colors available
+to them.
+2
+
+Now we show that the above algorithm gives a valid strong edge-coloring of G. To do that, we
+only need to show that each of the edges in {uv1,1, . . . , uvs,ts} can be colored. Consider uvi,j for
+1 ≤ i ≤ s and 1 ≤ j ≤ ti. It needs to get a color not on edges in N2(uvi,j). Note that N2(uvi,j)
+contains the edges incident to u1, u2, wi and the edges incident to the 2-neighbors of u; So the
+number of colored edges in N2(uvi,j), denoted as n2(uvi,j), is at most
+d(u1) + d(u2) + d(wi) + d(u) − 3 + d(u) − 2 − ti −
+i−1
+�
+p=1
+tp − (j − 1).
+We assume that n2(uvi,j) ≥ 5D − D1/2−ǫ + 2, for otherwise, uvi,j can be colored. Because of the
+way the edges being colored, we have some repeated colors on edges incident to the 2-neighbors of
+u, namely v1,1w1, v1,2w1, . . . , vs,tsws. The number of colors on N2(uvi,j) and edges incident to wi for
+i ∈ [s] (with repetition) is n2(uvi,j)+D1/2−ǫ·(s−1). Thus n2(uvi,j)+D
+1
+2 −ǫ(s−1)−(5D−D
+1
+2−ǫ+2)
+colors are repeated. As each wi may allow only one edge (for example, ti = 1) whose color is the
+same as other edges in N2(uvi,j), at least n2(uvi,j)+D
+1
+2 −ǫ(s−1)−(5D−D
+1
+2 −ǫ+2)
+D
+1
+2 −ǫ
+edges have the same
+colors as others. Since t1 < D1/2−ǫ and s ≥ d(u)−2
+t1
+≥
+d(u)−2
+D1/2−ǫ , the number of different colors in
+N2(uvi,j) is at most
+n2(uvi,j) − n2(uvi,j) + D
+1
+2−ǫ(s − 1) − (5D − D
+1
+2 −ǫ + 2)
+D
+1
+2−ǫ
+≤n2(uvi,j)(1 −
+1
+D
+1
+2 −ǫ) − (s − 1) + 5D1/2+ǫ − 1 +
+2
+D
+1
+2−ǫ
+≤(d(u1) + d(u2) + 2d(u) − 5 + d(wi) − t1)(1 −
+1
+D
+1
+2 −ǫ) − d(u) − 2
+D1/2−ǫ + 5D1/2+ǫ +
+2
+D
+1
+2−ǫ
+≤3∆(1 −
+1
+D
+1
+2−ǫ ) + d(u)(2 −
+3
+D
+1
+2−ǫ ) + 5D1/2+ǫ − 5 +
+9
+D
+1
+2 −ǫ
+≤3(D + D
+1
+2 −ǫ)(1 −
+1
+D
+1
+2 −ǫ) + D(2 −
+3
+D
+1
+2 −ǫ) + 5D1/2+ǫ − 5 +
+9
+D
+1
+2−ǫ
+≤5D + 3D
+1
+2 −ǫ − D1/2+ǫ − 8 +
+9
+D
+1
+2 −ǫ ≤ 5D − D1/2−ǫ + 1.
+The last inequality holds because D1/2+ǫ ≥ 4D1/2−ǫ and
+9
+D
+1
+2 −ǫ ≤ 9 when D ≥ 4
+1
+2ǫ . Therefore,
+the edge uvi,j can be colored, which implies that we can color all the uncolored edges.
+Case 2. t1 ≥ D1/2−ǫ. In this case, ∆ ≤ D + 2.
+Let G′ be the graph obtained from G by deleting the edge uv1,1 and adding up to two pendant
+edges notated {w1v′, w1v′′} so that w1 has at least three pendant edges and continues to have
+maximum degree at most D + 2.
+Note that for 1 ≤ i ≤ s and 1 ≤ j ≤ ts,
+(1) |N2(uvi,j)| ≤ d(u1) + d(u2) + d(wi) + d(u) − 3+ d(u) − 2− ti ≤ 3∆ + 2D − 5− ti ≤ 5D + 1− ti.
+We observe that G′ is 2-degenerate, has fewer vertices of degree at least 2, and contains at least
+d(v) − D vertices of degree one if d(v) > D. So, G′ can be colored with 5D − D1/2−ǫ + 2 colors.
+Keep the colors of the edges in both G and G′, we try to get a valid coloring of G.
+3
+
+Case 2.1. the color of edge v1,1w1 is different from ones on {uu1, uu2, uv1,2, . . . , uvs,ts}. By (1),
+we have at most 5D + 1 − D1/2−ǫ edges in N2(uv1,1). Since there are 5D − D1/2−ǫ + 2 different
+colors available, we can color uv1,1 with a color not on the edges in N2(uv1,1).
+Case 2.2. the color of edge v1,1w1 is the same as the color of edge uu1 (or edge uu2). In this
+case, we swap the color of v1,1w1 with the color of w1v′ or w1v′′. By (1) again, N2(uv1,1) has at
+most 5D +1−D1/2−ǫ edges in N2(uv1,1). Since there are 5D −D1/2−ǫ +2 different colors available,
+we can color uv1,1 with a color not on the edges in N2(uv1,1).
+Case 2.3. the color of edge v1,1w1 is the same as the color of edge uvi,j for some 2 ≤ i ≤ s. In
+this case, we uncolor the edges uvi,j, uv1,1, . . . , uv1,t1, and recolor them in the order. By (1), there
+are at most 5D + 1 − ti − (t1 − 1) ≤ 5D − D1/2−ǫ + 1 colors on edges in N2(uvi,j), so uvi,j can be
+colored. Similarly, for each j, there are at most 5D − D1/2−ǫ + 1 colors on edges in N2(uv1,j), so
+uv1,j can be colored as well.
+In any case, G can be colored with 5D − D1/2−ǫ + 2 colors. So χ′
+s(G) ≤ 5D − D1/2−ǫ + 2.
+□
+Observe that our main result follows from Theorem 3 with ∆ = D. Our proof does not work for
+corresponding result of list version.
+References
+[1] L. D. Andersen, The strong chromatic index of a cubic graph is at most 10, Discrete Math. 108 (1-3) (1992)
+231–252.
+[2] M. Bonamy, T. Perrett and L. Postle, Colouring graphs with sparse neighbourhoods: Bounds and applications,
+J. Combin. Theory Ser. B 155 (2022) 278–317.
+[3] H. Bruhn and F. Joos, A stronger bound for the strong chromatic index, Combin. Probab. Comput. 27 (1) (2018)
+21–43.
+[4] G. J. Chang and N. Narayanan, Strong chromatic index of 2-degenerate graphs, J. Graph Theory 73 (2) (2013)
+119–126.
+[5] I. Choi, J. Kim, A. V. Kostochka and A. Raspaud, Strong edge-colorings of sparse graphs with large maximum
+degree, European J. Combin. 67 (2018) 21–39.
+[6] F. R. K. Chung, A. Gy´arf´as, Z. Tuza and W. T. Trotter, The maximum number of edges in 2K2-free graphs of
+bounded degree, Discrete Math. 81 (2) (1990) 129–135.
+[7] K. Deng, G. Yu and X. Zhou, Recent progress on strong edge-coloring of graphs, Discrete Math. Algorithms
+Appl. 11 (5) (2019) 1950062.
+[8] P. Erd˝os and J. Neˇsetˇril, Irregularities of partitions, (G. Halasz, V.T. S´os (Eds.)), [Problem] (1989) 162-163.
+[9] J.-L. Fouquet and J.-L. Jolivet, Strong edge-colorings of graphs and applications to multi-k-gons, Ars Combin.
+16 (A) (1983) 141–150.
+[10] P. Hor´ak, Q. He and W. T. Trotter, Induced matchings in cubic graphs, J. Graph Theory 17 (2) (1993) 151–160.
+[11] M. Huang, M. Santana and G. Yu, Strong chromatic index of graphs with maximum degree four, Electron. J.
+Combin. 25 (3) (2018) Paper #3.31.
+[12] E. Hurley, R. de Joannis de Verclos and R. J. Kang, An improved procedure for colouring graphs of bounded
+local density, Adv. in Combinatorics, (2022), 7, 33pp.
+[13] R. Luo and G. Yu, A note on strong edge-colorings of 2-degenerate graphs, arXiv:1212.6092, December 25, 2012,
+https://arxiv.org/abs/1212.6092.
+[14] M. Molloy and B. Reed, A bound on the strong chromatic index of a graph, J. Combin. Theory Ser. B 69 (2)
+(1997) 103–109.
+[15] T. Wang, Strong chromatic index of k-degenerate graphs, Discrete Math. 330 (2014) 17–19.
+[16] G. Yu, Strong edge-colorings for k-degenerate graphs, Graphs Combin. 31 (5) (2015) 1815–1818.
+4
+

I9FOT4oBgHgl3EQfxTS9/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf,len=246
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='12924v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='CO]  30 Jan 2023  STRONG EDGE-COLORING OF 2-DEGENERATE GRAPHS  GEXIN YU1 AND RACHEL YU2  1Department of Mathematics, William & Mary, Williamsburg, VA 23185, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  2Jamestown High School, Williamsburg, VA 23185, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' A strong edge-coloring of a graph G is an edge-coloring in which every color class is an  induced matching, and the strong chromatic index χ′  s(G) is the minimum number of colors needed  in strong edge-colorings of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' A graph is 2-degenerate if every subgraph has minimum degree at  most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Choi, Kim, Kostochka, and Raspaud (2016) showed χ′  s(G) ≤ 5∆ + 1 if G is a 2-degenerate  graph with maximum degree ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' In this article, we improve it to χ′  s(G) ≤ 5∆ − ∆1/2−ǫ + 2 when  ∆ ≥ 41/(2ǫ) for any 0 < ǫ ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Introduction  A strong edge-coloring of a graph G is an edge-coloring in which every color class is an induced  matching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' that is, there are no 2-edge-colored triangles or paths of three edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' The strong chromatic  index χ′  s(G) is the minimum number of colors in a strong edge-coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' This notion was  introduced by Fouquet and Jolivet [9] and one of the main open problems was proposed by Erd˝os  and Neˇsetˇril [8] during a seminar in Prague:  Conjecture 1 (Erd˝os and Neˇsetˇril, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' If G is a simple graph with maximum degree ∆, then  χ′  s(G) ≤ 5∆2/4 if ∆ is even, and χ′  s(G) ≤ (5∆2 − 2∆ + 1)/4 if ∆ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  This conjecture is true for ∆ ≤ 3, see [1, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' For ∆ = 4, Huang, Santana and Yu [11] showed  that χ′  s(G) ≤ 21, one more than the conjectured upper bound 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Chung, Gy´arf´as, Trotter, and  Tuza (1990, [6]) confirmed the conjecture for 2K2-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Using probabilistic methods, Molloy  and Reed [14], Bruhn and Joos [3], Bonamy, Perrett, and Postle [2], and recently Hurley, Verclos,  and Kang [12] showed that χ′  s(G) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='772∆2 for sufficiently large ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Sparse graphs have also attracted a lot of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Interested readers may see the survey paper  [7] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' In this article, we are interested in k-degenerate graphs when k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' A graph  is k-degenerate if every subgraph has minimum degree at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Let G be a 2-degenerate graph with maximum degree ∆ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Chang and Narayanan [4] proved  that χ′  s(G) ≤ 10∆ − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Luo and Yu [13] improved it to χ′  s(G) ≤ 8∆ − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' For arbitrary values of  k, Wang [15] improved the result of Yu [16] and showed the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' If G is a k-degenerate graph with maximum degree ∆ ≥ k, then χ′  s(G) ≤ (4k −2)∆−  2k2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  This implies that χ′  s(G) ≤ 6∆ − 7 for 2-degenerate graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Choi, Kim, Kostochka, and  Raspaud [5] further improved it to χ′  s(G) ≤ 5∆ + 1 in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Many believe that the optimal bound  should be 4∆ + C for some constant C, but no progress has yet been made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  E-mail address: gyu@wm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  1  In this article, we show that for any 0 < ǫ ≤ 1/2, and ∆ ≥ 41/(2ǫ), χ′  s(G) ≤ 5∆ − ∆1/2−ǫ + 2 for  2-degenerate graph G with maximum degree ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Main result and its proof  A special vertex is a vertex with at most two neighbors of degree more than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Every 2-  degenerate graph contains special vertices, which are the new 2−-vertices after we remove all vertices  of degree at most two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Let G be a 2-degenerate graph and S be the set of special vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  For each u ∈ S, there exists a set Wu of vertices such that each w ∈ Wu shares some 2-neighbors  with u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' The capacity of special vertices of G is the maximum number of common 2-neighbors that  are shared by a vertex in S and other vertices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' A pendant edge is an edge incident with a leaf  (a vertex of degree one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Below is the main result of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' For any 0 < ǫ ≤ 1/2, let D be a positive integer when D ≥ 41/(2ǫ), for any 2-  degenerate graph G with maximum degree ∆, if a vertex u is adjacent to at least d(u) − D leaves  when it has d(u) > D, and  ∆ ≤ D + 2 when the capacity of special vertices is at least D1/2−ǫ, and  ∆ ≤ D + D1/2−ǫ when the capacity of special vertices is less than D1/2−ǫ,  then χ′  s(G) ≤ 5D − D1/2−ǫ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Let G be a counterexample with the fewest number of vertices of degree at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Since G is 2-degenerate, G must have a set S of special vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Let u ∈ S and Wu be the set  whose vertices share 2-neighbors with u such that the maximum number of common 2-neighbors  of u and vertices of Wu is the capacity of special vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Let Wu = {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' , ws} and u1, u2  be the two neighbors of u with degree more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' For each wi ∈ Wu, let Wi = {vi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' , vi,ti}  be the common 2-neighbors of wi and u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Then N(u) = {u1, u2} ∪ �s  i=1{vi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' , vi,ti}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' We assume  that t1 ≥ t2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' ≥ ts ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Then t1 is the capacity of special vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' It is not hard to see  that u has no neighbors of degree one and wi has degree at least two for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  For edge uv, let N2(uv) be the set of edges xy such that x or y is adjacent to u or v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' By definition,  uv should have a color different from the colors on edges in N2(uv) in a valid strong edge-coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' t1 < D1/2−ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' In this case, we have ∆ ≤ D + D1/2−ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Let G′ be the graph obtained from G−{uv1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' , uvs,ts} by adding up to D  1  2 −ǫ pendant neighbors  to each of {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' , ws} so that wi has degree at most D+D1/2−ǫ and wi has at least D  1  2 −ǫ pendant  neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Then the graph G′ has fewer vertices of degree at least 2 and can be colored with  5D − D1/2−ǫ + 2 colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' We modify the coloring of G′ to obtain a coloring of G according to the  following algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  (1) Keep the colors of edges that appear in both G and G′, but if wivi,j for some i, j in G has  the same color as uu1 or uu2, then we switch color of wivi,j with a color on other pendant  edges incident to wi in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  (2) For each i, if a color c appears on both a pendant edge incident to wi in G′ and an edge  incident to u1 or u2 (not including uu1 and uu2), then we switch the color of wivi,j for some  j with the color c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  (3) After (2), if a color c appears on pendant edges of two or more vertices in Wu in G′, then  we switch the color of wivi,j for some j with c for each such vertex wi ∈ Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  (4) After (2) and (3), we color the edges uv1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' , uvs,ts in reverse order with colors available  to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  2  Now we show that the above algorithm gives a valid strong edge-coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' To do that, we  only need to show that each of the edges in {uv1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' , uvs,ts} can be colored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Consider uvi,j for  1 ≤ i ≤ s and 1 ≤ j ≤ ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' It needs to get a color not on edges in N2(uvi,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Note that N2(uvi,j)  contains the edges incident to u1, u2, wi and the edges incident to the 2-neighbors of u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' So the  number of colored edges in N2(uvi,j), denoted as n2(uvi,j), is at most  d(u1) + d(u2) + d(wi) + d(u) − 3 + d(u) − 2 − ti −  i−1  �  p=1  tp − (j − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  We assume that n2(uvi,j) ≥ 5D − D1/2−ǫ + 2, for otherwise, uvi,j can be colored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Because of the  way the edges being colored, we have some repeated colors on edges incident to the 2-neighbors of  u, namely v1,1w1, v1,2w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' , vs,tsws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' The number of colors on N2(uvi,j) and edges incident to wi for  i ∈ [s] (with repetition) is n2(uvi,j)+D1/2−ǫ·(s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Thus n2(uvi,j)+D  1  2 −ǫ(s−1)−(5D−D  1  2−ǫ+2)  colors are repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' As each wi may allow only one edge (for example, ti = 1) whose color is the  same as other edges in N2(uvi,j), at least n2(uvi,j)+D  1  2 −ǫ(s−1)−(5D−D  1  2 −ǫ+2)  D  1  2 −ǫ  edges have the same  colors as others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Since t1 < D1/2−ǫ and s ≥ d(u)−2  t1  ≥  d(u)−2  D1/2−ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' the number of different colors in  N2(uvi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='j) is at most  n2(uvi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='j) − n2(uvi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='j) + D  1  2−ǫ(s − 1) − (5D − D  1  2 −ǫ + 2)  D  1  2−ǫ  ≤n2(uvi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='j)(1 −  1  D  1  2 −ǫ) − (s − 1) + 5D1/2+ǫ − 1 +  2  D  1  2−ǫ  ≤(d(u1) + d(u2) + 2d(u) − 5 + d(wi) − t1)(1 −  1  D  1  2 −ǫ) − d(u) − 2  D1/2−ǫ + 5D1/2+ǫ +  2  D  1  2−ǫ  ≤3∆(1 −  1  D  1  2−ǫ ) + d(u)(2 −  3  D  1  2−ǫ ) + 5D1/2+ǫ − 5 +  9  D  1  2 −ǫ  ≤3(D + D  1  2 −ǫ)(1 −  1  D  1  2 −ǫ) + D(2 −  3  D  1  2 −ǫ) + 5D1/2+ǫ − 5 +  9  D  1  2−ǫ  ≤5D + 3D  1  2 −ǫ − D1/2+ǫ − 8 +  9  D  1  2 −ǫ ≤ 5D − D1/2−ǫ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  The last inequality holds because D1/2+ǫ ≥ 4D1/2−ǫ and  9  D  1  2 −ǫ ≤ 9 when D ≥ 4  1  2ǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Therefore,  the edge uvi,j can be colored, which implies that we can color all the uncolored edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' t1 ≥ D1/2−ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' In this case, ∆ ≤ D + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Let G′ be the graph obtained from G by deleting the edge uv1,1 and adding up to two pendant  edges notated {w1v′, w1v′′} so that w1 has at least three pendant edges and continues to have  maximum degree at most D + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Note that for 1 ≤ i ≤ s and 1 ≤ j ≤ ts,  (1) |N2(uvi,j)| ≤ d(u1) + d(u2) + d(wi) + d(u) − 3+ d(u) − 2− ti ≤ 3∆ + 2D − 5− ti ≤ 5D + 1− ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  We observe that G′ is 2-degenerate, has fewer vertices of degree at least 2, and contains at least  d(v) − D vertices of degree one if d(v) > D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' So, G′ can be colored with 5D − D1/2−ǫ + 2 colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Keep the colors of the edges in both G and G′, we try to get a valid coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  3  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' the color of edge v1,1w1 is different from ones on {uu1, uu2, uv1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' , uvs,ts}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' By (1),  we have at most 5D + 1 − D1/2−ǫ edges in N2(uv1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Since there are 5D − D1/2−ǫ + 2 different  colors available, we can color uv1,1 with a color not on the edges in N2(uv1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' the color of edge v1,1w1 is the same as the color of edge uu1 (or edge uu2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' In this  case, we swap the color of v1,1w1 with the color of w1v′ or w1v′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' By (1) again, N2(uv1,1) has at  most 5D +1−D1/2−ǫ edges in N2(uv1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Since there are 5D −D1/2−ǫ +2 different colors available,  we can color uv1,1 with a color not on the edges in N2(uv1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' the color of edge v1,1w1 is the same as the color of edge uvi,j for some 2 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' In  this case, we uncolor the edges uvi,j, uv1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' , uv1,t1, and recolor them in the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' By (1), there  are at most 5D + 1 − ti − (t1 − 1) ≤ 5D − D1/2−ǫ + 1 colors on edges in N2(uvi,j), so uvi,j can be  colored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Similarly, for each j, there are at most 5D − D1/2−ǫ + 1 colors on edges in N2(uv1,j), so  uv1,j can be colored as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  In any case, G can be colored with 5D − D1/2−ǫ + 2 colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' So χ′  s(G) ≤ 5D − D1/2−ǫ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  □  Observe that our main result follows from Theorem 3 with ∆ = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Our proof does not work for  corresponding result of list version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  References  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Andersen, The strong chromatic index of a cubic graph is at most 10, Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content=' Bonamy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Perrett and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' B 155 (2022) 278–317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content=' Bruhn and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Joos, A stronger bound for the strong chromatic index, Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content='  [5] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content=' Kostochka and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' 67 (2018) 21–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Chung, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Gy´arf´as, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Tuza and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Trotter, The maximum number of edges in 2K2-free graphs of  bounded degree, Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' 81 (2) (1990) 129–135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content=' Deng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Yu and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Zhou, Recent progress on strong edge-coloring of graphs, Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Algorithms  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content=' Neˇsetˇril, Irregularities of partitions, (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' Halasz, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' S´os (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content=' )), [Problem] (1989) 162-163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content=' Fouquet and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content=' Jolivet, Strong edge-colorings of graphs and applications to multi-k-gons, Ars Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  16 (A) (1983) 141–150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
+page_content='  Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content=' de Joannis de Verclos and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}
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+page_content='  4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FOT4oBgHgl3EQfxTS9/content/2301.12924v1.pdf'}

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+Fast conformational clustering of extensive molecular dynamics simulation data∗
+Simon Hunkler,1 Kay Diederichs,1 Oleksandra Kukharenko,2, † and Christine Peter1, ‡
+1Department of Chemistry, University of Konstanz
+2Theory Department, Max Planck Institute for Polymer Research
+(Dated: January 12, 2023)
+We present an unsupervised data processing workflow that is specifically designed to obtain a
+fast conformational clustering of long molecular dynamics simulation trajectories. In this approach
+we combine two dimensionality reduction algorithms (cc analysis and encodermap) with a density-
+based spatial clustering algorithm (HDBSCAN). The proposed scheme benefits from the strengths
+of the three algorithms while avoiding most of the drawbacks of the individual methods.
