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+New results for thermal interquark bottomonium
+potentials using NRQCD from the HAL QCD method
+Thomas Spriggs,𝑎,∗ Chris Allton,𝑎 Timothy Burns𝑎 and Seyong Kim𝑏
+𝑎Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom
+𝑏Department of Physics, Sejong University, Seoul 143-747, Korea
+E-mail: {t.spriggs.996870,c.allton,t.burns}@swansea.ac.uk,
+[email protected]
+We report progress in the calculation of the thermal interquark potential of bottomonium using the
+HAL QCD method applied to bottom quarks in the non-relativistic approximation (i.e. NRQCD).
+We exploit the fast Fourier transform algorithm, using a momentum space representation, to
+efficiently calculate NRQCD correlation functions of non-local mesonic S-wave states, and thus
+obtain the potential for temperatures in both the hadronic and plasma phases. This work was
+performed on our anisotropic 2+1 flavour “Generation 2" FASTSUM ensembles.
+The 39th International Symposium on Lattice Field Theory, LATTICE2022 8th–13th August, 2022 Bonn,
+Germany
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.03320v1  [hep-lat]  9 Jan 2023
+
+Thermal interquark bottomonium potentials
+Thomas Spriggs
+1.
+Introduction
+The interquark potential of quarkonia was one of the first quantities studied in the quest for a
+deeper understanding of the nature of the strong interaction. Pioneering studies include [1] where
+the Cornell potential was used to calculate the spectrum of charmonium states using a quantum
+mechanical formalism. In thermal QCD, the temperature dependence of the interquark potential
+results in quarkonium states melting at different temperatures [2]. These considerations strongly
+motivate a study of the thermal behaviour of the quarkonia interquark potential.
+Slowly moving heavy quarks, interacting via QCD, can be studied using non-relativistic QCD
+(NRQCD) which allows significant benefits. For example, NRQCD calculations of bottomonia
+are typically accurate at the percent level or less and is an excellent ground for quantitative tests.
+In this work we use NRQCD to determine the interquark potential in bottomonia using the HAL
+QCD approach [3]: Correlation functions of bottomonia operators are studied where the quark
+and antiquark are spatially separated, and this allows an access to the Nambu-Bethe-Salpeter
+wavefunction in the quarkonium rest frame. Using this wavefunction in the Schrödinger equation
+leads to the interquark potential. We find indications of the weakening of the potential as the
+temperature increases, as expected. This work is a continuation of the work in [4] and extends
+previous studies of the interquark potential by the FASTSUM Collaboration in the charmonium
+system [5, 6]. Other work in this area includes [7].
+2.
+NRQCD and lattice setup
+NRQCD is an effective theory with a power counting in the heavy quark velocity, 𝑣. In this
+theory, the heavy quark and antiquark fields decouple and so virtual heavy quark-antiquark loops
+cannot form. The NRQCD quark propagator is calculated via an initial value problem, rather
+than via a boundary value problem (as is the case for relativistic quarks). NRQCD is particularly
+amenable for lattice simulations because NRQCD quarkonium correlation functions do not have
+“backward movers” which means the full extent of the lattice in the temporal direction can be used
+in the analysis.
+Our NRQCD formulation incorporates both O(𝑣4) and the leading spin-dependent corrections.
+The 𝑏-quark mass is tuned by setting the “kinetic” mass (i.e. from the dispersion relation) of the
+spin-averaged 1𝑆 states to its experimental value. Full details of our NRQCD setup appear in [8].
+All our results were obtained using our FASTSUM 𝑁 𝑓 = 2+1 flavour “Generation 2” ensembles
+which have the parameters listed in Table 1.
+𝑁𝜏
+16
+20
+24
+28
+32
+36
+40
+T [MeV]
+352
+281
+235
+201
+176
+156
+141
+𝑁configurations
+1050
+950
+1000
+1000
+1000
+500
+500
+Table 1: An overview of the FASTSUM Generation 2 correlation functions used in this work. Lattice
+volumes are (24𝑎𝑠)3 × (𝑁𝜏𝑎𝜏) with 𝑎𝑠 = 0.1227(8)fm and 𝑎𝜏 = 35.1(2)am. For these ensembles with a
+pion mass of 𝑀𝜋 = 384(4)MeV, the pseudo-critical temperature Tpc = 181(1)MeV [9].
+2
+
+Thermal interquark bottomonium potentials
+Thomas Spriggs
+3.
+Method
+3.1 The HAL QCD method
+To calculate the potential between two quarks in a bottomonium - the interquark potential, 𝑉(𝑟)
+- we use the method from the HAL QCD collaboration [3]. In brief, this method uses the point-split
+correlation function and the time independent Schrödinger equation to calculate the interquark
+potential.
+The point-split correlation function is defined by
+𝐶Γ(r, 𝜏) =
+∑︁
+x
+⟨𝐽Γ(x, 𝜏; r)𝐽†
+Γ(0; 0)⟩,
+(1)
+where the non-local mesonic operators are defined
+����Γ(𝑥; r) = ¯𝑞(𝑥)Γ𝑈(𝑥, 𝑥 + r)𝑞(𝑥 + r).
+(2)
+The quark and antiquark fields, 𝑞 and ¯𝑞, are separated in space by r. The gauge field 𝑈(𝑥, 𝑥 + r) is
+required to ensure gauge invariance and Γ signifies the channel being considered; in this work we
+consider vector and pseudoscalar S-wave states. The correlator in (1) is depicted in Figure 1.
+(0,0)
+(x,𝜏)
+(x+r,𝜏)
+𝐽†
+Γ(0; 0)
+𝐽Γ(x, 𝜏; r)
+Source
+Sink
+¯𝑏
+𝑏
+Figure 1: A representation of the point-split correlation function, as defined in (1)
+As usual, the correlation function can be expressed as a sum over eigenstates of the Hamiltonian,
+𝐶Γ(r, 𝜏) =
+∑︁
+𝑗
+Ψ𝑗(r)𝑒−𝐸𝑗 𝜏,
+(3)
+where 𝐸 𝑗 is the energy of a given state 𝑗, and the unnormalised wavefunction
+Ψ 𝑗(r) =
+𝜓∗
+𝑗(0)𝜓 𝑗(r)
+2𝐸 𝑗
+(4)
+is defined in terms of the Nambu-Bethe–Salpeter wavefunction 𝜓 𝑗(r).
+We introduce the time-independent Schrödinger equation,
+�
+−∇2
+𝑟
+2𝜇 + 𝑉Γ (𝑟)
+�
+Ψ 𝑗 (𝑟) = 𝐸 𝑗Ψ 𝑗 (𝑟) ,
+(5)
+where 𝑉Γ(𝑟) is the potential for the channel Γ and 𝜇 is the reduced quark mass. We apply the
+Schrödinger equation to the point-split correlation function in (3) through the following steps
+−𝜕𝐶Γ(r, 𝜏)
+𝜕𝜏
+=
+∑︁
+𝑗
+𝐸 𝑗Ψ 𝑗(r)𝑒−𝐸𝑗 𝜏 =
+∑︁
+𝑗
+�
+−∇2
+𝑟
+2𝜇 + 𝑉Γ (𝑟)
+�
+Ψ 𝑗 (𝑟) 𝑒−𝐸𝑗 𝜏
+=
+�
+−∇2
+𝑟
+2𝜇 + 𝑉Γ (𝑟)
+�
+𝐶Γ(r, 𝜏).
+(6)
+3
+
+Thermal interquark bottomonium potentials
+Thomas Spriggs
+This yields the form of the interquark potential for a given channel, 𝑉Γ, as
+𝑉Γ(𝑟) =
+1
+𝐶Γ(r, 𝜏)
+�∇2
+𝑟
+2𝜇 − 𝜕
+𝜕𝜏
+�
+𝐶Γ(r, 𝜏).
+(7)
+Note that in the continuum limit, we expect the potential to be function of 𝑟 = |r|. There is explicit
+time dependency in this form for the potential, and this will be studied in Section 4.1. Section 4.2
+will discuss how the reduced quark mass, 𝜇, is set.
+It is convenient to define the central potential, 𝑉C, obtained via the usual spin-average [10]
+𝑉C = 1
+4𝑉Pseudo Scalar + 3
+4𝑉Vector.
+(8)
+3.2 Using momentum space to reformulate the calculation
+This work is a continuation of [4] where more detail about the HAL QCD method can be
+found. We build upon [4] by using an efficient computation of the point-split correlation function,
+𝐶Γ(r, 𝜏).
+For each 𝜏, a direct calculation of (1) requires a loop over all lattice sites x for each value of r
+which is an expensive operation scaling as O(V2) where V is the spatial volume. What follows is
+a method to reduce the cost of this computation by introducing a momentum space representation
+for the propagator and correlation function, see the Appendix of [6].
+We introduce quark propagators, 𝐷−1(𝑥; 𝑦), by Wick contracting the quark fields in the point-
+split correlation function, (1),
+𝐶Γ(r, 𝜏) = −
+∑︁
+x
+⟨𝐷−1(x + r, 𝜏; 0, 0)Γ𝛾5
+�
+𝐷−1(x, 𝜏; 0, 0)
+�†
+𝛾5Γ†⟩.
+(9)
+Note that we have gauge fixed our configurations to the Coulomb gauge, and have replaced
+the gauge connection, 𝑈(𝑥, 𝑥 + r) in (2) by unity. We now implicitly define the corresponding
+momentum space quark propagator via
+𝐷−1(y, 𝜏; 0, 0) = 1
+𝑉
+∑︁
+p
+˜𝐷−1(p, 𝜏)𝑒𝑖y·p,
+(10)
+in terms of the 3-momentum, p, which is conjugate to the position y. Introducing this momentum-
+space quark propagator into (9) yields
+𝐶Γ(r, 𝜏) = 1
+𝑉
+∑︁
+p
+⟨ ˜𝐷−1(p, 𝜏)Γ𝛾5 ˜𝐷−1(−p, 𝜏)𝛾5Γ†⟩𝑒𝑖p·r,
+(11)
+which we will use to implicitly define the momentum-space correlator, ˜𝐶Γ(p, 𝜏), i.e.
+𝐶Γ(r, 𝜏) = 1
+𝑉
+∑︁
+p
+˜𝐶Γ(p, 𝜏)𝑒𝑖p·r.
+(12)
+We note that once we have calculated ˜𝐶Γ(p, 𝜏), we can determine the desired correlator 𝐶Γ(r, 𝜏)
+for any r using (12).
+4
+
+Thermal interquark bottomonium potentials
+Thomas Spriggs
+At first sight, the conversion to momentum space does not produce any savings, because the
+calculation of 𝐶Γ and ˜𝐷−1, defined via (10) and (12), are both O(V2) in the number of operations,
+i.e. the same as the direct method. However both (10) and (12) are Fourier transforms, and so
+significant speed-up for these steps can be achieved using the fast Fourier transform (FFT) algorithm
+which scales as O(V log V).
+4.
+Results
+For better comparison with [4], and as progress towards the treatment of 𝐶Γ(r, 𝜏) for all r, we
+consider here only the on-axis r data. Extensions to this will be discussed in Section 5.
+4.1 Time dependence
+The potential is defined in (7) where there is an apparent explicit dependence on time, 𝜏, from
+the correlation function. In Figure 2, the potential, 𝑉Vector, from (7) is plotted against 𝜏 for a variety
+of distances r for our two extreme temperatures, 𝑇 = 141 and 352 MeV. We can see a clear 𝜏
+dependence for small 𝜏 which increases with r. However, for various ranges of 𝜏 and r there are
+clear plateau.
+In addition, we note that we would like to uncover temperature effects in the potential. The
+most accurate way of doing this is to compare different temperatures’ potentials obtained with the
+same time window to avoid contamination by systematic artefacts.
+Based on these considerations, we restrict the range of 𝑟 and 𝜏 used in the determination of
+the potential to those listed in Table 2. Notice that in selecting a time window, there is a trade-off
+between the ranges of 𝑟 and 𝑇 for which the potential can be extracted: larger time windows give
+access to a larger range of 𝑟, but over a smaller range of 𝑇.
+In Figure 3 we show four determinations of the central potential, corresponding to the first four
+time windows identified in Table 2. In each plot we show the potentials for several temperatures, and
+since these have been obtained by averaging over the same range of 𝜏, the temperature dependence
+can be ascribed to temperature effects, rather than fitting artefacts.
+We find that the potential
+consistently flattens as the temperature increases above 𝑇pc, as expected. There is little thermal
+variation in the potential for 𝑇 ⪅ 𝑇pc.
+In Figure 3, the error bars show statistical errors only. The curves are fits to the Cornell
+potential, which will be discussed in Section 4.3.
+4.2 Quark mass dependence
+Equation (7) contains the reduced quark mass, 𝜇, which needs to be defined. In [7], the 1S
+and 2S states were used to determine the bottom quark mass 𝑚𝑏, and thus the reduced quark mass.
+In our simulations we do not have access to the 2S state. We instead use the simple argument:
+𝜇 ≡ 1
+2𝑚𝑏 ≈ 1
+2 𝑀Υ, with 𝑀Υ from [11]. We have tested the sensitivity of the potential on the quark
+mass and found that the variation (within sensible 𝜇 ranges) is minimal.
+5
+
+Thermal interquark bottomonium potentials
+Thomas Spriggs
+Figure 2: Time dependence in the potential restricting the range of 𝑟 that we can consider valid. Shown for
+two temperatures using the vector channel as an example.
+Time window [𝑎𝜏]
+𝑟 range [𝑎𝑠]
+𝑟 range [fm]
+Temperatures [MeV]
+13-14
+1-3
+0.12-0.37
+352-141
+17-18
+1-4
+0.12-0.49
+281-141
+19-22
+1-5
+0.12-0.61
+235-141
+21-26
+1-5
+0.12-0.61
+201-141
+24-30
+1-6
+0.12-0.74
+176-141
+24-33
+1-6
+0.12-0.74
+156-141
+Table 2: Range of displacements and temperatures allowed to best approximate time independence in𝑉(𝑟, 𝜏).
+Note that 𝑇pc = 181 MeV and thus the time windows below the solid line do not span this pseudocritical
+temperature.
+4.3 Cornell potential fits
+The Cornell potential [12] is a phenomenological description of a confining potential applicable
+to heavy quarks in QCD and is given by
+𝑉(𝑟) = −𝛼
+𝑟 + 𝜎𝑟 + 𝐷.
+(13)
+Fits using (13) to our potential data are shown as solid curves in Figure 3. As can be seen these
+reproduce the data well. When the string tension, 𝜎, in the Cornell potential is zero, this implies a
+deconfined potential. In all cases above 𝑇pc, we find that 𝜎 decreases with increasing temperature,
+confirming the expected thermal behaviour in the bottomonium system. Below𝑇pc the string tension
+does not change within statistical errors.
+5.
+Conclusion
+The temperature dependence of the central interquark potential in the bottomonium system
+using NRQCD quarks was explored. This work was an extension of [4] and use a momentum-space
+approach which can improve the efficiency of the calculation. Clear thermal effects in this potential
+6
+
+T = 141 MeV
+8
+Preliminary
+X
+6
+[GeV]
+*
+4
+*
+Vector
+来来
+2
+米
+10
+15
+20
+25
+30
+0
+5
+35
+40
+t/aT = 352 MeV
+8
+Preliminary
+6
+TI
+[GeV]
+*
+4
+2
+10
+15
+0
+5Thermal interquark bottomonium potentials
+Thomas Spriggs
+Figure 3: The central potential calculated from (7) (points), overlaid with a fit of these data to the Cornell
+potential (13) (curves). Each plot contains all temperatures and r ranges listed in Table 2.
+were observed using a method which decoupled systematic “time window” artefacts from physical,
+thermal effects. A systematic flattening of the potential with increasing temperature above 𝑇pc was
+observed, with no statistically significant variation in the potential for temperatures below 𝑇pc.
+This work will be extended in a number of directions. The potential will be calculated at all
+possible spatial separations, r, rather than just the on-axis values used here, and channels beyond
+the pseudoscalar and vector S-wave states will be included. Also, a more robust definition of the
+reduced quark mass will be developed. Finally, a direct comparison will be made between these
+bottomonium results and those obtained for the charmonium potential using the same ensembles in
+[6].
+Acknowledgments
+This work is supported by STFC grant ST/T000813/1. SK is supported by the National Research
+Foundation of Korea under grant NRF-2021R1A2C1092701andgrantNRF-2021K1A3A1A16096820,
+funded by the Korean government (MEST). This work used the DiRAC Extreme Scaling service at
+the University of Edinburgh, operated by the Edinburgh Parallel Computing Centre and the DiRAC
+7
+
+Time window: 13-14
+2.4
+- - T= 352 MeV
+2.2
+T = 281 MeV
+T = 235 MeV
+2.0
+T = 201 MeV
+1.8
+T = 176 MeV
+[GeV]
+T = 156 MeV
+1.6
+T= 141 MeV
+1.4
+C
+1.2
+1.0
+0.8
+Preliminary
+0.6
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+r [fm]Time window: 17-18
+2.4
+T = 281 MeV
+2.2
+T = 235 MeV
+T = 201 MeV
+2.0
+T = 176 MeV
+1.8
+T = 156 MeV
+[GeV]
+T = 141 MeV
+1.6
+1.4
+1.2
+1.0
+0.8
+Preliminary
+0.6
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+r [fm]Time window: 19-22
+2.4
+T = 235 MeV
+2.2
+T = 201 MeV
+T = 176 MeV
+2.0
+T = 156 MeV
+1.8
+T = 141 MeV
+[GeV]
+1.6
+1.4
+1.2
+1.0
+0.8
+Preliminary
+0.6
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+r [fm]Time window: 21-26
+2.4
+- T = 201 MeV
+2.2
+ T = 176 MeV
+ T = 156 MeV
+2.0
+ T = 141 MeV
+1.8
+[GeV]
+1.6
+1.4
+1.2
+1.0
+0.8
+Preliminary
+0.6
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+r [fm]Thermal interquark bottomonium potentials
+Thomas Spriggs
+Data Intensive service operated by the University of Leicester IT Services on behalf of the STFC
+DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BEIS capital funding via
+STFC capital grants ST/R00238X/1, ST/K000373/1 and ST/R002363/1 and STFC DiRAC Opera-
+tions grants ST/R001006/1 and ST/R001014/1. DiRAC is part of the UK National e-Infrastructure.
+This work was performed using PRACE resources at Cineca (Italy), CEA (France) and Stuttgart
+(Germany) via grants 2015133079, 2018194714, 2019214714 and 2020214714. We acknowledge
+the support of the Swansea Academy for Advanced Computing, the Supercomputing Wales project,
+which is part-funded by the European Regional Development Fund (ERDF) via Welsh Government,
+and the University of Southern Denmark and ICHEC, Ireland for use of computing facilities. We
+are grateful to the Hadron Spectrum Collaboration for the use of their zero temperature ensemble.
+References
+[1] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T.-M. Yan, Phys. Rev. D 17 (1978)
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+[2] T. Matsui and H. Satz, Phys. Lett. B 178 (1986) 416–422.
+[3] N. Ishii, S. Aoki and T. Hatsuda, Phys. Rev. Lett. 99 (2007) 022001, [nucl-th/0611096].
+[4] T. Spriggs, C. Allton, T. Burns and S. Kim, PoS LATTICE2021 (2022) 569,
+[arXiv:2112.09092].
+[5] P. W. M. Evans, C. R. Allton and J. I. Skullerud, Phys. Rev. D 89 (2014) 071502,
+[arXiv:1303.5331].
+[6] C. Allton, W. Evans, P. Giudice and J.-I. Skullerud, arXiv:1505.06616.
+[7] R. Larsen, S. Meinel, S. Mukherjee and P. Petreczky, Phys. Rev. D 102 (2020) 114508,
+[arXiv:2008.00100].
+[8] G. Aarts, C. Allton, T. Harris, S. Kim, M. P. Lombardo, S. M. Ryan et al., JHEP 07 (2014)
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+203.