+Here
+the cc analysis algorithm is for the first time applied to molecular simulation data. Encodermap
+complements cc analysis by providing an efficient way to process and assign large amounts of data
+to clusters. The main goal of the procedure is to maximize the number of assigned frames of a given
+trajectory, while keeping a clear conformational identity of the clusters that are found. In practice
+we achieve this by using an iterative clustering approach and a tunable root-mean-square-deviation-
+based criterion in the final cluster assignment. This allows to find clusters of different densities as
+well as different degrees of structural identity. With the help of four test systems we illustrate the
+capability and performance of this clustering workflow: wild-type and thermostable mutant of the
+Trp-cage protein (TC5b and TC10b), NTL9 and Protein B. Each of these systems poses individual
+challenges to the scheme, which in total give a nice overview of the advantages, as well as potential
+difficulties that can arise when using the proposed method.
+I.
+INTRODUCTION
+With the ever-growing power of computers over the
+last decades, researchers in the field of molecular dynam-
+ics (MD) have gotten access to increasingly long trajec-
+tories and thereby to increasingly large amounts of data.
+The introduction of supercomputers which are specifi-
+cally designed to generate MD trajectories (Anton [1]
+and Anton 2 [2]) is only the latest high point in this
+development. Furthermore, new sampling methods [3, 4]
+as well as distributed computing projects, such as Fold-
+ing@home [5], have contributed to a massive increase in
+generated simulation trajectories. With this increasing
+amount of data it is essential to have powerful analysis
+tools to process and understand underlying systems and
+processes.
+There is a rapid increase in application of unsupervised
+machine learning methods to analyze molecular simula-
+tion data.
+Two of the most used families of analysis
+techniques are clustering and dimensionality reduction
+(DR) algorithms.
+They help to find low-dimensional
+subspaces in which important aspects of the original
+data are preserved and to group the data based on a
+given similarity measure/metric and thereby gain a bet-
+ter overview and understanding.
+In practice, most of
+the times clustering and DR methods are used in com-
+bination.
+The DR algorithms can be roughly divided
+into:
+linear methods (the most known are principal
+component analysis (PCA) [6, 7] and time-lagged in-
+∗ Copyright 2023 Hunkler, Peter. This article is distributed under
+a Creative Commons Attribution (CC BY) License.
+† [email protected]
+‡ [email protected]
+dependent component analysis (TICA) [8, 9]), nonlin-
+ear methods (kernel and nonlinear PCA, multidimen-
+sional scaling (MDS) [10, 11] and MDS-based methods
+like sketch-map [12], isomap [13], diffusion maps [14, 15]
+or UMAP [16], etc.) and autoencoder-based approaches
+like (encodermap [17, 18], time-autoencoder [19], vari-
+ational autoencoders [20] and Gaussian mixture varia-
+tional autoencoders [21]).
+In terms of clustering algo-
+rithms, there are again a wide range of different methods:
+K-Means [22, 23], spectral-clustering [24], DBSCAN [25],
+density-peak clustering [26], CNN-clustering [27], root-
+mean-square deviation (RMSD) based clustering [28],
+neural-networks-based VAMPnets [29], etc. For a com-
+prehensive overview of unsupervised ML methods com-
+monly used to analyse MD simulation data we refer to
+Ref. 30.
+Even from this incomplete list of available methods
+it should become obvious that there are a lot of differ-
+ent clustering, as well as DR methods. All these meth-
+ods have their individual strengths and weaknesses. But
+there are still open challenges in the successful usage of
+the listed methods for processing simulation data with
+high spatial and temporal resolution. This is connected
+either to the proper choice of hyper-parameters (such as
+the number of dimensions for DR methods, the num-
+ber of expected states for some clustering algorithms,
+neural-networks architectures, different cut-offs, corre-
+lation times, etc.), expensive optimisation steps or the
+amount of data which could be processed simultaneously.
+In this work we present a data processing scheme which
+combines three different algorithms in one workflow to
+create a powerful clustering machinery. It tackles a num-
+ber of the mentioned challenges as it has a way to define
+an appropriate lower dimensionality of the data, does not
+require a priory information about the expected number
+arXiv:2301.04492v1  [physics.chem-ph]  11 Jan 2023
+
+2
+FIG. 1.
+Data processing routine presented in this article.
+of states and it is fast in processing extensive MD trajec-
+tories with both a very high dimensionality and a large
+number of observations. It is specifically designed to find
+conformational clusters in long molecular simulation data
+(Fig. 1).
+We use two different DR algorithms, namely an al-
+gorithm called “cc analysis” and the encodermap algo-
+rithm. The cc analysis method belongs to the family of
+the MDS-based techniques and was first introduced for
+the analysis of crystallographic data [31, 32]. Here it is
+used for the first time for projecting data of protein con-
+formations. The dimensionality of the cc analysis-space
+which is typically required is more than two (10 to 40
+for the systems shown in this work) and the amount of
+data, which can be efficiently projected simultaneously is
+limited by the available memory (about 50000 frames for
+a 72 GB workstation). To process much longer trajecto-
+ries and to obtain a two-dimensional representation we
+use the second DR algorithm – encodermap [33]. Its loss
+function however consist of two parts: the autoencoder
+loss and a MDS-like distance loss, which introduces an
+interpretability to the resulting 2D representation. More-
+over, once the encodermap network is trained, the en-
+coder function can be used to project data to the 2D
+map in an extremely efficient way. We use encodermap to
+project data into 2D and for a fast assignment of the addi-
+tional members to the clusters defined in the cc analysis
+space. Finally we use the HDBSCAN algorithm [34] to
+cluster the data in the cc analysis space and then visu-
+alize the resulting clusters in the 2D encodermap space.
+HDBSCAN is a combination of density and hierarchi-
+cal clustering, that can work efficiently with clusters of
+varying density, ignores sparse regions, and requires a
+minimum number of hyper parameters. We apply it in a
+non-classical iterative way with varying RMSD-cutoffs to
+extract the protein conformations of different similarities.
+The combination of these three algorithms allows us
+to leverage their different strengths, while avoiding the
+drawbacks of the individual methods. Subsequently we
+will show how the scheme performs on long MD trajecto-
+ries of wild-type and mutated Trp-cage with native and
+misfolded meta-stable states (208 µs and 3.2 µs long sim-
+ulations); really extensive simulations of NTL9 (1877 µs);
+and Protein B, where only a small percent of the simu-
+lation data (5%) is in the folded state (104 µs).
+II.
+METHODS
+A.
+cc analysis
+For dimensionality reduction, we use an cc analysis in-
+troduced in Ref. 31, 32. This algorithm was originally
+developed to analyse crystallographic data, where pres-
+ence of noise and missing observations pose a challenge to
+data processing in certain experimental situations. The
+method separates the inter-data-set influences of ran-
+dom error from those arising from systematic differences,
+and reveals the relations between high-dimensional in-
+put features by representing them as vectors in a low-
+dimensional space. Due to this property we expected it
+to be highly applicable to protein simulation data, where
+one seeks to ignore the differences arising from random
+fluctuations, and to separate the conformations based on
+systematic differences. In the course of the manuscript
+we show that this assumption proved to be correct.
+The cc analysis algorithm belongs to the family of
+MDS methods [10]. Its main distinction is that it min-
+imizes the sum of squared differences between Pearson
+correlation coefficients of pairs of high-dimensional de-
+scriptors and the scalar product of the low-dimensional
+vectors representing them (see Eq.
+(1)).
+The proce-
+dure places the vectors into a unit sphere within a low-
+dimensional space. Systematic differences between the
+high-dimensional features lead to differences in the an-
+gular directions of the vectors representing them, and
+purely random differences of data points lead to different
+vector lengths at the same angular direction. The algo-
+rithm minimizes, e.g. iteratively using L-BFGS [35], the
+
+Full trajectory
+Define high-D CVs
+Encodermap
+2D projection
+Expand clusters
+based on RMSD
+and 2D projection
+cc_analysis
+HDBSCAN
+Select random
+subset
+(up to 25000 frames)
+Remove
+For trajectories < 25000 frames
+clustered
+frames3
+expression
+Φ(x) =
+N−1
+�
+i=1
+N
+�
+j=i+1
+(rij − xi · xj)2
+(1)
+as a function of x, the column vector of the N low-
+dimensional vectors {xk}.
+Here rij is the correlation
+coefficient between descriptors i and j in the high-
+dimensional space and xi · xj denotes the dot product
+of the unit vectors xi and xj representing the data in the
+low-dimensional space; N is the number of observations,
+e.g. protein conformations. The output of cc analysis is
+the N low-dimensional vectors {xk}, and the eigenvalues
+of the xxT matrix.
+To understand why this is a sensible approach, one
+can think about obtaining the least squares solution of
+Eq. (1) algebraically by eigenanalysis of the matrix r =
+{rij}. In that case one would have to solve
+xxT = r
+where r is the matrix of the correlation coefficients rij.
+The n strongest eigenvalue/eigenvector pairs (eigenvec-
+tors corresponding to the largest eigenvalues) could then
+be used to reconstruct the N vectors xi, which are lo-
+cated in a n-dimensional unit sphere.
+The systematic
+differences between the input data are thereby shown by
+the different angular directions in this low-dimensional
+sphere.
+This approximation is meaningful because in
+general the Pearson correlation coefficient can be written
+as a dot product between two vectors (after subtraction
+of the mean and dividing by the standard deviation to
+scale the vectors to unit length) and is equal to the cosine
+of the angle between them. Hence, in an ideal scenario,
+�N
+i,j xi · xj can exactly reproduce the high-dimensional
+correlation coefficient matrix and Φ(x) in Eq. (1) would
+be equal to zero.
+The length of the vectors is less important than the
+angle between them. The latter has an inherent meaning:
+two high-dimensional feature vectors with a correlation
+coefficient of zero between them would be projected to
+unit vectors at 90◦ angles with respect to the origin, and
+two feature vectors with a correlation coefficient of one
+would have a corresponding angle of zero degrees.
+Despite the generality of the cc analysis approach, by
+now it was only applied to crystallographic data [36, 37])
+and protein sequence grouping [38]. Here we present a
+first application of cc analysis for protein simulation data
+analysis.
+B.
+Encodermap
+To accelerate the processing of large datasets, e.g. from
+extensive simulations, in addition to cc analysis, we make
+use of one more dimensionality reduction technique – en-
+codermap.
+It was developed by Lemke and Peter [33]
+and is used here for fast assignment of data points to
+clusters as will be explained in details in Sec. II D. The
+method combines the advantages of a neural network au-
+toencoder [17] with a MDS contribution, here the loss
+function from the sketch-map algorithm [12] (Fig.
+2).
+This combination is exceptionally efficient for projecting
+large simulation data to the two-dimensional representa-
+tions: the sketch-map loss function allows to concentrate
+only on relevant dissimilarities between conformations
+(ignoring thermal fluctuations and coping with the large
+dissimilarity values caused by the data’s high dimension-
+ality). Furthermore the autoencoder approach allows to
+avoid complex minimisation steps of the sketch-map pro-
+jection and to process huge amounts of data in a very
+short time.
+FIG. 2.
+Schematic description of encodermap. It has an
+architecture of the classic autoencoder consisting of two neu-
+ral networks (encoder and decoder) with the same number of
+layers and neurons in each layer connected through the bottle-
+neck layer with two neurons. In addition to autoencoder loss
+La(X, ˜
+X) encodermap loss has a term with the sketch-map
+loss function Ls(X, x), which improves the quality of two-
+dimensional projection obtained in the bottle-neck layer (see
+Eq. (2)).
+The encodermap loss function Lencodermap (Eq. (2)) is
+a weighted sum of the autoencoder loss Lauto (Eq. (3))
+and the sketch-map loss function Lsketch (Eq. (4)), which
+emphasizes mid-range distances by transforming all dis-
+tances via a sigmoid function (Eq. (5)).
+Lencodermap = kaLauto + ksLsketch + Reg,
+(2)
+Lauto = 1
+N
+N
+�
+i=1
+D(Xi, ˜Xi),
+(3)
+Lsketch = 1
+N
+N
+�
+i̸=j
+[SIGh(D(Xi, Xj)) − SIGl(D(xi, xj))]2,
+(4)
+where ka, ks are adjustable weights, Reg is a regular-
+ization used to prevent overfitting; N is a number of
+data points to be projected; D(·, ·) is a distance be-
+tween points, X is a high-dimensional input, x is a low-
+dimensional projection (the bottleneck layer); SIGh and
+SIGl are sigmoid functions of the form shown in Eq. (5).
+SIGσ,a,b(D) = 1 − (1 + (2
+a
+b − 1)(D
+σ )a)− b
+a ,
+(5)
+
+CVs
+CVs
+2D
+projection
+neural
+neural
+network
+network
+encoder
+decoder
+X4
+FIG. 3.
+Application of HDBSCAN on a toy data set with
+three clusters. i) Example for the computation of the MRD
+for two points (red and blue). The red and blue circles in-
+dicate the farthest distance to the 5 nearest neighbours for
+both points. One can see that the distance between the red
+and blue points (green line) is larger than both the radii of
+the blue and the red circle. Therefore in this case the green
+line distance is chosen as MRD. ii) The minimum spanning
+tree based on the MRDs. iii) The cluster hierarchy. iv) The
+condensed clustering.
+where a, b and σ are parameters defining which distances
+to preserve.
+C.
+Hierarchical Density-Based Spatial Clustering
+of Applications with Noise (HDBSCAN)
+The HDBSCAN [34, 39] can be approached from
+two different sides: it can be described as a hierarchi-
+cal implementation of a new formulation of the origi-
+nal DBSCAN [25] algorithm called DBSCAN* by J. G.
+B. Campello et al. [34] or it can be formulated as a ro-
+bust version of single-linkage clustering with a sophisti-
+cated method to obtain a flat clustering result, as done
+by McInnes et al. [39]. Here we describe it through the
+second approach.
+In the first step the algorithm introduces the so-called
+mutual reachability distance (MRD) (Eq.
+(6)), which
+transforms the space to make sparse points even sparser
+but does not significantly change the distance between
+already dense points.
+Dmreach−k(xi, xj) =
+max{corek(xi), corek(xj), D(xi, xj)},
+(6)
+where x are points being clustered, k is a constant which
+determines a number of nearest neighbouring points,
+corek(x) is a function, which finds the maximum distance
+between a point x and its k nearest neighbours; D(·, ·) is
+a distance between two points. The maximum of three
+distances is selected as the MRD (Fig. 3 i)).
+In the next step the minimum spanning tree based on
+the MRDs is build via Prim’s algorithm [40] (see Fig. 3
+ii)). This is done by starting with the lowest MRD in
+the data set and connecting the two points by a straight
+line. In the following steps always the next nearest data
+point to the existing tree, which is not yet connected, is
+added to the tree.
+Once the minimum spanning tree is generated the clus-
+ter hierarchy can be built. This is done by first, sorting
+the edges of the tree by distance. Then the algorithm
+iterates over the edges, always merging the clusters with
+the smallest MRD. The result of this procedure can be
+seen in Fig. 3 iii).
+In order to extract a flat clustering form this hierarchy,
+a final step is needed. In this step the cluster hierarchy
+is condensed down, by defining a minimum cluster size
+and checking at each splitting point if the new forming
+cluster has at least the same amount of members as the
+minimum cluster size.
+If that is the case, then a new
+cluster is accepted, if not then the data points splitting
+off are considered noise.
+The condensed tree of a toy
+system can be seen in Fig. 3 iv).
+D.
+Introduction of a new clustering workflow
+In this article we present a data processing routine
+which we found to be extremely efficient for large molec-
+ular dynamics simulation trajectories.
+It relies on the
+three algorithms introduced above. A schematic descrip-
+tion is given in Fig.
+1. In this workflow a given data
+set is clustered iteratively until either a specified amount
+of data points are assigned to clusters or a maximum
+number of iterations have been reached.
+Fig.
+1 illustrates the sequence of data processing
+steps along the clustering workflow.
+In the first step
+a high-dimensional collective variable (CV) is chosen.
+For all systems that are shown in this article all pair-
+wise distances between the Cα atoms were selected. Af-
+ter a CV has been chosen, for trajectories containing
+more than 25,000 frames, encodermap is trained on all
+data. Thereby we obtain a function which can be used
+to project data very efficiently to the newly generated
+2D space. In parallel, a random subset from the entire
+data set is generated.
+The reason to use such a sub-
+set is a limitation that comes with the cc analysis di-
+mensionality reduction. As mentioned in Sec. II A the
+cc analysis algorithm works with the correlation matrix.
+This means that the Pearson correlation coefficients of
+the selected CV (here the pairwise c-alpha distances) are
+calculated for all unique pairs of frames, and used as in-
+put to cc analysis. However the larger a data set is, the
+larger the correlation coefficient matrix will be, until it
+is no longer efficient to work with that matrix due to
+very long computation times as well as memory issues.
+Therefore a subset is created, by randomly selecting up
+to 25,000 data points from the entire data set. This sub-
+set is then used in the cc analysis dimensionality reduc-
+tion to project the high dimensional CVs (between 190
+and 1081 dimensions for the systems in this article) to a
+
+ii)
+dmreach
+0.25
+云
+reachal
+0.10
+Mutual
+0.05
+iii)
+iv)
+0
+100
+0.25
+6
+5
+80
+0.20
+5
+points
+4
+lue
+10
+60
+of
+val
+m
+Number
+0.10
+^ 15
+40
+2
+0.05
+20
+20
+1
+0.00
+25
+0
+05
+lower dimensional subspace (20 to 30 dimensions for the
+systems in this article). The choice of the appropriate
+amount of reduced dimensions is done by searching for
+a spectral gap among the cc analysis eigenvalues. Once
+the cc analysis space has been identified, a clustering is
+generated by applying the HDBSCAN algorithm to that
+lower dimensional data. A detailed description on how
+to choose the dimensionality for cc analysis and the pa-
+rameters for HDBSCAN is given in the supporting infor-
+mation (SI), Sec. S-I.
+We use two different DR algorithms in the workflow
+due to the following reasons. For once, the cc analysis
+algorithm is used to project the smaller subsets to a
+still comparably high-dimensional subspace, which holds
+more information compared to the 2D projection of en-
+codermap. This higher dimensional subspace is therefore
+very well suited to be the clustering space.
+Once the
+data subset is clustered in the cc analysis space, the 2D
+encodermap space is used to assign the points that were
+not a part of the subset to the found clusters. The 2D
+projection is very well suited to do a fast assignment of
+additional points from the data set, as well as to serve for
+visualization purposes. Additionally, encodermap is able
+to project huge data sets very time-efficiently.
+Hence,
+the generated 2D projection of a given data set can be
+used to avoid the main disadvantage of the cc analysis
+algorithm in the way we use the algorithm here, which
+is having to use subsets of the data due to memory is-
+sues. In order to circumvent this disadvantage, we build
+a convex hull in the 2D space for each cluster that was
+found in the cc analysis space. If an unassigned point lies
+inside a convex hull, the RMSD to the central conforma-
+tion of that cluster is computed. In case the RMSD is
+inside a given cutoff, the data point is considered to be
+part of that cluster, else it is not assigned to the clus-
+ter. This RMSD cutoff is chosen by taking the weighted
+mean of all average internal cluster RMSDs 1 of the first
+clustering iteration. We found that this procedure gen-
+erates structurally quite well defined clusters with a low
+internal cluster RMSD since the RMSD criterion is based
+on well defined conformational states that emerged from
+cc analysis combined with HDBSCAN. However there is
+also the possibility to identify more fuzzy clusters that
+only share a general structural motif by using a larger
+RMSD cutoff for the assignment. An example of the iden-
+tification of such fuzzy clusters is described in Sec. III B.
+By introducing a RMSD criterion in the last step, we
+force the clustering to only include structurally very sim-
+ilar conformations in the respective clusters. There are of
+course various other clustering algorithms, which group
+MD data sets based on their RMSD values, e.g. an imple-
+mentation [28] in the GROMACS software package [41].
+Such RMSD-based clustering algorithms have been used
+in the MD community for at least 20 years now and they
+1 By the average internal cluster RMSD we mean the average
+RMSD of all conformations to the cluster centroid.
+are a very obvious choice for conformational clusterings
+of MD trajectories. They directly compare the positions
+of specified atoms in various conformations of a molecule
+and then group the individual conformations along the
+trajectory using a cutoff value. However these methods
+generally rely on the full RMSD matrix of a given data
+set. For larger trajectories it becomes almost infeasible
+to compute these matrices due to extremely long com-
+putation times as well as memory issues that arise when
+working with such large matrices. In our workflow we
+can circumvent these issues by only having to compute
+the RMSD between the coordinates of Cα atoms of the
+central conformations of each cluster and the data points
+that lie inside the convex hull of the respective clusters
+in the 2D space.
+In case a given system has less then about 50,000
+frames, it is in principle also possible to omit the en-
+codermap part, since the cc analysis algorithm is able to
+handle the entire data set. If this approach is chosen,
+the user can either entirely skip the RMSD criterion, or
+the members of clusters that are found in the cc analysis
+space can still be accepted/rejected based on a RMSD
+cutoff. An advantage of using both the cc analysis algo-
+rithm and the encodermap algorithm together is the pos-
+sibility to check the dimensionality reduction steps on the
+fly. Since the clustering is done in one DR space, but vi-
+sualized in the other, narrow and well defined clusters in
+the 2D space indicate that the 2D map separates the dif-
+ferent conformational clusters nicely and that therefore
+the chosen encodermap parameters were well selected.
+Our clustering scheme is not very dependent on the
+quality of encodermap projection, as it is used only to as-
+sign additional structures to already identified clusters.
+Since the clustering itself is done in the higher dimen-
+sional cc analysis space and the final cluster assignment
+uses a RMSD cutoff.
+The only requirement that the
+scheme poses towards the 2D map is that similar con-
+formations are located close to each other in the map.
+This is achieved by the MDS-like distance loss part of
+the overall loss function of encodermap.
+Remaining points which were not assigned to any clus-
+ter after one clustering iteration are then used as a new
+pool of data, from which the new random subset is build.
+This whole cycle is repeated until a certain amount of
+data points are assigned to clusters or until a certain
+number of clustering iterations are performed. To decide
+on a stopping point for the iterative procedure we rely
+on two possible convergence criteria: either a percentage
+of assigned conformations or average cluster sizes found
+at an iteration. If we observe a plateau in the percent-
+age of unassigned data points during several successive
+iterations the clustering procedure is stopped.
+Due to
+the design of our workflow, the average cluster size of
+newly added clusters will decrease with each iteration.
+Therefore, the average size of newly added clusters or
+the convergence of this property during successive itera-
+tions can also be used as a stopping criterion. Examples
+are shown in SI, Sec. S-II, Fig. S2.
+
+6
+Trp-cage RE (TC5b) Trp-cage Anton (TC10b)
+NTL9
+Protein B
+Trajectory length in µs
+3.2
+208
+1877
+104
+Number of frames
+1,577,520
+1,044,000
+9,389,654
+520,250
+Input CVs dimensionality
+190
+190
+703
+1081
+Number of cc analysis dimensions
+20
+20
+20
+30
+Average iteration time
+on our local workstation
+(see SI, Sec. S-V) [min]
+15
+18
+55
+12
+Average iteration time
+over all used
+CPU threads [min]
+24 x 15
+= 360
+24 x 18
+= 432
+24 x 55
+= 1320
+24 x 12
+= 288
+Frames assigned to clusters
+after 10 iterations
+60%
+33.1%
+80.9%
+20%
+Total CPU time
+over all iterations [min]
+3600
+4320
+13200
+2880
+TABLE I: Proteins analysed in this study and performance overview of the clustering scheme.
+III.
+RESULTS AND DISCUSSION
+A.
+Description of the proteins’ trajectories used
+for the analysis
+In order to illustrate the capability and performance of
+the proposed scheme, we chose four test systems: 40 tem-
+perature replica exchange (RE) trajectories of the Trp-
+cage protein (TC5b) analysed in the original encodermap
+paper [33]; the other three systems are long trajectories of
+Trp-cage (TC10b), NTL9 and Protein B simulated by the
+Shaw group on the Anton supercomputer [42] and gen-
+erously provided by them. The four systems are listed in
+Table I. For all the systems we chose distances between
+Cα atoms as the input collective variables.
+The first protein we analyse in this work is the Trp-
+cage system (TC5b) (Trp-cage RE). It is a comparatively
+small protein (20 residues) which has a very stable native
+state when simulated at room temperature. The combi-
+nation of 40 temperature replica exchange trajectories
+(temperature range from 300 to 570 K, 3.2 µs of simu-
+lation time, 1,577,520 frames) give a very diverse mix-
+ture of structures including trajectories where the sys-
+tem is very stable and barely moves away from the na-
+tive state, as well as highly disordered trajectories where
+high-energy conformations are visited. This combination
+of conformations makes the data set extremely diverse
+and complicated for the analysis due to the high num-
+ber of expected clusters with extremely varying size and
+density.
+Secondly we consider the K8A mutant of the ther-
+mostable Trp-cage variant TC10b (Trp-cage Anton) sim-
+ulated by Lindorff-Larsen et al. [42] (208 µs; 1,044,000
+frames). This simulation was run at 290 K and produced
+a much more disordered trajectory compared to the low
+temperature replica simulations of the TC5b system. De-
+spite the fact that TC5b and the K8A mutant of TC10b
+have slightly different amino acid sequences, we use the
+same trained encodermap to project both systems in the
+same 2D map (see Fig. 4 and Fig. 5), since both sys-
+tems have the same number of residues and therefore the
+same dimensionality of CVs. This offers the opportunity
+to demonstrate that different systems can be compared
+to each other very nicely when projected to the same 2D
+space.
+Next we probed our clustering scheme with extremely
+long (1877 µs 2; 9,389,654 frames) simulations [42] of the
+larger (39 amino acids) N-terminal fragment of ribosomal
+protein L9 (NTL9) which has an incredibly stable native
+state. Besides the possibility to show how the algorithm
+deals with this extremely large data set, the system has
+also been studied by several other researchers [29, 44].
+This allows us to compare our results to their findings.
+Schwantes and Pande [44] reported on very low pop-
+ulated states which involve register-shifts between the
+residues that are involved in the formation of the beta
+sheet structures of NTL9. This opens the opportunity
+to show whether our clustering workflow is able to iden-
+tify both very large states, as well as extremely lowly
+populated states in the same data set.
+Lastly we chose to analyse the protein B simulations
+(104 µs; 520,250 frames) [42]. Compared to the afore-
+2 We used the trajectories 0, 2 and 3 according to the nomenclature
+of Ref. 42. We have not used trajectory 1 because the topology
+file for this specific trajectory differs slightly form the other three
+in terms of the order and the numbering of the atoms. This issue
+has also been reported by other researchers [43].
+
+中7
+FIG. 4.
+Trp-Cage TC5b (40 temperature RE trajectories): Exemplary conformations of the most populated clusters found
+in each of the areas indicated by coloured circles and their populations in percentages. The cluster representatives show the
+average secondary structure over the entire cluster. The clusters are coloured randomly, the colours repeat. Therefore clusters
+that have the same colour but are separated in the 2D space contain different conformations. The depicted clusters hold 36.5%
+of all conformations. Most of the remaining 24% of conformations that have been assigned to clusters are slight variations of
+the native structure and are not shown here due to visibility reasons. The cluster that is referred to by an arrow is one of the
+fuzzy clusters that were generated by increasing the RMSD cutoff. Top right: a histogram of the 2D encodermap space.
+mentioned proteins protein B does not have a single very
+stable state, instead three helices can move quite easily
+against each other. This leads to a broad conformational
+space, where the energy barriers between the individual
+states are very small. Therefore the individual confor-
+mational states are not as easily separable and rather
+fade/transition into each other. Taking into account the
+long simulation time this system is very hard to cluster
+conformationally.
+To demonstrate how our clustering scheme works we
+chose to apply it to these four systems that pose very
+diverse challenges (e.g. an extremely large data set, both
+highly and very lowly populated states in the same data,
+differences in the amount of folded/unfolded conforma-
+tions along the trajectories). For each of the systems we
+initially conducted the same amount of clustering itera-
+tions (10) and then evaluated the resulting clustering and
+decided whether for a given system additional iterations
+were needed.
+B.
+Trp-cage
+a.
+TC5b.
+For the RE simulations of the Trp-cage
+the clustering scheme was run over 10 iterations and as-
+signed 60.5% of all conformations to clusters.
+Fig.
+4
+shows an encodermap projection of all 40 replicas with
+some of the most populated clusters found after 10 it-
+erations and representative conformations of these clus-
+ters.
+Similar conformations are grouped together and
+rare structures are spread out across the map. For ex-
+ample, the native conformation of Trp-cage RE (33.4%
+
+0.1%
+0.1%
+0.2%
+0.2%
+0.1%
+0.2%
+0.3%
+0.1%
+0.1%
+<0.1%
+0.1%
+<0.1%
+<0.1%
+native; 33.4%
+1.5%
+<0.1%8
+FIG. 5.
+The most populated clusters and respective conformations of Trp-Cage TC10b [42] projected to the same 2D
+encodermap space as TC5b (Fig. 4).bTop right: a histogram of the 2D projection.
+of all conformations) is shown in the bottom right of the
+2D map in Fig.
+4. On the bottom left conformations
+with one turn near the middle of the backbone are lo-
+cated.
+The two parts of the backbone chain of these
+conformations lie right next to each other and partially
+form beta-sheet structures.
+Using a larger cutoff distance in the RMSD-based as-
+signment of structures to the clusters (the other clusters
+were generated by applying a 1.8 ˚A RMSD cutoff to the
+central conformation) we obtained larger and quite dif-
+fuse clusters of extended conformations (one of these clus-
+ters is shown in the left part of the projection in Fig. 4
+where it is referred to by an arrow). An appropriate size
+of this RMSD cutoff was defined for each fuzzy cluster
+individually by computing the mean value of the largest
+20% of the RMSD values between the centroid and cluster
+members of the cluster identified in the current iteration
+(it is equal to 5.5 ˚A for the cluster shown here). Before we
+identify fuzzy clusters, we first continuously assign struc-
+tures based on a fixed RMSD cutoff (1.8 ˚A for TC5b)
+until one of the stopping points defined in Sec.
+II D is
+reached (average cluster size for TC5b). Once this stop-
+ping point is reached, the RMSD cutoff is adjusted in
+the way explained above and fuzzy clusters are obtained.
+Thereby one ensures that all conformations that can be
+assigned to well-defined clusters are removed from con-
+sideration before identifying fuzzy clusters. The usage of
+such a varying cutoff can be very helpful in order to iden-
+tify diffuse clusters, where the members share a certain
+structural motif but do not converge to a very defined
+conformation, just like the cluster shown here.
+From the clustering results shown in Fig.
+4 one can
+see that the proposed clustering workflow manages to ef-
+ficiently identify structurally very well defined clusters
+for the TC5b system. Over 10 clustering iterations it as-
+signed 60.5% of all conformations to 260 clusters. Besides
+the highly populated native state (33.4%), the algorithm
+also finds very ”rare” states, which contain only a very
+small amount of conformations (≤0.1%) but show never-
+theless a very defined structural identity.
+b.
+TC10b.
+Fig. 5 shows the same analysis applied
+to the trajectory of the K8A mutant of TC10b Trp-cage.
+
+8.2%
+0.5%
+0.1%
+0.5%
+0.1%
+<0.1%
+0.7%
+0.02%
+0.1%
+0.2%
+<0.1%
+<0.1%
+1.7%
+0.7%
+native; 12%9
+We used the encodermap which we trained on TC5b to
+project the trajectories to the same 2D space. The iden-
+tification of clusters however is of course entirely inde-
+pendent and unique for both cases, since the clustering
+is done in the higher dimensional cc analysis space.
+Notably, the backbone conformation of the native state
+of this mutant is extremely similar to the one in the TC5b
+system. However this biggest cluster only contains 12%
+of all conformations along the trajectory compared to
+the 33.4% in the case of the TC5b system. If all clus-
+ters whose central conformation are within a 2 ˚A RMSD
+to the native conformation are combined, we get native
+conformation percentage of 16.9%. This is in excellent
+agreement with the native cluster sizes reported by Deng
+et al. [45], Ghorbani et al. [46] who analysed the same
+Trp-cage trajectories provided by Lindorff-Larsen et al.
+[42]. Furthermore our 33.4% of assigned conformations
+coincide very well with the reporting of Sidky et al. [47].
+They found a total of 31% of conformations distributed
+over eight metastable macrostates and the remaining 69%
+as one big ”molten globule” state.
+The TC10b trajectory is more disordered, this can be
+seen by the more homogeneous projection in 2D space
+(upper right plot in Fig.
+5) and the RMSD values to
+the native conformation in SI, Sec. S-III, Fig. S3. This is
+also the reason why the clustering scheme assigned only
+33.4% of all conformations to clusters after 10 iterations.