+8
+

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+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' NRQCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  We exploit the fast Fourier transform algorithm, using a momentum space representation, to  efficiently calculate NRQCD correlation functions of non-local mesonic S-wave states, and thus  obtain the potential for temperatures in both the hadronic and plasma phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' This work was  performed on our anisotropic 2+1 flavour “Generation 2" FASTSUM ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  The 39th International Symposium on Lattice Field Theory, LATTICE2022 8th–13th August, 2022 Bonn,  Germany  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='03320v1  [hep-lat]  9 Jan 2023  Thermal interquark bottomonium potentials  Thomas Spriggs  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  Introduction  The interquark potential of quarkonia was one of the first quantities studied in the quest for a  deeper understanding of the nature of the strong interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Pioneering studies include [1] where  the Cornell potential was used to calculate the spectrum of charmonium states using a quantum  mechanical formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' In thermal QCD, the temperature dependence of the interquark potential  results in quarkonium states melting at different temperatures [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' These considerations strongly  motivate a study of the thermal behaviour of the quarkonia interquark potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  Slowly moving heavy quarks, interacting via QCD, can be studied using non-relativistic QCD  (NRQCD) which allows significant benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' For example, NRQCD calculations of bottomonia  are typically accurate at the percent level or less and is an excellent ground for quantitative tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  In this work we use NRQCD to determine the interquark potential in bottomonia using the HAL  QCD approach [3]: Correlation functions of bottomonia operators are studied where the quark  and antiquark are spatially separated, and this allows an access to the Nambu-Bethe-Salpeter  wavefunction in the quarkonium rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Using this wavefunction in the Schrödinger equation  leads to the interquark potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' We find indications of the weakening of the potential as the  temperature increases, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' This work is a continuation of the work in [4] and extends  previous studies of the interquark potential by the FASTSUM Collaboration in the charmonium  system [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Other work in this area includes [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  NRQCD and lattice setup  NRQCD is an effective theory with a power counting in the heavy quark velocity, 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' In this  theory, the heavy quark and antiquark fields decouple and so virtual heavy quark-antiquark loops  cannot form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' The NRQCD quark propagator is calculated via an initial value problem, rather  than via a boundary value problem (as is the case for relativistic quarks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' NRQCD is particularly  amenable for lattice simulations because NRQCD quarkonium correlation functions do not have  “backward movers” which means the full extent of the lattice in the temporal direction can be used  in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  Our NRQCD formulation incorporates both O(𝑣4) and the leading spin-dependent corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  The 𝑏-quark mass is tuned by setting the “kinetic” mass (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' from the dispersion relation) of the  spin-averaged 1𝑆 states to its experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Full details of our NRQCD setup appear in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  All our results were obtained using our FASTSUM 𝑁 𝑓 = 2+1 flavour “Generation 2” ensembles  which have the parameters listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  𝑁𝜏  16  20  24  28  32  36  40  T [MeV]  352  281  235  201  176  156  141  𝑁configurations  1050  950  1000  1000  1000  500  500  Table 1: An overview of the FASTSUM Generation 2 correlation functions used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Lattice  volumes are (24𝑎𝑠)3 × (𝑁𝜏𝑎𝜏) with 𝑎𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='1227(8)fm and 𝑎𝜏 = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='1(2)am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' For these ensembles with a  pion mass of 𝑀𝜋 = 384(4)MeV, the pseudo-critical temperature Tpc = 181(1)MeV [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  2  Thermal interquark bottomonium potentials  Thomas Spriggs  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  Method  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='1 The HAL QCD method  To calculate the potential between two quarks in a bottomonium - the interquark potential, 𝑉(𝑟)  we use the method from the HAL QCD collaboration [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' In brief, this method uses the point-split  correlation function and the time independent Schrödinger equation to calculate the interquark  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  The point-split correlation function is defined by  𝐶Γ(r, 𝜏) =  ∑︁  x  ⟨𝐽Γ(x, 𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' r)𝐽†  Γ(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' 0)⟩,  (1)  where the non-local mesonic operators are defined  𝐽Γ(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' r) = ¯𝑞(𝑥)Γ𝑈(𝑥, 𝑥 + r)𝑞(𝑥 + r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  (2)  The quark and antiquark fields, 𝑞 and ¯𝑞, are separated in space by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' The gauge field 𝑈(𝑥, 𝑥 + r) is  required to ensure gauge invariance and Γ signifies the channel being considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' in this work we  consider vector and pseudoscalar S-wave states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' The correlator in (1) is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  (0,0)  (x,𝜏)  (x+r,𝜏)  𝐽†  Γ(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' 0)  𝐽Γ(x, 𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' r)  Source  Sink  ¯𝑏  𝑏  Figure 1: A representation of the point-split correlation function, as defined in (1)  As usual, the correlation function can be expressed as a sum over eigenstates of the Hamiltonian,  𝐶Γ(r, 𝜏) =  ∑︁  𝑗  Ψ𝑗(r)𝑒−𝐸𝑗 𝜏,  (3)  where 𝐸 𝑗 is the energy of a given state 𝑗, and the unnormalised wavefunction  Ψ 𝑗(r) =  𝜓∗  𝑗(0)𝜓 𝑗(r)  2𝐸 𝑗  (4)  is defined in terms of the Nambu-Bethe–Salpeter wavefunction 𝜓 𝑗(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  We introduce the time-independent Schrödinger equation,  �  −∇2  𝑟  2𝜇 + 𝑉Γ (𝑟)  �  Ψ 𝑗 (𝑟) = 𝐸 𝑗Ψ 𝑗 (𝑟) ,  (5)  where 𝑉Γ(𝑟) is the potential for the channel Γ and 𝜇 is the reduced quark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' We apply the  Schrödinger equation to the point-split correlation function in (3) through the following steps  −𝜕𝐶Γ(r, 𝜏)  𝜕𝜏  =  ∑︁  𝑗  𝐸 𝑗Ψ 𝑗(r)𝑒−𝐸𝑗 𝜏 =  ∑︁  𝑗  �  −∇2  𝑟  2𝜇 + 𝑉Γ (𝑟)  �  Ψ 𝑗 (𝑟) 𝑒−𝐸𝑗 𝜏  =  �  −∇2  𝑟  2𝜇 + 𝑉Γ (𝑟)  �  𝐶Γ(r, 𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  (6)  3  Thermal interquark bottomonium potentials  Thomas Spriggs  This yields the form of the interquark potential for a given channel, 𝑉Γ, as  𝑉Γ(𝑟) =  1  𝐶Γ(r, 𝜏)  �∇2  𝑟  2𝜇 − 𝜕  𝜕𝜏  �  𝐶Γ(r, 𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  (7)  Note that in the continuum limit, we expect the potential to be function of 𝑟 = |r|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' There is explicit  time dependency in this form for the potential, and this will be studied in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='2  will discuss how the reduced quark mass, 𝜇, is set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  It is convenient to define the central potential, 𝑉C, obtained via the usual spin-average [10]  𝑉C = 1  4𝑉Pseudo Scalar + 3  4𝑉Vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  (8)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='2 Using momentum space to reformulate the calculation  This work is a continuation of [4] where more detail about the HAL QCD method can be  found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' We build upon [4] by using an efficient computation of the point-split correlation function,  𝐶Γ(r, 𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  For each 𝜏, a direct calculation of (1) requires a loop over all lattice sites x for each value of r  which is an expensive operation scaling as O(V2) where V is the spatial volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' What follows is  a method to reduce the cost of this computation by introducing a momentum space representation  for the propagator and correlation function, see the Appendix of [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  We introduce quark propagators, 𝐷−1(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' 𝑦), by Wick contracting the quark fields in the point-  split correlation function, (1),  𝐶Γ(r, 𝜏) = −  ∑︁  x  ⟨𝐷−1(x + r, 𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' 0, 0)Γ𝛾5  �  𝐷−1(x, 𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' 0, 0)  �†  𝛾5Γ†⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  (9)  Note that we have gauge fixed our configurations to the Coulomb gauge, and have replaced  the gauge connection, 𝑈(𝑥, 𝑥 + r) in (2) by unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' We now implicitly define the corresponding  momentum space quark propagator via  𝐷−1(y, 𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' 0, 0) = 1  𝑉  ∑︁  p  ˜𝐷−1(p, 𝜏)𝑒𝑖y·p,  (10)  in terms of the 3-momentum, p, which is conjugate to the position y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Introducing this momentum-  space quark propagator into (9) yields  𝐶Γ(r, 𝜏) = 1  𝑉  ∑︁  p  ⟨ ˜𝐷−1(p, 𝜏)Γ𝛾5 ˜𝐷−1(−p, 𝜏)𝛾5Γ†⟩𝑒𝑖p·r,  (11)  which we will use to implicitly define the momentum-space correlator, ˜𝐶Γ(p, 𝜏), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  ��Γ(r, 𝜏) = 1  𝑉  ∑︁  p  ˜𝐶Γ(p, 𝜏)𝑒𝑖p·r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  (12)  We note that once we have calculated ˜𝐶Γ(p, 𝜏), we can determine the desired correlator 𝐶Γ(r, 𝜏)  for any r using (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  4  Thermal interquark bottomonium potentials  Thomas Spriggs  At first sight, the conversion to momentum space does not produce any savings, because the  calculation of 𝐶Γ and ˜𝐷−1, defined via (10) and (12), are both O(V2) in the number of operations,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' the same as the direct method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' However both (10) and (12) are Fourier transforms, and so  significant speed-up for these steps can be achieved using the fast Fourier transform (FFT) algorithm  which scales as O(V log V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  Results  For better comparison with [4], and as progress towards the treatment of 𝐶Γ(r, 𝜏) for all r, we  consider here only the on-axis r data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Extensions to this will be discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='1 Time dependence  The potential is defined in (7) where there is an apparent explicit dependence on time, 𝜏, from  the correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' In Figure 2, the potential, 𝑉Vector, from (7) is plotted against 𝜏 for a variety  of distances r for our two extreme temperatures, 𝑇 = 141 and 352 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' We can see a clear 𝜏  dependence for small 𝜏 which increases with r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' However, for various ranges of 𝜏 and r there are  clear plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  In addition, we note that we would like to uncover temperature effects in the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' The  most accurate way of doing this is to compare different temperatures’ potentials obtained with the  same time window to avoid contamination by systematic artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  Based on these considerations, we restrict the range of 𝑟 and 𝜏 used in the determination of  the potential to those listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Notice that in selecting a time window, there is a trade-off  between the ranges of 𝑟 and 𝑇 for which the potential can be extracted: larger time windows give  access to a larger range of 𝑟, but over a smaller range of 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  In Figure 3 we show four determinations of the central potential, corresponding to the first four  time windows identified in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' In each plot we show the potentials for several temperatures, and  since these have been obtained by averaging over the same range of 𝜏, the temperature dependence  can be ascribed to temperature effects, rather than fitting artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  We find that the potential  consistently flattens as the temperature increases above 𝑇pc, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' There is little thermal  variation in the potential for 𝑇 ⪅ 𝑇pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  In Figure 3, the error bars show statistical errors only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' The curves are fits to the Cornell  potential, which will be discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='2 Quark mass dependence  Equation (7) contains the reduced quark mass, 𝜇, which needs to be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' In [7], the 1S  and 2S states were used to determine the bottom quark mass 𝑚𝑏, and thus the reduced quark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  In our simulations we do not have access to the 2S state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' We instead use the simple argument:  𝜇 ≡ 1  2𝑚𝑏 ≈ 1  2 𝑀Υ, with 𝑀Υ from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' We have tested the sensitivity of the potential on the quark  mass and found that the variation (within sensible 𝜇 ranges) is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  5  Thermal interquark bottomonium potentials  Thomas Spriggs  Figure 2: Time dependence in the potential restricting the range of 𝑟 that we can consider valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Shown for  two temperatures using the vector channel as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  Time window [𝑎𝜏]  𝑟 range [𝑎𝑠]  𝑟 range [fm]  Temperatures [MeV]  13-14  1-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='12-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='37  352-141  17-18  1-4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='12-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='49  281-141  19-22  1-5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='12-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='61  235-141  21-26  1-5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='12-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='61  201-141  24-30  1-6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='12-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='74  176-141  24-33  1-6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='12-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='74  156-141  Table 2: Range of displacements and temperatures allowed to best approximate time independence in𝑉(𝑟, 𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  Note that 𝑇pc = 181 MeV and thus the time windows below the solid line do not span this pseudocritical  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='3 Cornell potential fits  The Cornell potential [12] is a phenomenological description of a confining potential applicable  to heavy quarks in QCD and is given by  𝑉(𝑟) = −𝛼  𝑟 + 𝜎𝑟 + 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  (13)  Fits using (13) to our potential data are shown as solid curves in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' As can be seen these  reproduce the data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' When the string tension, 𝜎, in the Cornell potential is zero, this implies a  deconfined potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' In all cases above 𝑇pc, we find that 𝜎 decreases with increasing temperature,  confirming the expected thermal behaviour in the bottomonium system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Below𝑇pc the string tension  does not change within statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  Conclusion  The temperature dependence of the central interquark potential in the bottomonium system  using NRQCD quarks was explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' This work was an extension of [4] and use a momentum-space  approach which can improve the efficiency of the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Clear thermal effects in this potential  6  T = 141 MeV  8  Preliminary  X  6  [GeV] 4 Vector  来来  2  米  10  15  20  25  30  0  5  35  40  t/aT = 352 MeV  8  Preliminary  6  TI  [GeV] 4  2  10  15  0  5Thermal interquark bottomonium potentials  Thomas Spriggs  Figure 3: The central potential calculated from (7) (points), overlaid with a fit of these data to the Cornell  potential (13) (curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Each plot contains all temperatures and r ranges listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  were observed using a method which decoupled systematic “time window” artefacts from physical,  thermal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' A systematic flattening of the potential with increasing temperature above 𝑇pc was  observed, with no statistically significant variation in the potential for temperatures below 𝑇pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  This work will be extended in a number of directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' The potential will be calculated at all  possible spatial separations, r, rather than just the on-axis values used here, and channels beyond  the pseudoscalar and vector S-wave states will be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Also, a more robust definition of the  reduced quark mass will be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Finally, a direct comparison will be made between these  bottomonium results and those obtained for the charmonium potential using the same ensembles in  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  Acknowledgments  This work is supported by STFC grant ST/T000813/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' SK is supported by the National Research  Foundation of Korea under grant NRF-2021R1A2C1092701andgrantNRF-2021K1A3A1A16096820,  funded by the Korean government (MEST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' This work used the DiRAC Extreme Scaling service at  the University of Edinburgh, operated by the Edinburgh Parallel Computing Centre and the DiRAC  7  Time window: 13-14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='4  - T= 352 MeV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='2  T = 281 MeV  T = 235 MeV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='0  T = 201 MeV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='8  T = 176 MeV  [GeV]  T = 156 MeV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='6  T= 141 MeV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='4  C  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='8  Preliminary  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='7  r [fm]Time window: 17-18  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='4  T = 281 MeV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='2  T = 235 MeV  T = 201 MeV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='0  T = 176 MeV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='8  T = 156 MeV  [GeV]  T = 141 MeV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='8  Preliminary  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='7  r [fm]Time window: 19-22  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='4  T = 235 MeV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='2  T = 201 MeV  T = 176 MeV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='0  T = 156 MeV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='8  T = 141 MeV  [GeV]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
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+page_content='7  r [fm]Thermal interquark bottomonium potentials  Thomas Spriggs  Data Intensive service operated by the University of Leicester IT Services on behalf of the STFC  DiRAC HPC Facility (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
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+page_content=' This equipment was funded by BEIS capital funding via  STFC capital grants ST/R00238X/1, ST/K000373/1 and ST/R002363/1 and STFC DiRAC Opera-  tions grants ST/R001006/1 and ST/R001014/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' DiRAC is part of the UK National e-Infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  This work was performed using PRACE resources at Cineca (Italy), CEA (France) and Stuttgart  (Germany) via grants 2015133079, 2018194714, 2019214714 and 2020214714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' We acknowledge  the support of the Swansea Academy for Advanced Computing, the Supercomputing Wales project,  which is part-funded by the European Regional Development Fund (ERDF) via Welsh Government,  and the University of Southern Denmark and ICHEC, Ireland for use of computing facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' We  are grateful to the Hadron Spectrum Collaboration for the use of their zero temperature ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
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+page_content='  [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Eichten, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Gottfried, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Kinoshita, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Lane and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Yan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content=' D 21 (1980)  203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}
+page_content='  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'}

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+Early universe nucleosynthesis in massive
+conformal gravity
+F. F. Faria ∗
+Centro de Ciˆencias da Natureza,
+Universidade Estadual do Piau´ı,
+64002-150 Teresina, PI, Brazil
+Abstract
+We study the dynamics of the early universe in massive conformal
+gravity. In particular, we show that the theory is consistent with the
+observed values of the primordial abundances of light elements if we
+consider the existence of right-handed sterile neutrinos.
+PACS numbers: 04.62.+v, 04.60-m, 12.60.-i
+* [email protected]
+arXiv:2301.11954v1  [physics.gen-ph]  27 Jan 2023
+
+1
+Introduction
+It is well known that the standard ΛCDM cosmological model is consistent
+with most observations of the universe at both early and late times [1, 2].
+However, for this consistency to occur, a very small value for the cosmological
+constant (Λ) is required, which by far does not match with the huge value pre-
+dicted by quantum field theory (see [3] for a nice review). This discrepancy
+between the cosmological and quantum values of Λ is known as the cosmolog-
+ical constant problem [4]. Another important problem of ΛCDM is that the
+primordial lithium abundance from the early universe nucleosynthesis pre-
+dicted by it differs by about a factor of three from the observed abundance
+[5], which is known as the lithium problem. Despite several attempts over
+the years, no alternative cosmological model has succeeded in solving these
+two problems and being consistent with other cosmological observations at
+the same time.
+One of such models comes from massive conformal gravity (MCG), which
+is a conformally invariant theory of gravity in which the gravitational action
+is the sum of the Weyl action with the Einstein-Hilbert action conformally
+coupled to a scalar field [6]. Among so many cosmological models, we chose
+the MCG model because it fits well with the Type Ia supernovae (SNIa)
+data without the cosmological constant problem [7]. In addition, the theory
+is free of the van Dam-Veltman-Zakharov (vDVZ) discontinuity [8], can re-
+produce the orbit of binaries by the emission of gravitational waves [9] and
+is consistent with solar system observations [10]. Furthermore, MCG is a
+power-counting renormalizable [11, 12] and unitary [13] quantum theory of
+gravity.
+In this paper, we want to see if the MCG cosmology is consistent with
+the observed primordial abundances of light elements without the lithium
+problem. In Sec. 2, we describe the MCG cosmological equations. In Sec.
+3, we derive the matter energy-momentum tensor used in the theory. In Sec.
+4, we study the dynamics of the early MCG universe. In Sec. 5, we compare
+the early universe nucleosynthesis of MCG with cosmological observations. In
+Sec. 6, we analyze the evolution of the baryon density of the MCG universe.
+Finally, in Sec. 7, we present our conclusions.
+1
+
+2
+Massive conformal gravity
+The total MCG action is given by1 [8]
+S =
+�
+d4x √−g
+�
+ϕ2R + 6∂µϕ∂µϕ −
+1
+2α2CαβµνCαβµν
+�
++ 1
+c
+�
+d4xLm,
+(1)
+where ϕ is a scalar field called dilaton, α is a coupling constant,
+CαβµνCαβµν = RαβµνRαβµν − 4RµνRµν + R2 + 2
+�
+RµνRµν − 1
+3R2
+�
+(2)
+is the Weyl tensor squared, Rαµβν = ∂βΓα
+µν + · · · is the Riemann tensor,
+Rµν = Rαµαν is the Ricci tensor, R = gµνRµν is the scalar curvature, and
+Lm = Lm(gµν, Ψ) is the Lagrangian density of the matter field Ψ. It is worth
+noting that besides being invariant under coordinate transformations, the
+action (1) is also invariant under the conformal transformations
+˜Φ = Ω(x)−∆ΦΦ,
+(3)
+where Ω(x) is an arbitrary function of the spacetime coordinates, and ∆Φ is
+the scaling dimension of the field Φ, whose values are −2 for the metric field,
+0 for gauge bosons, 1 for scalar fields, and 3/2 for fermions.
+The variation of (1) with respect to gµν and ϕ gives the MCG field equa-
+tions
+ϕ2Gµν +6∂µϕ∂νϕ−3gµν∂ρϕ∂ρϕ+gµν∇ρ∇ρϕ2 −∇µ∇νϕ2 −α−2Wµν = 1
+2cTµν,
+(4)
+�
+∇µ∇µ − 1
+6R
+�
+ϕ = 0,
+(5)
+where
+Wµν
+=
+∇ρ∇ρRµν − 1
+3∇µ∇νR − 1
+6gµν∇ρ∇ρR + 2RρσRµρνσ − 1
+2gµνRρσRρσ
+−2
+3RRµν + 1
+6gµνR2
+(6)
+1This action is obtained from the action of Ref. [8] by rescaling ϕ →
+��
+32πG/3
+�
+ϕ
+and considering m =
+�
+3/64πGα.
+2
+
+is the Bach tensor,
+Gµν = Rµν − 1
+2gµνR
+(7)
+is the Einstein tensor,
+∇ρ∇ρϕ =
+1
+√−g∂ρ �√−g∂ρϕ
+�
+(8)
+is the generally covariant d’Alembertian for a scalar field, and
+Tµν = −
+2
+√−g
+δLm
+δgµν
+(9)
+is the matter energy-momentum tensor.
+Before we proceed, it is important to note that both the symmetries of
+the theory allow us to introduce in (1) a quartic self-interacting term of the
+dilaton λ
+� √−gϕ4 as well as interaction terms of the dilaton with the matter
+fields. In the case of the dilaton self-interaction term, we do not include it
+in the MCG action because this inclusion makes the flat metric no longer a
+solution of the field equations, which invalidates the S-matrix formulation.
+Although such a term is reintroduced in the effective action by quantum
+corrections, we can consider the renormalized value of the coupling constant
+λ equal zero so that the self-interacting term is present in the renormalized
+action only to cancel out the corresponding divergent term. In addition, we
+neglect the couplings between the dilaton and the matter fields because they
+make the field equation (5) no longer valid. This equation is fundamental to
+cancel non-renormalizable divergent terms that appear in the effective action
+[14].
+At scales below the Planck scale, the dilaton field acquires a spontaneously
+broken constant vacuum expectation value ϕ0 [15]. In this case, the field
+equations (4) and (5) become
+ϕ2
+0Gµν − α−2Wµν = 1
+2cTµν,
+(10)
+R = 0.
+(11)
+In addition, for ϕ = ϕ0, the MCG line element ds2 = (ϕ/ϕ0)2 gµνdxµdxν
+reduces to
+ds2 = gµνdxµdxν.
+(12)
+The full dynamics of the MCG universe can be described by (10)-(12) without
+loss of generality.
+3
+
+3
+Dynamical perfect fluid
+In order to find the MCG matter energy-momentum tensor, we consider the
+conformally invariant matter Lagrangian density [16]
+Lm = −√−gc
+�
+S2R+6∂µS∂µS+λS4+ i
+2ℏ
+�
+ψγµDµψ − Dµψγµψ
+�
+−ℏµSψψ
+�
+,
+(13)
+where S is a scalar Higgs field2, λ and µ are coupling constants, ψ = ψ†γ0 is
+the adjoint fermion field, Dµ = ∂µ + [γν, ∂µγν]/8 − [γν, γλ]Γλµν/8 (Γλµν is the
+Levi-Civita connection), and γµ are the general relativistic Dirac matrices,
+which satisfy the anti-commutation relation {γµ, γν} = 2gµν.
+By varying (13) with respect to S, ψ and ψ, we obtain the field equations
+12∇µ∇µS − 2RS − 4λS3 + ℏµψψ = 0,
+(14)
+iγµDµψ − µSψ = 0,
+(15)
+iDµψγµ + µSψ = 0.
+(16)
+Additionally, the substitution of (13) into (9) gives
+Tµν
+c
+=
+12∂µS∂νS − 6gµν∂ρS∂ρS + 2gµν∇ρ∇ρS2 − 2∇µ∇νS2
++ 2S2Gµν − gµν
+�
+λS4 + i
+2ℏ
+�
+ψγρDρψ − Dρψγρψ
+�
+− ℏµSψψ
+�
++ i
+4ℏ
+�
+ψγµDνψ − Dνψγµψ + ψγνDµψ − Dµψγνψ
+�
+.
+(17)
+Then, using (14)-(16) and ∇µ∇νS2 = 2(S∇µ∇νS + ∂µS∂νS) in (17), we find
+the energy-momentum tensor
+Tµν
+=
+c (8∂µS∂νS − 2gµν∂ρS∂ρS − 4S∇µ∇νS + gµνS∇ρ∇ρS)
++ 2cS2
+�
+Rµν − 1
+4gµνR
+�
++ T f
+µν,
+(18)
+where
+T f
+µν = i
+4cℏ
+�
+ψγµDνψ − Dνψγµψ + ψγνDµψ − Dµψγνψ
+�
+− 1
+4gµνcℏµSψψ (19)
+2Although the Higgs field is actually a doublet, and it is more likely that we must have
+two more scalar fields to get the correct quantum phenomenology at low energies [17],
+considering only a scalar Higgs field will not change the classical results of the theory.
+4
+
+is the fermion energy-momentum tensor.
+Considering that, at scales below the electroweak scale, the Higgs field
+acquires a spontaneously broken constant vacuum expectation value S0, and
+making some algebra, we find that (15) and (18) become
+�
+DµDµ −
+�mc
+ℏ
+�2�
+ψ = 0,
+(20)
+Tµν(S0, gµν) = 2cS2
+0
+�
+Rµν − 1
+4gµνR
+�
++ T f
+µν(S0, gµν),
+(21)
+where
+T f
+µν(S0, gµν) = i
+4cℏ
+�
+ψγµDνψ−Dνψγµψ+ψγνDµψ−Dµψγνψ
+�
+− 1
+4gµνmc2ψψ,
+(22)
+with m = µS0ℏ/c being the fermion mass. In flat spacetime, is not difficult
+to see that (20) and (22) reduce to
+�
+∂µ∂µ −
+�mc
+ℏ
+�2�
+ψ = 0,
+(23)
+T f
+µν(S0, ηµν) = i
+4cℏ
+�
+ψγµ∂νψ − ∂νψγµψ + ψγν∂µψ − ∂µψγνψ
+�
+− 1
+4ηµνmc2ψψ,
+(24)
+where now the Dirac matrices satisfy the anti-commutation relation {γµ, γν} =
+2ηµν.
+The normalized plane wave solution to (23) is given by
+ψ =
+1
+√V Ek
+uk eikµxµ,
+(25)
+where V is the volume, Ek =
+√
+k2c2 + m2c4 is the energy, uk is a spinor
+which satisfies [γµkµ + mc/ℏ] uk = 0, and kµ = (Ek/cℏ,⃗k/ℏ) is the wave
+vector, with ⃗k being the momentum and k = |⃗k|. By substituting (25) and
+its adjoint into (24), and using ukuk = −mc2, we obtain
+T f
+µν(S0, ηµν) =
+� c2ℏ2
+V Ek
+�
+kµkν +
+� m2c4
+4V Ek
+�
+ηµν.
+(26)
+5
+
+Incoherently adding to (26) the individual contributions of a set of six plane
+waves moving in the ± x, ± y and ± z directions, all with the same Ek and
+k, we can write the energy-momentum tensor (26) in the perfect fluid form
+T f
+µν(S0, ηµν) =
+�
+ρ + p
+c2
+�
+uµuν + ηµνp + ηµνc2ρΛ,
+(27)
+where
+c2ρ = 6Ek
+V
+(28)
+is the energy density of the fluid,
+p = 2k2c2
+V Ek
+(29)
+is the pressure of the fluid,
+c2ρΛ = 3m2c4
+2V Ek
+(30)
+is the vacuum energy (dark energy) density, and uµ is the four-velocity of the
+fluid, which is normalized to uµuµ = −c2. It follows from (28)-(30) that
+p = 0,
+ρΛ = 1
+4ρ,
+(31)
+for a non-relativistic perfect fluid (k2c2 ≪ m2c4), and
+p = 1
+3c2ρ,
+ρΛ = 0,
+(32)
+for a relativistic perfect fluid (k2c2 ≫ m2c4).
+In curved spacetime, the perfect fluid energy-momentum tensor (27) is
+generalized to
+T f
+µν(S0, gµν) =
+�
+ρ + p
+c2
+�
+uµuν + gµνp + gµνc2ρΛ.
+(33)
+Finally, the insertion of (33) into (21) gives the energy-momentum tensor of
+a dynamical perfect fluid
+Tµν(S0, gµν) = 2cS2
+0
+�
+Rµν − 1
+4gµνR
+�
++
+�
+ρ + p
+c2
+�
+uµuν +gµνp+gµνc2ρΛ. (34)
+6
+
+Taking the trace of (34), and substituting into the trace of (10), whose left
+hand side is zero due to the field equation (11) and the tracelessness of the
+Bach tensor (W = gµνWµν = 0), we arrive at
+T = gµνTµν = 3p − c2ρ + 4c2ρΛ = 0.