+If more frames should be assigned to clusters, more clus-
+tering iterations can be performed, the RMSD cutoff can
+be increased or both can be done simultaneously (for the
+Protein B system we show the results of this approach
+later in the article).
+However the clusters in the very center of the map
+(dark blue circle) are much more compact and collapsed
+compared to the clusters that were found in the similar
+area of Trp-cage RE’s 2D projection. Also some of the
+clusters that were found in the very bottom of the left
+hand side of the map in case of the replica trajectories
+(light blue circle) were not found at all in the TC10b
+trajectory. The very large and diffuse cluster on the left
+side of the map is present in both systems as well.
+c.
+Clustering directly in 2D space of TC5b.
+The
+clustering discussed above was done in a 20 dimensional
+space after applying the cc analysis algorithm and only
+displayed at a 2D projection done with encodermap. In
+order to demonstrate the advantages of our approach we
+also directly clustered the 2D encodermap space using the
+HDBSCAN. The encodermap space that we used for this
+clustering is the same space that we used to visualize the
+cc analysis clustering in Fig. 4 and Fig. 5. The results
+of this clustering and a few chosen clusters can be seen
+in Fig. 6. In total this clustering assigned 13.5% of all
+conformations to 362 clusters. The biggest cluster that
+was found is the native cluster, however it only contains
+0.8% of all conformations compared to the 33.4% that
+were found by clustering the cc analysis space. The clus-
+tering in the 2D space identifies some structurally very
+well defined clusters, such as the clusters 0, 1 and 3, but
+FIG. 6.
+2D encodermap space of TC5b clustered with HDB-
+SCAN. Representations of chosen clusters that have the same
+location in the 2D map as clusters found with the clustering
+scheme in Fig. 4 are shown.
+also a lot of very diffuse and inhomogeneous clusters. To
+quantify this inhomogeneity we computed the average of
+the internal cluster RMSDs. For the TC5b system our
+clustering workflow resulted in an average cluster RMSD
+of 1.34 ˚A and a weighted average RMSD of 1.03 ˚A, where
+weights are defined as the fraction of each cluster to all
+clustered data. The average RMSD for the direct cluster-
+ing in the 2D space is 2.25 ˚A and the weighted average
+RMSD is 2.73 ˚A. This clearly shows that the internal
+cluster RMSD variance is on average much larger when
+clustering directly in the 2D space. Furthermore the clus-
+tering in the 2D space itself naturally highly depends on
+the quality of the 2D map.
+Other than the much clearer conformational identity
+of the individual clusters (shown via internal cluster
+RMSDs), our clustering scheme also manages to assign
+60.5% of all conformations to different clusters.
+Com-
+pared to that the clustering in the 2D projection only
+assigned 9-14% of all conformations depending on the
+choice of clustering parameters.
+d.
+Comparison to other clustering approaches.
+For a
+further assessment of our clustering scheme we have also
+applied a frequently used clustering routine to the TC5b
+data. In Si, Sec. S-IV and Figs. S4 and S5 the results
+of applying the k-means algorithm to an 11 dimensional
+PCA projection of the same CVs (pairwise Cα distances
+of TC5b) are shown.
+In summary, the scheme identified both structurally
+very defined as well as quite diffuse clusters in considered
+systems. Even though the combination of the 40 RE tra-
+jectories produces a very diverse data set, the clustering
+scheme manages to assign a large amount of the confor-
+mations to clusters (60%). Our clustering results for the
+TC10b are in a very good agreement with the findings
+of other researchers [45–47]. Furthermore the compar-
+ison to a clustering in the 2D space clearly shows the
+superiority of using more dimensions obtained with the
+cc analysis algorithm in HDBSCAN over just relying on
+a low-dimensional representation alone.
+
+Cluster 4
+Cluster 3
+Cluster 2
+Cluster 5
+Cluster 1
+Cluster O
+Cluster 610
+FIG. 7.
+The 2D encodermap projection of NTL9. The projection can be approximately divided into three parts: the upper part
+with the most dense areas (where the native-like states are located); the lower left and right planes divided by an unpopulated
+vertical gap. The left side includes various conformations with a singular beta sheet formed mostly between the beginning
+and the end of the protein. In contrast on the right side lie mostly extended conformations with multiple helices along the
+backbone. Exemplary conformations of some of the most populated clusters found in each of the marked areas in the map and
+their populations are shown. All clusters in the yellow circle are extremely similar to the native cluster and can be summed up
+to a total of 76% of all conformations. The structures that are shown here make up 78.4% of all conformations. Top right:
+Histogram of the 2D encodermap space.
+C.
+NTL9
+Next we examined very long (1877 µs) simulations of
+NTL9 [42]. With 9.38 million frames to cluster, this sys-
+tem is an ideal candidate to demonstrate how the pro-
+posed algorithm copes with large amounts of data. Af-
+ter 10 iterations 81% of all conformations were assigned
+to clusters.
+Fig.
+7 shows a 2D projection made with
+encodermap, where points are colored according to the
+clusters found after ten iterations of the scheme and a
+histogram of the 2D space in the upper right corner. In
+total we found 157 clusters and assigned them 81% of all
+conformations over 10 clustering iterations.
+A comparison of the timeseries of the RMSD values to
+the folded state to the respective data of the Trp-cage
+Anton simulations (SI, Sec. S-III, Fig. S3) reveals that
+the two systems exhibit very different dynamics. While in
+the Trp-cage case the RMSDs show the disordered nature
+of the system, in the case of the NTL9 trajectories the
+RMSDs are predominantly quite low and only spike up to
+larger values for rather short time periods. This suggests
+that the NTL9 system resides in a native-like state for
+the majority of the simulated time. This is confirmed
+during the very first iteration of the clustering scheme.
+There we found two clusters which make up for 75.8% of
+all conformations.
+This example also nicely illustrates how the iterative
+clustering approach can be efficient in identifying clus-
+ters of very different size and density (highly populated
+native states and low populated clusters). After finding
+
+1.3%
+0.2%
+0.2%
+0.1%
+1.4%
+0.4%
+>0.01%
+native; 74.5%
+0.3%
+>0.01%
+>0.01%
+cumulative ~0.1%!
+0.03%
+0.03%
+cumulative ~0.01%
+>0.01%11
+FIG. 8.
+Register-shifted states found in the NTL9 trajecto-
+ries 0, 2 and 3. The residues which form the beta sheets in
+the native state are colored based on their residue ID.
+and removing the first two clusters (75.8% of the data)
+the clustering algorithm becomes much more sensitive
+towards the less dense areas in the CV-space in the fol-
+lowing clustering iterations.
+We compared our clustering results with other publi-
+cations analyzing the NTL9 trajectories from Ref. [42].
+Mardt et al. [29] applied the VAMPnets to trajectory 0
+and found in total 89.1% of folded, native like confor-
+mations. If we take the clusters we found by analysing
+the trajectories 0, 2 and 3 and evaluate the conforma-
+tions stemming from trajectory 0 (trajectory 0 resides
+in the native-like state for a larger fraction of the simu-
+lated time; see RMSD plots in SI, Sec. S-III, Fig. S3, the
+amount of folded, native-like conformations we find is in
+very good agreement with [29]. Furthermore Schwantes
+and Pande [44] reported the finding of three “register-
+shifted” states, which are very low populated and there-
+fore very hard to find. “Register-shifted” refers to the
+identity of the specific residues involved in forming the
+beta sheet structure in the native-like states (residues 1-
+6, 16-21 and 35-39). With our method we identified six
+different register-shifted states in the NTL9 trajectories
+0, 2 and 3 (see Fig. 8).
+The states 0, 1 and 2 are the ones which were also
+found in [44]. To our knowledge states 3, 4 and 5 have
+not been reported yet.
+In state 0 the central of the
+three beta-sheet strands is shifted downwards, whereas
+in state 2 the rightmost strand is shifted downwards.
+In state 1 both the middle and the rightmost strands
+are dislocated compared to the native state. State 3 is
+similar to state 1 in the fact that both the middle and
+the rightmost strands are shifted, however in state 3 the
+rightmost strand is shifted upwards and not downwards
+like in state 1. Among these six states state 4 is unique
+since there the rightmost strand is turned by 180 degrees.
+Finally state 5 differ from other states in having an extra
+helix along the chain between the leftmost and the mid-
+dle strand. Because of this additional helix the leftmost
+strand is extremely shifted compared to the native state.
+The identification of these register-shifted states high-
+lights one asset of the proposed workflow. It is able to
+find both very large states (native, 74.5%) as well as very
+low populated clusters (<0.001%) in the same data set.
+D.
+Protein-B
+The last system we analysed is Protein B. This sys-
+tem does not have a very stable native state, instead
+the three helices can move against each other relatively
+freely. This can be seen in the timeseries of the RMSD to
+the closest experimental homologue (1PRB) shown in SI,
+Sec. S-III, Fig. S3. There are no extended periods where
+the values are stable over some time, meaning there are
+no large free-energy barriers separating the various acces-
+sible conformations and thus the system constantly tran-
+sitions into different conformations. This has also been
+found in [42], where authors stated that they were un-
+able to identify a free-energy barrier between folded and
+unfolded states for Protein B (tested over many different
+reaction coordinates).
+Such a highly dynamic system is very challenging for a
+conformational clustering. Here we want to show where
+our algorithm has its limitations and what can be done
+to get a satisfactory clustering result. Fig.
+9 gives an
+overview of some of the clusters found after ten iterations
+of the scheme. These clusters include only 20% of the
+Protein B trajectory and thus 80% of all conformations
+are still unclustered.
+In order to have more data assigned to clusters two pa-
+rameters can be adjusted. First, the RMSD cutoff value
+can be increased and thereby more conformations can be
+assigned to the found clusters. In this specific case this
+adjustment is justified, since due to the low free-energy
+barriers between different states, the individual clusters
+are not as sharply defined in terms of their conforma-
+tions. In the 10 clustering iterations which are shown in
+Fig. 9 we used a RMSD cutoff of 3.0 ˚A. In a second run
+we increased it to 3.5 ˚A. This resulted in an assignment
+of 31% of all conformations to generally more loosely de-
+fined clusters.
+A second approach is to increase the amount of clus-
+tering iterations. For the first ten clustering iterations of
+previously analysed systems, we tuned the clustering pa-
+rameters manually. This includes the choice of the num-
+ber of cc analysis dimensions, as well as the min samples
+and min cluster size parameters of HDBSCAN. However
+such a manual adjustment of the parameters is of course
+not feasible for automating the script in order to perform
+many more clustering iterations.
+Since the amount of
+cc analysis dimensions needs to be very rarely changed
+once a suitable amount has been identified in the first
+clustering iteration, the automation of the script only re-
+lies on the choice of the HDBSCAN parameters. Once the
+amount of clusters found in a single iteration falls below a
+certain threshold (10 clusters in this case), the numerical
+
+Native State
+State 4: ~0.001%
+5
+State 1: r0.1%
+State 3: 0.01%
+State 0: ~0.1%
+State 2; ~0.01%
+State 5; ~0.1%12
+FIG. 9.
+Protein B: Exemplary conformations of some of the most populated clusters found for the Protein B system after 10
+clustering iterations and their populations; Top right: Histogram of the 2D encodermap space.
+values of the min samples and min cluster size parame-
+ters of HDBSCAN are slightly decreased. This leads to
+the detection of smaller clusters that have not been iden-
+tified before. By applying this automation approach after
+the first 10 iterations to Protein B and using a RMSD
+cutoff of 3.5 ˚A, we could assign 44.3% of all conforma-
+tions to clusters over 100 iterations, which took roughly
+15 hours on our workstation.
+IV.
+DISCUSSION
+The Trp-cage system (TC5b) is a relatively small pro-
+tein which has a quite stable native conformation. The
+combination of 40 temperature RE trajectories however
+gives a very diverse data set including (under standard
+conditions) very improbable high-energy conformations.
+Over ten iterations the algorithm managed to assign
+60.5% of all conformations to clusters, which took on av-
+erage 360 min per iteration over all CPU threads (15 min
+per iteration on a standard office machine with 24 CPU
+threads). Table I shows the clustering performance for
+the four systems discussed here. By switching the gen-
+erally static RMSD cutoff to a varying cutoff we could
+show that the algorithm can both generate conforma-
+tionally very defined clusters as well as quite diffuse.
+The conformations assigned to such loose clusters share
+a general structural motif. The ability to identify both
+of these cluster types is one of the advantages of the
+proposed algorithm. Furthermore we demonstrate that
+the clustering workflow is able to directly compare dif-
+ferent systems (even if they slightly differ structurally),
+by projecting them to the same 2D map using the en-
+codermap algorithm.
+This enables a direct and visual
+comparison of the sampled phase-spaces of different tra-
+jectories and their respective identified states. By com-
+paring the clustering result where the clustering is done
+in a 20-dimensional cc analysis space and then projected
+to a two-dimensional space to a clustering where the
+clusters are purely found in a 2D encodermap space, we
+prove an advantage using more dimensions and combine
+cc analysis with encodermap. The scheme created clus-
+
+0.1%
+0.3%
+0.1%
+>0.1%
+>0.1%
+0.4%
+0.3%
+3.2%
+1.5%
+most populated; 5.2%13
+ters with a much clearer structural identity (lower RMSD
+variance), while being much less dependent on the quality
+of the 2D map.
+We analysed long (9.38 million frames) trajectories of
+NTL9 to show how the proposed scheme copes with very
+large amounts of data. On average the algorithm needed
+1320 min of computation time over all CPU threads per
+iteration (55 min per iteration on our office machine).
+Since this system also has one hugely populated native-
+state, it is also a nice example to demonstrate an ad-
+vantage of the iterative clustering.
+After the clusters
+with the native states are removed from consideration,
+the algorithm becomes much more sensitive towards less
+populated areas in the following iterations.
+Applying
+this approach we could identify three very low popu-
+lated register-shifted states, which have been reported
+before [44], and three not yet seen register-shifted states.
+Lastly we looked at is Protein B, which is a highly
+dynamic system.
+To analyse this 1.04 million frames
+trajectory it took on average 288 min of computation
+time per iteration (12 min per iteration on our office
+machine).
+This system has no large free-energy barri-
+ers separating the various conformations, which makes
+it very difficult to cluster. This was confirmed by the
+fact that after ten clustering iterations only 20% of all
+conformations could be assigned to clusters. However by
+increasing the RMSD cutoff from 3.0 ˚A to 3.5 ˚A we could
+already increase the amount of assigned conformations to
+31%, which of course resulted in slightly less structurally
+defined clusters. It is also possible to automate the clus-
+tering and run until a certain amount of conformations
+are assigned to clusters or until a given number of itera-
+tions is reached. In this specific case we ran the scheme
+for 100 automated iterations (≈15 hours), during which
+44.3% of the conformations were assigned to clusters.
+For all considered systems the proposed workflow was
+able to identify defined clusters at the cost of leaving
+some amount of the trajectories unassigned. As we have
+shown here, the rest of the structures does not belong to
+any specific clusters and can be considered as unfolded
+or transition states. We intentionally do not propose any
+additional steps to assign or classify those conformations
+as it is highly dependant on the intended application of
+the data. For example in case the data is used to build
+subsequent kinetic models the rest of the points can be
+assigned to the nearest (e.g. in simulation time) cluster
+using methods such as PCCA+ analysis [48], or defined
+as a metastable transition state as in Ref. 47. It can also
+be defined as noise and used as discussed in Ref. 49.
+All performance data is shown in Table I and was ob-
+tained by running the clustering scheme script on the
+office workstation described in SI, Sec. S-V. The pro-
+posed workflow is, however, highly parallelizable, since
+the computationally most expensive step is the assign-
+ment of additional data points to the initially identified
+clusters in the small subset based on the convex hull and
+the RMSD criterion.
+If a large amount of CPU cores
+are available, the 2D encodermap projection array can
+be split by the amount of cores and the assignment can
+thereby be run in parallel which leads to a significant
+speed up.
+The convex hull around the clusters identified in the
+small subset is used to reduce the amount of RMSD com-
+putations that have to be performed when assigning ad-
+ditional conformations in each clustering iteration. This
+however might in principle lead to the exclusion of data
+points that might otherwise have been assigned to some
+of the clusters. In order to get an idea of the magnitude
+of this “loss” of potential cluster members, we computed
+the RMSD of all data which was labeled as noise (623,000
+conformations; 39.5%) to each of the cluster centers of
+TC5b (260 clusters). This computationally very expen-
+sive task took an additional 5 hours on our working ma-
+chine. We found that 42,000 conformations (2.7%) were
+not assigned to the identified clusters due to the con-
+vex hull criterion. When keeping in mind that the entire
+10 iteration clustering process took 2.5 hours, the ”loss”
+of 2.7% of unclustered data can be considered a worthy
+trade-off.
+Another point to consider is that due to the convex hull
+criterion clusters can be split. If data points that would
+be assigned to a certain cluster by reason of the RMSD
+criterion lie outside of the convex hull, they could be iden-
+tified as another cluster in one of the following clustering
+iterations. In such cases it can make sense to merge these
+clusters in hindsight, due to their very similar structural
+identity. In order to showcase such a merge, we again
+analysed TC5b. We computed the RMSDs between all
+of the 260 central cluster conformations and merged all
+clusters that had a RMSD of ≤ 1 ˚A. This resulted in a re-
+duction to 201 clusters with only very marginal influence
+on the average internal cluster RMSDs.
+The code for the encodermap algorithm is avail-
+able on the following github page https://github.
+com/AG-Peter/encodermap.
+The
+cc analysis
+code
+can
+be
+found
+under
+https://strucbio.biologie.
+uni-konstanz.de/xdswiki/index.php/Cc_analysis.
+V.
+CONCLUSION
+We
+developed
+a
+clustering
+scheme
+which
+com-
+bines two different dimensionality reduction algorithms
+(cc analysis and encodermap) and the HDBSCAN in an
+iterative approach to perform fast and accurate clus-
+tering of molecular dynamics simulations’ trajectories.
+The cc analysis dimensionality reduction method was
+first applied to protein simulation data.
+The method
+projects collective variables to a usually relatively high-
+dimensional (∼10-40 dim) unit sphere, separating noise
+and fluctuations from important structural information.
+Then the data can be efficiently clustered by density
+based clustering methods, such as HDBSCAN. The it-
+erative application of HDBSCAN allows to account for
+the inhomogeneity in population and density of the pro-
+jected points, which is very typical for protein simulation
+
+14
+data. As cc analysis relies on the calculations of correla-
+tion matrices between each frame, this drastically limits
+the amount of data one can project simultaneously. To
+allow processing of long simulation trajectories we in-
+cluded encodermap to the scheme.
+In addition to the
+obvious advantage of the two-dimensional visualisation
+it is used – in combination with a RMSD-based accep-
+tance criterion – for a fast structure-based assignment of
+additional points to the clusters initially identified in the
+higher dimensional projection done with cc analysis. To
+demonstrate the accuracy and performance of the pro-
+posed scheme we applied the clustering scheme to four
+test systems: replica exchange simulations of Trp-cage
+and three long trajectories of a Trp-cage mutant, NTL9
+and Protein B generated on the Anton supercomputer.
+By applying the scheme to these four test systems we
+could show that: the algorithm can efficiently handle
+very large amounts of data, that it can be used to com-
+pare the clusters of structurally different systems in one
+2D map, and that it can also be applied to cluster sys-
+tems which do not have very stable native states and
+are therefore intrinsically very difficult to cluster confor-
+mationally.
+Furthermore the algorithm is able to find
+clusters independent of their size. By varying a RMSD
+cutoff both conformationally very well defined clusters,
+as well as fuzzy clusters, whose members only share an
+overall structural motive, can be identified.
+VI.
+SUPPORTING INFORMATION
+Supporting Information (PDF) includes:
+(S-I): Methods to chose parameters for cc analysis and
+HDBSCAN.
+(S-II): Stopping criteria for the clustering workflow.
+(S-III): RMSD plots of trajectories for Trp-cage, Pro-
+tein B and NTL9.
+(S-IV): Comparison of the proposed clustering work-
+flow to PCA and k-means clustering for Trp-cage (TC5b).
+(S-V): Workstation specifications.
+VII.
+ACKNOWLEDGEMENTS
+This work was supported by the DFG through
+CRC 969. We also greatly appreciate the computing time
+on bwHPC clusters which was used to produce the Trp-
+cage TC5b trajectories. Furthermore we would like to
+thank the D.E. Shaw research group for providing the
+Trp-cage, NTL9 and Protein B trajectories.
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+

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+arXiv:2301.02453v1  [cs.IT]  6 Jan 2023
+1
+Delay-Doppler Domain Tomlinson-Harashima
+Precoding for OTFS-based Downlink
+MU-MIMO Transmissions: Linear Complexity
+Implementation and Scaling Law Analysis
+Shuangyang Li, Member, IEEE, Jinhong Yuan, Fellow, IEEE,
+Paul Fitzpatrick, Senior Member, IEEE, Taka Sakurai, Member, IEEE, and
+Giuseppe Caire, Fellow, IEEE
+Abstract
+Orthogonal time frequency space (OTFS) modulation is a recently proposed delay-Doppler (DD) do-
+main communication scheme, which has shown promising performance in general wireless communications,
+especially over high-mobility channels. In this paper, we investigate DD domain Tomlinson-Harashima
+precoding (THP) for downlink multiuser multiple-input and multiple-output OTFS (MU-MIMO-OTFS)
+transmissions. Instead of directly applying THP based on the huge equivalent channel matrix, we propose a
+simple implementation of THP that does not require any matrix decomposition or inversion. Such a simple
+implementation is enabled by the DD domain channel property, i.e., different resolvable paths do not share
+the same delay and Doppler shifts, which makes it possible to pre-cancel all the DD domain interference in
+a symbol-by-symbol manner. We also study the achievable rate performance for the proposed scheme
+by leveraging the information-theoretical equivalent models. In particular, we show that the proposed
+scheme can achieve a near optimal performance in the high signal-to-noise ratio (SNR) regime. More
+importantly, scaling laws for achievable rates with respect to number of antennas and users are derived,
+which indicate that the achievable rate increases logarithmically with the number of antennas and linearly
+with the number of users. Our numerical results align well with our findings and also demonstrate a
+significant improvement compared to existing MU-MIMO schemes on OTFS and orthogonal frequency-
+division multiplexing (OFDM).
+Part of the paper was presented at IEEE Global Communications Conference 2022 [1].
+
+2
+Index Terms
+OTFS, MU-MIMO, THP, delay-Doppler domain communication, scaling law
+I. INTRODUCTION
+Orthogonal time frequency space (OTFS) modulation has received much attention in the past few
+years since its invention in [2], thanks to its capability of providing highly reliable communications
+over complex transmission scenarios, such as high-mobility channels [3], [4]. Compared to the cur-
+rently deployed orthogonal frequency-division multiplexing (OFDM) modulation, OTFS modulation
+has demonstrated high-Doppler resilience and robust communication performance against various
+channel conditions [3], [5]–[7]. Therefore, OTFS modulation has been recognized as a potential
+solution to supporting the heterogeneous requirements of beyond fifth-generation (B5G) wireless
+systems, especially in high-mobility scenarios [3], [5].
+The success of OTFS originates from the delay-Doppler (DD) domain signal processing [8],
+[9], guided by the elegant mathematical theory of the Zak transform [10], [11]. The Zak transform
+gives rise to the DD domain symbol placement, which potentially enables pulse localization without
+violating Heisenberg’s uncertainty principle [2], [5]. Furthermore, the DD domain symbol placement
+allows the information symbols to directly interact with the DD domain channel response, resulting
+in a much simpler input-output relationship compared to that of OFDM modulation over complex
+channels such as the high-mobility channel. More importantly, it can be shown that with DD domain
+modulation, each information symbol principally experiences the whole fluctuations of the time-
+frequency (TF) channel over an OTFS frame. Thus, the OTFS modulation offers the potential of
+achieving full TF diversity [12]–[16].
+The DD domain channel response has several appealing properties including compactness, quasi-
+stationarity, separability, and sparsity [17], [18], which enables simple channel estimation and
+reduced-complexity detection approaches. For example, an embedded pilot scheme for OTFS chan-
+nel estimation was proposed in [19], where a sufficiently large guard interval is applied around
+the pilot to improve the acquisition of delay and Doppler responses. Such a scheme can permit
+a direct channel estimation by simply checking the received signal’s value around the DD grid
+of the embedded pilot. In [20], a sparse Bayesian-learning-assisted channel estimation approach
+was presented, where both on-grid and off-grid (due to the virtual sampling) delay and Doppler
+components are used to perform sparse signal recovery in order to estimate the delay and Doppler
+
+3
+responses. A message passing algorithm (MPA) was proposed in [21], where the Gaussian ap-
+proximation is applied to model the characteristic of DD domain interference. This algorithm and
+its variants, such as [22], [23], and [24], take advantage of the DD domain sparsity, such that
+fewer iterations over the graphical model are sufficient to obtain a good error performance. The
+aforementioned algorithms and many other excellent works [25], [26] have laid a strong foundation
+for single-input and single-output (SISO)-OTFS transceiver designs. However, related investigations
+on multiple-input and multiple-output (MIMO)-OTFS systems are only in the their infancy.
+MIMO technology is an important candidate to meet the stringent requirements of the achievable
+rate for B5G wireless systems [27]. Research on MIMO-OTFS, especially multiuser MIMO-OTFS
+(MU-MIMO-OTFS), is important to determine whether OTFS modulation can be applied in practical
+multiple-antenna systems [28]. Unfortunately, the design of MU-MIMO-OTFS is challenging. This
+is because OTFS modulation does not guarantee interference-free transmission like OFDM modula-
+tion in static channels. In fact, the DD domain received symbols generally contain interference [21]
+in the multi-path transmission, as the result of the “twisted convolution” between the transmitted
+symbols and the DD domain channel responses1 [5]. Consequently, most of the designs of MU-
+MIMO-OTFS will face an equivalent channel matrix with a huge size, e.g., number of delay bins
+times number of Doppler bins times number of antennas. With such an enormous matrix size,
+conventional precoding/equalization techniques, such as zero forcing and minimum mean square
+error (MMSE), cannot be directly applied due to the extremely high computational complexity
+introduced by the channel inversion. As a result, most of the existing works for downlink MU-
+MIMO-OTFS rely on simple precoding approaches, such as maximum ratio transmission (MRT)
+precoding [29], or approximation of channel inversion, such as [30], with an aim to reduce the
+computational complexity by trading off performance.
+In this paper, we consider the precoding design for downlink MU-MIMO-OTFS from a different
+perspective by using the Tomlinson-Harashima precoding (THP) [31], [32]. THP is a classic
+non-linear precoding scheme that has been widely applied in practice, whose core idea is to
+pre-cancel/pre-subtract the known interference before transmission. THP has shown promising
+performance in terms of the achievable rate. In particular, it has been shown in [33] that the
+constant “shaping loss” is the only loss of the achievable rate for THP at high signal-to-noise ratios
+1The term “twisted convolution” comes from the first OTFS paper [2], which is similar to the circular convolution but with an
+additional phase term.
+
+4
+(SNRs) [33]. Thus, we postulate that the application of THP in MU-MIMO-OTFS would result
+in a promising rate performance. Note that the conventional implementation of THP requires QR
+decomposition [31], [32], such that the decomposed channel matrix has a triangular structure. How-
+ever, with a huge matrix size in the MU-MIMO-OTFS transmission, such a decomposition could
+be computationally expensive. In contrast to the existing works, we do not aim to design precoding
+directly based on the huge equivalent channel matrix. Instead, we propose to perform interference
+pre-cancellation directly in the DD domain without any channel decomposition or inversion. This is
+possible by exploiting the fact that different resolvable paths must be distinguishable in at least one
+dimension of delay and Doppler, and consequently cannot share both the same delay and Doppler
+shifts at the same time2 [17], [34]. The major contributions of this paper can be summarized as
+follows.
+• We derive a concise input-output relation for downlink MU-MIMO-OTFS with beamforming
+(BF) in the matrix form, which lays the foundations for our digital precoder designs and later
+performance analysis.
+• Using the derived system model, we conduct a detailed analysis on the DD domain interference
+pattern and compare it to the TF domain interference pattern for the OFDM counterpart.
+In particular, we show that the DD domain received symbols suffer from three types of
+interference, namely multi-path self-interference (MPSI), inter-beam interference (IBI), and
+crosstalk interference (CTI). We unveil the physical meanings of those interference terms, and
+show that IBI can be ignored by considering user grouping or user scheduling, while MPSI
+can be mitigated by BF in practical systems.
+• We propose a DD domain THP design that only entails linear complexity without any matrix
+decomposition or inversion based on the characteristics of DD domain channel responses.
+In particular, we show that the DD domain interference pattern contains several cycles. The
+existence of the cycles suggests that the interference pre-cancellation can start from any DD
+grid in the cycle and all the interference can be cancelled out in a symbol-by-symbol manner.
+• We study the sum-rate of the proposed scheme by deriving the representative information-
+theoretical equivalent models according to the property of the modulo operation. Based on the
+2Physical channels can have multiple paths sharing the same or very similar delay and Doppler responses. However, due to the
+limited capability of distinguishing delay and Doppler for practical receivers, those paths cannot be fully resolved or separated.
+Consequently, the receiver only sees one multi-path component (DD response) due to the combining of these paths [34].
+
+5
+TABLE I
+LIST OF MAIN SYSTEM PARAMETERS.
+Parameters
+Definitions
+K
+Number of users
+M
+Number of delay bins/subcarriers
+N
+Number of Doppler bins/time slots
+P
+Number of resolvable paths
+NBS
+Number of antennas at BS
+∆f
+Subcarrier spacing
+T
+Time slot duration
+L
+Number of interference terms considered for cancellation
+h(k)
+p
+Channel coefficient for the p-th path of the k-th user
+l(k)
+p
+and k(k)
+p
+Delay and Doppler indices for the p-th path of the k-th user
+g(i)
+p [j]
+Spatial interference power of the j-th beam on i-th user’s p-th path
+X(i)
+DD [l, k] and Y (i)
+DD [l, k]
+(l, k)-th DD domain transmitted and received symbol of the i-th user
+derived sum-rate, we show that the proposed scheme can achieve a near-optimal performance
+that only has a constant rate loss (the shaping loss) compared to the optimal interference-
+free transmission. Furthermore, we investigate the sum-rate performance with respect to the
+number of antennas at the base station (BS) NBS and the number of users K, respectively. In
+particular, we show that the sum-rate of the proposed scheme increases linearly with K and
+logarithmically with NBS.
+Notations: The blackboard bold letters A, E, and C denote the constellation set, the expectation
+operator, and the complex number field, respectively; the notations (·)T and (·)H denote the transpose
+and the Hermitian transpose for a matrix, respectively; vec(·) denotes the vectorization operation;
+diag{·} denotes the diagonal matrix; “⊗” denotes the Kronecker product operator; min (·) returns
+the minimum value of a function; I (·; ·) and h (·) denote the mutual information and the differential
+entropy, respectively; [·]x denotes the modulo operation with respect to x. FN and IM denote the
+discrete Fourier transform (DFT) matrix of size N × N and the identity matrix of size M × M;
+the big-O notation O (·) describes the asymptotic growth rate of a function. For the sake of clarity,
+the main system parameters are summarized in Table I.
+
+6
+ISFFT
+IFFT
+( )
+tx
+g
+t
+[
+]
+DD
+,
+X
+l k
+DD
+X
+[
+]
+TF
+,
+X
+m n
+TF
+X
+( )
+s t
+TD
+x
+Heisenberg Transform
+OTFS Modulation
+[
+]
+TD
+x
+m
+nM
++
+FFT
+SFFT
+Wigner Transform
+OTFS Demodulation
+( )
+rx
+g
+t
+( )
+r t
+TD
+y
+[
+]
+TD
+y
+m
+nM
++
+[
+]
+TF
+,
+y
+m n
+TF
+Y
+DD
+Y
+[
+]
+DD
+,
+y
+l k
+Channel
+Fig. 1. The transmitter structure of SISO-OTFS transmissions.
+II. SYSTEM MODEL
+In this section, we will derive a concise system model for MU-MIMO-OTFS transmissions.
+Before going into the details of MU-MIMO-OTFS transmissions, we will briefly review some
+preliminaries on SISO-OTFS transmissions, which will then be used for the related discussions on
+MU-MIMO-OTFS transmissions.
+A. Preliminaries on SISO-OTFS Transmissions
+Without loss of generality, let us consider the OTFS transmitter shown in Fig. 1. Let M be the
+number of delay bins/subcarriers and N be the number of Doppler bins/time slots, respectively.
+The corresponding subcarrier spacing and time slot duration are given by ∆f and T, respectively.
+Let xDD ∈ AMN be the DD domain information symbol vector of length MN. In particular, the
+information symbol vector xDD can be arranged as a two-dimensional (2D) information symbol
+matrix XDD ∈ AM×N, i.e., xDD
+∆= vec (XDD), and the (l, k)-th element of XDD, XDD [l, k], is the
+information symbol at the l-th delay grid and the k-th Doppler grid [2], for 0 ≤ k ≤ N − 1, 0 ≤
+l ≤ M − 1. As indicated by Fig. 1, the TF domain transmitted symbol XTF [m, n] , 0 ≤ m ≤
+M − 1, 0 ≤ n ≤ N − 1 can be obtained from XDD via the inverse symplectic finite Fourier
+transform (ISFFT) [35], i.e.,
+XTF
+∆= FMXDDFH
+N,
+(1)
+where XTF [m, n] is the (m, n)-th element in XTF, and FM and FN are the normalized DFT
+matrices of size M × M and N × N defined in the Notations. It is also convenient to write the
+corresponding vector form of (1), which is given by [36]
+xTF
+∆= vec (XTF) =
+�
+FH
+N ⊗ FM
+�
+xDD.
+(2)
+The transmitted OTFS signal s (t) can be obtained by performing the Heisenberg transform [2] to
+XTF with the transmitter shaping pulse gtx(t), as shown in Fig. 1. In particular, the Heisenberg
+
+7
+transform can be interpreted as a multicarrier modulator and a popular choice for implementing the
+Heisenberg transform is to apply the OFDM modulator [3]. According to the OFDM modulation,
+the Heisenberg transform can be implemented by an inverse fast Fourier transform (IFFT) module
+and transmit pulse shaping, in which case the resultant transmitted OTFS signal s (t) is given by
+s (t) =
+N−1
+�
+n=0
+M−1
+�
+m=0
+XTF [m, n] gtx (t − nT) ej2πm∆f(t−nT).
+(3)
+Based on (3), it is useful to define the time-delay (TD) domain transmitted symbol vector xTD of
+length MN. Considering the energy-normalized rectangular shaping pulse gtx (t), xTD is defined
+by [35]
+xTD
+∆= vec
+�
+FH
+MXTF
+�
+=
+�
+FH
+N ⊗ IM
+�
+xDD.
+(4)
+Let hDD (τ, ν) be the DD domain channel response given by
+hDD (τ, ν) =
+P
+�
+p=1
+hpδ (τ − τp) δ (ν − νp) ,
+(5)
+where hp, τp, and νp are the fading coefficient, the delay shift, and the Doppler shift associated
+with the p-th path.
+According to [35], the corresponding TD domain channel response of (5) can be equivalently
+represented in a matrix form in the case of rectangular filtering pulse grx (t), reduced CP structure,
+and non-fractional delay and Doppler shifts, such that
+HTD =
+P
+�
+p=1
+hpΠlp∆kp,
+(6)
+where Π is the permutation matrix (forward cyclic shift), i.e.,
+Π =
+
+
+0
+· · ·
+0
+1
+1
+...
+0
+0
+...
+...
+...
+...
+0
+· · ·
+1
+0
+
+
+MN×MN
+,
+(7)
+and ∆ = diag{γ0, γ1, ..., γMN−1} is a diagonal matrix with γ
+∆= e
+j2π
+MN [35]. In (6), the terms lp and
+kp are the indices of delay and Doppler, respectively, associated with the p-th path, respectively,
+where
+τp =
+lp
+M∆f ,
+and
+νp = kp
+NT ,
+(8)
+and we have lp ≤ lmax and −kmax ≤ kp ≤ kmax, for 1 ≤ p ≤ P, with lmax and kmax denoting the
+largest delay index and Doppler index, respectively. It should be noted that the system model in (6)
+
+8
+only considers the integer delay and Doppler case, which is only valid with a sufficiently large
+signal bandwidth and a sufficiently long frame duration [21]. However, it is reported in [37] that
+the effects of fractional Doppler could be mitigated by adding TF domain windows. Furthermore,
+some recent developments of OTFS have shown that the pulse shaping could improve the DD
+domain sparsity [8], [9], [38]–[40]. As the main focus of this paper is on the application of THP
+to MU-MIMO-OTFS transmissions, we restrict ourselves to the case of integer delay and Doppler.