+(35)
+We can see from (31) and (32) that both non-relativistic and relativistic
+perfect fluids satisfies the tracelessness relation (35). For simplicity, we could
+isolate ρΛ in (35) and replace it in (34) as done in Ref. [7]. In this case, it is
+made clear that the vacuum energy density does not contribute directly to
+the dynamic evolution of the MCG universe, which solves the cosmological
+constant problem found in the ΛCDM model. However, here we will keep ρΛ
+so we don’t miss any physical details during the calculations.
+By substituting (34) into (10), and considering (11), we find
+�
+ϕ2
+0 − S2
+0
+�
+Rµν − α−2Wµν = 1
+2c
+��
+ρ + p
+c2
+�
+uµuν + gµνp + gµνc2ρΛ
+�
+,
+(36)
+which is the field equation that we will use in the study of the dynamics of
+the early MCG universe in the next section. But before that, it is important
+to compare MCG with another conformally invariant theory of gravity called
+conformal gravity (CG)3, whose action is given by [18]
+S = − 1
+2α2
+�
+d4x √−g
+�
+CαβµνCαβµν
+�
++ 1
+c
+�
+d4xLm.
+(37)
+By varying (37) with respect to gµν, we obtain the field equation
+− α−2Wµν = 1
+2cTµν,
+(38)
+where Tµν is given by (18). We can easily see the difference between the two
+theories by comparing (38) with (10) and (11). Just to stay within the scope
+of this paper, it is worth noting that CG does not pass the early universe
+nucleosynthesis test [19].
+3Although the difference between the two theories is quite obvious, as we will readily
+show next, MCG is often confused with CG. Perhaps this is because CG is much older
+and known than MCG.
+7
+
+4
+Early universe
+As usual, we consider that the geometry of the universe is described by the
+Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) line element
+ds2 = −c2dt2 + a(t)2
+�
+dr2
+1 − Kr2 + r2dθ2 + r2 sin2 θdφ2
+�
+,
+(39)
+where a = a(t) is the scale factor and K = -1, 0 or 1 is the spatial curvature.
+By substituting (39) and the fluid four-velocity uµ = (c, 0, 0, 0) into (36), we
+obtain4
+¨a
+a = −
+c
+6 (ϕ2
+0 − S2
+0)
+�
+c2ρ − c2ρΛ
+�
+,
+(40)
+¨a
+a + 2
+� ˙a
+a
+�2
++ 2Kc2
+a2
+=
+c
+2 (ϕ2
+0 − S2
+0)
+�
+p + c2ρΛ
+�
+,
+(41)
+where the dot denotes d/dt.
+Subtracting (40) from (41), and considering that5
+ϕ2
+0 =
+3c3
+32πG ≫ S2
+0,
+(42)
+we obtain
+� ˙a
+a
+�2
+= 8πG
+9c2
+�
+c2ρ + 3p + 2c2ρΛ
+�
+− Kc2
+a2 .
+(43)
+The combination of (43) with (40) then gives the energy continuity equation
+c2 ˙ρ + 3 ˙a
+a
+�
+c2ρ + p
+�
+− c2 ˙ρΛ = 0,
+(44)
+which can also be obtained by the conservation law ∇µT f
+µν = 0, with T f
+µν
+being the perfect fluid energy-momentum tensor (33).
+Using either (31) or (32) in (44), we get
+˙ρ + 4 ˙a
+aρ = 0,
+(45)
+4It is worth noting that Wµν = 0 for the FLRW spacetime.
+5This value of ϕ0 is necessary for the theory to be consistent with solar system obser-
+vations [10].
+8
+
+which, consequently, is valid for both non-relativistic and relativistic dynam-
+ical perfect fluids. As usual, we can write the solution to (45) in the form
+ρ = ρ0
+�a0
+a
+�4
+,
+(46)
+where, from now on, the subscript 0 denotes values at the present time t0.
+In the case of the early universe, which is composed by a very hot plasma
+dominated by relativistic particles (radiation), we find that (43) becomes
+˙a2 = 16πGa4
+0
+9a2
+ρr0 − Kc2.
+(47)
+where we used (32) and (46), with ρr being the mass density of the radiation.
+Since a is small in the early universe, we can neglect the curvature term on
+the right hand side of (47) and write it in the approximate form
+˙a2 = 16πGa4
+0
+9a2
+ρr0,
+(48)
+whose solution is given by
+a(t) =
+�64πGa4
+0ρr0
+9
+�1/4
+t1/2.
+(49)
+Finally, inserting (49) into the Hubble constant
+H = ˙a
+a,
+(50)
+we obtain
+H = 1
+2t,
+(51)
+which is the same relation between the Hubble constant and time that occurs
+in the early ΛCDM universe. However, since the MCG scale factor (49) is
+equal 0.9 times the value of the ΛCDM scale factor, the expansion of the early
+MCG universe is slower than the expansion of the early ΛCDM universe,
+which will give a difference in the values of the two Hubble constants, as we
+will show in the next section.
+9
+
+5
+Nucleosynthesis
+The abundances of light chemical elements in the early universe are mainly
+determined by one cosmological parameter, namely, the baryon-to-photon
+ratio η = nb/nγ, where nb and nγ are the number densities of baryons and
+photons in the universe. As usual, to find η we must first write the Hubble
+constant in function of temperature T using the Stefan-Boltzmann law
+ρr =
+�g∗aB
+2c2
+�
+T 4,
+(52)
+where aB is the radiation energy constant and g∗ counts the number of rela-
+tivistic particle species determining the energy density in radiation. Substi-
+tuting (52) and (49) into (46), we obtain
+t =
+�
+9c2
+32πGg∗aB
+�1/2 1
+T 2.
+(53)
+It then follows from (51) and (53) that
+H =
+�8πGg∗aB
+9c2
+�1/2
+T 2,
+(54)
+which is equal 0.82 times the value of the ΛCDM Hubble constant.
+In order to describe the thermal history of the early MCG universe, we
+must compare the Hubble constant in the form (54) with the collision rate
+of particle interactions
+Γ = nσv,
+(55)
+where n is the number density of particles, σ is their interaction cross section
+and v is the average velocity of the particles. A specific temperature that is of
+particular importance for the outcome of the early universe nucleosynthesis
+(EUN) is the one at which the thermal equilibrium between neutrons and
+protons begins to break down, which happens when H ∼ Γν, where
+Γν ≈ G2
+F
+c6ℏ7(kBT)5
+(56)
+is the collision rate of a neutrino with electrons or positrons, with GF being
+the Fermi constant and kB the Boltzmann constant.
+10
+
+By equating (54) with (56), and assuming that at the onset of the electron-
+positron annihilation the remaining relativistic particles are photons, elec-
+trons, positrons and left-handed neutrinos, for which g∗ = 10.75, we obtain
+kBTeq = 0.75 MeV.
+(57)
+We can see from (57) that the thermal equilibrium between neutrons and
+protons is maintained at temperatures above Teq = 8.7 × 109 K in the early
+MCG universe. At that time, the neutron-to-proton ratio was
+�nn
+np
+�
+eq
+= e−Q/kBTeq = 0.178,
+(58)
+where we used (57) and the neutron-proton energy difference Q = 1.239 MeV.
+Using (58), we can make a rough estimate that the final freeze-out neutron
+abundance is given by
+X∞
+n ∼ Xeq
+n =
+e−Q/kBTeq
+1 + e−Q/kBTeq = 0.15.
+(59)
+Including the neutron decay in our calculation, we find
+Xn(t) = X∞
+n e−t/τn = 0.15 e−t/τn,
+(60)
+where τn = 879.4 s is the neutron mean lifetime [20].
+The first light element formed in the early universe was deuterium (D),
+whose ratio to proton is approximately given by
+nD
+np
+≈ 6.9η
+� kBT
+mnc2
+�3/2
+exp
+� BD
+kBT
+�
+,
+(61)
+where we used (58) and BD = 2.2 MeV is the binding energy of deuterium.
+Noting that the EUN starts when nD ∼ np, it follows from (61) that
+6.9ηEUN
+�kBTEUN
+mnc2
+�3/2
+exp
+�
+BD
+kBTEUN
+�
+≈ 1,
+(62)
+where ηEUN and TEUN are the baryon-to-photon ratio and temperature of
+the EUN. We can see from (62) that we need the value of TEUN to find
+11
+
+ηEUN. Fortunately, we can find such value from the primordial helium (4He)
+abundance
+YP ≡ 4n4He
+nH
+=
+2Xn(tEUN)
+1 − Xn(tEUN),
+(63)
+where tEUN is the time of the EUN.
+The substitution of (60) and the observed value of the helium abundance
+YP = 0.245 [21] into (63) gives
+tEUN ≈ 279.7 s.
+(64)
+Then, by inserting (64) into (53), and considering that the electrons and
+protons are no longer relativistic after their annihilation, which gives g∗ =
+3.36, we obtain
+TEUN ≈ 8.8 × 108 K.
+(65)
+Finally, using (65) in (62), we arrive at
+ηEUN ≈ 5.12 × 10−8,
+(66)
+which produces abundances of other light elements besides helium orders
+of magnitude below the primordial abundances inferred from current obser-
+vations [22].
+However, this result does not automatically rule out MCG.
+If we consider that the theory has low energy (≲ eV) right-handed sterile
+neutrinos6, then we must replace g∗ = 10.75 by g∗ = 16.125 prior to the
+electron-positron annihilation and g∗ = 3.36 by g∗ = 5.04 after the electron-
+positron annihilation due to the contribution of the sterile neutrinos to the
+relativistic energy content of the universe. These replacements lead to the
+standard value
+ηEUN ≈ 6 × 10−10,
+(67)
+which is consistent with the observed abundances of all light elements with
+the exception of lithium7.
+6The existence of such neutrinos is allowed by the symmetries of the theory and may
+be responsible for the small masses of the left-handed neutrinos found in nature [23].
+7It is possible that the decay of the sterile neutrinos solves the inconsistency between
+the predicted and observed values of the lithium abundance [24].
+12
+
+6
+Baryon density
+Another important cosmological parameter that is determined by η is the
+baryon mass density ρb of the universe. In order to find the relation between
+these two parameters in the MCG universe, we start from the definitions of
+the baryon and photon number densities
+nb = ρb
+mN
+,
+(68)
+nγ = 2ζ(3)8π
+c3
+�kBT
+h
+�3
+≈ 2 × 107T 3,
+(69)
+where mN is the nucleons mass. The combination of (68), (69) and (52),
+with g∗ = 2, then gives the relation
+η =
+aB
+2 × 107mNc2
+ρb
+ργ
+T,
+(70)
+which is valid for any cosmological model. Noting that both ρb and ργ obey
+(46) in MCG, we can write (70) in the form
+η =
+aB
+2 × 107mNc2
+ρb0
+ργ0
+T,
+(71)
+which means that the baryon-to-photon ratio evolves over time in the MCG
+universe8, different to what happens in the ΛCDM universe where η is con-
+stant after the EUN.
+Using the current temperature of the universe T0 = 2.73 K in (52), with
+g∗ = 2, we find
+ργ0 = 4.65 × 10−31 kg/m3.
+(72)
+In addition, the use of (67) in (62), with 6.9 replaced by 6.5 due to the
+different value of (58) which leads to (67), gives
+TEUN ≈ 7.56 × 108 K.
+(73)
+8It would be important to check if (71) at the time of recombination is consistent with
+the value of η measured by cosmic microwave background (CMB) anisotropies. However,
+a theory for the growth of inhomogeneities in MCG has not yet been developed due to the
+complexity generated by the contribution of the Bach tensor in (10). Therefore, we will
+leave this analysis for future works.
+13
+
+Finally, substituting (67), (72) and (73) into (71), we obtain the current
+baryon mass density
+ρb0 = 1.46 × 10−36 kg/m3.
+(74)
+Since ρr and ρb evolve at the same rate in MCG, it follows from (72) and
+(74) that radiation always dominates the MCG universe.
+In fact, the scale factor is big at late times such that we can neglect
+the density term on the right hand side of (47), which makes the late MCG
+universe curvature dominated. In this case, we must impose K = −1, which
+gives the approximated solution
+a(t) = ct
+(75)
+in the late MCG universe. It is not difficult to show that for an open uni-
+verse with the scale factor (75) such as the late MCG universe, we have the
+luminosity distance
+dL(z) = c
+H0
+�(1 + z)2 − 1
+2
+�
+,
+(76)
+which fits well to SNIa data9 [6]. We intend to check if (75) provides good
+fits to other low redshift data in future works.
+Just to finish, it is important to note that the evolution of the baryon-
+to-photon ratio (71) causes the number of baryons Nb to decrease over time
+in the MCG universe. We can see this explicitly by substituting (46) and
+V ∼ a3 in
+Nb = nbV = ρbV
+mN
+,
+(77)
+which gives
+Nb ∼ ρb0a4
+0
+mNa.
+(78)
+Using (75), we find that the number of baryons evolves over time according
+to
+Nb ∼
+�ρb0c3t4
+0
+mN
+�
+t−1
+(79)
+in the late MCG universe.
+It follows from the energy continuity equation (44) that
+˙ρb + 3Hρb = ˙ρΛ.
+(80)
+9It is worth noting that the density term has not been neglected in Ref. [6], which in
+practice does not change the SNIa data fitting.
+14
+
+By comparing (80) with the standard adiabatic conservation equation, and
+noting that ˙ρΛ < 0, we conclude that the decrease in the number of baryons
+(79) is due to the decay of the baryons into dynamic vacuum10, which clearly
+leads to a violation of the conservation of the quantum numbers. However,
+we can see from (79) that the variation of the number of baryons should only
+be significant on cosmological time scales, which makes the decay of baryons
+into vacuum not observable in the laboratory.
+On the other hand, the non-conservation of baryons can have an im-
+portant impact on the evolution of inhomogeneous structures of the universe
+from the end of recombination until today. Due to the decrease in the amount
+of baryons in the MCG universe, it is expected that the formation of struc-
+tures happen much later than is observed or not happen at all. However,
+the evolution of cosmological structures does not depend only on baryons
+but also on dark matter, whose existence is necessary in MCG to explain the
+galaxy rotation curves and the deflection of light by galaxies [10]. Therefore,
+although the theory possibly has an extra scalar field that is a good candidate
+for dark matter [14], much still has to be studied to find out if the evolution
+of cosmological structures predicted by MCG is consistent with observations
+or not.
+7
+Final remarks
+Here we have shown that the abundances of light elements, including lithium,
+predicted by the early MCG cosmology are consistent with the observed val-
+ues provided the theory has right-handed sterile neutrinos, which is allowed
+by the symmetries of the theory. Even though we still need to check the
+existence of such neutrinos in experiments like the Mini Booster Neutrino
+Experiment (MiniBooNE) [25], this result is quite encouraging for us to con-
+tinue with the study of the theory.
+In addition, it was shown in this paper that the baryon-to-photon ratio
+of the MCG universe evolves over time. Although further studies are needed
+to verify whether this evolution is consistent with the value of the baryon-
+to-photon ratio determined by the CMB anisotropies, who knows it solves
+other early universe problems found in the ΛCDM model such as the baryon
+asymmetry problem. We intend to study this and other MCG cosmological
+predictions in future works.
+10This decaying process can be accounted by the Yukawa interaction µSψψ in (19).
+15
+
+References
+[1] A.G. Riess et al., Astron. J. 116, 1009 (1998); S. Perlmutter et al., ApJ
+517, 565 (1999).
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+cal parameters, Astron. Astrophys. 641, A6 (2020); Astron. Astrophys.
+652, C4 (2021).
+[3] S. E. Rugh and H. Zinkernagel, Stud. Hist. Phil. Sci. B 33, 663 (2002).
+[4] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
+[5] R. H. Cyburt, B. D. Fields, K. A. Olive and T.-H. Yeh, Rev. Mod. Phys.
+88, 015004 (2016).
+[6] F. F. Faria, Adv. High Energy Phys. 2014, 520259 (2014).
+[7] F. F. Faria, Mod. Phys. Lett. A 36, 2150115 (2021).
+[8] F. F. Faria, Adv. High Energy Phys. 2019, 7013012 (2019).
+[9] F. F. Faria, Eur. Phys. J. C 80, 645 (2020).
+[10] F. F. Faria, Mod. Phys. Lett. A 37, 2250033 (2022).
+[11] F. F. Faria, Eur. Phys. J. C 76, 188 (2016).
+[12] F. F. Faria, Eur. Phys. J. C 77, 11 (2017).
+[13] F. F. Faria, Eur. Phys. J. C 78, 277 (2018).
+[14] F. F. Faria, arXiv:1903.04893 [hep-th].
+[15] N. Matsuo, Gen. Relativ. Gravit. 22, 561 (1990).
+[16] P. D. Mannheim, Gen. Relativ. Gravit. 22, 289 (1990).
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+113 (2017).
+[18] P. D. Mannheim, Prog. Part. Nucl. Phys. 56, 340 (2006).
+[19] L. Knox and A. Kosowsky, arXiv:9311006 [astro-ph].
+16
+
+[20] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001
+(2018).
+[21] E. Aver et al., JCAP 03, 027 (2021).
+[22] P. A. Zyla et al. (Particle Data Group), PTEP 2020, 083C01 (2020).
+[23] K. A. Meissner and H. Nicolai, Phys. Lett. B 648, 312 (2007).
+[24] L. Salvati et al., JCAP 08, 022 (2016).
+[25] A. A. Aguilar-Arevalo et al. (MiniBooNE Collaboration), Phys. Rev.
+Lett. 121, 221801 (2018).
+17
+

4tFKT4oBgHgl3EQf9i5P/content/tmp_files/load_file.txt

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+page_content='Early universe nucleosynthesis in massive  conformal gravity  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Faria ∗  Centro de Ciˆencias da Natureza,  Universidade Estadual do Piau´ı,  64002-150 Teresina, PI, Brazil  Abstract  We study the dynamics of the early universe in massive conformal  gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In particular, we show that the theory is consistent with the  observed values of the primordial abundances of light elements if we  consider the existence of right-handed sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  PACS numbers: 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='+v, 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='60-m, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='-i  felfrafar@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='com  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='11954v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='gen-ph]  27 Jan 2023  1  Introduction  It is well known that the standard ΛCDM cosmological model is consistent  with most observations of the universe at both early and late times [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  However, for this consistency to occur, a very small value for the cosmological  constant (Λ) is required, which by far does not match with the huge value pre-  dicted by quantum field theory (see [3] for a nice review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' This discrepancy  between the cosmological and quantum values of Λ is known as the cosmolog-  ical constant problem [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Another important problem of ΛCDM is that the  primordial lithium abundance from the early universe nucleosynthesis pre-  dicted by it differs by about a factor of three from the observed abundance  [5], which is known as the lithium problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Despite several attempts over  the years, no alternative cosmological model has succeeded in solving these  two problems and being consistent with other cosmological observations at  the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  One of such models comes from massive conformal gravity (MCG), which  is a conformally invariant theory of gravity in which the gravitational action  is the sum of the Weyl action with the Einstein-Hilbert action conformally  coupled to a scalar field [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Among so many cosmological models, we chose  the MCG model because it fits well with the Type Ia supernovae (SNIa)  data without the cosmological constant problem [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In addition, the theory  is free of the van Dam-Veltman-Zakharov (vDVZ) discontinuity [8], can re-  produce the orbit of binaries by the emission of gravitational waves [9] and  is consistent with solar system observations [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Furthermore, MCG is a  power-counting renormalizable [11, 12] and unitary [13] quantum theory of  gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  In this paper, we want to see if the MCG cosmology is consistent with  the observed primordial abundances of light elements without the lithium  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' 2, we describe the MCG cosmological equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  3, we derive the matter energy-momentum tensor used in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  4, we study the dynamics of the early MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' 5, we compare  the early universe nucleosynthesis of MCG with cosmological observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' 6, we analyze the evolution of the baryon density of the MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' 7, we present our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  1  2  Massive conformal gravity  The total MCG action is given by1 [8]  S =  �  d4x √−g  �  ϕ2R + 6∂µϕ∂µϕ −  1  2α2CαβµνCαβµν  �  + 1  c  �  d4xLm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (1)  where ϕ is a scalar field called dilaton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' α is a coupling constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  CαβµνCαβµν = RαβµνRαβµν − 4RµνRµν + R2 + 2  �  RµνRµν − 1  3R2  �  (2)  is the Weyl tensor squared,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Rαµβν = ∂βΓα  µν + · · · is the Riemann tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  Rµν = Rαµαν is the Ricci tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' R = gµνRµν is the scalar curvature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' and  Lm = Lm(gµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Ψ) is the Lagrangian density of the matter field Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' It is worth  noting that besides being invariant under coordinate transformations, the  action (1) is also invariant under the conformal transformations  ˜Φ = Ω(x)−∆ΦΦ,  (3)  where Ω(x) is an arbitrary function of the spacetime coordinates, and ∆Φ is  the scaling dimension of the field Φ, whose values are −2 for the metric field,  0 for gauge bosons, 1 for scalar fields, and 3/2 for fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  The variation of (1) with respect to gµν and ϕ gives the MCG field equa-  tions  ϕ2Gµν +6∂µϕ∂νϕ−3gµν∂ρϕ∂ρϕ+gµν∇ρ∇ρϕ2 −∇µ∇νϕ2 −α−2Wµν = 1  2cTµν,  (4)  �  ∇µ∇µ − 1  6R  �  ϕ = 0,  (5)  where  Wµν  =  ∇ρ∇ρRµν − 1  3∇µ∇νR − 1  6gµν∇ρ∇ρR + 2RρσRµρνσ − 1  2gµνRρσRρσ  −2  3RRµν + 1  6gµνR2  (6)  1This action is obtained from the action of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' [8] by rescaling ϕ →  ��  32πG/3  �  ϕ  and considering m =  �  3/64πGα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  2  is the Bach tensor,  Gµν = Rµν − 1  2gµνR  (7)  is the Einstein tensor,  ∇ρ∇ρϕ =  1  √−g∂ρ �√−g∂ρϕ  �  (8)  is the generally covariant d’Alembertian for a scalar field, and  Tµν = −  2  √−g  δLm  δgµν  (9)  is the matter energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  Before we proceed, it is important to note that both the symmetries of  the theory allow us to introduce in (1) a quartic self-interacting term of the  dilaton λ  � √−gϕ4 as well as interaction terms of the dilaton with the matter  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In the case of the dilaton self-interaction term, we do not include it  in the MCG action because this inclusion makes the flat metric no longer a  solution of the field equations, which invalidates the S-matrix formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  Although such a term is reintroduced in the effective action by quantum  corrections, we can consider the renormalized value of the coupling constant  λ equal zero so that the self-interacting term is present in the renormalized  action only to cancel out the corresponding divergent term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In addition, we  neglect the couplings between the dilaton and the matter fields because they  make the field equation (5) no longer valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' This equation is fundamental to  cancel non-renormalizable divergent terms that appear in the effective action  [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  At scales below the Planck scale, the dilaton field acquires a spontaneously  broken constant vacuum expectation value ϕ0 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In this case, the field  equations (4) and (5) become  ϕ2  0Gµν − α−2Wµν = 1  2cTµν,  (10)  R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (11)  In addition, for ϕ = ϕ0, the MCG line element ds2 = (ϕ/ϕ0)2 gµνdxµdxν  reduces to  ds2 = gµνdxµdxν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (12)  The full dynamics of the MCG universe can be described by (10)-(12) without  loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  3  3  Dynamical perfect fluid  In order to find the MCG matter energy-momentum tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' we consider the  conformally invariant matter Lagrangian density [16]  Lm = −√−gc  �  S2R+6∂µS∂µS+λS4+ i  2ℏ  �  ψγµDµψ − Dµψγµψ  �  −ℏµSψψ  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (13)  where S is a scalar Higgs field2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' λ and µ are coupling constants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' ψ = ψ†γ0 is  the adjoint fermion field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Dµ = ∂µ + [γν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' ∂µγν]/8 − [γν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' γλ]Γλµν/8 (Γλµν is the  Levi-Civita connection),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' and γµ are the general relativistic Dirac matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  which satisfy the anti-commutation relation {γµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' γν} = 2gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  By varying (13) with respect to S, ψ and ψ, we obtain the field equations  12∇µ∇µS − 2RS − 4λS3 + ℏµψψ = 0,  (14)  iγµDµψ − µSψ = 0,  (15)  iDµψγµ + µSψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (16)  Additionally, the substitution of (13) into (9) gives  Tµν  c  =  12∂µS∂νS − 6gµν∂ρS∂ρS + 2gµν∇ρ∇ρS2 − 2∇µ∇νS2  + 2S2Gµν − gµν  �  λS4 + i  2ℏ  �  ψγρDρψ − Dρψγρψ  �  − ℏµSψψ  �  + i  4ℏ  �  ψγµDνψ − Dνψγµψ + ψγνDµψ − Dµψγνψ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (17)  Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' using (14)-(16) and ∇µ∇νS2 = 2(S∇µ∇νS + ∂µS∂νS) in (17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' we find  the energy-momentum tensor  Tµν  =  c (8∂µS∂νS − 2gµν∂ρS∂ρS − 4S∇µ∇νS + gµνS∇ρ∇ρS)  + 2cS2  �  Rµν − 1  4gµνR  �  + T f  µν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (18)  where  T f  µν = i  4cℏ  �  ψγµDνψ − Dνψγµψ + ψγνDµψ − Dµψγνψ  �  − 1  4gµνcℏµSψψ (19)  2Although the Higgs field is actually a doublet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' and it is more likely that we must have  two more scalar fields to get the correct quantum phenomenology at low energies [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  considering only a scalar Higgs field will not change the classical results of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  4  is the fermion energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  Considering that, at scales below the electroweak scale, the Higgs field  acquires a spontaneously broken constant vacuum expectation value S0, and  making some algebra, we find that (15) and (18) become  �  DµDµ −  �mc  ℏ  �2�  ψ = 0,  (20)  Tµν(S0, gµν) = 2cS2  0  �  Rµν − 1  4gµνR  �  + T f  µν(S0, gµν),  (21)  where  T f  µν(S0, gµν) = i  4cℏ  �  ψγµDνψ−Dνψγµψ+ψγνDµψ−Dµψγνψ  �  − 1  4gµνmc2ψψ,  (22)  with m = µS0ℏ/c being the fermion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In flat spacetime, is not difficult  to see that (20) and (22) reduce to  �  ∂µ∂µ −  �mc  ℏ  �2�  ψ = 0,  (23)  T f  µν(S0, ηµν) = i  4cℏ  �  ψγµ∂νψ − ∂νψγµψ + ψγν∂µψ − ∂µψγνψ  �  − 1  4ηµνmc2ψψ,  (24)  where now the Dirac matrices satisfy the anti-commutation relation {γµ, γν} =  2ηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  The normalized plane wave solution to (23) is given by  ψ =  1  √V Ek  uk eikµxµ,  (25)  where V is the volume, Ek =  √  k2c2 + m2c4 is the energy, uk is a spinor  which satisfies [γµkµ + mc/ℏ] uk = 0, and kµ = (Ek/cℏ,⃗k/ℏ) is the wave  vector, with ⃗k being the momentum and k = |⃗k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' By substituting (25) and  its adjoint into (24), and using ukuk = −mc2, we obtain  T f  µν(S0, ηµν) =  � c2ℏ2  V Ek  �  kµkν +  � m2c4  4V Ek  �  ηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (26)  5  Incoherently adding to (26) the individual contributions of a set of six plane  waves moving in the ± x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' ± y and ± z directions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' all with the same Ek and  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' we can write the energy-momentum tensor (26) in the perfect fluid form  T f  µν(S0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' ηµν) =  �  ρ + p  c2  �  uµuν + ηµνp + ηµνc2ρΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (27)  where  c2ρ = 6Ek  V  (28)  is the energy density of the fluid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  p = 2k2c2  V Ek  (29)  is the pressure of the fluid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  c2ρΛ = 3m2c4  2V Ek  (30)  is the vacuum energy (dark energy) density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' and uµ is the four-velocity of the  fluid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' which is normalized to uµuµ = −c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' It follows from (28)-(30) that  p = 0,  ρΛ = 1  4ρ,  (31)  for a non-relativistic perfect fluid (k2c2 ≪ m2c4), and  p = 1  3c2ρ,  ρΛ = 0,  (32)  for a relativistic perfect fluid (k2c2 ≫ m2c4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  In curved spacetime, the perfect fluid energy-momentum tensor (27) is  generalized to  T f  µν(S0, gµν) =  �  ρ + p  c2  �  uµuν + gµνp + gµνc2ρΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (33)  Finally, the insertion of (33) into (21) gives the energy-momentum tensor of  a dynamical perfect fluid  Tµν(S0, gµν) = 2cS2  0  �  Rµν − 1  4gµνR  �  +  �  ρ + p  c2  �  uµuν +gµνp+gµνc2ρΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' (34)  6  Taking the trace of (34), and substituting into the trace of (10), whose left  hand side is zero due to the field equation (11) and the tracelessness of the  Bach tensor (W = gµνWµν = 0), we arrive at  T = gµνTµν = 3p − c2ρ + 4c2ρΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (35)  We can see from (31) and (32) that both non-relativistic and relativistic  perfect fluids satisfies the tracelessness relation (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' For simplicity, we could  isolate ρΛ in (35) and replace it in (34) as done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In this case, it is  made clear that the vacuum energy density does not contribute directly to  the dynamic evolution of the MCG universe, which solves the cosmological  constant problem found in the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' However, here we will keep ρΛ  so we don’t miss any physical details during the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  By substituting (34) into (10), and considering (11), we find  �  ϕ2  0 − S2  0  �  Rµν − α−2Wµν = 1  2c  ��  ρ + p  c2  �  uµuν + gµνp + gµνc2ρΛ  �  ,  (36)  which is the field equation that we will use in the study of the dynamics of  the early MCG universe in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' But before that, it is important  to compare MCG with another conformally invariant theory of gravity called  conformal gravity (CG)3, whose action is given by [18]  S = − 1  2α2  �  d4x √−g  �  CαβµνCαβµν  �  + 1  c  �  d4xLm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (37)  By varying (37) with respect to gµν, we obtain the field equation  − α−2Wµν = 1  2cTµν,  (38)  where Tµν is given by (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' We can easily see the difference between the two  theories by comparing (38) with (10) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Just to stay within the scope  of this paper, it is worth noting that CG does not pass the early universe  nucleosynthesis test [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  3Although the difference between the two theories is quite obvious, as we will readily  show next, MCG is often confused with CG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Perhaps this is because CG is much older  and known than MCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  7  4  Early universe  As usual, we consider that the geometry of the universe is described by the  Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) line element  ds2 = −c2dt2 + a(t)2  �  dr2  1 − Kr2 + r2dθ2 + r2 sin2 θdφ2  �  ,  (39)  where a = a(t) is the scale factor and K = -1, 0 or 1 is the spatial curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  By substituting (39) and the fluid four-velocity uµ = (c, 0, 0, 0) into (36), we  obtain4  ¨a  a = −  c  6 (ϕ2  0 − S2  0)  �  c2ρ − c2ρΛ  �  ,  (40)  ¨a  a + 2  � ˙a  a  �2  + 2Kc2  a2  =  c  2 (ϕ2  0 − S2  0)  �  p + c2ρΛ  �  ,  (41)  where the dot denotes d/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  Subtracting (40) from (41), and considering that5  ϕ2  0 =  3c3  32πG ≫ S2  0,  (42)  we obtain  � ˙a  a  �2  = 8πG  9c2  �  c2ρ + 3p + 2c2ρΛ  �  − Kc2  a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (43)  The combination of (43) with (40) then gives the energy continuity equation  c2 ˙ρ + 3 ˙a  a  �  c2ρ + p  �  − c2 ˙ρΛ = 0,  (44)  which can also be obtained by the conservation law ∇µT f  µν = 0, with T f  µν  being the perfect fluid energy-momentum tensor (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  Using either (31) or (32) in (44), we get  ˙ρ + 4 ˙a  aρ = 0,  (45)  4It is worth noting that Wµν = 0 for the FLRW spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  5This value of ϕ0 is necessary for the theory to be consistent with solar system obser-  vations [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  8  which, consequently, is valid for both non-relativistic and relativistic dynam-  ical perfect fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' As usual, we can write the solution to (45) in the form  ρ = ρ0  �a0  a  �4  ,  (46)  where, from now on, the subscript 0 denotes values at the present time t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  In the case of the early universe, which is composed by a very hot plasma  dominated by relativistic particles (radiation), we find that (43) becomes  ˙a2 = 16πGa4  0  9a2  ρr0 − Kc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (47)  where we used (32) and (46), with ρr being the mass density of the radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  Since a is small in the early universe, we can neglect the curvature term on  the right hand side of (47) and write it in the approximate form  ˙a2 = 16πGa4  0  9a2  ρr0,  (48)  whose solution is given by  a(t) =  �64πGa4  0ρr0  9  �1/4  t1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (49)  Finally, inserting (49) into the Hubble constant  H = ˙a  a,  (50)  we obtain  H = 1  2t,  (51)  which is the same relation between the Hubble constant and time that occurs  in the early ΛCDM universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' However, since the MCG scale factor (49) is  equal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='9 times the value of the ΛCDM scale factor, the expansion of the early  MCG universe is slower than the expansion of the early ΛCDM universe,  which will give a difference in the values of the two Hubble constants, as we  will show in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  9  5  Nucleosynthesis  The abundances of light chemical elements in the early universe are mainly  determined by one cosmological parameter, namely, the baryon-to-photon  ratio η = nb/nγ, where nb and nγ are the number densities of baryons and  photons in the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' As usual, to find η we must first write the Hubble  constant in function of temperature T using the Stefan-Boltzmann law  ρr =  �g∗aB  2c2  �  T 4,  (52)  where aB is the radiation energy constant and g∗ counts the number of rela-  tivistic particle species determining the energy density in radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Substi-  tuting (52) and (49) into (46), we obtain  t =  �  9c2  32πGg∗aB  �1/2 1  T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (53)  It then follows from (51) and (53) that  H =  �8πGg∗aB  9c2  �1/2  T 2,  (54)  which is equal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='82 times the value of the ΛCDM Hubble constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  In order to describe the thermal history of the early MCG universe, we  must compare the Hubble constant in the form (54) with the collision rate  of particle interactions  Γ = nσv,  (55)  where n is the number density of particles, σ is their interaction cross section  and v is the average velocity of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' A specific temperature that is of  particular importance for the outcome of the early universe nucleosynthesis  (EUN) is the one at which the thermal equilibrium between neutrons and  protons begins to break down, which happens when H ∼ Γν, where  Γν ≈ G2  F  c6ℏ7(kBT)5  (56)  is the collision rate of a neutrino with electrons or positrons, with GF being  the Fermi constant and kB the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  10  By equating (54) with (56), and assuming that at the onset of the electron-  positron annihilation the remaining relativistic particles are photons, elec-  trons, positrons and left-handed neutrinos, for which g∗ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='75, we obtain  kBTeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='75 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (57)  We can see from (57) that the thermal equilibrium between neutrons and  protons is maintained at temperatures above Teq = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='7 × 109 K in the early  MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' At that time, the neutron-to-proton ratio was  �nn  np  �  eq  = e−Q/kBTeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='178,  (58)  where we used (57) and the neutron-proton energy difference Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='239 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  Using (58), we can make a rough estimate that the final freeze-out neutron  abundance is given by  X∞  n ∼ Xeq  n =  e���Q/kBTeq  1 + e−Q/kBTeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (59)  Including the neutron decay in our calculation, we find  Xn(t) = X∞  n e−t/τn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='15 e−t/τn,  (60)  where τn = 879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='4 s is the neutron mean lifetime [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  The first light element formed in the early universe was deuterium (D),  whose ratio to proton is approximately given by  nD  np  ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='9η  � kBT  mnc2  �3/2  exp  � BD  kBT  �  ,  (61)  where we used (58) and BD = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='2 MeV is the binding energy of deuterium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  Noting that the EUN starts when nD ∼ np, it follows from (61) that  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='9ηEUN  �kBTEUN  mnc2  �3/2  exp  �  BD  kBTEUN  �  ≈ 1,  (62)  where ηEUN and TEUN are the baryon-to-photon ratio and temperature of  the EUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' We can see from (62) that we need the value of TEUN to find  11  ηEUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Fortunately, we can find such value from the primordial helium (4He)  abundance  YP ≡ 4n4He  nH  =  2Xn(tEUN)  1 − Xn(tEUN),  (63)  where tEUN is the time of the EUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  The substitution of (60) and the observed value of the helium abundance  YP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='245 [21] into (63) gives  tEUN ≈ 279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='7 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (64)  Then, by inserting (64) into (53), and considering that the electrons and  protons are no longer relativistic after their annihilation, which gives g∗ =  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='36, we obtain  TEUN ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='8 × 108 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (65)  Finally, using (65) in (62), we arrive at  ηEUN ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='12 × 10−8,  (66)  which produces abundances of other light elements besides helium orders  of magnitude below the primordial abundances inferred from current obser-  vations [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  However, this result does not automatically rule out MCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  If we consider that the theory has low energy (≲ eV) right-handed sterile  neutrinos6, then we must replace g∗ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='75 by g∗ = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='125 prior to the  electron-positron annihilation and g∗ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='36 by g∗ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='04 after the electron-  positron annihilation due to the contribution of the sterile neutrinos to the  relativistic energy content of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' These replacements lead to the  standard value  ηEUN ≈ 6 × 10−10,  (67)  which is consistent with the observed abundances of all light elements with  the exception of lithium7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  6The existence of such neutrinos is allowed by the symmetries of the theory and may  be responsible for the small masses of the left-handed neutrinos found in nature [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  7It is possible that the decay of the sterile neutrinos solves the inconsistency between  the predicted and observed values of the lithium abundance [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  12  6  Baryon density  Another important cosmological parameter that is determined by η is the  baryon mass density ρb of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In order to find the relation between  these two parameters in the MCG universe, we start from the definitions of  the baryon and photon number densities  nb = ρb  mN  ,  (68)  nγ = 2ζ(3)8π  c3  �kBT  h  �3  ≈ 2 × 107T 3,  (69)  where mN is the nucleons mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' The combination of (68), (69) and (52),  with g∗ = 2, then gives the relation  η =  aB  2 × 107mNc2  ρb  ργ  T,  (70)  which is valid for any cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Noting that both ρb and ργ obey  (46) in MCG, we can write (70) in the form  η =  aB  2 × 107mNc2  ρb0  ργ0  T,  (71)  which means that the baryon-to-photon ratio evolves over time in the MCG  universe8, different to what happens in the ΛCDM universe where η is con-  stant after the EUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  Using the current temperature of the universe T0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='73 K in (52), with  g∗ = 2, we find  ργ0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='65 × 10−31 kg/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (72)  In addition, the use of (67) in (62), with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='9 replaced by 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='5 due to the  different value of (58) which leads to (67), gives  TEUN ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='56 × 108 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (73)  8It would be important to check if (71) at the time of recombination is consistent with  the value of η measured by cosmic microwave background (CMB) anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' However,  a theory for the growth of inhomogeneities in MCG has not yet been developed due to the  complexity generated by the contribution of the Bach tensor in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Therefore, we will  leave this analysis for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  13  Finally, substituting (67), (72) and (73) into (71), we obtain the current  baryon mass density  ρb0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='46 × 10−36 kg/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (74)  Since ρr and ρb evolve at the same rate in MCG, it follows from (72) and  (74) that radiation always dominates the MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  In fact, the scale factor is big at late times such that we can neglect  the density term on the right hand side of (47), which makes the late MCG  universe curvature dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' In this case, we must impose K = −1, which  gives the approximated solution  a(t) = ct  (75)  in the late MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' It is not difficult to show that for an open uni-  verse with the scale factor (75) such as the late MCG universe, we have the  luminosity distance  dL(z) = c  H0  �(1 + z)2 − 1  2  �  ,  (76)  which fits well to SNIa data9 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' We intend to check if (75) provides good  fits to other low redshift data in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  Just to finish, it is important to note that the evolution of the baryon-  to-photon ratio (71) causes the number of baryons Nb to decrease over time  in the MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' We can see this explicitly by substituting (46) and  V ∼ a3 in  Nb = nbV = ρbV  mN  ,  (77)  which gives  Nb ∼ ρb0a4  0  mNa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (78)  Using (75), we find that the number of baryons evolves over time according  to  Nb ∼  �ρb0c3t4  0  mN  �  t−1  (79)  in the late MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  It follows from the energy continuity equation (44) that  ˙ρb + 3Hρb = ˙ρΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  (80)  9It is worth noting that the density term has not been neglected in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' [6], which in  practice does not change the SNIa data fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  14  By comparing (80) with the standard adiabatic conservation equation, and  noting that ˙ρΛ < 0, we conclude that the decrease in the number of baryons  (79) is due to the decay of the baryons into dynamic vacuum10, which clearly  leads to a violation of the conservation of the quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' However,  we can see from (79) that the variation of the number of baryons should only  be significant on cosmological time scales, which makes the decay of baryons  into vacuum not observable in the laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  On the other hand, the non-conservation of baryons can have an im-  portant impact on the evolution of inhomogeneous structures of the universe  from the end of recombination until today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Due to the decrease in the amount  of baryons in the MCG universe, it is expected that the formation of struc-  tures happen much later than is observed or not happen at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' However,  the evolution of cosmological structures does not depend only on baryons  but also on dark matter, whose existence is necessary in MCG to explain the  galaxy rotation curves and the deflection of light by galaxies [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Therefore,  although the theory possibly has an extra scalar field that is a good candidate  for dark matter [14], much still has to be studied to find out if the evolution  of cosmological structures predicted by MCG is consistent with observations  or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  7  Final remarks  Here we have shown that the abundances of light elements, including lithium,  predicted by the early MCG cosmology are consistent with the observed val-  ues provided the theory has right-handed sterile neutrinos, which is allowed  by the symmetries of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Even though we still need to check the  existence of such neutrinos in experiments like the Mini Booster Neutrino  Experiment (MiniBooNE) [25], this result is quite encouraging for us to con-  tinue with the study of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  In addition, it was shown in this paper that the baryon-to-photon ratio  of the MCG universe evolves over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Although further studies are needed  to verify whether this evolution is consistent with the value of the baryon-  to-photon ratio determined by the CMB anisotropies, who knows it solves  other early universe problems found in the ΛCDM model such as the baryon  asymmetry problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' We intend to study this and other MCG cosmological  predictions in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  10This decaying process can be accounted by the Yukawa interaction µSψψ in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content='  15  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
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+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Cosmologi-  cal parameters, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
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+page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
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+page_content=' Faria, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
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+page_content=' Faria, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
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+page_content=' Faria, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
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+page_content=' Faria, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'}
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@@ -0,0 +1,1640 @@
+’0x / N.N. and N.N.
+(Guest Editors)
+Volume 0 (200x), Number 0
+Interactive Control over Temporal Consistency
+while Stylizing Video Streams
+Sumit Shekhar1∗
+, Max Reimann1∗
+, Moritz Hilscher1, Amir Semmo1,2
+,
+Jürgen Döllner1, and Matthias Trapp1
+1Hasso Plattner Institute for Digital Engineering, University of Potsdam, Germany
+2Digital Masterpieces GmbH, Germany
+(“*” denotes equal contribution)
+Abstract
+With the advent of Neural Style Transfer (NST), stylizing an image has become quite popular. A convenient way for extending
+stylization techniques to videos is by applying them on a per-frame basis. However, such per-frame application usually lacks
+temporal consistency expressed by undesirable flickering artifacts. Most of the existing approaches for enforcing temporal
+consistency suffers from one or more of the following drawbacks: They (1) are only suitable for a limited range of techniques, (2)
+typically do not support live processing as they require the complete video as input, (3) cannot provide consistency for the task
+of stylization, or (4) do not provide interactive consistency-control. Note that existing consistent video-filtering approaches aim
+to completely remove flickering artifacts and thus do not respect any specific consistency-control aspect. For stylization tasks,
+however, consistency-control is an essential requirement where a certain amount of flickering can add to the artistic look and
+feel. Moreover, making this control interactive is paramount from a usability perspective. To achieve the above requirements, we
+propose an approach that can stylize video streams while providing interactive consistency-control. For achieving interactive
+performance, we develop a lite optical-flow network that operates at 80 Frames per second (FPS) on desktop systems with
+sufficient accuracy. Further, we employ an adaptive combination of local and global consistent features and enable interactive
+selection between the two. By objective and subjective evaluation, we show that our method is superior to state-of-the-art
+approaches.
+CCS Concepts
+• Computing methodologies ,..., Image-based rendering; Non-photorealistic rendering; Image processing;
+1. Introduction
+For thousands of years, paintings have served as a tool for vi-
+sual communication and expression. However, it was not until
+the late 20th century that computers were used to simulate paint-
+ings [Hae90]. In the course of following decades, the field of artistic
+stylization [KCWI13] has significantly developed and extended by
+NSTs [SID17,JYF∗20]. Even though a large number of image styl-
+ization techniques exist, extending these to video remains challeng-
+ing. A major obstacle in this regard is the enforcement of temporal
+coherence between stylized video frames. With the proliferation of
+video streaming applications, stylizing video streams has become
+popular, however, the requirements of low-latency processing add
+additional challenges. Most of the existing methods, to address the
+above, can be classified into one of the following four categories:
+Style Specific. A common approach is to develop a specific
+method for a particular artistic style and exploit its characteris-
+tics for temporal coherency [BNTS07]. Such methods work ef-
+fectively for the specific target style, however, do not generalize
+well. Many of these specialized approaches have been discussed
+by Bénard et al. [BTC13].
+Coherent Noise. Another class of techniques adopt and transform
+a generic, temporally-coherent noise function to yield a visually
+plausible stylized output [BLV∗10, KP11]. Compared to target-
+based coherence enforcement [BNTS07], these are applicable to
+a wider range of techniques but are limited for scenarios with
+rapid temporal changes.
+Stylization by Example. More recently, authors have adopted a
+stylization-by-example approach to support a wide range of styl-
+ization techniques [BCK∗13, JST∗19, TFK∗20, FKL∗21]. How-
+ever, this approach requires the paring of the complete video and
+keyframe marking. Thus, by design it is not applicable to video
+streams.
+Consistent Video Filtering. One can also enable stylization of
+video streams using consistent video filtering techniques. Exist-
+ing approaches are either not well-suited for Image-based Artis-
+tic Rendering (IB-AR) [BTS∗15,YCC17] (Fig. 1) or do not pro-
+vide interactive consistency control [LHW∗18,TDKP21], which
+submitted to 200x.
+arXiv:2301.00750v1  [cs.GR]  2 Jan 2023
+
+2
+S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams
+Table 1: Comparing existing consistent video filtering methods with ours with regards to consistency-control. Here, the color green denotes
+the aspect which is favourable to interactive consistency-control while the color red denotes otherwise (“NA” stands for Not-Applicable).
+Bonneel et al. [BTS∗15]
+Yao et al. [YCC17]
+Lai et al. [LHW∗18]
+Shekhar et al. [SST∗19]
+Thiomonier et al. [TDKP21]
+Ours
+Requires pre-processing?
+No
+Yes
+No
+Yes
+No
+No
+Provides consistency-control at inference time?
+Yes
+No
+No
+Yes
+No
+Yes
+Is the consistency-control interactive?
+No
+NA
+NA
+Yes
+NA
+Yes
+(a) Input
+(b) Processed
+(c) Ours
+(d) Lai et al. [LHW∗18]
+(e) Bonneel et al. [BTS∗15]
+Figure 1: For the top-row: first two columns depicts (a) input and (b) processed result for frame-24, column three to five depict the correspond-
+ing consistent output using (c) Ours (d) Lai’s, and (e) Bonneel’s method. For the mid-row: depict the corresponding results for frame-80.
+For the bottom-row: we show the Temporal Slice Image (TSI) for the entire video sequence depicting long-term temporal similarity with the
+per-frame processed output. Note, that our method is able to preserve the look and feel of the per-frame processed result in comparison to
+the method of Lai et al. which suffers from color bleeding artifacts while the stylized textures are lost for the output of Bonneel et al. . Please
+see the supplementary material for video results.
+is an essential requirement for artistic rendering [FLJ∗14]. Cur-
+rently, the only method that provides interactive consistency-
+control is limited to offline processing and requires pre-
+processing [SST∗19].
+We aim to develop a temporal-consistency enforcement ap-
+proach for artistic stylization techniques that provides (1) interac-
+tive consistency-control and (2) online processing to facilitate the
+application to video streams.
+A determining factor towards the slow performance of ex-
+isting online and interactive consistent video filtering tech-
+nique [BTS∗15] is the costly step of optic-flow computation. Pre-
+vious works using learning-based methods are able to achieve a
+considerable accuracy for optic-flow estimation [TD20, JCL∗21].