+Following on from (6), the received time-delay (TD) domain symbol vector yTD is given by
+yTD = HTDxTD + w,
+(9)
+where w is the corresponding additive white Gaussian noise (AWGN) sample vector in the TD
+domain with one-sided power spectral density (PSD) N0. The OTFS demodulation can be interpreted
+as the concatenation of the Wigner transform and the SFFT [2]. Based on (9), the DD domain
+received symbol vector is given by3 [35],
+yDD = (FN ⊗ IM) yTD = HDDxDD + w,
+(10)
+where HDD is the corresponding equivalent DD domain channel matrix of the form [14]
+HDD
+∆=
+P
+�
+p=1
+hp (FN ⊗ IM)Πlp∆kp �
+FH
+N ⊗ IM
+�
+.
+(11)
+For ease of derivation, it is useful to derive a DD domain symbol-wise input-output relation
+based on (10). In fact, (11) has a direct connection to the inverse discrete Zak transform (IDZT),
+which gives rise to the following lemma.
+Lemma 1 (DD Domain Input-Output Relation via IDZT): Let YDD be the corresponding matrix
+representation of yDD, i.e., yDD
+∆= vec (YDD). Then, in the case of integer Doppler indices and
+rectangular shaping pulses, the input-output relation for OTFS transmissions with the reduced-CP
+structure can be characterized by
+YDD [l, k] =
+P
+�
+p=1
+hpej2π
+kp(l−lp)
+MN
+αl,lp,k,kpXDD
+�
+[l − lp]M, [k − kp]N
+�
+,
+(12)
+where αl,lp,k,kp is a phase offset as the result of the quasi-periodicity property of the IDZT, and it
+is given by
+αl,lp,k,kp =
+
+
+
+1,
+l − lp ≥ 0,
+e−j2π k−kp
+N ,
+l − lp < 0.
+(13)
+3In (10), we use the same notation for the AWGN samples in both TD and DD domains, because they follow the same distribution.
+
+9
+THP
+OTFS Modulation
+OTFS Modulation
+BF
+( )
+1
+DD
+s
+(
+)
+DD
+K
+sM
+(
+)
+DD
+K
+x
+( )
+1
+DD
+x
+M
+( )
+1
+TD
+x
+(
+)
+TD
+K
+x
+BS
+N
+z
+1z
+M
+OTFS 
+Demodulation
+OTFS 
+Demodulation
+( )
+1
+DD
+y
+(
+)
+DD
+K
+y
+( )
+1
+TD
+y
+(
+)
+TD
+K
+y
+M
+Fig. 2. The block diagram of considered THP-based downlink MU-MIMO-OTFS transmissions.
+Proof: The proof is straightforward by invoking the IDZT. Furthermore, derivations without
+applying IDZT can also be found in Section 4.6.2 of [41].
+Despite the fact that Lemma 1 has already appeared in the literature [41], we still want to
+emphasize the importance of those results here because of the following two reasons. Firstly,
+the symbol-wise DD domain input-output relation for OTFS has not been widely considered and
+understood in the literature. Secondly, the results of Lemma 1 will be frequently used in the later
+part of this paper as the building block for our derivations. Based on the above descriptions of
+SISO-OTFS transmissions, we will der ive the system model of MU-MIMO-OTFS transmissions
+in the following subsection.
+B. Derivations of the System Model for MU-MIMO-OTFS Transmissions
+Without loss of generality, let us consider the downlink MU-MIMO-OTFS transmission for K
+users, where the BS is equipped with K radio-frequency (RF) chains and NBS antennas with
+NBS ≥ K, while each user is equipped with only one antenna, as shown in Fig. 2. For notational
+consistency, we will extend the related notations from the above subsection by adding superscripts
+or subscripts to specify the underlying users or antennas. Denote by s(k)
+DD ∈ AMN×1 the DD domain
+information symbol vector of length MN for the k-th user, where 1 ≤ k ≤ K. In particular, the
+DD domain information symbol vectors for the K users can be arranged into a 2D matrix SDD
+of size MN × K, whose k-th column is s(k)
+DD. As indicated by Fig. 2, we apply THP to SDD and
+the resultant symbol matrix after precoding is XDD of size MN × K, whose k-th column is the
+DD domain symbol vector for the k-th user after precoding, denoted by x(k)
+DD. After passing x(k)
+DD
+through the OTFS modulator, the TD domain symbol vector for the k-th user can be obtained by
+
+10
+x(k)
+TD =
+�
+FH
+N ⊗ IM
+�
+x(k)
+DD according to (4). Thus, we can write
+XTD =
+�
+FH
+N ⊗ IM
+�
+XDD,
+(14)
+where XTD of size MN ×K is the TD domain symbol matrix after OTFS modulation, and its k-th
+column is x(k)
+TD. For ease of derivation, let us consider the vectorized version of XTD by stacking
+each column of XTD into a vector, such as
+xTD
+∆=
+��
+x(1)
+TD
+�H
+,
+�
+x(2)
+TD
+�H
+, ...,
+�
+x(K)
+TD
+�H�H
+= vec (XTD) =
+�
+IK ⊗ FH
+N ⊗ IM
+�
+xDD,
+(15)
+where xDD
+∆= vec (XDD) is the DD domain symbol vector of size KMN × 1. We consider
+conventional BF for the downlink transmission as indicated in Fig. 2. Let VBF of size K × NBS
+be the BF matrix adopted. Then, the transmitted symbol matrix Z after BF is given by
+Z = XTDVBF,
+(16)
+where the n-th column of Z, zn, is the transmitted symbol vector on the n-th antenna at the BS,
+for 1 ≤ n ≤ NBS. Similar to (15), we can write the corresponding vector form of (16), which is
+given by
+z
+∆=
+�
+zH
+1 , zH
+2 , ..., zH
+NBS
+�H = vec (Z) =
+�
+VT
+BF ⊗ IMN
+�
+xTD =
+�
+VT
+BF ⊗ FH
+N ⊗ IM
+�
+xDD.
+(17)
+Now let us turn our attention to the wireless channel for MU-MIMO transmissions. Without
+loss of generality, we assume that the antenna array at the BS is in the form of a uniform linear
+array (ULA). We further assume that the underlying channel between the BS and each user has
+P independent resolvable paths, where the angle-of-departure (AoD) for the p-th path of the k-th
+user, for 1 ≤ p ≤ P and 1 ≤ k ≤ K, is given by ϕ(k)
+p , and ϕ(k)
+p
+̸= ϕ(k′)
+p′ , for p ̸= p′ or k ̸= k′.
+Then, according to the far field assumption [27] and the DD domain channel characteristics in (5),
+the DD domain channel for the n-th antenna and the k-th user can be modeled by
+h (n, k, τ, ν) =
+P
+�
+p=1
+h(k)
+p exp
+�
+jπ (n − 1) sin
+�
+ϕ(k)
+p
+��
+δ
+�
+τ − τ (k)
+p
+�
+δ
+�
+ν − ν(k)
+p
+�
+,
+(18)
+where we assume that the distance between adjacent antennas is equal to half of the wavelength.
+In (18), h(k)
+p
+∈ C, τ (k)
+p , and ν(k)
+p
+are the fading coefficient, the delay shift, and the Doppler shift
+corresponding to the p-th path of the k-th user, respectively. According to (18), let us denote by
+l(k)
+p
+and k(k)
+p
+the delay and Doppler indices corresponding to the p-th path of the k-th user, i.e.,
+τ (k)
+p
+=
+l(k)
+p
+M∆f ,
+ν(k)
+p
+= k(k)
+p
+NT .
+(19)
+
+11
+Let us further define the effective TD domain channel matrix for the p-th path of the k-th user based
+on (6) by ˜Hk,p
+TD = h(k)
+p Πl(k)
+p ∆k(k)
+p . Similarly, based on (11), the effective DD domain channel matrix
+for the p-th path of the k-th user is defined by ˜Hk,p
+DD
+∆= h(k)
+p (FN ⊗ IM)Πl(k)
+p ∆k(k)
+p �
+FH
+N ⊗ IM
+�
+. After
+some derivations, we can write the TD domain received symbol vector y(k)
+TD for the k-th user by
+y(k)
+TD =
+�
+NBS
+P
+�
+p=1
+�
+aT �
+ϕ(k)
+p
+�
+⊗ ˜Hk,p
+TD
+�
+z + w(k),
+(20)
+where
+a
+�
+ϕ(k)
+p
+� ∆=
+1
+√NBS
+�
+1, exp
+�
+jπ sin ϕ(k)
+p
+�
+, ..., exp
+�
+jπ (NBS − 1) sin ϕ(k)
+p
+��T,
+(21)
+is the normalized steering vector for the p-th path of the k-th user, and w(k) is the AWGN sample
+vector with one-sided PSD N0. Next, by considering (17), (20) can be further expanded as
+y(k)
+TD =
+�
+NBS
+P
+�
+p=1
+��
+aT �
+ϕ(k)
+p
+�
+VT
+BF
+�
+⊗
+�
+˜Hk,p
+TD
+�
+FH
+N ⊗ IM
+���
+xDD + w(k)
+=
+�
+NBS
+P
+�
+p=1
+�
+˜Hk,p
+TD
+�
+FH
+N ⊗ IM
+��
+XDD
+�
+VBFa
+�
+ϕ(k)
+p
+��
++ w(k),
+(22)
+where the second equation is due to the properties of the Kronecker product. Considering (22), it is
+convenient to define the effective spatial domain channel vector g(k)
+p
+∆= VBFa
+�
+ϕ(k)
+p
+�
+to characterize
+the interference from different data streams to the received symbols of the k-th user from the p-th
+path. Finally, by performing OTFS demodulation to y(k)
+TD, the DD domain received symbol vector
+y(k)
+DD for the k-th user can be written by
+y(k)
+DD =
+�
+NBS
+P
+�
+p=1
+�
+(FN ⊗ IM) ˜Hk,p
+TD
+�
+FH
+N ⊗ IM
+��
+XDDg(k)
+p
++ w(k)
+=
+�
+NBS
+P
+�
+p=1
+˜Hk,p
+DDXDDg(k)
+p
++ w(k).
+(23)
+So far, we have derived the system model of the MU-MIMO-OTFS transmissions. In the following
+section, we will develop our digital THP scheme based on (23) by adopting a simple BF matrix
+according to the steering vectors, where the k-th row of VBF is the Hermitian transpose of the
+steering vector associated with the strongest path of the k-th user.
+
+12
+III. DD DOMAIN THP FOR DOWNLINK MU-MIMO-OTFS TRANSMISSIONS
+In this section, we will discuss the proposed DD domain THP. It should be noted that the direct
+application of THP by employing QR decomposition may require high complexity since the size
+of the equivalent channel matrix is KMN × KMN. Therefore, we propose a DD domain THP
+scheme that does not require the decomposition of channel matrices. In particular, we assume
+that the channel state information (CSI) is available at the transmitter, which can be achieved by
+exploiting the DD domain reciprocity [17] based on uplink channel estimation.
+A. DD Domain Interference Pattern Analysis
+Let us first have a close look at the interference pattern in the DD domain. To provide some
+insights, let us rewrite (23) as
+y(i)
+DD =
+�
+NBS
+P
+�
+p=1
+K
+�
+j=1
+g(i)
+p [j] ˜Hi,p
+DDx(j)
+DD + w(i),
+(24)
+where g(i)
+p [j] denotes the j-th element of g(i)
+p
+implying the contribution from the j-th beam to the
+i-th user via the i-th user’s p-th path. As implied by (24), the DD domain received symbol vector
+of the i-th user is related to the DD domain transmitted symbols of each user. Furthermore, by
+considering (12), (24) can be expanded as
+Y (i)
+DD [l, k] =
+P
+�
+p=1
+K
+�
+j=1
+˜g(i,j)
+l,l(i)
+p ,k,k(i)
+p ,pX(j)
+DD
+��
+l − l(i)
+p
+�
+M,
+�
+k − k(i)
+p
+�
+N
+�
++ w(i) [l, k],
+(25)
+where Y (i)
+DD [l, k] denotes the (l, k)-th symbol of the received symbol matrix Y(i)
+DD of the i-th user,
+i.e., y(i)
+DD
+∆= vec
+�
+Y(i)
+DD
+�
+, and ˜g(i,j)
+l,l(i)
+p ,k,k(i)
+p ,p characterizes the symbol-wise effective channel coefficient,
+including the angular domain interference from the j-th user/beam to the i-th user/beam, the fading
+coefficient from the p-th path of the i-th user, and the phase rotation due to the twisted convolution,
+and is given by4
+˜g(i,j)
+l,l(i)
+p ,k,k(i)
+p ,p=
+
+
+
+
+
+
+
+√NBSg(i)
+p [j] h(i)
+p exp
+�
+j2π
+k(i)
+p
+�
+l−l(i)
+p
+�
+MN
+�
+, l − l(i)
+p ≥0
+√NBSg(i)
+p [j] h(i)
+p exp
+�
+j2π
+k(i)
+p
+�
+l−l(i)
+p
+�
+MN
+�
+exp
+�
+−j2π
+�
+k−k(i)
+p
+�
+N
+�
+, l − l(i)
+p <0
+.
+(26)
+To further characterize the interference pattern, let us assume that the channel strengths, i.e.,
+absolute values of fading coefficients, associated to each user are sorted in descending order, i.e.,
+4The additional phase term in the second line of (26) is the consequence of the quasi-periodicity of the Zak transform [10].
+
+13
+���h(i)
+1
+��� ≥
+���h(i)
+2
+��� ≥ ... ≥
+���h(i)
+P
+���, for 1 ≤ i ≤ K, without loss of generality. In this case, the BS forms
+multi-beams towards the directions of the first paths of all users. We henceforth refer to the first
+path of each user as the BF path, while the other paths are called non-BF paths. With these in
+mind, we can expand (25) to yield
+Y (i)
+DD [l, k] = ˜g(i,i)
+l,l(i)
+1 ,k,k(i)
+1 ,1X(i)
+DD
+��
+l − l(i)
+1
+�
+M,
+�
+k − k(i)
+1
+�
+N
+�
+�
+��
+�
+Desired signal
++
+P
+�
+p=2
+˜g(i,i)
+l,l(i)
+p ,k,k(i)
+p ,pX(i)
+DD
+��
+l − l(i)
+p
+�
+M,
+�
+k − k(i)
+p
+�
+N
+�
+�
+��
+�
+MPSI
++
+K
+�
+j=1
+j̸=i
+˜g(i,j)
+l,l(i)
+1 ,k,k(i)
+1 ,1X(j)
+DD
+��
+l − l(i)
+1
+�
+M,
+�
+k − k(i)
+1
+�
+N
+�
+�
+��
+�
+IBI
++
+P
+�
+p=2
+K
+�
+j=1
+j̸=i
+˜g(i,j)
+l,l(i)
+p ,k,k(i)
+p ,pX(j)
+DD
+��
+l − l(i)
+p
+�
+M,
+�
+k − k(i)
+p
+�
+N
+�
+�
+��
+�
+CTI
++w(i) [l, k].
+(27)
+From (27), we notice that the value of Y (i)
+DD [l, k] is composed of several terms with different physical
+meanings. We can characterize those signals based on their physical meanings as follows:
+• Desired signal: The first term in (27) is the desired signal. The desired signal contains the
+information of the desired user and it is transmitted from the BF path.
+• MPSI: The second term in (27) is the MPSI. The MPSI contains the interference from the
+desired user caused by the multi-path transmissions from the non-BF paths of the desired user.
+• IBI: The third term in (27) is the IBI. The IBI contains the interference from other users
+caused by the superposition among different beams, as each user has a distinctive beam.
+• CTI: The fourth term in (27) is the CTI. The CTI contains the interference from other users
+caused by the unintended alignment between the other users’ BF directions and the desired
+user’s non-BF paths.
+A brief diagram characterizing the interference pattern is given in Fig. 3, where both the IBI and
+CTI are clearly indicated. As implied by the interference descriptions above, we notice that the
+interference terms have different characteristics. However, it should be noted that not all those
+interference terms make a significant contribution to the received symbol Y (i)
+DD [l, k]. In particular,
+
+14
+CTI
+IBI
+CTI
+BF path
+User 1
+User 2
+BS
+Fig. 3. The brief diagram of the interference pattern for downlink MU-MIMO-OTFS transmissions.
+user scheduling is usually performed at the BS before transmitting the downlink signals. One
+of the objectives of performing user scheduling is to avoid severe interference among different
+beams, which is enabled by grouping users with diverse spatial characteristics, e.g., AoDs [42].
+Furthermore, thanks to the nature of BF, the impact of MPSI is generally small. This is because
+the BS only forms narrow beams towards the BF paths of each user, and consequently the residual
+power on the non-BF paths is low. However, it can be shown that the CTI could have a high
+impact if the BF path of one user overlaps with one of the non-BF paths from a different user.
+This is because the transmitted signal after BF usually has a large power towards the BF direction.
+Therefore, even though the non-BF path may not have a large channel gain, the overall received
+power is still non-negligible as the transmitted power towards this direction is large.
+B. Approximations with User Grouping
+As indicated by the discussions in the previous subsection, the interference terms have different
+characteristics. In the following subsection, we will develop a DD domain THP scheme by exploiting
+the nature of those interference terms with the aid of user grouping. Let us consider the following
+assumption for user grouping:
+• Assumption 1: We assume that the beams formulated for different users in the group are
+sufficiently separated (orthogonal) in the angular domain by having NBS ≫ K. With this
+assumption, it is reasonable to ignore the IBI between different users.
+
+15
+Furthermore, it should be noted that the AoDs of different paths associated to the same user are
+usually separated, especially for a sufficiently large number of transmit antennas. On top of that,
+the non-BF paths usually have much lower channel gain compared to the BF paths in practical
+settings thanks to the BF. Those two observations give rise to the following assumption:
+• Assumption 2: We assume that the non-BF paths associated to the same user are relatively
+separated in the angular domain, where the channel gains are much lower compared to that
+of the BF path. With this assumption, it is reasonable to ignore the MPSI of each user.
+We henceforth refer to the transmission where both assumptions 1 and 2 hold as the favorable
+propagation conditions, which is realizable with NBS ≫ K. Under the favorable propagation
+conditions, (27) becomes
+Y (i)
+DD [l, k] ≈˜g(i,i)
+l,l(i)
+1 ,k,k(i)
+1 ,1X(i)
+DD
+��
+l − l(i)
+1
+�
+M,
+�
+k − k(i)
+1
+�
+N
+�
++
+L
+�
+p=1
+˜g(i,Bi[p])
+l,l(i)
+Pi[p],k,k(i)
+Pi[p],Pi[p]X(Bi[p])
+DD
+��
+l − l(i)
+Pi[p]
+�
+M,
+�
+k − k(i)
+Pi[p]
+�
+N
+�
++ w(i) [l, k] ,
+(28)
+where the MPSI, IBI are ignored and only L CTI terms are considered with 1 ≤ L ≤ (P − 1) (K − 1).
+Here, the term L is the number of CTI terms with significant power that will be considered in the
+precoding. The introduction of L aims to strike a balance between the error performance and the
+computational complexity of the precoder. In (28), we define Bi of length L as the CTI beam vector
+for the i-th user and Pi of length L as the CTI path vector for the i-th user, respectively. The CTI
+beam vector contains the beam indices that correspond to the L CTI terms with the most significant
+power for the i-th user, while the CTI path vector contains the indices of paths for the i-th user
+that spatially overlap with the beams with indices given in the CTI beam vector. In other words,
+with a descending power order of the CTI terms, the p-th CTI term, for 1 ≤ p ≤ L, is caused by
+Bi [p]-th beam overlaping with the Pi [p]-th path of the i-th user. In particular, by examining (26),
+the elements of Bi and Pi can be determined based on the absolute values of hi [p] g(i)
+p [j], for
+2 ≤ p ≤ P and 1 ≤ j ≤ K, j ̸= i.
+The approximated input-output relation in (28) has an important property. For each DD domain
+received symbol, all the related DD domain transmitted symbols that contribute to the interference
+of this received symbol are from different DD grids of other users, as indicated in Fig. 4(a). This is
+quite different from the OFDM counterpart, where all the related TF domain transmitted symbols
+that contribute to a specific received TF domain symbol are from the same TF grid of different
+users, as indicated in Fig. 4(b). The rationale behind this observation is that the TF domain channel
+
+16
+operation can be characterized by an element-wise product [27], while the DD domain channel
+operation is characterized by the twisted convolution [8]. In fact, this property is the key enabler
+for a reduced-complexity THP for downlink MU-MIMO transmissions, which will be introduced
+in detail in the coming subsection.
+Rx User 2
+Rx User 1
+Tx User 2
+Tx User 1
+t
+n
+Rx User 2
+Rx User 1
+Tx User 2
+Tx User 1
+(a) MU-MIMO-OTFS transmission.
+Rx User 2
+Rx User 1
+Tx User 2
+Tx User 1
+Rx User 2
+Rx User 1
+Tx User 2
+Tx User 1
+t
+f
+(b) MU-MIMO-OFDM transmission.
+Fig. 4. A diagram characterizing the difference of interference patterns between MU-MIMO-OTFS and MU-MIMO-OFDM, where
+two users are considered. In particular, the red arrow denotes the BF path, while the blue dashed line implies the CTI.
+C. DD Domain THP
+The core idea of THP is to pre-cancel the interference before transmission, where a modulo
+operation is applied to control the transmitted signal power [32], [33]. Before introducing the
+considered DD domain THP, let us consider the following example as shown in Fig. 5, where
+M = N = 3, P = 2, and K = 2, respectively. There are in total 9 DD grids for each user and we
+use the capital letters A to I with different colors to refer to the DD domain transmitted symbols
+associated to each DD grid, where the subscripts for the capital letters denote the corresponding
+user indices. Furthermore, we use the solid and dashed arrows at the “transmitter” part indicating
+the DD shift corresponding to each resolvable path, where we assume that l(1)
+1
+= 0, k(1)
+1
+= 0, and
+l(1)
+2
+= 0, k(1)
+2
+= −1 for user 1, while l(2)
+1
+= 1, k(2)
+1
+= 0, and l(2)
+2
+= 0, k(2)
+2
+= 1 for user 2. Here,
+we assume that the positive delay and Doppler indices shift the symbol up and to the left, while
+the negative delay and Doppler indices shift the symbol down and to the right. The interference
+pattern corresponding to (28) is shown in the “receiver” part of Fig. 5, where the symbols on the
+left hand side in each DD grid is the desired signal (marked in red), while the symbols on the right
+hand side are the interference (marked in blue). For a better illustration, we also use dashed circles
+
+17
+BF 
+path
+Non-BF 
+path
+A1 C2  B1 A2 
+C1 B2  
+D1 F2  E1 D2 
+F1 E2 
+G1 I2  H1 G2  I1 H2  
+D2 B1
+E2 C1  F2 A1
+G2 E1 
+H2 F1  
+I2 D1  
+A2 H1  B2 I1  
+C2 G1  
+User 1
+User 2
+A2
+D2
+E2
+F2
+G2
+H2
+I2
+B2
+C2
+User 2
+User 1
+A1
+D1
+E1
+F1
+G1
+H1
+I1
+B1
+C1
+Non-BF 
+path
+BF 
+path
+BF 
+path
+Transmitter:
+Receiver:
+Fig. 5. An example of the interference pattern for MU-MIMO-OTFS, where M = N = 3, P = 2, and K = 2, respectively.
+highlighting the iteration between the users, where the color of each dashed circle corresponds to
+the DD grid from which the interference comes.
+It is interesting to note from Fig. 5 that there is a possibility that we can directly pre-cancel
+all the interference in the DD domain by exploiting the different delay and Doppler responses
+associated to different paths. For example, the received value of the first DD grid for user 1 only
+consists of the desired signal A1 and the interference from C2. Therefore, the interference for A1
+can be perfectly canceled if we know the exact value of C2. Similarly, the interference for C2 can
+be canceled if we know the exact value of G1. So on and so forth, it can be shown that there are
+DD domain cycles that contain several DD domain symbols for the interference cancellation, e.g.,
+A1 → C2 → G1 → I2 → D1 → F2 → A1. However, it should be noted that the pre-cancellation
+could change the value of the corresponding DD domain transmitted symbols. Consequently, due to
+the DD domain cycles, the pre-cancellation of interference cannot be directly applied. For instance,
+in the considered example, to pre-cancel the interference for A1, it is required to know the value of
+A1 after interference cancellation as suggested by the cycle, which is a non-causal operation and
+
+18
+BF 
+path
+Non-BF 
+path
+  
+2 
+2  
+1 
+2 
+2  
+  
+  
+  
+2 
+1 
+1  
+  
+  
+0 C2  
+B1 A2 
+0 B2  
+D1 F2  
+E1 0 
+F1 E2 
+G1 I2  H1 G2  I1 H2  
+0 B1
+E2 0  
+F2 0
+G2 E1 
+H2 F1  
+I2 D1  
+A2 H1  B2 I1  
+C2 G1  
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+User 1
+User 2
+User 1
+User 2
+User 1
+User 2
+Fig. 6. The application of DD domain THP for the example given in Fig. 5.
+cannot be implemented in practice.
+To solve this problem, we propose to assign known symbols to specific DD grids in order to break
+the DD domain cycles. For example, if we assign a zero to the symbol A1, then the pre-cancellation
+for F2 can be conducted. Following the DD domain cycle, the interference can be pre-cancelled
+step by step, such as A1 → F2 → D1 → I2 → G1 → C2. The corresponding pre-cancelation is
+illustrated in Fig. 6, where there are in total 3 DD domain cycles. We use numbers with different
+colors to represent the schedule of interference cancellation for each DD domain cycle, where we
+set A1, D2, and C1 as zeros and use zeros to represent the start of the pre-cancelation for each
+DD domain cycle. It is not hard to see that the considered pre-cancellation can indeed cancel all
+the interference without any matrix decomposition or inversion via intentionally assigning known
+symbols.
+Based on the above example, we are ready to present the implementation of DD domain THP.
+Note that the proposed THP follows a symbol-by-symbol pre-cancelation, and for each DD domain
+symbol, it is required to know where the interference comes from and which symbol should be
+pre-canceled next. Let us denote by ˆB of length K the interfered beam vector for all the users and
+ˆP of length K the interfered path vector for all the users. In particular, the i-th element of ˆB is the
+index of the user, to whom the i-th beam (the transmitted signal of the i-th user) causes the most
+significant CTI, and the i-th element of ˆP is the corresponding path index, from which the ˆB[i]-th
+user receives the CTI due to the i-th beam. Those terms indicate the precoding schedule for the
+considered THP scheme, as the most significant CTI from the i-th beam is likely to be included
+in the CTI beam vector of the ˆB [i]-th user. In this case, the symbols in the i-th beam after pre-
+cancellation are likely to be used for the pre-cancellation for the ˆB [i]-th user, thereby reducing the
+
+19
+overhead. In particular, by observing (26), we have ˆB [i]
+∆= arg max
+i
+���hj [p] g(j)
+p [i]
+���, for 2 ≤ p ≤ P
+and 1 ≤ j ≤ K, j ≤ i, and ˆP [i]
+∆= arg max
+p
+����h ˆB[i] [p] g( ˆB[i])
+p
+[i]
+����, for 2 ≤ p ≤ P. Corresponding
+to the above discussions, the details of DD domain THP are summarized in Algorithm 1, where
+mod [·] denotes the modulo operation in the conventional THP. Some discussions on the modulo
+threshold will be presented in the coming section.
+As implied by Algorithm 1, the L most significant CTI will be pre-cancelled via THP for each
+DD domain symbol. Therefore, according to (28) and the principle of THP, the receiver side applies
+a single-tap equalization together with a modulo operation to recover the DD domain transmitted
+symbols [33]. In particular, we have
+ˆY (i)
+DD [l, k] = mod
+
+
+1
+˜g(i,i)
+l,l(i)
+1 ,k,k(i)
+1 ,1
+Y (i)
+DD [l, k]
+
+ .
+(29)
+Based on ˆY (i)
+DD [l, k], a straightforward demodulation could be applied to recover the transmitted
+information for each user.
+D. Complexity and Signaling Overhead
+We will discuss the computational complexity and the required signaling overhead for the
+considered THP in this subsection. As indicated by Algorithm 1, there are at most L times of pre-
+cancellation for each DD domain transmitted symbol. Thus, the overall computational complexity
+is linear to the number of transmitted symbols with a linearity coefficient L, i.e., O (LKMN). It
+should be noted that such a linear complexity is lower than most of the existing precoding schemes
+for MU-MIMO-OTFS, including the ones in [29], [43], because the proposed THP does not rely
+on the complex channel decomposition or inversion.
+On the other hand, it can be observed that the signaling overhead for the proposed THP depends
+on the value of L, and the channel conditions, such as the number of paths, number of users, and
+delay and Doppler responses. Furthermore, the pre-cancellation order is also of great importance
+for the signaling overhead. Note that Algorithm 1 is a performance-centric implementation of DD
+domain THP, where the algorithm aims to pre-cancel all the interference terms without considering
+the required overhead. Consequently, the total number of assigned known symbols increases if
+the corresponding interference symbols have not yet been pre-cancelled, e.g., line 9 to 13 in
+Algorithm 1. In contrast, there could also be an overhead-centric implementation, where the pre-
+cancellation is performed with the priority to the symbols, to whom the corresponding interference
+
+20
+Algorithm 1 DD Domain THP for Downlink MU-MIMO-OTFS Transmissions
+Input: ˜g(i,j)
+l,l(i)
+p ,k,k(i)
+p ,p, S(i)
+DD, l(i)
+p , k(i)
+p , Pi, Bi, ˆB, and ˆP,
+for 0 ≤ l ≤ M − 1, 0 ≤ k ≤ N − 1, 1 ≤ p ≤ P, 1 ≤ i, j ≤ K.
+Initialization: Set Indicator mtx[l, k, i] = 0, for 0 ≤ l ≤ M − 1, 0 ≤ k ≤ N − 1, 1 ≤ i ≤ K.
+Set Overhead mtx[l, k, i] = 0, for 0 ≤ l ≤ M − 1, 0 ≤ k ≤ N − 1, 1 ≤ i ≤ K.
+Steps:
+1: for l′ from 0 to M − 1 do
+2:
+for k′ from 0 to N − 1 do
+3:
+for i′ from 1 to K do
+4:
+Set l = l′, k = k′, and i = i′.
+5:
+while Indicator mtx[l, k, i] = 0 do
+6:
+X(i)
+DD [l, k] = S(i)
+DD [l, k].
+7:
+for p from 1 to L do
+8:
+Set delay idx =
+��
+l − l(i)
+1
+�
+M + l(i)
+Pi[p]
+�
+M and Doppler idx =
+��
+k − k(i)
+1
+�
+N + k(i)
+Pi[p]
+�
+N.
+9:
+if Indicator mtx[delay idx, Doppler idx, Bi[p]] = 0 do
+10:
+Set X(Bi[p])
+DD
+[delay idx, Doppler idx] = 0.
+11:
+Set Indicator mtx[delay idx, Doppler idx, Bi[p]] = 1.
+12:
+Set Overhead mtx[delay idx, Doppler idx, Bi[p]] = 1.
+13:
+end if
+14:
+X(i)
+DD [l, k] = X(i)
+DD [l, k] −
+˜g(i,Bi[p])
+l,l(i)
+Pi[p],k,k(i)
+Pi[p],Pi[p]
+˜g(i,i)
+l,l(i)
+1
+,k,k(i)
+1
+,1
+X(Bi[p])
+DD
+[delay idx, Doppler idx].
+15:
+end for
+16:
+X(i)
+DD [l, k] = mod
+�
+X(i)
+DD [l, k]
+�
+.
+17:
+Set Indicator mtx[l, k, i] = 1.
+18:
+Set l =
+��
+l − l( ˆ
+B[i])
+ˆ
+P [i]
+�
+M
++ l( ˆ
+B[i])
+1
+�
+M
+, k =
+��
+k − k( ˆ
+B[i])
+ˆ
+P [i]
+�
+N
++ k( ˆ
+B[i])
+1
+�
+N
+, and i = ˆB [i].
+19:
+end while
+20:
+end for
+21:
+end for
+22: end for
+23: Return ˆX(i)
+DD, for 1 ≤ i ≤ K.
+symbols have already been pre-cancelled, e.g., line 14 in Algorithm 1, in order to minimized the
+required overhead. However, the reduced overhead implementation is currently still an open problem
+and we are unable to discuss this issue in detail due to the space limitation. But it should be pointed
+out that the searching algorithms for tree- and trellis-based graphical models may shed light on
+
+21
+MU-MIMO-
+OTFS
+S
+a
+w
+ˆY
+1
+g -%
+S
+ˆY
+a
+w%
+Mod-d
+Mod-d
+Mod-d
+(a) Equivalent diagram of the system model in Fig. 2.
+MU-MIMO-
+OTFS
+S
+ˆY
+w%
+Mod-
+Mod-
+Mod-d
+(b) Simplified diagram of Fig. 7(a).
+Fig. 7. Equivalent and simplified system models corresponding to Fig. 2.
+this issue [44], [45].
+IV. ACHIEVABLE RATE ANALYSIS
+We discuss the achievable rates of the proposed THP scheme in this section. Without loss
+of generality, we consider the quadrature amplitude modulation (QAM) constellation set5 A. In
+particular, we focus on the average achievable rate for each DD domain symbol under favorable
+propagation conditions by assuming that NBS ≫ K. For ease of derivation, we provide an equivalent
+diagram of the proposed THP-based MU-MIMO-OTFS characterizing the corresponding processing
+between S(i)
+DD [l, k] and ˆY (i)
+DD
+��
+l + l(i)
+1
+�
+M,
+�
+k + k(i)
+1
+�
+N
+�
+in Fig. 7(a), where we neglect the symbol
+indices for notational brevity. Specifically, we use the term α in Fig. 7(a) to describe the pre-
+cancellation of THP. As indicated by this diagram, an arbitrary DD domain symbol S after pre-
+cancellation with term α and modulo operation with threshold d is transmitted over the MU-MIMO-
+OTFS channel. The received channel observation contains the corruption from the AWGN sample w,
+which is used for symbol detection after an single tap equalization with ˜g−1, e.g.,
+�
+˜g(i,i)
+l,l(i)
+1 ,k,k(i)
+1 ,1
+�−1
+,
+and applying the modulo operation with threshold d. Those descriptions are consistent with our
+system model in Section II. In particular, the above processing can be described by the following
+equation
+ˆY = mod
+�1
+˜g (˜g (mod [S + α]) + η + w)
+�
+= mod
+�
+mod [S + α] + 1
+˜g (η + w)
+�
+,
+(30)
+where η denotes the interference term due to the MU-MIMO-OTFS transmission as suggested
+in (27). Note that mod [mod [a] + b] = mod [a + b]. Thus, (30) can be further simplified to
+ˆY = mod
+�
+S + α + 1
+˜gη + 1
+˜gw
+�
+.
+(31)
+5Although we only focus on QAM constellation here, the related discussions can be straightforwardly extended to the case of
+general constellations, e.g., pulse amplitude modulation (PAM).
+
+22
+Furthermore, as implied by Line 14 of Algorithm 1, the interference term η/˜g will be cancelled by
+pre-cancellation, e.g., term α, with a sufficiently large number of L, in the case of user grouping
+and BF. Therefore, we can further approximate (31) by
+ˆY ≈ mod
+�
+S + 1
+˜gw
+�
+.
+(32)
+The corresponding diagram to (32) is presented in Fig. 7(b), where ˜w = 1
+˜gw denotes the equivalent
+AWGN sample with one-sided PSD N0
+�
+|˜g|2.
+Now we focus on the achievable rate for the considered scheme based on (32). In particular, the
+mutual information between S and ˆY is given by [33], [46]
+I
+�
+S; ˆY
+�
+∆= h
+�
+ˆY
+�
+− h
+�
+ˆY |S
+�
+≈ h
+�
+mod
+�
+S + 1
+˜gw
+��
+− h
+�
+mod
+�1
+˜gw
+��
+.
+(33)
+Notice that the modulo operation strictly limits the signal value from
+�
+−d
+2, d
+2
+�
+for both the real and
+imaginary dimensions, and the maximum entropy probability distribution for a random variable
+with support constrained to an interval is the independent and identically distributed (i.i.d.) uniform
+distribution [46]. Thus, (33) can be approximately upper-bounded by
+I
+�
+S; ˆY
+�
+≲ 2 log2 (d) − h
+�
+mod
+�1
+˜gw
+��
+.
+(34)
+Note that the values of AWGN samples are generally small in the high SNR regime. Thus, in
+the high SNR regime (e.g., the real/imaginary part of the noise sample is within the range of
+�
+−d
+2, d
+2
+�
+), (34) can be shown to converge to [33]
+I
+�
+S; ˆY
+�
+≲ 2 log2 (d) − h
+�1
+˜gw
+�
+= 2log2 (d) − log2
+�
+πe N0
+|˜g|2
+�
+= log2
+�
+d2|˜g|2
+πeN0
+�
+.
+(35)
+Based on (35), we are ready to investigate the sum-rate performance for the considered THP scheme.