+However, we argue that such a high accuracy is not particularly
+necessary to enforce temporal consistency for artistic stylization
+tasks. To validate our conjecture, we conduct a user study, wherein
+the participants prefer the final consistent video output generated
+using our flow network as compared to that being obtained us-
+ing State-of-the-art (SOTA) approaches. We define artistic styl-
+ization as the adaptation of colors, textures, and strokes. While
+our approach is effective for most image-based stylization tech-
+niques (e.g., NSTs, algorithmic filtering), it is not able to han-
+dle significant shape or content inconsistencies between frames in-
+troduced by semantically-driven image synthesis (e.g., image-to-
+image diffusion-based models [RBL∗22]), as flow-based warping
+is insufficient to enforce consistency in these cases.
+In contrast to accuracy, little attention has been paid to improve
+the run-time performance of optic-flow estimation, but which is
+essential for online-interactive editing. To this end, we develop a
+lite optic-flow neural network that runs at a high-speed (approx.
+80 FPS on mid-tier desktop GPUs) while maintaining sufficient
+accuracy. The compact network is also deployable on mobile de-
+vices (iPhones and iPads) where it runs at interactive frame rates
+(24 FPS on iPad Pro 2020). We use the optic-flow output from the
+above network to enforce warping-based consistency at interactive
+frame rates. Moreover, we construct an adaptive consistency prior
+which allows for global and local temporal-consistency control. To
+summarize we present the following contributions:
+1. A novel approach for making per per-frame stylized videos tem-
+porally consistent via adaptive combination of local and global
+consistency features which allows for interactive consistency-
+control.
+2. A lite optic-flow network, to achieve interactive performance,
+that runs at 80 FPS on a mid-tier desktop PC and at 24 FPS on a
+mobile device while achieving reasonable accuracy.
+2. Background & Related Work
+Consistent Video Filtering. Lang et al.
+[LWA∗12] propose
+a solution to enforce temporal consistency for a large-class of
+optimization-based problems via iterative filtering along the mo-
+tion path. Dong et al. [DBZY15] address the problem of temporal
+inconsistency for enhancement algorithms by dividing individual
+video frames into multiple regions and performing a region-based
+spatio-temporal optimization. Bonneel et al.
+[BTS∗15] was the
+submitted to 200x.
+
+S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams
+3
+𝐼𝑡−1
+𝐼𝑡−1
+𝐼𝑡𝐼𝑡
+𝐼𝑡+1
+𝐼𝑡+1
+𝑃𝑡
+𝑃𝑡
+𝑃𝑡−1
+𝑃𝑡−1
+𝑃𝑡+1
+𝑃𝑡+1
+𝑂𝑡−1
+𝑂𝑡−1
+𝑤𝑝
+𝑤𝑛
+ϴ𝑙
+ϴ𝑙
+Linear 
+combination
+ϴ𝑔
+ϴ𝑔
+Linear 
+combination
+𝐴𝑡
+𝐴𝑡
+Optimization 
+Solving
+𝑂𝑡
+𝑂𝑡
+𝑤𝑝
+Use 𝑤𝑝 and 𝑤𝑛
+for combining
+1
+2
+3
+4
+5
+Estimate 𝑤𝑐
+for optimization
+Figure 2: Schematic overview of our approach: (1) We start by calculating the warping weights wp and wn using Eqn. 3. (2) The computed
+weights are used to linearly combine Pt, Pt−1, and Pt+1 to obtain the locally consistent image Lt, see Eqn. 2. (3) To obtain the globally
+consistent version Gt we warp the output at previous time instance Ot−1 as depicted in Eqn. 4. (4) The local and global consistent images, Lt
+and Gt, are linearly combined to obtain a temporally smooth version At, see Eqn. 5. (5) To include high-frequency details from the per-frame
+processed result, At and Pt is adaptively combined via the optimization in Eqn. 1 using the weights wc (Eqn. 7) to obtain the final result Ot.
+first to present a generalized approach for consistent video filter-
+ing which is agnostic to the type of filtering applied on individ-
+ual video-frames. The method combines gradient-based character-
+istics of the per-frame processed result with the warped version of
+the previous-frame output using a gradient-domain based optimiza-
+tion scheme. Yao et al. [YCC17] propose a similar approach how-
+ever considers multiple key-frames for warping-based consistency
+to avoid problems due to occlusion. Both of the approaches assume
+that the gradient of the processed video is similar to that of the
+input video and thus cannot handle artistic rendering tasks where
+new gradients resembling brush strokes are generated as part of
+the stylization process. Moreover, due to slow optic-flow computa-
+tion they are non-interactive in nature. Shekhar et al. [SST∗19]
+employs a similar formulation as Bonneel et al. , with the dif-
+ference of using a temporally denoised version of the current-
+frame for consistency guidance. However, the temporal denois-
+ing requires the complete video as input making the method of-
+fline in nature. Lai et al. [LHW∗18] propose the first learning-
+based technique in this context. The authors employ perceptual
+loss to enforce similarity with the processed frames and for con-
+sistency make use of short-term and long-term temporal losses.
+Thimonier et al. [TDKP21] employ a ping-pong loss and a cor-
+responding training procedure for temporal consistency. Both the
+learning based technique are faster than their optimization-based
+counterpart since they do not perform optic-flow computation at
+test time. However, these learning based techniques do not allow
+to control the degree of consistency in the final output which is
+vital for the task of stylization. Thus, the above discussed meth-
+ods are either non-interactive/offline or do not provide any con-
+sistency control at inference time. Our approach addresses these
+limitations (Tab. 1).
+Optic Flow for Consistent Filtering. Both Booneel et al. and
+Yao et al. use the PatchMatch algorithm [BSFG09] for flow-based
+warping, however, the slow performance of PatchMatch makes
+them non-interactive. Lai et al. use FlowNet 2.0 [IMS∗17] for flow-
+based warping to design their short-term and long-term temporal
+consistency losses. FlowNet 2.0 is on par with the quality of state-
+of-the-art classical methods, however, due to large number of pa-
+rameters and operations, achieves only interactive frame rates even
+on high-end desktop Graphical Processing Units (GPUs). An im-
+proved compact optic-flow Convolutional Neural Network (CNN)
+is proposed by Sun et al. [SYLK18] – PWC-Net. It combines
+coarse-to-fine estimation with pyramidal image features, correla-
+tion, warping, and CNN-based estimation. Furthermore, a refine-
+ment CNN is stacked at the end to improve the final flow estimate.
+PWC-Net is orders of magnitude smaller than FlowNet 2.0, runs
+at real-time frame rates using desktop GPUs. Liu et al. [LZH∗20]
+employ their approach to train a similar architecture in an un-
+supervised setting and achieve reasonable accuracy – ARFlow.
+LiteFlowNet and its successor LiteFlowNet2, both proposed by
+Hui et al. [HTL18, HTL20], have similar compact architectures.
+Further improvement in accuracy is achieved by models using iter-
+ative refinement, such as RAFT [TD20] and transformer modules
+such as GMA [JCL∗21], however they heavily trade runtime for ac-
+curacy. Based on a runtime-accuracy comparison (see Sec. 3.2), we
+select PWC-Net as a base network to develop a "Lite" flow network
+with improved performance for interactive consistent filtering.
+Temporal
+Consistency
+for
+Video
+Stylization. Litwinow-
+icz [Lit97] describes a technique to apply an impressionist effect
+on images and videos. For enforcing temporal coherence, optical
+flow was used to transform the brush strokes from one frame to
+the next. Winnemöller et al. [WOG06] develop a real-time video
+and image abstraction framework. The authors employ soft quan-
+tization that spreads over a larger area, thus significantly reducing
+temporal incoherence. Bousseau et al. [BNTS07] advects texture
+in forward and backward direction using optical flow for coherent
+water-colorization of videos. Numerous such specialized video-
+based approaches have been discussed by Bénard et al. [BTC13].
+The above classical IB-AR techniques approximate rendering
+primitives by modifying traditional image filters. Most often,
+they use low-level image features for modeling and fail to model
+structures resembling a particular style. Recently, deep CNNs
+were successfully used to transfer high-level style attributes from
+a painting onto a given image [GEB16]. Various methods have
+been proposed to extend the above for videos [HWL∗17,CLY∗17,
+GJAF17,RDB18,LLKY19,PP19,DTD∗21]. Ruder et al. [RDB18]
+submitted to 200x.
+
+4
+S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams
+propose novel initialization technique and loss functions for
+consistent stylized output even in cases with large motion and
+strong occlusion. The methods of Gupta et al.
+[GJAF17],
+Chen et al. [CLY∗17], and Huang et al. [HWL∗17] enforce con-
+sistency via certain formulation of temporal loss and use optical
+flow based warping only during the training phase thus achieving
+fast performance. Puy and Pérez [PP19] develop a flexible deep
+CNN for controllable artistic style transfer that allows for addition
+of a temporal regularizer at testing time to remove the flickering
+artefacts. The above method comes closest in terms of providing
+some consistency control at test time for NST-based methods.
+However, they cannot handle classical stylization techniques.
+Stylization by example [BCK∗13, JST∗19, TFK∗20, FKL∗21]
+caters to both (classical and neural) paradigms via priors involving
+keyframe-based warping but can only be applied as an offline
+process. We aim to propose a generic solution which is agnostic
+to the type of stylization and provides online performance and
+interactive consistency-control.
+3. Method
+3.1. Temporal Consistency Enforcement
+Given an input video stream ...It−1, It, It+1,... and its per-frame
+processed version ...Pt−1, Pt, Pt+1,..., we seek to find a tempo-
+rally consistent output ...Ot−1, Ot, Ot+1 .... Our method is ag-
+nostic to the stylization technique f applied to each frame, where
+Pt = f(It). However, it is necessary for f to not introduce signifi-
+cant shape or content inconsistencies between consecutive frames,
+as the changes in the stylized frames should correspond to the op-
+tical flow (calculated based on the content). We initialize the con-
+sistent output for the first frame as its per-frame processed result
+i.e., O1 = P1. To obtain the output for subsequent frames (Ot at any
+given instance t) we require only a snippet of input (It−1,It,It+1)
+and processed streams (Pt−1,Pt,Pt+1), and the consistent output at
+the previous instance Ot−1. For enforcing consistency, we solve the
+following gradient-domain optimization scheme:
+E(Ot) =
+�
+Ω
+�
+||∇Ot −∇Pt||2
+�
+��
+�
+data
++ wc||Ot −At||2
+�
+��
+�
+smoothness
+�
+dΩ.
+(1)
+where Ω represents the image domain. The data term in this opti-
+mization enforces similarity with the per-frame processed result Pt
+in the gradient-domain. Thus, high-frequency details are taken from
+Pt and the smoothness term enforces temporal-consistency where
+low-frequency content is taken from the image At. The optimiza-
+tion formulation in Eqn. 1 is commonly known as screened Pois-
+son equation and has been successfully employed for various image
+Table 2: Constituent elements of smoothness term in Eqn. 1
+for different methods. Here, ws and Td refers to saliency-based
+weights and temporally-denoised image respectively, introduced by
+Shekhar et al.
+Method
+Weight
+Consistent Image
+Ours
+wc
+At
+Boneell et al. [BTS∗15]
+wp
+Γ(Ot−1)
+Shekhar et al. [SST∗19]
+ws
+Td
+editing applications [BCCZ08,BZCC10]. In the context of consis-
+tent video filtering, it was first used by Bonneel et al. [BTS∗15]
+followed by Shekhar et al. [SST∗19] (Tab. 2). However, our nov-
+elty is the way in which we construct our smoothness term which,
+unlike previous approaches, considers both global and local consis-
+tency aspects. Our novel smoothness term is able to better preserve
+the color and textures in the stylized output while providing both
+short-term and long-term temporal consistency.
+Local Consistency. For enforcing temporal consistency at a local
+level, we use optic-flow to warp neighboring per-frame processed
+results to the current time instance t. This is perfomred by comput-
+ing an adaptive combination of (1) warped previous per-frame pro-
+cessed image Γ(Pt−1), (2) warped next per-frame processed image
+Γ(Pt+1), and (3) the current per-frame processed image Pt, where
+Γ is the warping function. By including both backward and for-
+ward warping in our formulation, we are able to significantly re-
+duce artefacts due to occlusion and flow inaccuracies. The linear
+combination of (1), (2), and (3) gives us a locally consistent ver-
+sion Lt where,
+Lt = (1−(wp+wn))·Pt + wp·Γ(Pt−1) + wn·Γ(Pt+1).
+(2)
+The weights wp and wn capture the inaccuracies in the warping of
+previous and next frames respectively and are defined as follows:
+wp = exp
+�
+−α||It −Γ(It−1)||2�
+and
+wn = exp
+�
+−α||It −Γ(It+1)||2�
+.
+(3)
+In order to also incorporate contribution from Pt, we clamp the
+weights wp and wn as follows:
+�
+wp
+�
+= k1 and
+�
+wn
+�
+= k2, where
+k1 and k2 are two constants. The locally consistent image sequence
+given by Lt has improved temporal consistency over the per-frame
+processed output, however, it still has visible flickering artifacts.
+Thus, the reduction in flickering due to warping of only one tempo-
+ral neighbor is not sufficient. To further improve consistency, one
+can warp more neighboring frames around the current time instance
+t. As we increase the temporal window-size for such an adaptive
+combination it has a denoising effect leading to further reduction in
+flickering. The temporal denoising for enforcing consistency, per-
+formed by Shekhar et al. [SST∗19] can be considered as an specific
+example of the above scenario. However, for interactive stylization
+warping more frames to the current instance is not feasible due to
+time constraint. Moreover, in case of video streams we do not have
+frames to warp from the forward temporal direction.
+Global Consistency. In order to overcome this limitation, exist-
+ing approaches [BTS∗15, LHW∗18] adopt a global approach. For
+global consistency, one can consider the previous stabilized output
+Ot−1 and enforce similarity with its warped version Gt where,
+Gt = Γ(Ot−1).
+(4)
+To enforce only global temporal smoothness, we replace At with Gt
+in Eqn. 1. Further, in order to compensate for optic-flow inaccura-
+cies, the smoothness term is weighted using wp (i.e., wc = wp) in
+Eqn. 1. However, considering only global consistency for flicker
+reduction leads to loss of stylization and local temporal varia-
+tions in the final output. Moreover, in this case any warping-error
+(due to flow-inaccuracies) or noise (as part of stylization process)
+submitted to 200x.
+
+S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams
+5
+keeps getting propagated to future frames. Due to the above fac-
+tors, such an approach only gives plausible results where the gra-
+dients of the original video are similar to the gradients of the pro-
+cessed video. The above does not hold for the task of stylization
+where stylistic elements such as brush strokes, textures or stroke
+textons [ZGWX05], in general, can vary largely between frames
+even for small changes in gradient.
+Combining Global and Local Consistency. For preserving local
+temporal variations (in terms of look and feel) while significantly
+reducing the flickering artifacts, we linearly combine globally and
+locally consistent images Gt and Lt respectively,
+At = wp·Gt + (1−wp)·Lt.
+(5)
+We use the adaptively combined image At as our reference for
+consistency while enforcing temporal smoothness in Eqn. 1. The
+�
+wp
+�
+can be increased to increase the influence of global-temporal
+smoothness and vice versa. Further, the influence of the smoothness
+term is controlled by per-pixel consistency weights wc. We would
+like to invoke the smoothness term only when the warping accuracy
+is sufficiently high. To this end, we construct a warped version of
+the input image similar to Lt as,
+AIt = (1−(wp+wn))·It + wp·Γ(It−1) + wn·Γ(It+1).
+(6)
+Only when the input image It is similar to AIt, the smoothness term
+is invoked. To measure this similarity, we use the weight wc,
+wc = λ·exp
+�
+−α||It −AIt||
+2�
+.
+(7)
+The parameter λ is used to scale up or down the weight wc.
+Consistency Control Modes. The above adaptive combination
+of local and global consistency provides two different ways of
+consistency-control in the final output. By increasing
+�
+wp
+�
+we can
+increase the proportion of global consistency in the adaptively com-
+bined image At and vice versa. On the other hand the optimization
+parameter λ dictates how close the output Ot will be to the adap-
+tively combined image At. Thus, the level of consistency in the
+final output can be controlled in two different ways: (1) by set-
+ting up the limit of parameter wp, i.e.,
+�
+wp
+�
+or (2) by scaling the
+weight parameter λ. For lower values of
+�
+wp
+�
+(Fig. 6b), the consis-
+tency enforced is negligible and the final result resembles the per-
+frame processed output (Fig. 6f). However, for higher values we
+start observing noisy ghosting artefacts (Fig. 6e). The lower values
+of
+�
+wp
+�
+translates to using only global consistency which results in
+accumulation of flow inaccuracies visualized as ghosting artefacts.
+Similarly, for lower values of λ (Fig. 6g), the final result is visually
+similar to the per-frame processed output (Fig. 6f). However, for
+higher values the optimization becomes unstable resulting in noisy
+optimization-based artefacts. (Fig. 6j).
+Optimization Solver. The energy terms in Eqn. 1 are smooth and
+convex in nature, which allows a straightforward energy minimiza-
+tion with respect to Ot. To this end, we employ an iterative ap-
+proach thus avoiding – storage of a large matrix in memory and
+further estimating its inverse. Moreover, an iterative approach al-
+lows us to stop the solver once we have achieved visually plau-
+sible results. An iterative update Otk+1 is obtained by employing
+43
+                (a)                                            (b)                                           (c)   
+Refinement
+Flow 
+Estimation 
+Modules
+Feature
+Extraction
+Input Frames
+Output Flow
+Figure 3: Modification of the PWC-Net [SYLK18] architecture for
+real-time performance. We apply following network compression
+steps: (a) Replace DenseNet connections with light ones, (b) Re-
+duce the number of flow estimators, and (c) Replace dense connec-
+tions in the refinement module with separable convolutions.
+Stochastic Gradient Descent (SGD) with momentum [Qia99],
+Otk+1 = Otk −η∇E(Otk)+κ(Otk −Otk−1).
+(8)
+where η and κ are the step size parameters, ∇E is the energy gradi-
+ent with respect to Ot, and k is the iteration count. For most of our
+experiments, η = 0.15 and κ = 0.2 yield plausible results. We con-
+sider the trade-off between performance vs. accuracy as a stopping
+criteria and do not compute energy residue for this purpose. To ob-
+tain a consistent output while having interactive performance, we
+empirically determine 150 iterations to be sufficient. The optimiza-
+tion is stable for the given parameter settings and early stopping is
+only employed for computational gain.
+An integral aspect common to both our local and global consis-
+tency is the warping function Γ. Apart from the number of solver
+iterations, for interactive performance the above warping should
+also happen at a fast rate – which in turn necessitates fast optic-
+flow estimation.
+3.2. Lite Optic-Flow Network
+We aim to obtain a flow network capable of running at high-speed
+on consumer hardware with reasonable accuracy. To this end, we
+start by selecting an existing CNN-based optical flow estimation
+technique, based on accuracy vs. run-time analysis. After the se-
+lection of a base network, we perform further optimization steps to
+increase the performance as outlined in Fig. 3.
+Base Network Selection for Compression. In Fig. 4, we com-
+pare several well-known optical methods to find a base network
+candidate that best matches our runtime/accuracy requirements.
+We employ the following models for this: FlowNet 2.0 [IMS∗17],
+SpyNet [RB17], LiteFlowNet2 [HTL20], PWCNet [SYLK18],
+ARFlow [LZH∗20], VCN [YR19], RAFT [TD20] and finally
+GMA [JCL∗21] (state-of-the-art in terms of EPE-based accuracy).
+Our experiments are carried out on a Nvidia RTX 2070 GPU,
+which we deem to be a good representative of a current mid-to
+submitted to 200x.
+
+6
+S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams
+0px
+2px
+4px
+6px
+8px
+0
+10
+20
+30
+40
+flownet2
+spynet
+pwcnet
+arflow
+liteflownet2
+vcn
+raft
+gma
+Sintelfinal-test EPE (lower=better)
+FPS (higher=better)
+Figure 4: Accuracy vs. run-time performance of existing methods
+measured on Sintel Final (Test set) [BWSB12]. The Endpoint Er-
+ror (EPE) metric measures Euclidean distance (in pixels) between
+ground-truth and predicted optical flow vectors.
+higher-end consumer GPU. Under a constraint of interactive perfor-
+mance on consumer hardware, LiteFlowNet2 [HTL20] and PWC-
+Net [SYLK18] offer the best trade-off between run-time perfor-
+mance and accuracy (Fig. 4). LiteFlowNet2 [HTL20] is already an
+optimized version of FlowNet 2.0 [IMS∗17], in comparison PWC-
+Net [SYLK18] has more potential for optimization/compression.
+Moreover, recently it has been shown that PWC-Net can achieve
+similar accuracy to RAFT when trained on a large-scale synthetic
+dataset [SVH∗21] and that PWC-Net achieves favourable trade-offs
+vs. other state-of-the-art methods when selecting for runtime per-
+formance or higher image resolutions [SHR∗22]. Hence, we select
+PWC-Net for further compression.
+Optimized Network Architecture. We start with the base archi-
+tecture of PWC-Net. As the first compression step we reduce the
+computationally expensive DenseNet [HLvdMW17] connections
+in the flow estimators to retain connections only in the last two
+layers ("-light" in Fig. 5b). Similar to LiteFlowNet2 [HTL20], we
+remove the fifth flow estimator – operating on the highest resolu-
+tion – as it heavily trades off run-time for only marginal increase
+in accuracy (compare "4light" vs "5light" in Fig. 5b). We replace
+the standard convolutions in the refinement by depthwise separable
+convolutions [HZC∗17] ("-sepref" in Fig. 5b). Moreover, we also
+explore reducing the number of channels [HZC∗17], but find that
+reducing channels results in a worse trade-off as compared to other
+optimizations.
+Training. For
+training,
+we
+follow
+the
+original
+PWC-
+Net
+[SYLK18]
+schedule.
+However,
+we
+find
+that
+weight-
+ing
+the
+multi-scale
+losses
+equally,
+instead
+of
+exponen-
+tially [SYLK18, HTL18, HTL20, YR19], improves accuracy. For
+our experiments on the desktop system, we use PyTorch [PGM∗19]
+and take inspiration from the implementation by Niklaus [Nik18].
+Similar to PWC-Net [SYLK18], we train our mobile architecture
+on the training dataset schedule FlyingChairs [FDI∗15] → Fly-
+ingThings3D [MIH∗16]→ Sintel [BWSB12]. In the supplementary
+material, we provide training settings for each stage in detail. We
+Table 3: Runtime performance in milliseconds per frame. We mea-
+sure the total processing time (without disk IO) and the individual
+stages for a mid-tier GPU (Nvidia GTX 1080Ti) and a higher-end
+GPU (Nvidia RTX 3090), results are averaged over 100 runs.
+Task
+Optical flow
+Stabilization
+Total
+↓ Res. / GPU
+1080Ti 3090
+1080Ti 3090
+1080Ti 3090
+1920×1080 px
+66.8
+40.0
+184.1
+42.7
+250.8
+82.7
+1280×720 px
+31.3
+19.7
+86.5
+21.1
+117.8
+40.8
+640×480 px
+12.6
+6.2
+20.6
+6.3
+33.2
+12.5
+employ a multi-scale loss [SYLK18] applied to each flow estimator
+and optimize using the AdamW optimizer [LH19] with β1 = 0.09,
+β2 = 0.99, and l2 weight regularization with trade-off γ = 0.0004.
+Furthermore, extensive dataset augmentation is applied to prevent
+model overfitting. We refer to the supplementary material for more
+details.
+Our Final Model. We analyze various optimization options and
+chose “our-4light-sepref” as our final model for desktop systems
+as it provides the best trade-off between accuracy vs. run-time. As
+depicted in Fig. 5a, our method improves run-time performance of
+PWC-Net from 30 FPS to 85 FPS – a speed-up of factor 2.8. For
+Sintel training data the accuracy drops by ≈ 0.5px in EPE terms,
+however for test data the drop in accuracy is significant where the fi-
+nal EPE is 7.43. Nevertheless, the accuracy is sufficient enough for
+enforcing warping-based consistency. To validate our design deci-
+sions, we conduct an extensive ablation study in which we vary the
+architectural and training choices – please see the supplementary
+for details. Furthermore, we tune our architecture for optical flow
+calculation on mobile devices using channel pruning and quantiza-
+tion, which we also detail in the supplementary material. Here, we
+improve run-time performance from 2.8 FPS to 24 FPS (iPad Pro
+2020), and 1.5 FPS to 13 FPS (iPad Air) – an improvement of factor
+8. Next to showing the general applicability of optical flow CNNs
+on mobile devices, this demonstrates that real-time on-device sta-
+bilization of videos using our presented approach will become fea-
+sible with a further moderate increase in mobile GPU computing
+power. A fast optic-flow based warping enables our framework to
+interactively control the degree of consistency and generate visu-
+ally plausible results.
+4. Experimental Results
+4.1. Implementation Details
+All our experiments were performed on an consumer PC with an
+AMD Ryzen 1920X 12-Core CPU, 48 GB of RAM, and a Nvidia
+GTX 1080Ti and RTX 3090 graphics cards with VRAMs of 11
+GB and 24 GB respectively. We implement a real-time video-
+consistency framework in C++, using ONNXRuntime for cross-
+platform acceleration of our lite optical-flow network and imple-
+ment the stabilization code using Nvidia CUDA (v11.4). In Tab. 3,
+we measure the runtime performance of our system. We find that
+an incoming stream of frames can be stabilized at real-time perfor-
+mance for VGA resolution even on low- and mid-tier GPUs and
+higher-tier GPUs (such as a RTX 3090) can stabilize HD at com-
+mon video frame rates (approx. 24 FPS) and full-HD resolutions at
+submitted to 200x.
+
+S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams
+7
+2px
+3px
+4px
+5px
+0
+20
+40
+60
+80
+100
+flownet2
+liteflownet2
+pwcnet
+our-5light
+our-5light-5sep
+our-5light-2sep
+our-4light-1sep
+our-5light-c50
+our-5light-c75
+our-4light
+our-4light-sepref
+(a) Sintelfinal-train EPE (lower=better)
+FPS (higher=better)
+Modifier
+Description
+Default
+-Nlight
+N light [LZH∗20] flow esti-
+mators.
+5 dense [SYLK18]
+-Msep
+last M flow estimators use
+depthwise separable convo-
+lutions [HZC∗17].
+standard convs.
+-sepref
+refinement
+uses
+depth-
+wise
+separable
+convolu-
+tions [HZC∗17].
+standard convs.
+-cP
+use P% of channels.
+100%
+(b) Legend of our CNN variants.
+Figure 5: Accuracy vs. run-time performance of our CNN variants on desktop, measured on Sintel Final (Train) [BWSB12]. Optimization
+steps that lead to significant improvement in run-time are connected by a line. Our architectural modifications to PWC-Net [SYLK18] are
+detailed on the right, e.g., our-4light-sepref denotes a 4 light flow estimators and refinement using depthwise separable convolutions.
+(a) Input
+(b)
+�
+wp
+�
+= 0.3
+(c)
+�
+wp
+�
+= 0.5
+(d)
+�
+wp
+�
+= 0.7
+(e)
+�
+wp
+�
+= 0.9
+(f) Processed
+(g) λ = 0.1
+(h) λ = 1.0
+(i) λ = 5.0
+(j) λ = 7.06
+Figure 6: The level of consistency in the final output can be controlled via parameters
+�
+wp
+�
+and λ. Here we show how the final result vary
+by increasing these, for lower values the consistency is negligible and the results (Fig. 6b and Fig. 6g) visually look similar to the per-frame
+processed output (Fig. 6b). For higher values we start observing artefacts due to ghosting and/or optimization (Fig. 6e and Fig. 6j).
+interactive frame rates (> 10 FPS) for different parameter settings
+(Tab. 3).
+4.2. Parameter Settings
+Initially, we tune the parameters of our consistency framework to-
+wards achieving a low warping error (Tab. 5). We refer to this set-
+ting as Ours-objective with the following parameter values k1 =
+k2 = 0.3, α = 10 × 103, and λ = 0.7. However, we observed that
+even though the warping error indicated a good temporal stabil-
+ity, subjectively flickering and artefacts were noticeable. Unlike ex-
+isting approaches, our framework allows for interactive parameter
+adjustment. Thus, a parameter set that subjectively produces well-
+stabilized results on a broad range of tasks and videos was obtained
+experimentally. As our final version, we use the values of k1 = 0.3,
+k2 = 0.5, α = 6.5 × 103, and λ = 2.0 to generate all the images in
+the paper and the videos provided in the supplementary. We fur-
+ther compare Ours-objective settings with our final version as part
+of our user study to validate our parameter choices. The consistent
+outputs obtained using the above parameter settings are compared
+against state of the art approaches thereby showcasing its efficacy.
+4.3. Consistent Outputs
+We use videos from DAVIS [PPTM∗16] dataset and other open
+source videos (taken from [Vid] and [Pex]) for comparison. For per-
+frame stylization, we employ the following stylization techniques:
+Fast NST [JAFF16], WCT [LFY∗17], and CycleGAN [ZPIE17].
+The results for the method of Lai et al. and Bonneel et al. on videos
+taken from DAVIS [PPTM∗16] and Videvo ( [Vid]) are borrowed
+from the results dataset provided by Lai et al. . For other videos
+we employ the source code provided by the authors to generate
+submitted to 200x.
+
+8
+S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams
+132
+128
+127
+39
+43
+44
+0
+20
+40
+60
+80
+100
+120
+140
+Lai
+Bonneel
+Ours-obj.
+Others
+Ours
+Figure 7: Statistics of the user study results on removal of temporal
+flickering from per-frame stylized videos. For 19 participants and
+9 different videos we compare our method against Bonneel et al. ,
+Lai et al. , and Ours-objective through a total of 171 randomized
+A/B tests.
+the results. We compare our consistent outputs with that of Bon-
+neel et al. [BTS∗15] and Lai et al. [LHW∗18] in Fig. 8. Among
+the three competing methods Bonneel et al. is the least effective in
+preserving the underlying style for the final output (compare sec-
+ond column with the fourth one in Fig. 8). Hyper-parameter tun-
+ing in the above method (with only global consistency) can pro-
+vide a certain degree of consistency-control. However, by employ-
+ing both global and local consistency we achieve finer consistency-
+control while being similar to the per-frame-processed result. For
+the method of Lai et al. , we observe some color bleeding or dark-
+ening in the output frames (compare second column with the third
+one in Fig. 8). In comparison we are able to preserve the style, color
+and textures, while being consistent (Fig. 7).
+4.4. Optic Flow Results
+We visualize optical flow on frames from the Sintel [BWSB12]
+dataset in Fig. 9 and compare to state-of-the-art methods. All de-
+picted methods have been fine-tuned on Sintel. We find that our
+optimized method has more blurry motion boundaries and misses
+to estimate certain details accurately (e.g., the hand in the first
+row, however, PWCNet also fails at this), but still captures over-
+all motion direction of objects correctly with a smooth flow field.
+Fig. 10 shows results for real-world videos on the DAVIS dataset
+[PTPC∗17] (no ground-truth flow available). We find that some
+real-world image phenomena, such as complex/ambiguous occlu-
+sions (e.g., bus behind tree) are not well-handled by state-of-the-art
+methods like RAFT [TD20] or PWC-Net [SYLK18], and thus re-
+sults are degraded for our optimized method as well. Besides the
+stronger blurred motion boundaries, we find that our network gen-
+erally performs well and is also robust for real-world videos.
+5. Evaluation
+5.1. Quantitative
+Following Lai et al. [LHW∗18], we measure the similarity between
+per-frame processed output and stabilized results, and the temporal
+warping error between consecutive stabilized frames.
+For the fomer, we report the similarity in form of the SSIM met-
+ric in Tab. 4. We achieve significantly higher similarity scores than
+the methods of Bonneel et al. [BTS∗15] and Lai et al. [LHW∗18].
+Following [BTS∗15] and [LHW∗18], we also measure the tempo-
+ral warping error between a frame Vt and the warped consecutive
+frame ˆVt+1, defined as:
+Ewarp (Vt,Vt+1) =
+1
+∑N
+i=1 M(i)
+t
+N
+∑
+i=1
+M(i)
+t
+���V (i)
+t
+− ˆV (i)
+t+1
+���
+1 ,
+(9)
+where Mt ∈ {0,1} is a non-occlusion mask [LHW∗18,RDB18], in-
+dicating non-occluded regions. The warped frame ˆVt+1 is obtained
+by calculating the optical flow (using GMA [JCL∗21]) between
+frames Vt,Vt+1, and applying a backwards warping to frame Vt+1.
+We compute Ewarp for every frame of a video and then average to
+obtain the warping error of a video Ewarp(V). In Tab. 5 we report the
+average warping error per dataset (see the supplementary for a per-
+task breakdown). We find that the warping error is slightly higher
+than that of Bonneel et al. [BTS∗15] and Lai et al. [LHW∗18].
+However, as Lai et al. [LHW∗18] notes, results with high temporal
+stability (expressed by a low warping error) can also be achieved
+via temporally smoothing the video, which can be seen in vari-
+ous results of Bonneel et al. [BTS∗15]. Our qualitative results in
+form of a user study Sec. 5.2 further substantiate the divide between
+warping error (as a stability metric) and perceived stability.
+5.2. Qualitative
+For qualitative evaluation we perform a subjective user study where
+we ask participants to compare the temporally-consistent result ob-
+tained using our method with that of Lai et al. , Bonneel et al. ,
+and Ours-objective – a different parameter setting of ours. We use
+9 different videos for this purpose: 3 from DAVIS [PPTM∗16], 3
+from Videvo [Vid], and 3 from Pexels [Pex] datasets respectively.
+For each of the above video we stylize them using either the Fast
+NST [JAFF16] (in the styles of udnie, rain-princess, and mosaic)
+or WCT [LFY∗17] (in the styles of wave and antimono) or Cycle-
+GAN (in the styles of photo2vangogh and photo2ukiyoe). For each
+sample, we show the input video and its per-frame stylized version
+on the top row of user-study interface for inference. In the bottom
+row we show two different version of the temporally stabilized out-
+put where one of them is ours. We ask the participants to select
+the output which best preserves: (i) temporally consistency and (ii)
+similarity with the per-frame processed video. For 9 videos and 3
+other competing methods each user sees a total of 27 blind A/B
+tests which are shown in a randomized order to each participant.
+In total, 19 persons (3 female and 16 male) within the ages of 22
+to 43 years participated in the study. Fig. 7 shows that our method
+surpasses all others by a large margin. It was interesting to observe
+that for certain cases the method of Bonneel et al. which degrades
+the processed style significantly was still preferred by users over
+others due to its high consistency quality.
+submitted to 200x.
+
+S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams
+9
+(a) Input
+(b) Processed
+(c) Ours
+(d) Lai et al. [LHW∗18]
+(e) Bonneel et al. [BTS∗15]
+Figure 8: Comparing our results with Lai et al. [LHW∗18] and Bonneel et al. [BTS∗15] for three different video sequences: Cow (top two
+rows), Farming (mid two rows), and Woman (last two rows). Note how the consistent output for Lai et al. and Bonneel et al. look different
+from the corresponding per-frame processed results.
+5.3. Using other Optical Flows
+We also tested other optical flow methods within our pipeline
+which were either faster [KTDVG16] or more accurate [TD20].
+For the fast optical method by Kroeger et al. [KTDVG16](DIS)
+the final output is less consistent than ours in both objective and
+subjective metrics. Using DIS for our stabilization, the average
+warp-error over DAVIS is 0.05 (vs. 0.046 ours) and perceptual-
+similarity with the per-frame processed result is 0.9 in SSIM terms
+(vs. 0.923 ours). Visually, DIS-stabilized results show significantly
+more flickering, validating our design choice for the optical-flow.
+A much more accurate optic flow is given by the method of
+Teed et al. [TD20] (RAFT) at the cost of slow computation. The
+stabilized results obtained using RAFT look visually indistinguish-
+able to the one obtained using our flow; the average warp-error over
+DAVIS is 0.045, the perceptual-similarity is 0.923.
+6. Discussion
+Our approach takes a video pair as an input: (i) the original and
+(ii) its per-frame stylized version. We assume that the stylization
+is based on the input image-gradients and appears as variations in
+the form of colors and/or textures. Thereby, we employ the origi-
+nal video as a guide for enforcing consistency. However, for text-
+guided generative arts such as recent diffusion model-based ap-
+proaches [RDN∗22, RBL∗22] the stylized frames are often only
+weakly correlated with the original input, we cannot handle such
+cases.
+For the evaluation we mainly use CNN-based stylization tech-
+niques. However our approach can also handle classical stylization
+approaches [KCWI13], we show few such examples in the supple-
+mentary. Our local-consistency component comprising of convex
+combination of temporal neighbors can be seen as crude form of
+submitted to 200x.
+
+10
+S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams
+(a) Frame Overlay
+(b) Ground-truth
+(c) RAFT [TD20]
+(d) PWC-Net [SYLK18]
+(e) Ours
+Figure 9: Optical flow estimated using the synthetic Sintel dataset [BWSB12].
+(a) Frame Overlay
+(b) RAFT [TD20]
+(c) PWC-Net [SYLK18]
+(d) Ours
+Figure 10: Optical flow estimated for the real-world dataset DAVIS [PTPC∗17].
+local temporal denoising. Previously it has been shown that tem-
+poral denoising is effective in enforcing consistency [SST∗19]. We
+conjecture that efficient temporal-denoising combined with flow-
+based warping can further improve temporal stabilization not only
+for stylization but also for other tasks.
+We start with the assumption that temporal flickering is not com-
+pletely undesirable for the task of stylization and thus we pro-
+vide interactive consistency control. However, during the subjec-
+tive user study we observed that participants had different toler-
+ance levels for flickering in the foreground as compared to that in
+the background. As part of future work, one can use depth-based
+or saliency-based masks to vary the consistency control parameters
+spatially for a more visually pleasing result.
+Limitation: Our approach tends to have ghosting artifacts for
+fast moving objects where the object motion between consecutive
+frames is large (Fig. 11). The above can be reduced by reducing
+the value of
+�
+wp
+�
+, however such a reduction also reduces consis-
+(a)
+�
+wp
+�
+= 0.5
+(b)
+�
+wp
+�
+= 0.1
+Figure 11: The ghosting artifacts on the rear wheel of the scooter
+is significant in the final output for
+�
+wp
+�
+= 0.5, however it reduces
+significantly for
+�
+wp
+�
+= 0.1.
+tency in the final output. We argue that since we provide interactive
+control of parameters the above trade off between artifacts vs. con-
+sistency will not hinder its usability significantly.
+submitted to 200x.
+
+EPE: 0.000EPE: 0.000EPE: 0.629EPE: 0.171EPE: 1.283EPE: 0.339EPE: 2.091EPE: 0.520S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams
+11
+Table 4: Quantitative evaluation on perceptual distance using SSIM (higher = more similar to per-frame processed result).
+DAVIS
+VIDEVO
+Task
+[BTS∗15]
+[LHW∗18]
+Ours
+[BTS∗15]
+[LHW∗18]
+Ours
+CycleGAN/photo2ukiyoe [ZPIE17]
+0.693
+0.781
+0.978
+0.626
+0.743
+0.980
+CycleGAN/photo2vangogh [ZPIE17]
+0.707
+0.792
+0.961
+0.679
+0.789
+0.965
+fast-neural-style/rain-princess [JAFF16]
+0.553
+0.799
+0.921
+0.491
+0.796
+0.920
+fast-neural-style/udnie [JAFF16]
+0.597
+0.785
+0.956
+0.579
+0.747
+0.959
+WCT/antimonocromatismo [LFY∗17]
+0.389
+0.811
+0.915
+0.388
+0.761
+0.914
+WCT/asheville [LFY∗17]
+0.329
+0.801
+0.904
+0.348
+0.771
+0.901
+WCT/candy [LFY∗17]
+0.289
+0.763
+0.882
+0.310
+0.738
+0.885
+WCT/feathers [LFY∗17]
+0.418
+0.863
+0.891
+0.415
+0.848
+0.888
+WCT/sketch [LFY∗17]
+0.370
+0.845
+0.923
+0.370
+0.833
+0.922
+WCT/wave [LFY∗17]
+0.358
+0.700
+0.902
+0.352
+0.637
+0.899
+Average
+0.470
+0.794
+0.923
+0.456
+0.766
+0.923
+Table 5: Flow warping error average over tasks shown in Tab. 4.
+A per-task breakdown is shown in the supplementary. Note that
+the slightly higher warping error (lower is better) of our method is
+subjectively not noticeable as we show in a user study.
+Dataset
+Vp
+[BTS∗15]
+[LHW∗18]
+Ours
+DAVIS
+0.056
+0.034
+0.040
+0.046
+VIDEVO
+0.051
+0.036
+0.036
+0.042
+7. Conclusions
+We propose an approach that makes per-frame stylized videos tem-
+porally coherent irrespective of the underlying stylization applied
+on individual frames. At this, we introduce a novel temporal con-
+sistency prior which combines both local and global consistency
+aspects. We maintain similarity with the per-frame processed result
+by minimizing the difference in the gradient-domain. Unlike previ-
+ous approaches we provide interactive consistency control by com-
+puting optic-flow on the incoming video stream with only sufficient
+accuracy but at high speed. Fats optic-flow inference is achieved
+by developing a lightweight flow network architecture based on
+PWC-Net. The entire optimization solving is GPU-based and runs
+at real-time frame-rates for HD resolution. We showcase that our
+temporally consistent output is preferred over the output of com-
+peting methods by conducting a user study. As part of future work
+we would like to employ learning-based temporal denoising to fur-
+ther improve quality of results. Moreover, we would like to ex-
+plore the usage of depth-based and saliency-based masks to spa-
+tially vary consistency parameters according to perceptual princi-
+ples. We hope that our design paradigm of interactive consistency
+control will potentially make per-frame video stylization more user
+friendly.
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+

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@@ -0,0 +1,1005 @@
+arXiv:2301.08625v1  [physics.plasm-ph]  20 Jan 2023
+Interaction of thin tungsten and tantalum films with ultrashort laser pulses:
+calculations from first principles
+N. A. Smirnov∗
+Federal State Unitary Enterprise, Russian Federal Nuclear Center - Zababakhin
+All-Russian Research Institute of Technical Physics, 456770, Snezhinsk, Russia
+(Dated: January 23, 2023)
+The interaction of ultrashort laser pulses with thin tungsten and tantalum films is investigated
+through the full-potential band-structure calculations. Our calculations show that at relatively low
+absorbed energies (the electron temperature Te≲7 kK), the lattice of tantalum undergoes noticeable
+hardening. The hardening leads to the change of the tantalum complete melting threshold under
+these conditions. Calculations suggest that for the isochorically heated Ta film, if such hardening
+really occurs, the complete melting threshold will be at least 25% higher. It is also shown that
+the body-centered cubic structures of W and Ta crystals become dynamically unstable when the
+electronic subsystem is heated to sufficiently high temperatures (Te>22 kK). This lead to their
+complete melting on the sub-picosecond time scale.
+PACS numbers:
+I.
+INTRODUCTION
+As shown in a number of experimental studies, the
+melting of different materials after their interaction with
+ultrashort (femtosecond) laser pulses have their specific
+features [1–5].
+Absorption of this radiation leads to a
+strongly non-equilibrium heating of the system where the
+temperatures of its electronic and ionic subsystems are
+very much different, Te≫Ti. This state may keep for tens
+of picoseconds and even longer [5]. Under these condi-
+tions, semiconductors, for example, undergo the so-called
+nonthermal melting caused not by their lattice heating
+due to heat transfer from hot electrons to cold ions but
+by a dramatic change in the shape of the potential energy
+surface and hence dynamic lattice destabilization [1–3].
+In semimetallic bismuth, the situation seems to be sim-
+ilar [4]. The determining factor here is the estimate of
+the electron-phonon coupling factor G, which defines the
+rate of heat transfer from the electronic to ionic subsys-
+tem. For bismuth, the theoretical estimates of G strongly
+differ [6–8], leaving room for disputes on the presence of
+nonthermal melting in this metal after interaction with
+ultrashort laser pulses [9].
+On the other hand, the change of the shape of the po-
+tential energy surface may also lead, under certain con-
+ditions, to the hardening of irradiated crystal [10–12],
+thus increasing the time of its melting and causing its
+strong overheating. Despite some claims that the lattice
+hardening has been experimentally observed [13], there
+is still no evidence of its reliable detection in experiments
+[5, 12, 14].
+The experimental work reported in Ref. [15] aimed to
+explore the possibility of the nonthermal melting of tung-
+sten by measuring reflectivity of the metal surface after
+its irradiation. The experiments show that above a cer-
+∗Electronic address: [email protected]
+tain value of absorbed excitation fluence, ablation of the
+metal surface proceeds in a sub-picosecond time interval.
+The revealed effect may be indicative of the ultrafast non-
+thermal melting because in the normal thermal scenario
+of ablation, the characteristic times of this process must
+be much higher than those obtained in experiment [15].
+Ab initio calculations [16] show that the heating of
+the electronic subsystem of tungsten to Te above 20 kK
+may lead to a structural transition from bcc to fcc phase.
+The transition is also caused by the abrupt change in the
+shape of the potential energy surface, leading to fcc sta-
+bilization at high values of Te [16]. In its turn, the bcc
+structure may lose dynamic stability under these condi-
+tions. It is however difficult to detect this transition in
+experiment because of the possibility of sub-picosecond
+nonthermal melting. Just this was shown in molecular
+dynamics (MD) calculations [17] where the interaction
+of femtosecond laser pulses with thin tungsten film was
+investigated. The nuclei of the new fcc phase were only
+able to form mainly on the surface of the film before the
+sample melted during about 0.8 ps. On whole, MD re-
+sults [17] suggest that the detection probability for the
+nonthermal melting of tungsten is much higher than for
+the structural transition predicted in Ref. [16].
+As mentioned above, an important factor of detecting
+nonthermal phenomena in metals is the electron-phonon
+coupling factor G. Its values for metals are usually high
+[18], meaning that the nonthermal character of processes
+that occur after irradiation can hardly be recognized.
+There are different approaches to the theoretical deter-
+mination of G (see, for example, [12, 18, 19]). In our re-
+search we will follow methodology described in Ref. [12],
+but also discuss results obtained with other approaches.
+This paper studies the interaction of femtosecond laser
+pulses with thin (a few tens of nanometers thick) tung-
+sten and tantalum films. The physical quantities required
+for calculations with a two-temperature model [20] were
+obtained from first principles. The issues discussed in-
+clude the processes involved in the nonthermal melting
+
+2
+of the metals and the possibility of detecting tantalum
+lattice hardening at moderate absorbed energies.
+Our
+results are compared with available experimental data
+and other calculations.
+II.
+CALCULATION METHOD
+In this work, the temperature evolution of electronic
+and ionic subsystems with time after irradiation by ul-
+trashort laser pulses is determined using a well-known
+two-temperature model [20]. Since the thin (∼10 nm)
+films of W and Ta are considered, the two-temperature
+model equations can be written as
+Ce(Te)∂Te
+∂t = −(Te − Ti)G(Te) + S(t),
+(1)
+Ci(Ti)∂Ti
+∂t = (Te − Ti)G(Te),
+(2)
+where S(t) is the time dependent radiation source func-
+tion [17], Ce(Te) and Ci(Ti) are electron and lattice heat
+capacities, and G(Te) is the electron-phonon coupling fac-
+tor. Here we neglect lattice (κi) and electron (κe) ther-
+mal conductivities because, on the one hand, κe≫κi in
+our case, and on the other hand, in thin foils, ballistic
+electrons bring the electronic subsystem to thermody-
+namic equilibrium over a time about a pulse duration τp
+[21, 22]. So, no significant gradients in temperature oc-
+cur in the target. The method to calculate Ce, Ci, and G
+as functions of electron and ion temperatures from first
+principles is described in rather detail in Ref. [12]. Here
+we only provide the key formula for the electron-phonon
+coupling factor. It reads as
+G(Te) =
+2πℏ
+(Tl − Te)
+∞
+�
+0
+ΩdΩ
+∞
+�
+−∞
+N(ε)α2F(ε, Ω)
+× S(ε, ε + ℏΩ)dε.
+(3)
+where N(ε) is the electronic density of states (DOS),
+α2F(ε,Ω) is the electron-phonon spectral function, ε
+and ℏΩ are, respectively, electron and phonon energies,
+S(ε,ε + ℏΩ)=[fe(ε)-fe(ε + ℏΩ)][n(ℏΩ,Ti)-n(ℏΩ,Te)] with
+fe standing for the Fermi distribution function and n for
+the Bose-Einstein distribution function.
+Another formula which is often used to determine
+G(Te) has some simplifications as compared to (3) and
+reads as [18]
+G(Te) = πℏkBλ⟨ω2⟩
+N(EF )
+∞
+�
+−∞
+N 2(ε)
+�
+−∂fe
+∂ε
+�
+dε.
+(4)
+Here λ is the electron-phonon mass enhancement param-
+eter, ⟨Ω⟩2 is the second moment of the phonon spectrum
+[23], and EF is the Fermi energy. Formula 4 is derived un-
+der the assumption that in the interaction with phonon,
+the scattering probability matrix elements is independent
+of the initial {k, i} and final {k′, j} electronic states. The
+authors of Ref. [18] determined the values of λ and ⟨Ω⟩2
+from experimental evaluation, not from first-principles
+calculations.