+Notice that there is no joint decoding among different users. Thus, with favorable propagation
+conditions, the sum-rate for the considered downlink MU-MIMO-OTFS can be formulated by
+Rsum
+∆=
+K
+�
+i=1
+I
+�
+S(i)
+DD [l, k] ; ˆY (i)
+DD [l, k]
+�
+=
+K
+�
+i=1
+log2
+
+
+
+
+
+d2
+����˜g(i,i)
+l,l(i)
+1 ,k,k(i)
+1 ,1
+����
+2
+πeN0
+
+
+
+
+.
+(36)
+Furthermore, by substituting (26) into (36), we have
+Rsum =
+K
+�
+i=1
+log2
+
+
+
+d2NBS
+���h(i)
+1
+���
+2
+πeN0
+
+
+.
+(37)
+
+23
+As implied by (37), the sum-rate is related to the choice of modulo threshold d. According to [33],
+the average power for transmitted symbol X(i)
+DD converges to d2/12 and d2/6 for PAM and QAM
+constellations, respectively. Thus, with QAM constellations, the total transmit power for a given time
+slot is Kd2/6. Based on the total transmit power, we can define the SNR for the THP transmission
+by SNR
+∆= Kd2
+6N0 . Finally, we obtain the sum-rate at high SNRs by
+Rsum =
+K
+�
+i=1
+log2
+� 6
+πe
+NBS
+K
+���h(i)
+1
+���
+2
+SNR
+�
+.
+(38)
+Next, we discuss some important insights based on the previous analysis. In particular, we
+restrict ourselves to the high SNR regime, where the sum-rate is characterized by (38). Let us
+first characterize the sum-rate gap of the proposed scheme to the optimal transmission scenario,
+where there is only one resolvable path between the BS and each user with sufficiently separated
+(orthogonal) angular features. The latter transmission scenario is optimal in the sense that it does
+not have neither MPSI, IBI, nor CTI, and therefore maximizes the throughput of the downlink
+transmission. The following lemma shows the sum-rate in the optimal transmission scenario.
+Lemma 2 (Optimal Sum-rate): In the optimal transmission scenario, where there is only one
+resolvable path between the BS and each user without IBI, the sum-rate is given by
+Ropt
+sum =
+K
+�
+i=1
+log2
+�
+1 + NBS
+K
+���h(i)
+1
+���
+2
+SNR
+�
+.
+(39)
+Proof : By considering the uniform power allocation among different users, (39) can be derived
+by following the capacity calculation for parallel Gaussian channels with independent noise [46].
+The detail derivations are omitted here due to the space limitation.
+■
+Based on Lemma 2, the following theorem characterizes the sum-rate gap between the proposed
+scheme and the optimal case in the high SNR regime.
+Theorem 1 (Shaping Loss): For sufficiently large L (perfect pre-cancellation of interference)
+and NBS ≫ K, the proposed scheme only has a constant rate loss for each user compared to the
+optimal transmission scenario in the high SNR regime.
+Proof :
+Ropt
+sum − Rsum =
+K
+�
+i=1
+log2
+
+
+
+1 + NBS
+K
+���h(i)
+1
+���
+2
+SNR
+6
+πe
+NBS
+K
+���h(i)
+1
+���
+2
+SNR
+
+
+ ≈
+K
+�
+i=1
+log2
+�πe
+6
+�
+,
+(40)
+where the approximation holds in the high SNR regime. Note that 1
+2log2
+� πe
+6
+�
+≈ 0.255, which is
+the well-known “shaping loss” for general PAM constellations in the THP literature.
+■
+
+24
+As implied by Theorem 1, the proposed scheme can obtain a promising rate performance that
+only has a constant gap to the optimal transmission. As pointed out by [47], this performance loss
+is the “shaping loss”, which is caused by the peak limitation introduced by precoding. Next, we
+will discuss the growth rate of the sum-rate with respect to different parameters. The following
+theorem shows the scaling law of the proposed scheme.
+Theorem 2 (Scaling Law for Sum-rate): For sufficiently large L (perfect pre-cancellation of
+interference) and NBS ≫ K, the sum-rate of the proposed scheme scales linearly with the number
+of users K under favorable propagation conditions at the asymptotically high SNRs.
+Proof : Based on (38), we have
+lim
+SNR→∞
+Rsum
+log2 (SNR) =
+lim
+SNR→∞
+K�
+i=1
+log2
+�
+6
+πe
+NBS
+K
+���h(i)
+1
+���
+2�
++ Klog2 (SNR)
+log2 (SNR)
+= K,
+(41)
+which indicates that the sum-rate growth is linear in K.
+■
+The conclusion in Theorem 2 is not unexpected. Note that the proposed scheme contains NBS
+antennas and K RF, where NBS > K. Thus, it can be shown that the degree-of-freedom (DoF) of
+the proposed scheme is limited by K instead of NBS [48], which in fact determines the maximum
+sum-rate growth rate (the pre-log factor) as shown in Theorem 2. Next, we study the sum-rate
+performance with respect to the number of antennas at BS NBS.
+Theorem 3 (Sum-Rate vs. NBS): For sufficiently large L (perfect pre-cancellation of interference)
+and NBS ≫ K, the sum-rate of the proposed scheme for a given K increases logarithmically with
+the number of antennas at BS under favorable propagation conditions.
+Proof : Based on (38), we have
+lim
+SNR→∞
+Rsum
+log2 (NBS) =
+lim
+SNR→∞
+K
+�
+i=1
+log2
+�
+6
+πe
+���h(i)
+1
+���
+2
+K
+SNR
+�
++ Klog2 (NBS)
+log2 (NBS)
+= K,
+(42)
+which indicates that the sum-rate growth increases logarithmically with NBS.
+■
+The conclusion in Theorem 3 aligns with Theorem 2. As the DoF is determined by the number
+of users K, a larger number of NBS can only provide the SNR gain, which is consistent with the
+general conclusions for MU-MIMO [48]. The correctness of the above theorems will be verified
+in the coming section.
+
+25
+V. NUMERICAL RESULTS
+In this section, we will use numerical results to verify the effectiveness of the proposed schemes.
+We consider MU-MIMO-OTFS transmissions with M = 32 and N = 16, where we set the
+maximum delay and Doppler indices to lmax = 5 and kmax = 7, respectively. The delay and
+Doppler indices are assumed to be integer values unless otherwise specified. The fading coefficients
+are generated based on the exponential power delay profile with a path loss exponent of 2.76.
+The signal constellation is the quadrature phase shift keying (QPSK) constellation. Furthermore,
+we present the results under both favorable propagation and practical channel conditions. For the
+favorable propagation case, the received signals are generated based on (27), where both the MPSI
+and IBI are ignored. For the practical case, the received signals are generated based on (25), and
+a user grouping strategy is applied such that the maximum spatial correlation between different
+users is no larger than 0.1, i.e., g(i)
+p [j] ≤ 0.1, for i ̸= j. Meanwhile, we assume that the different
+resolvable paths have AoDs that are at least 5 degrees away from each other.
+A. Numerical Results under Favorable Propagation Conditions
+We first present the sum-rate performance of the proposed scheme with respect to different
+numbers of antennas NBS in Fig. 8(a), where we set K = 2, P = 2, and L = 1. As shown in
+the figure, the sum-rate increases by K bits/s/Hz when doubling the number of antennas, which
+indicates a logarithmical increase of the sum-rate with with the number of antennas NBS as indicated
+by Theorem 3. The sum-rate performance for different numbers of users is presented in Fig. 8(b),
+where we set P = 3 and L = 1. In particular, we apply a fixed ratio ρ = 2 between the number
+of antennas NBS and number of users K. It can be seen that the sum-rate appears to increase first
+with SNR and then slightly saturate in the very high SNR regime. This is because L = 1 is not
+sufficient to perfectly cancel out the CTI for the considered case. But we still observe that the
+sum-rate exhibits a strong increasing trend at practical SNRs, e.g., SNR from 10 dB to 30 dB.
+Furthermore, we also notice that with a fixed ratio ρ, the sum-rate is doubled if the number of
+users is doubled. This observation suggests a linear increase of the sum-rate with respect to the
+number of users K, and it is consistent with our findings in Theorem 2.
+In Fig. 8(c), the sum-rate performance with different values of L is considered, where we set
+NBS = 8, K = 4, P = 3. The performance bounds given in both (38) and (39) are also drawn in
+the figure. As can be observed from the figure, the proposed scheme outperforms the no precoding
+
+26
+0
+10
+20
+30
+40
+50
+SNR (dB)
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Sum-rate (bits/s/Hz)
+NBS = 4
+NBS = 8
+NBS = 16
+NBS = 32
+NBS = 64
+Logarithmically increasing
+(a) Sum-rate performance for K = 2 and different NBS.
+0
+10
+20
+30
+40
+50
+SNR (dB)
+0
+10
+20
+30
+40
+50
+60
+Sum-rate (bits/s/Hz)
+K = 2, NBS = 4
+K = 3, NBS = 6
+K = 4, NBS = 8
+K = 5, NBS = 10
+Linearly increasing
+(b) Sum-rate performance for different K and NBS.
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+SNR (dB)
+-10
+0
+10
+20
+30
+40
+50
+60
+70
+Sum-rate (bits/s/Hz)
+No precoding
+L = 1
+L = 2
+L = 3
+Bound in (38)
+Bound in (39)
+(c) Sum-rate performance for different values of L.
+0
+5
+10
+15
+20
+25
+30
+SNR (dB)
+10-4
+10-3
+10-2
+10-1
+100
+BER
+NBS = 8, K = 2, P = 2
+NBS = 16, K = 2, P = 2
+NBS = 8, K = 4, P = 4
+NBS = 16, K = 4, P = 4
+(d) BER performance for different K, NBS, and L.
+Fig. 8. The sum-rate and BER performances of the proposed scheme with respect to different numbers of users K and antennas
+NBS and different values of L.
+benchmark in terms of the sum-rate. Furthermore, we also observe that the sum-rate increases with
+a larger L, but the rate saturation appears at very high SNRs. This is not unexpected because
+the number of CTI terms is large with a small antenna-to-user ratio and many resolvable paths.
+Consequently, a large L is required to fully cancel the interference. On the other hand, it should be
+noticed that the sum-rate of the proposed scheme still shows a good increasing rate with imperfect
+cancellation at practical SNRs, e.g., SNR from 10 dB to 30 dB, as evidenced by the bounds. The
+choice of L is important for the system designs, and more discussions on how to choose L will be
+given later in Remark 1.
+The bit error rate (BER) performance with various numbers of users, antennas, and resolvable
+paths is presented in Fig. 8(d), where we set L = 1. As indicated by the figure, the BER
+
+27
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+SNR (dB)
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Sum-rate (bits/s/Hz)
+NBS = 8, K = 2, P = 3
+NBS = 12, K = 3, P = 3
+NBS = 16, K = 4, P = 3
+NBS = 20, K = 5, P = 3
+(a) Sum-rate performance for different K and NBS.
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+SNR (dB)
+0
+5
+10
+15
+20
+25
+30
+Sum-rate (bits/s/Hz)
+Fractional delay Doppler, with MPSI and IBI
+Integer delay Doppler, with MPSI and IBI
+Integer delay Doppler, without MPSI and IBI
+(b) Sum-rate comparison between various channel conditions.
+0
+5
+10
+15
+20
+25
+30
+35
+SNR (dB)
+10-4
+10-3
+10-2
+10-1
+BER
+NBS = 20, K = 4, P = 2, OTFS + THP
+NBS = 20, K = 4, P = 2, OTFS + MRT
+NBS = 20, K = 4, P = 2, OFDM + ZF
+(c) BER of THP, MRT [29], and OFDM with ZF.
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+SNR (dB)
+0
+5
+10
+15
+20
+25
+30
+Sum-rate (bits/s/Hz)
+OTFS + THP, without overhead
+OTFS + THP, with overhead
+OTFS + MRT
+OFDM + ZF
+(d) Sum-rates of THP, MRT [29], and OFDM with ZF.
+Fig. 9. The sum-rate performance of the proposed scheme with different parameters and benchmark technologies.
+performance with various channel conditions does not show a noticeable error floor at practical
+SNRs. Furthermore, we notice that increasing P and K could degrade the BER performance. This
+observation is consistent with the fact that more interference terms are introduced with an increasing
+number of resolvable paths and users. On the other hand, we also observe that the BER performance
+improves with an increasing number of BS antennas NBS. This observation is also consistent with
+our conclusions from Fig. 8(a).
+B. Numerical Results under Practical Channel Conditions
+In this subsection, we present the numerical results of the proposed scheme under more realistic
+channel conditions, where both the MPSI and IBI are considered. We compare the sum-rate
+performance for different K and NBS in Fig. 9(a), where P = 3 and L = 1. As can be observed
+
+28
+from the figure, the sum-rate improves roughly linearly with the increase of K at mid-to-high
+SNRs, but saturates when the SNR is larger than 30 dB. This rate saturation is mainly caused by
+the MPSI and IBI.
+We examine the proposed scheme with more complex channel conditions in Fig. 9(b), where we
+consider NBS = 8, K = 3, P = 4, and L = 1. In particular, we present the sum-rate performance
+with favorable propagation (no MPSI and IBI), practical channel (with MPSI and IBI), and practical
+channel having fractional delay and Doppler. It can be observed that the proposed scheme enjoys
+a sum-rate increase with the growth of SNR even in the presence of fractional delay and Doppler.
+However, it suffers from a noticeable rate degradation, because the inter-Doppler and inter-delay
+interferences are treated as noise in the case of fractional delay and Doppler. It should be noted
+that the fractional delay and Doppler can be and should be dealt with by baseband filtering, such
+as windowing [37], and pulse shaping [8], [9], [38]–[40]. On the other hand, we observe that the
+influence of MPSI and IBI becomes more severe at high SNRs, which aligns with the rate saturation
+observed from Fig. 9(a).
+A performance comparison between the proposed scheme, the MRT precoding in [29], and
+OFDM with zero-forcing (ZF) precoding is presented in Fig. 9(c) and Fig. 9(d). To have a fair
+comparison, the OFDM also applies a reduced-CP structure, where no CP is appended between
+the adjacent OFDM symbols. But we apply a large ZF precoder of size KN × KN on each
+subcarrier to mitigate the intersymbol interference and multiuser interference. In Fig. 9(c), the BER
+performance of those schemes are presented, where we consider NBS = 20, K = 4, P = 2, and
+L = 1. It can be observed from the figure that the proposed scheme outperforms the MRT scheme
+and the OFDM with ZF at mid-to-high SNRs. This observation validates the advantage of the
+proposed THP over existing schemes. This advantage can also be demonstrated by the achieved
+sum-rate gain shown in Fig. 9(d), where we consider NBS = 8, K = 4, P = 3, and L = 1. In
+particular, we also include the sum-rate results of the proposed THP with and without considering
+the required overhead in Fig. 9(d). It can be noticed that even though the overhead reduces the sum-
+rate, the proposed scheme is still advantageous in terms of the sum-rate over the existing schemes.
+However, it should be noted that the required overhead can be reduced as discussed in Section
+III-D, which is a topic for future research. More importantly, the proposed THP only requires a
+linear complexity of O (LKMN), while the MRT in [29] requires matrix/vector superposition and
+multiplication, thus having a complexity of O (KM2N2). Furthermore, the ZF precoded OFDM
+
+29
+TABLE II
+OVERHEAD VS. DIFFERENT NUMBERS OF USERS AND RESOLVABLE PATHS.
+K = 2, L = 1
+K = 3, L = 1
+K = 3, L = 2
+P = 2
+2.9%
+24.1%
+34.9%
+P = 3
+9.6%
+25.2%
+37.8%
+P = 4
+12.9%
+25.7%
+39.0%
+requires the matrix inversion and has a complexity of O (MK3N3). The superior performance and
+the low implementation complexity make our proposed THP a promising candidate for downlink
+MU-MIMO transmissions.
+Remark 1: The pre-cancellation term L is a key parameter for our proposed THP, which
+determines how many CTI interference terms are pre-cancelled in the precoding. Note that the value
+of L should be selected considering the channel condition, operating SNR, and the cancellation
+strategy discussed in Section III-D. In our simulations, we intentionally use small values of L, such
+as L = 1, because this is the most straightforward application of the proposed THP and it also
+requires the least overhead. As extensively discussed in our numerical results, L = 1 performs quite
+well under various channel conditions. We argue that this is not a coincidence. Instead, this is an
+expected result due to the careful user grouping strategy. The important insight here is that the CTI
+interference is only severe when the BF path of one user has a direction that is sufficiently close
+to the non-BF path of a different user, as depicted in Fig. 3. Therefore, it is almost impossible
+that the BF paths of different users have similar AoDs overlapping with the same non-BF path
+of a specific user after a reasonable user grouping. Furthermore, the possibility of multiple users’
+BF paths overlapping with different non-BF paths of the same user is generally low, and this case
+can also be avoided by smart grouping strategy. Therefore, we can safely choose a relatively small
+value of L in practical systems facilitated by a carefully grouping of users.
+Remark 2: It is important to evaluate the required overhead of the proposed scheme. In Table II,
+we compute the overhead of the proposed scheme with NBS = 16 and different K and L. The
+overhead is calculated as the ratio between the number of assigned known symbols in the DD
+domain and the number of DD grids in total, i.e., KMN, which is represented in the form of a
+percentage. We observe that the overhead generally increases with more resolvable paths and users,
+due to the increase of interference terms. On the other hand, we also notice that a larger value of L
+also increases the overhead. However, we have discussed in Remark 1 that a relatively small value
+
+30
+of L is sufficient in practical systems, which is also consistent with our numerical results in this
+section. Furthermore, it should be noted that the overhead performance can be further improved by
+considering the scheduling of pre-cancellation as discussed in Section III-D.
+VI. CONCLUSIONS
+In this paper, we investigated the DD domain THP for MU-MIMO-OTFS. In particular, the
+proposed THP implementation exploits the DD domain channel characteristics and does not require
+any matrix decomposition or inversion. Furthermore, we analyzed performance for the proposed
+scheme in terms of the achievable rates and investigated the scaling factors for the number of BS
+antennas and users. Our derivations implied that the sum-rate increases logarithmically with the
+number of antennas and linearly with the number of users (under the same antenna-to-user ratio).
+Our derivations were verified by numerical results. Our future work may investigate overhead
+reduction approaches for DD domain THP.
+ACKNOWLEDGEMENT
+The authors would like to express their thanks to the inventor of OTFS modulation, Prof. Ronny
+Hadani, for his enlightening speech on MU-MIMO-OTFS, which motivates this work.
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