+One more way to calculate G(Te) is based on the calcu-
+lation of the electron-ion collision integral Ie−i
+nm with the
+use of an approximate tight-binding model to calculate
+the band structure, combined with MD simulation [19].
+The expression for Ie−i
+nm is written as
+Ie−i
+nm = 2π
+ℏ |Me−i(εn, εm)|2
+�
+fe(εn)[2 − fe(εm)] − fe(εm)[2 − fe(εn)]e−∆ε/Ti;
+for n>m
+fe(εm)[2 − fe(εn)]e−∆ε/Ti − fe(εn)[2 − fe(εm)];
+otherwise ,
+(5)
+where ∆ε=εn − εm is the energy difference between two
+states, and Me−i is the electron-ion scattering matrix el-
+ement. The electron-phonon coupling factor can be writ-
+ten as
+G(Te) =
+1
+V (Te − Ti)
+�
+n,m
+εmIe−i
+nm ,
+(6)
+here V is the specific volume. It should be noted here
+that our method for determining G(Te) (by formula (3))
+does not use any experimentally determined parameters
+or approximations which simplify the scattering proba-
+bility matrix element, as it is done in Ref. [18], or serious
+simplifications related to particle interactions in the sys-
+tem, as it is done in the tight-binding model [19].
+In this work, first-principles calculations were done
+with the all-electron full-potential linear muffin-tin or-
+
+3
+0
+2
+4
+6
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+2
+4
+6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+PDOS (arb. units)
+Frequency (THz)
+W
+Frequency (THz)
+Ta
+FIG. 1: Tungsten and tantalum phonon spectra at the equi-
+librium experimental specific volume from calculations done
+in this work for zero temperature (red lines) and from exper-
+iment at room temperature [30] (circles connected by a line).
+bital method (FP-LMTO) [24]. We consider here pro-
+cesses at a constant specific volume, i.e.
+the isochoric
+heating of targets.
+Within the scope of density func-
+tional theory the FP-LMTO method calculates the elec-
+tron structure, internal and free energies, phonon spec-
+trum and other material properties [12, 24–26]. Phonon
+spectrum and electron-phonon spectral function calcu-
+lations for the metals of interest were done with lin-
+ear response theory implemented in the FP-LMTO code
+[24, 25].
+Integration over the Brillouin zone was done
+with an improved tetrahedron method [27]. Meshes in
+k-space corresponded to equidistant spacing 30×30×30.
+For integration over the q-points of the phonon spectrum,
+a 10×10×10 mesh appeared quite sufficient (see [26] for
+more details on meshes). The cutoff energy for repre-
+senting the basis functions as a set of plane waves in the
+interstitial region was taken to be 900 eV. The basis set
+included MT-orbitals with moments to lb
+max=5. Charge
+density and potential expansions in terms of spherical
+harmonics were done to lw
+max=7. The internal FP-LMTO
+parameters such as the linearization energy, tail energies,
+and the radius of the MT-sphere were chosen using an
+approach similar to that one used in Ref. [28].
+The valence electrons in our calculations were 5s, 5p,
+4f, 5d, and 6s. For better comparison with calculations
+by other authors, the exchange-correlation potential was
+chosen to be similar to that one used in Ref. [17], i.e.,
+PBE [29]. This functional reproduces well the different
+properties of tungsten and tantalum. For example, the
+equilibrium volume V0 from calculation differs by no more
+than 2% from experiment for both the metals. Figure 1
+shows the phonon densities of states (PDOS) from calcu-
+lation in comparison with experimental data [30]. They
+are seen to be in quite a good agreement.
+The entropy of the electronic subsystem was deter-
+0
+15
+30
+45
+-0.6
+-0.3
+0.0
+0.3
+0.6
+0.9
+0
+15
+30
+45
+ 
+ W  
+ T
+e
+=1 kK
+ T
+e
+=10 kK
+ T
+e
+=20 kK
+ 
+ 
+N (states/Ry/atom)
+ Ta 
+ 
+ 
+N (states/Ry/atom)
+E-
+ (Ry)
+FIG. 2: Electronic DOS for W (top) and Ta (bottom) at equi-
+librium specific volume and zero temperature (black lines).
+The green, blue and red lines are the Fermi distribution func-
+tions at different electron temperatures.
+mined as
+Se(Te) = −kB
+� ∞
+−∞
+dεN(ε)[feln(fe) + (1 − fe)ln(1 − fe)].
+(7)
+With the known entropy Se(Te) and internal energy
+Ee(Te) of electrons, it is easy to obtain the free energy
+Fe=Ee − TeSe of the electron gas.
+The phonon spectrum of tungsten and tantalum was
+determined within quasiharmonic approximation [12].
+The melting temperature Tm of crystal W and Ta versus
+electron temperature was estimated in the same manner
+as it was done in Ref. [31] with the well performing Lin-
+demann criterion.
+III.
+RESULTS
+Let’s first compare the electronic structures of tungsten
+and tantalum. Figure 2 shows their electronic densities
+of states versus energy at V =V0 and T =0 calculated in
+this work. It is seen that the chemical potential µ which
+coincides with the Fermi energy at zero temperature is
+near the minimum of the DOS for tungsten, while for
+tantalum, the density of states at ε=µ is much higher
+compared to W. For Ta, the Fermi level is near the peak
+of the DOS. Compared to tantalum, the electronic struc-
+ture of tungsten is very much depleted in states in the
+vicinity of µ. Calculations show that as Te grows to ∼15
+kK, the values of N(µ) increase for tungsten and decrease
+for tantalum. This causes certain differences in the be-
+havior of these metals at elevating electron temperatures.
+Now consider how the free energy of electrons depends
+on the lattice parameter c/a (i.e., the Bain path) at dif-
+ferent temperatures Te.
+Figures 3 and 4 show results
+
+4
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+-8
+-4
+0
+4
+8
+12
+ W  
+ T
+e
+=8.7 kK
+ T
+e
+=14.5 kK
+ T
+e
+=29 kK
+fcc
+F
+e
+-F
+0
+ (mRy/atom)
+c/a
+bcc
+FIG. 3: Free electron energy versus lattice parameter c/a at
+different Te for tungsten (V =V0). The vertical lines show the
+values of c/a which correspond to its bcc and fcc structures.
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+-5
+0
+5
+10
+15
+20
+ Ta 
+ 
+ 
+F
+e
+-F
+0
+ (mRy/atom)
+c/a
+ T
+e
+=1 kK
+ T
+e
+=5.8 kK
+ T
+e
+=17.4 kK
+ T
+e
+=34.8 kK
+bcc
+fcc
+FIG. 4: Free electron energy versus lattice parameter c/a at
+different Te for tantalum (V =V0). The vertical lines show the
+values of c/a which correspond to its bcc and fcc structures.
+obtained for W and Ta, respectively. In both metals, the
+fcc structure is seen to be dynamically unstable at low
+electron temperatures. With the increasing temperature
+it stabilizes and at Te>15 kK it becomes thermodynam-
+ically more preferable than bcc. It is seen that tantalum
+behaves very much like tungsten but requires somewhat
+higher temperatures for stabilization of the fcc structure.
+On the other hand, with the increasing Te the bcc struc-
+ture becomes dynamically unstable both in tungsten and
+in tantalum. These changes must lead to a bcc→fcc tran-
+sition when the electronic subsystem is heated. As how-
+ever mentioned in paper [17], in such conditions their
+melting is more probable. On whole, our calculations for
+tungsten agree well with results presented in Ref. [16].
+One more feature of tantalum should be noted here. It
+is seen from Fig. 4 that there exists a limited interval of
+temperatures at relatively low values of Te (see Te=5.8
+0
+2
+4
+6
+0.0
+0.2
+0.4
+0.6
+0.8
+0
+2
+4
+6
+0.0
+0.4
+0.8
+1.2
+ T
+e
+=300 K
+ T
+e
+=5.8 kK
+ T
+e
+=11.6 kK
+PDOS (arb. units)
+Frequency (THz)
+W
+Frequency (THz)
+Ta
+FIG. 5: Phonon densities of states in tungsten (left) and tan-
+talum (right) at different electron temperatures (V =V0).
+kK), where the bcc lattice hardens. The free energy curve
+runs steeper near the minimum corresponding to the bcc
+phase. This feature is absent in tungsten. Figure 5 shows
+the densities of phonon states for W and Ta we calculated
+in this work for different electron temperatures. It is seen
+that with the increasing Te tungsten gradually softens
+and its phonon frequencies reduce. The phonon frequen-
+cies of tantalum first increase with the growing Te and
+cause bcc lattice hardening. Then the tendency changes
+– the high-frequency part of the spectrum goes on to
+harden, while the low-frequency part begins to soften re-
+ducing its frequencies (see Fig. 5, Te=11.6 kK). At Te
+above 20 kK the bcc structure in both metals loses its
+dynamic stability. It happens at about 22 kK in tung-
+sten and 29 kK in tantalum. The hardening of the Ta
+lattice at relatively low electron temperatures leads to a
+sudden effect we will consider later.
+Figures 6 and 7 show the electron-phonon coupling fac-
+tor G as a function of electron temperature at V =V0,
+calculated in this work for tungsten and tantalum, re-
+spectively. The dependences G(Te) are provided for bcc
+and fcc structures in their stability regions.
+The val-
+ues of G for the structures are seen to be close to each
+other and it is quite possible to approximate our results
+by a continuous line. The figures also show data from
+low-temperature experiments [32–34]. For tungsten, our
+results are seen to agree quite well with experiment. For
+tantalum, experimental data from Ref. [34] provides only
+the lower boundary of G, which does not contradict our
+calculations. Figures 6 and 7 also show results from some
+other calculations. It is seen that compared to our re-
+sults, calculations by Lin et al. [18] for W give overesti-
+mated values of G for the increasing temperature (Fig. 6).
+Such a behavior has earlier been observed in other metals
+[12] and can be related to the more correct account for the
+energy dependence of α2F(ε,Ω) in formula (3). In turn,
+the values of G(Te) from Ref. [19] are much lower than
+our results and the experimental data available.
+Note
+that the presence of adjustable parameters in the calcu-
+lation method may reduce the accuracy of results if they
+
+5
+0
+10
+20
+30
+40
+0
+3
+6
+9
+12
+fcc
+ 
+ 
+G (10
+17
+ W/m
+3
+/K)
+T
+e
+ (kK)
+bcc
+ W  
+FIG. 6: Electron-phonon coupling factor versus Te for tung-
+sten from our calculation (solid, dashed lines for bcc and fcc,
+respectively), from calculations reported in papers [18] (dot-
+ted line) and [19] (dashed-dotted line), and from experiments
+[32] and [33] (the circle and the triangle, respectively). The
+vertical line shows the approximate value of Te above which
+the fcc phase becomes more energetically favorable than bcc.
+0
+10
+20
+30
+40
+0
+2
+4
+6
+8
+10
+ 
+ 
+G (10
+17
+ W/m
+3
+/K)
+T
+e
+ (kK)
+ Ta 
+bcc
+fcc
+FIG. 7: Electron-phonon coupling factor versus Te for tan-
+talum from our calculations using formula (3) (solid, dashed
+lines for bcc and fcc, respectively) and by a formula (4) (dot-
+ted line). Other calculations: dashed-dotted line - Ref. [19],
+dashed-dotted-dotted line - Ref. [34] by a formula from
+Ref. [18] (see the text). The triangle shows the lower bound-
+ary of G from experiment [34]. The vertical line shows the
+approximate value of Te above which the fcc phase is more
+energetically preferable than bcc.
+are adjusted to conditions (for example, at T =0) different
+from what we are having here.
+For tantalum (fig. 7), our calculations by expression (4)
+(the dotted line) had one distinction from those reported
+in paper [18]: the values of λ and ⟨Ω⟩2 were determined
+from first-principles calculations rather than from experi-
+mental evaluation. It is seen that in this case, approaches
+[18] and [12] give close values for G(Te), the differences
+0
+4
+8
+12
+16
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+ W  
+I / I
+0
+t (ps)
+FIG. 8: Intensity of diffraction peak (211) versus time for
+tungsten for absorbed energy density 0.8 MJ/kg from our
+calculation (the solid line), calculations with a constant G
+[33] (the dashed line), calculations with G(Te) from Ref. [18]
+(the dashed-dotted line), and measurements [33] (circles).
+are minimal. In Ref. [34], the electron-phonon coupling
+factor was also calculated with formula (4) but with the
+electronic DOS determined from MD calculations. But
+here deviations from our results come, first of all, from
+the underestimated parameter λ.
+The authors of [34]
+used the empirical value from Ref. [23], λ=0.65. Our cal-
+culations from first principles gave λ=0.88 in the case of
+tantalum. For tungsten, the difference between the em-
+pirical [23] and calculated values of λ is not so large; they
+agree within ∼3%.
+Let’s consider the accuracy of our calculations in com-
+parison with other experimental results. The authors of
+paper [33] measured how evolved the intensity of the Laue
+diffraction peak (211) after a 30-nm-thick tungsten film
+deposited on a silicon nitride substrate was irradiated by
+400-nm laser pulses with τp=130 fs. The absorbed energy
+density Eabs was about 0.8 MJ/kg. Figure 8 compares
+experimental data with calculations performed in three
+variants (see [12] for calculation details). In addition to
+our computation with use of formula (4), it shows cal-
+culations with G(Te) taken from Ref. [18] and with con-
+stant G=2 · 1017 W/m3/K and ΘD=312 K [33]. The re-
+sults obtained with expression (3) are seen to agree quite
+well with experiment. The use of G(Te) from Ref. [18]
+slightly worsens the agreement and the calculation with
+the constant G markedly underestimates the change of
+the diffraction peak intensity at times below 10 ps.
+Figure 9 presents ion temperature versus electron tem-
+perature for tungsten, calculated by solving equations
+(1)-(2).
+We reproduced experimental conditions from
+Ref. [33] but did calculations for several values of Eabs.
+The possibility of the bcc→fcc transition was not consid-
+ered because ultrafast melting was here more probable
+[17]. Figure 9 also shows the melting temperature of W
+versus Te, obtained in this work and by Murphy et al.
+
+6
+0
+5
+10
+15
+20
+25
+30
+0
+1
+2
+3
+4
+5
+6
+T
+m
+(T
+e
+)
+0.91 MJ/kg
+2.77 MJ/kg
+ W  
+T
+i
+ (kK)
+T
+e
+ (kK)
+0.8 MJ/kg
+T
+m
+0
+FIG. 9: Calculated evolution of electron and ion tempera-
+tures (isochoric heating) after irradiation of the 30-nm-thick
+tungsten film by a 130-fs pulse for different absorbed energy
+densities (dashed, dashed-dotted, and dashed-dotted-dotted
+lines). The solid line shows the melting temperature Tm as a
+function of Te from our calculation, the circles show Tm(Te)
+from Ref. [17] (non-isochoric conditions), and the dotted line
+shows the normal melting temperature of W.
+[17] from MD calculations. Remind that our Tm(Te) was
+calculated with the Lindemann criterion. As seen from
+Fig. 9, the melting temperature of tungsten decreases
+with the increasing Te due to lattice softening (Fig. 5).
+The resulted dependence Tm(Te) agrees rather well with
+data from Ref. [17] despite the essentially different ap-
+proaches to its determination. Some discrepancy comes
+from the fact that our calculation corresponded to the
+isochore V =V0, while in MD simulation [17], the sample
+could expand along the axis normal to the target surface.
+In paper [33], a threshold value Em
+abs required for the
+complete melting of tungsten was determined. For the
+conditions of that experiment, it was found to be 0.9
+MJ/kg. Our calculations give a very close value of 0.91
+MJ/kg (details of calculation can be found in paper [12]).
+Complete melting occurs after the temperature Tm is
+reached and the lattice gets sufficient heat to overcome
+the latent heat of fusion, ∆Hm [35]. The absorbed en-
+ergy density of 0.8 MJ/kg is not enough to completely
+melt the target [33]. It is seen from Fig. 9 that at high
+Eabs (>2.5 MJ/kg) the lattice temperature Ti reaches Tm
+even earlier than Ti(Te) reaches its maximum. At high
+Te, the melting temperature of tungsten becomes much
+lower than the normal melting temperature determined
+at ambient pressure, T 0
+m≈3.7 kK. MD calculations and
+analytic equations of state [36, 37], including that one
+for tungsten, suggest that the heat of fusion changes un-
+der the action of external conditions and it will reduce as
+Tm decreases. This will also influence the time of melt-
+ing. Usually, Te reaches a maximum after irradiation by
+ultrashort pulses at a time of about a few τp.
+There-
+fore at sufficiently high Eabs (>2.5 MJ/kg) tungsten will
+0
+5
+10
+15
+20
+25
+30
+35
+0
+1
+2
+3
+4
+5
+6
+7
+1.12 MJ/kg
+3.2 MJ/kg
+ Ta 
+1 MJ/kg
+T
+i
+ (kK)
+T
+e
+ (kK)
+T
+m
+0
+T
+m
+(T
+e
+)
+FIG. 10: Calculated evolution of electron and ion temper-
+atures (isochoric heating) after irradiation of a 30-nm-thick
+tantalum film by a 130-fs-pulse for different absorbed energy
+densities (dashed, dashed-dotted, and dashed-dotted-dotted
+lines). The solid line shows Tm versus Te from our calculation
+and the dotted line shows the normal melting temperature of
+Ta.
+melt during sub-picosecond times which is also proved by
+calculations [17].
+Now consider tantalum. Figure 10 demonstrates the
+Ti(Te) dependence for Ta similarly to tungsten. Irradia-
+tion conditions and target thickness are the same as for
+W. It is seen that the melting curve Tm(Te) reaches a
+maximum approximately at Te=7.3 kK due to the hard-
+ening of the Ta crystal lattice at these temperatures, as
+mentioned earlier (see Fig. 5). Unlike gold, whose melt-
+ing temperature begins to increase only at Te>15 kK (re-
+maining almost constant at lower Te) [12], for tantalum
+this growth of Tm starts right after the electron temper-
+ature increases.
+At Te higher than 7.3 kK, its lattice
+begins to gradually soften. Like tungsten, tantalum at
+sufficiently high values of Eabs (>3 MJ/kg) must melt on
+the sub-picosecond time scale due to the loss of dynamic
+stability by its lattice (Fig. 10). We do not consider the
+bcc→fcc transition here also. The high electron-phonon
+coupling factor of tantalum signals a higher probability
+of its ultrafast melting. However, the existence of a max-
+imum of Tm(Te) at relatively low electron temperatures
+gives an interesting effect. If such hardening really oc-
+curs, it should lead to an increase in the melting thresh-
+old Em
+abs for Ta metal. As shown in calculations, Em
+abs
+will be at least 25% higher. For tantalum normal melt-
+ing temperature, T 0
+m=3.29 kK, the threshold value �Em
+abs
+equals 0.74 MJ/kg. If the crystal lattice hardens, then,
+under isochoric heating, an absorbed energy density of
+∼1.12 MJ/kg is required for complete melting. For non-
+isochoric conditions, the threshold may be lower, about
+0.93 MJ/kg. However, the value is still rather far from
+normal �Em
+abs=0.74 MJ/kg and can be determined quite
+reliably in experiment (see, for example, [5]). In addi-
+
+7
+tion, the growth of Tm make the latent heat of fusion
+higher which will also delay the complete melting.
+A similar maximum of Tm(Te) at relatively low heating
+(Te∼5 kK) is also present in platinum [12]. As shown by
+calculations from first principles, its electronic structure
+is also characterized by a high electronic density of states
+N(µ) on the Fermi level [18], which strongly reduces with
+the increasing Te. Our calculations show that the effect
+of lattice hardening is a bit lower here and the melting
+threshold increases by about 18%.
+But since �Em
+abs for
+platinum at the normal melting temperature T 0
+m is quite
+small (∼0.39 MJ/kg), the detection of its increase in ex-
+periment may be limited by experimental accuracy.
+IV.
+CONCLUSIONS
+The paper studied the interaction of femtosecond laser
+pulses with thin tungsten and tantalum films through cal-
+culations from first principles. Calculated results shows
+the body-centered cubic structure of both the metals to
+lose its dynamic stability at rather high electron tem-
+peratures. This effect must lead to their melting on the
+sub-picosecond time scale when the electronic subsystem
+is heated above 22 kK. It is also demonstrated that the
+metals have rather high values of the electron-phonon
+coupling factor (∼ several units per 1017 W/m3/K) at
+electron temperatures from room temperature to ∼45
+kK. In addition, unlike tungsten, the crystal lattice of
+tantalum hardens at relatively low values of Te (≲7 kK).
+The hardening changes the value of the complete melt-
+ing threshold. Our calculations show that the melting
+threshold will be at least 25% higher if hardening re-
+ally occurs.
+We suppose that this effect for tantalum
+can be detected quite reliably by modern experimental
+techniques used to study the interaction of matter with
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+HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN
+Adam Kania 1 Artur Kasymov 1 Maciej Zieba 2 Przemysław Spurek 1
+Abstract
+Recently, generative models for 3D objects are
+gaining much popularity in VR and augmented
+reality applications. Training such models us-
+ing standard 3D representations, like voxels or
+point clouds, is challenging and requires com-
+plex tools for proper color rendering. In order to
+overcome this limitation, Neural Radiance Fields
+(NeRFs) offer a state-of-the-art quality in synthe-
+sizing novel views of complex 3D scenes from a
+small subset of 2D images.
+In the paper, we propose a generative model
+called HyperNeRFGAN, which uses hypernet-
+works paradigm to produce 3D objects repre-
+sented by NeRF. Our GAN architecture leverages
+a hypernetwork paradigm to transfer gaussian
+noise into weights of NeRF model. The model
+is further used to render 2D novel views, and a
+classical 2D discriminator is utilized for training
+the entire GAN-based structure. Our architecture
+produces 2D images, but we use 3D-aware NeRF
+representation, which forces the model to produce
+correct 3D objects. The advantage of the model
+over existing approaches is that it produces a ded-
+icated NeRF representation for the object without
+sharing some global parameters of the rendering
+component. We show the superiority of our ap-
+proach compared to reference baselines on three
+challenging datasets from various domains.
+1. Introduction
+Generative Adversarial Nets (GANs) (Goodfellow et al.,
+2014) allow us to generate high-quality 2D images (Yu
+et al., 2017; Karras et al., 2017; 2019; 2020; Struski et al.,
+2022). On the other hand, maintaining similar quality for
+*Equal contribution
+1Faculty of Mathematics and Computer
+Science, Jagiellonian University 6 Lojasiewicza Street, 30-348
+Krak´ow, Poland 2Department of Artificial Intelligence, Univer-
+sity of Science and Technology Wyb. Wyspianskiego 27, 50-370,
+Wrocław, Poland. Correspondence to: Przemysław Spurek <prze-
+[email protected]>.
+Figure 1. HyperNeRFGAN architecture leverages a hypernetwork
+paradigm to transfer gaussian noise into weights of NeRF model.
+After that, we render 2D novel views by NeRF and use a classical
+2D discriminator. Our architecture produces 2D images, but we
+use 3D-aware NeRF representation, which forces the model to
+produce correct 3D objects.
+3D objects is challenging. It is mainly caused by using
+3D representations like voxels and point clouds that require
+massive deep architectures and have problems with proper
+color rendering.
+We can solve this problem by operating directly on 2D
+image space. We expect our approach to extract informa-
+tion from unlabeled 2D views to obtain 3D shapes. To
+obtain such effects, we can use Neural Radiance Fields
+(NeRFs) (Mildenhall et al., 2021), which allow synthesizing
+novel views of complex 3D scenes from a small subset of
+2D images. Based on the relations between those base im-
+ages and computer graphics principles, such as ray tracing,
+this neural network model can render high-quality images
+of 3D objects from previously unseen viewpoints.
+Unfortunately, it is not trivial how to use NeRF represen-
+tation with GAN-type architecture. The most challenging
+problem is connected with the conditioning mechanism (Re-
+bain et al., 2022) dedicated to NeRF. Therefore, most models
+arXiv:2301.11631v1  [cs.CV]  27 Jan 2023
+
+[x,y,z]
+Weights
+Generator
+NeRE
+Training Data
+True
+Discriminator
+FalseHyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN
+Figure 2. Comparison of HyperNeRFGAN and HoloGAN, GRAF, π-GAN on CARLA. We obtain similar results to π-GAN, but we have
+a better value of FID score, see Tab 2.
+use SIREN (Sitzmann et al., 2020) instead of NeRF, where
+we can naturally add conditioning. But the quality of the 3D
+object is slice worst than in NeRF. In GRAF (Schwarz et al.,
+2020) and π-GAN (Chan et al., 2021), authors propose a
+model which uses SIREN and a conditioning mechanism to
+produce implicit representation. Such solutions give promis-
+ing results, but it is not trivial how to use NeRF instead of
+SIREN in such solutions. In Fig. 2 we present a qualita-
+tive comparison between our model, GRAF (Schwarz et al.,
+2020) and π-GAN (Chan et al., 2021). As we can see, our
+model can model the transparency of glass.
+In the paper, we propose a generative model called HyperN-
+eRFGAN1, which combines the hypernetworks paradigm
+and NeRF representation. Hypernetworks, introduced in
+(Ha et al., 2016), are defined as neural models that generate
+weights for a separate target network solving a specific task.
+Our GAN-based model leverages a hypernetwork paradigm
+to transfer gaussian noise into weights of NeRF (see Fig. 1).
+After that, we render 2D novel views by NeRF and use a
+classical 2D discriminator to train the entire GAN-based
+structure in implicit form. Our architecture produces 2D
+images, but we use 3D-aware NeRF representation, which
+forces the model to produce correct 3D objects.
+Our contributions to this paper include the following:
+• We introduce the NeRF-based implicit GAN architec-
+ture - the first GAN model for generating high-quality
+3D NeRF representations.
+• We show that utilizing the hypernetwork paradigm for
+NeRFs leads to a better quality of 3D representations
+1The source code is available at: https://github.com/
+gmum/HyperNeRFGAN
+than SIREN-based architectures.
+• Our model allows 3D-aware image synthesis from un-
+supervised 2D images.
+2. Related Work
+Neural representations and rendering
+3D objects can
+be represented by using many different approaches, includ-
+ing voxel grids (Choy et al., 2016), octrees (H¨ane et al.,
+2017), multi-view images (Arsalan Soltani et al., 2017; Liu
+et al., 2022), point clouds (Achlioptas et al., 2018; Shu
+et al., 2022; Yang et al., 2022), geometry images (Sinha
+et al., 2016), deformable meshes (Girdhar et al., 2016), and
+part-based structural graphs (Li et al., 2017).
+The above representations are discreet, which causes some
+problems in real-life applications. Contrary, we can repre-
+sent 3D objects as a continuous function (Dupont et al.,
+2022). In practice implicit occupancy (Chen & Zhang,
+2019; Mescheder et al., 2019; Peng et al., 2020), distance
+field (Michalkiewicz et al., 2019; Park et al., 2019) and
+surface parametrization (Yang et al., 2019; Spurek et al.,
+2020; 2022; Cai et al., 2020) models use a neural network to
+parameterize a 3D object. In such a case, we do not have a
+fixed number of voxels, points, or vertices, but we represent
+shapes as a continuous function.
+These models are limited by their requirement of access to
+ground truth 3D geometry. Implicit neural representations
+(NIR) have been proposed to solve such a problem. Such
+architectures can reconstruct 3D structures from multi-view
+2D images (Mildenhall et al., 2021; Niemeyer et al., 2020;
+Tewari et al., 2020).
+The two most important methods are NeRF (Mildenhall
+
+HoloGAN
+GRAF
+pi-GANHyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN
+et al., 2021) and SIREN (Sitzmann et al., 2020). NeRF
+uses volume rendering (Kajiya & Von Herzen, 1984) for
+reconstructing a 3D scene using neural radiance and den-
+sity fields to synthesize novel views. SIREN replaced the
+popular ReLU activation function with sine functions with
+modulated frequencies. Most NeRF and SIREN-based meth-
+ods focus on a single 3D object or scene. In practice, we
+overfit individual objects or scenes. In our paper, we focus
+on generating 3D models represented by NeRF.
+Single-View Supervised 3D-Aware GANs
+Generative
+Adversarial Nets (GANs) (Goodfellow et al., 2014) allow
+for the generation of high-quality images (Yu et al., 2017;
+Karras et al., 2017; 2019; 2020; Struski et al., 2022). How-
+ever, GANs operate on 2D images and ignore the 3D nature
+of our physical world. Therefore, it is important to use 3D
+structures of objects to generate images and 3D objects.
+The first approach for 3D-aware image syntheses like Visual
+Object Networks (Zhu et al., 2018) and PrGANs (Gadelha
+et al., 2017) first generating a voxelized 3D shape using a
+3D-GAN (Wu et al., 2016) and then projecting it into 2D.
+HoloGAN (Nguyen-Phuoc et al., 2019) and Block-
+GAN (Nguyen-Phuoc et al., 2020) work in a similar fusion
+but use implicit 3D representation for modeling 3D represen-
+tation of the world. Unfortunately, using an explicit volume
+representation has constrained their resolution (Lunz et al.,
+2020). In (Szab´o et al., 2019), authors propose using meshes
+to represent 3D geometry. On the other hand, in (Liao et al.,
+2020) uses collections of primitives for image synthesis.
+In GRAF (Schwarz et al., 2020) and π-GAN (Chan et al.,
+2021), authors use implicit neural radiance fields for 3D-
+aware image and geometry generation. In our work, we
+use NeRF instead of SIREN and hypernetwork paradigm
+instead of a conditioning procedure.
+Authors use a shading-guided pipeline in ShadeGAN (Pan
+et al., 2021), and in GOF (Xu et al., 2021), they gradu-
+ally shrink the sampling region of each camera ray. GI-
+RAFFE (Niemeyer & Geiger, 2021) we first generate low-
+resolution feature maps. In the second step, we passed
+representation to a 2D CNN to generate outputs at a higher
+resolution.
+In StyleSDF (Or-El et al., 2022), authors merge an SDF-
+based 3D representation with a StyleGAN2 for image gen-
+eration. In (Chan et al., 2022), authors use StyleGAN2
+generator and tri-plane representation of 3D objects. Such
+methods outperform other methods in the quality of gener-
+ated objects but are extremely hard to train.
+Hypernetworks + generative modeling
+Combining hy-
+pernetworks and generative models is not new. In (Ratzlaff
+& Fuxin, 2019; Henning et al., 2018) authors built GANs
+to generate parameters of a neural network dedicated to
+regression or classification tasks. HyperVAE (Nguyen et al.,
+2020) is designated to encode any target distribution by
+producing generative model parameters given distribution
+samples. HCNAF (Oechsle et al., 2019) is a hypernetwork
+that produces parameters for a conditional autoregressive
+flow model (Kingma et al., 2016; Oord et al., 2018; Huang
+et al., 2018). In (Skorokhodov et al., 2021) authors proposed
+INR-GAN (Skorokhodov et al., 2020) uses a hypernetwork
+to produce a continuous representation of images. The hy-
+pernetwork can modify the shared weights by the low-cost
+mechanism of factorized multiplicative modulation.
+3. HyperNeRFGAN: Hypernetwork for
+Generating NeRF representions
+In this section, we present HyperNeRFGAN - a novel gener-
+ative model for 3D objects. The main idea of the proposed
+approach is that generator serves as a hypernetwork (Ha
+et al., 2016) and transforms the noise vector sampled from
+the known distribution to the weights of the target model.
+Compared to previous works (Skorokhodov et al., 2020),
+the target model is represented by NeRF (Mildenhall et al.,
+2021) 3D representation of the object. Consequently, it is
+possible to generate many images of the object from various
+perspectives in a controllable manner. Moreover, thanks to
+the NeRF-based image rendering, the discriminator operates
+on 2D images generated from multiple perspectives, com-
+pared to GAN-based models fed by complex 3D structures.
+In this section, we first briefly discuss the basic concepts
+used in our approach, and further, we focus on the architec-
+ture and training details.
+Hypernetwork
+Hypernetworks, introduced in (Ha et al.,
+2016), are defined as neural models that predict weights
+for a different target network designed to solve a specific
+task. This approach reduces the number of trainable param-
+eters compared to standard methods that inject additional
+information into the target model using a single embedding.
+A significant reduction of the size of the target model can
+be achieved since it is not sharing the global weights, but
+they are returned by the hypernetwork. Making an analogy
+between Hypernetworks and generative models, the authors
+of (Sheikh et al., 2017), use this mechanism to generate
+a diverse set of target networks approximating the same
+function.
+Hypernetworks are widely used in many domains, including
+few-shot problems (Sendera et al., 2023) or probabilistic
+regression scenarios (Zieba et al., 2020). Various methods
+also use them to produce a continuous representation of 3D
+objects (Spurek et al., 2020; 2022). For instance, Hyper-
+Cloud (Spurek et al., 2020) represents a 3D point cloud as a
+classical MLP that serves as a target model and transforms
+
+HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN
+Figure 3. Elements generated by model train on three classes of ShapeNet (car, plane, chairs).
+points from a uniform distribution on the gaussian ball to the
+point clouds that represent the desired shape. , In (Spurek
+et al., 2022), the target model is represented by a Continuous
+Normalizing Flow (Grathwohl et al., 2018), the generative
+model that creates the point cloud from the assumed base
+distribution in 3D space.
+GAN
+is a framework for training deep generative models
+using a minimax game. The goal is to learn a generator
+distribution PG(x) that matches the real data distribution
+Pdata(x). GAN learns a generator network G that produces
+samples from the generator distribution PG by transform-
+ing a noise variable z ∼ Pnoise(z) (usually Gaussian noise
+N(0, I)) into a sample G(z). The generator learns by play-
+ing against an adversarial discriminator network D aiming
+to distinguish between samples from the true data distri-
+bution Pdata and the generator’s distribution PG. More
+formally, the minimax game is given by the following ex-
+pression:
+minG maxD[V (D, G) =
+Ex∼Pdata[log D(x)] + Ez∼Pnoise[log(1 − D(G(z)))]].
+The main advantage over other models is producing sharp
+images indistinguishable from real ones. GANs are impres-
+sive regarding the visual quality of images sampled from the
+model, but the training process is often challenging and un-
+stable. This phenomenon is caused by direct optimization of
+the training objective is intractable, and the model is usually
+trained by optimizing the parameters of the discriminator
+and generator in alternating steps.
+In recent years, many researchers focused on modifying
+the vanilla GAN procedure to improve the stability of the
+training process. Some modifications were based on chang-
+ing the objective function to Wasserstein distance (WGAN)
+(Arjovsky et al., 2017), restrictions on the gradient penal-
+ties (Gulrajani et al., 2017; Kodali et al., 2017), Spectral
+Normalization (Miyato et al., 2018), or imbalanced learning
+rate for generator and discriminator(Gulrajani et al., 2017;
+
+HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN
+Figure 4. Linear interpolation examples generated with models trained on images of cars, planes, and chairs from ShapeNet (three first
+lines) and CARL data set (last two rows).
+Miyato et al., 2018). The model architectures were also
+more deeply explored by utilizing self-attention mechanisms
+SAGAN (Zhang et al., 2018), and progressively growing
+ProGAN (Karras et al., 2017) and style-gan architectures
+StyleGAN (Karras et al., 2019).
+INR-GAN
+Implicit Neural Representation GAN (Sko-
+rokhodov et al., 2020) is a variant of the GAN-based model
+that utilizes hypernetworks to generate parameters for the
+target model instead of direct image generation. The tar-
+get model, represented by simple MLP, returns the color in
+RGB format for a given pixel location. The model is very
+close architecturally to StyleGAN2 (Karras et al., 2020) and
+has clear advantages over the direct approach, mainly be-
+cause using INR-GAN enables generating images without
+assuming the arbitrarily given resolution.
+NeRF representation of 3D objects
+NeRFs (Mildenhall
+et al., 2021) represent a scene using a fully-connected archi-
+tecture. As the input, NeRF takes a 5D coordinate (spatial
+location x = (x, y, z) and viewing direction d = (θ, ψ))
+and it outputs an emitted color c = (r, g, b) and volume
+density σ.
+A vanilla NeRF uses a set of images for training. In such a
+scenario, we produce many rays passing through the image
+and a 3D object represented by a neural network. NeRF
+approximates this 3D object with a MLP network:
+FΘ : (x, d) → (c, σ),
+and optimizes its weights Θ to map each input 5D coordinate
+to its corresponding volume density and directional emitted
+color.
+The loss of NeRF is inspired by classical volume rendering
+(Kajiya & Von Herzen, 1984). We render the color of all
+rays passing through the scene. The volume density σ(x)
+can be interpreted as the differential probability of a ray. The
+expected color C(r) of camera ray r(t) = o + td (where o
+is ray origin and d is direction) can be computed with an
+integral.
+In practice, this continuous integral is numerically estimated
+using a quadrature. We use a stratified sampling approach
+where we partition our ray [tn, tf] into N evenly-spaced
+bins and then draw one sample uniformly at random from
+within each bin:
+ti ∼ U[tn + i − 1
+N
+(tf − tn), tn + i
+N (tf − tn)].
+We use these samples to estimate C(r) with the quadrature
+rule discussed in the volume rendering review by Max (Max,
+1995):
+ˆC(r) =
+N
+�
+i=1
+Ti(1 − exp(−σiδi))ci,
+where T(t) = exp
+�
+�−
+i−1
+�
+j=1
+σiδi
+�
+� ,
+where δi = ti+1 − ti is the distance between adjacent sam-
+ples. This function for calculating ˆC(r) from the set of
+(ci, σi) values is trivially differentiable.
+We then use the volume rendering procedure to render the
+color of each ray from both sets of samples. Contrary to the
+baseline NeRF (Mildenhall et al., 2021), where two ”coarse”
+
+HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN
+and ”fine” models were simultaneously trained, we use only
+the ”coarse” architecture.
+3.1. HyperNeRFGAN
+In this work, we propose a novel GAN architecture, HyperN-
+eRFGAN, for generating 3D representations. The proposed
+approach utilizes INR-GAN, the implicit approach for gen-
+erating samples. We postulate using the NeRF model as a
+target network compared to standard INR-GAN architec-
+ture, which uses the MLP model to create the output image.
+Thanks to that approach, the generator creates a specific
+3D representation of the scene or object by delivering the
+specific NeRF parameters.
+The architecture of our model is provided in Fig. 1. The
+generator G takes the sample from the assumed base dis-
+tribution (Gaussian) and returns the set of parameters Θ.
+These parameters are further used inside the NeRF model
+FΘ to transform the spatial location x = (x, y, z) to emitted
+color c = (r, g, b) and volume density σ. Instead of stan-
+dard linear architecture, FΘ uses factorized multiplicative
+modulation (FMM) layers.
+The FMM layer with input of size nin and output of size
+nout can be defined as:
+y = W ⊙ (A × B) · xin + b = ˜W · xin + b,
+where W and b are matrices that share the parameters across
+3D representations, and A, B are two modulation matrices
+of shapes nout × k, k × nin respectively, created by the
+generator. The parameter k controls the rank of A × B.
+Higher values of k increase the expressiveness of the FMM
+layer but also increase the amount of memory required by
+the hypernetwork. We always use k = 10.
+The INR model FΘ is a simplified version of the baseline
+NeRF. To make training less computationally expensive,
+we do not optimize two networks as the original NeRF. We
+reject the bigger ”fine” network and only employ the smaller
+“coarse” network. Additionally, we reduce the size of the
+”coarse” network by decreasing the number of channels in
+each hidden layer from 256 to 128. In some experiments,
+we also decrease the number of layers from 8 to 4.
+We differ from the baseline NeRF in one more aspect, as we
+don’t use the viewing direction. That’s because the images
+used for training don’t have view-dependent features like
+reflections. Even though the viewing direction is not used
+in our architecture, there is no reason why it couldn’t be
+enabled for datasets that would benefit from it.
+Our NeRF is a single MLP, which takes only the spatial
+location as input:
+FΘ : x → (c, σ),
+In this work, we utilize the StyleGAN2 architecture, follow-
+ing the design patterns of INR-GAN. The entire model is
+trained using the StyleGanv2 objective in a similar way as
+in INR-GAN. In each training iteration, the noise vector is
+sampled and transformed using generator G to obtain the
+weights of the target NeRF model FΘ. The target model is
+further used to render 2D images from various angles. The
+generated 2D images further serve as fake images for the
+discriminator D. The role of the generator G is to create the
+3D representation that enables to render 2D images that will
+fool the discriminator. The discriminator aims to distinguish
+between fake renders and authentic 2D images from the data
+distribution.
+4. Experiments
+In this section, we first evaluate the quality of generating 3D
+objects by HyperNeRFGAN. We use a data set containing
+2D images of 3D objects obtained from ShapeNet (Zimny
+et al., 2022). The data set contains 50 images of each ele-
+ment from the plane, chair, and car classes. It is the most
+suitable data set for our purpose since each object has a few
+images of each element. Then we use CARLA (Dosovitskiy
+et al., 2017), which contains images of cars. In such a case,
+we have only one image per object, but still, we have photos
+of objects from all sides. We can produce full 3D objects,
+which can be used in VR or augmented reality. In the end,
+we use classical CelebA (Liu et al., 2015) dataset, which
+contains faces. From a 3D generation point of view, it is
+challenging since we only have fronts of faces. In practice,
+3D based generative model can be used to 3D-aware image
+synthesis (Chan et al., 2022).
+4.1. 3D object generation from ShapeNet
+In our first experiments, we use a ShapeNet base data set
+containing 50 images of each element from the plane, chair,
+and car classes. Such representation is perfect for training
+3D models since each element has been seen from many
+views. The data was taken from (Zimny et al., 2022), where
+authors train an autoencoder-based generative model. In
+Fig. 3, we present objects generated from our model. In
+Fig. 4, we also present linear interpolation of objects. As
+we can see, objects are of very good quality, see Tab 1.
+ShapeNet
+cars
+planes
+chairs
+Points2NeRF
+82.1
+239
+129.3
+HyperNeRFGAN
+29.6
+33.4
+22.0
+Table 1. Competition of HyperNeRFGAN and autoencoder based
+model by using FID. Competition between GAN and autoencoder
+and GAN is difficult. But we can obtain a better FID score.
+
+HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN
+Figure 5. Examples from a model trained on CARLA.
+Figure 6. Meshes generated with a model trained on CARLA
+dataset and with models trained on planes and chairs from
+ShapeNet.
+4.2. 3D object generation from CARLA data set
+In the second experiment, we compare our model on
+CARLA dataset with other GAN-based models: Holo-
+GAN (Nguyen-Phuoc et al., 2019), GRAF (Schwarz et al.,
+2020) and π-GAN (Chan et al., 2021). CARLA (Dosovit-
+skiy et al., 2017) contains images of cars. We have only
+one image per object, but still, we have photos of objects
+from all sides. Consequently, full 3D objects can be used
+in VR or augmented reality. The visual comparison we
+present in Fig. 2. As illustrated, we can effectively model
+the transparency of glass in cars, see Fig. 5. In Tab. 2 we
+present a numerical comparison of the Frechet Inception
+Distance (FID), Kernel Inception Distance (KID), and In-
+ception Score (IS). As can be seen, we obtain better results
+than the π-GAN model.
+In the case of NeRF representation, we can produce meshes,
+see Fig. 6.
+CARL
+FID
+KID
+IS
+HoloGAN
+67.5
+3.95
+3.52
+GRAF
+41.7
+2.43
+3.60
+π−GAN
+29.2
+1.36
+4.27
+HyperNeRFGAN
+20.5
+0.78
+4.20
+Table 2. FID, KID mean×100, and IS for CARLA dataset.
+4.3. 3D-aware image synthesis from CelebA
+In our third experiment, we further compare the same mod-
+els as in the second experiment by changing the setup to
+face generation. For this task, we utilize the CelebA (Liu
+et al., 2015) dataset, which contains 200,000 high-resolution
+face images of 10,000 different celebrities. We crop the im-
+ages from the top of the hair to the bottom of the chin and
+resize them to 128x128 resolution as π-GAN authors did.
+We present quantitative results in Tab. 3. As you can notice,
+HyperNeRFGAN and π-GAN achieve similar performance,
+which can also be seen in Fig. 7.
+5. Conclusions
+In this work, we presented a novel approach to generating
+NeRF representation from 2D images. Our model leverages
+
+HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN
+CelebA
+FID
+KID
+IS
+HoloGAN
+39.7
+2.91
+1.89
+GRAF
+41.1
+2.29
+2.34
+π−GAN
+14.7
+0.39
+2.62
+HyperNeRFGAN
+15.04
+0.66
+2.63
+Table 3. FID, KID mean×100, and IS for CelebA dataset.
+HyperNeRFGAN
+π-GAN
+Figure 7. Comparison between HyperNeRFGAN (first 3 columns)
+and π-GAN uncurated generated faces
+a hypernetwork paradigm and NeRF representation of the
+3D scene. HyperNeRFGAN take a Gaussian noise and
+return the weights of a NeRF network that reconstructs 3D
+objects from 2D images. In training, we use only unlabeled
+images and a StyleGan2 discriminator. Such representation
+gives several advantages over the existing approaches. First
+of all, we can use NeRF instead of SIREN representation
+in the GAN type algorithm. Secondly, our model is simple
+and can be effectively trained on 3D objects. Finally, our
+model directly produces NeRF objects without sharing some
+global parameters of the rendering component.
+Limitations
+The main limitation of HyperNeRFGAN is
+the fact that we use only 2D images instead any knowledge
+about 3D object representation. In future work, we plan to
+add some information about the structure of 3D meshes.
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+arXiv:2301.01989v1  [math.AG]  5 Jan 2023
+Construction of tropical morphisms from tropical
+modifications of nonhyperelliptic genus 3 metric graphs
+with tree gonality 3 to metric trees
+Hamdi D¨ervodeli
+Abstract
+In this article, we look into the tree gonality of genus 3 metric graphs
+Γ which is defined as the minimum of degrees of all tropical morphisms
+from any tropical modification of Γ to any metric tree. It is denoted by
+tgon(Γ) and is at most 3. We define hyperelliptic metric graphs in terms
+of tropical morphisms and tree gonality. Let Γ be a genus 3 metric graph
+with tgon(Γ) = 3 which is not hyperelliptic. In this paper, for such metric
+graphs Γ, we construct a tropical modification Γ′ of Γ, a metric tree T
+and a tropical map ϕ : Γ′ → T of degree 3.
+Contents
+1
+Introduction
+2
+2
+Preliminaries
+3
+2.1
+Metric graphs.
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+2.2
+Harmonic maps and tropical morphisms. . . . . . . . . . . . . . .
+6
+2.3
+Tree gonality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2.4
+Hyperelliptic metric graphs. . . . . . . . . . . . . . . . . . . . . .
+8
+3
+Construction of tropical morphisms
+10
+1
+
+1
+Introduction
+We look into the tree gonality of metric graphs. Its motivation comes from
+the striking interplay between graphs and algebraic curves discovered over the
+last two decades. For example, there exists a good theory of divisors on graphs
+(see BN07]) (also including such notions as linear systems, linear equivalences,
+canonical divisors, degrees, and ranks), and maps between metric graphs with
+suitable balancing conditions that behave similarly to morphisms between curves
+(see BN07]).
+Recall that the gonality of an algebraic curve C is the minimum of degrees
+of all non-constant morphisms from C to the projective line P1.
+There are
+two notions of graph gonality in the literature, which are both inspired by
+the gonality of an algebraic curve. They are tree (or geometric) gonality and
+divisorial gonality e.g., studied for ordinary or metric graphs (see Bak08]). Yet
+another variant is stable gonality, which is the infimum of the divisorial gonality
+over all subdivisions of an ordinary graph (see CKK15]).
+We study a tropical version of gonality, where the roles of algebraic curves
+and the projective line are played by metric graphs and metric trees, respectively,
+and the morphisms are replaced by the tropical morphisms (see BN07], Cap14],
+Mik17], BBM11], Cha13]). The tree gonality of a metric graph Γ is defined as
+minimum of degrees of all tropical morphisms from any tropical modification
+of Γ to any metric tree. The tree gonality of any metric graph of genus g is at
+most
+� g
+2
+�
++ 1 (see Theorem 1, DV19]). Its proof is entirely combinatorial and
+provides an explicit method to construct divisors with degree
+� g
+2
+�
++ 1 and rank
+1 on genus-g metric graphs. In this article, we are interested in constructing
+a degree-(
+� g
+2
+�
++ 1) tropical morphism from a tropical modification of Γ to a
+metric tree, where Γ is of genus g with tree gonality
+� g
+2
+�
++ 1.
+Interest for
+such method dates back to (Bak08], Remark 3.13). In this regard, our modest
+contribution is on the case where g = 3 and Γ is not hyperelliptic, i.e., given
+a nonhyperelliptic genus 3 metric graph Γ with tree gonality 3, we construct
+a tropical modification Γ′, a metric tree T , and a degree 3 tropical morphism
+φ : Γ′ → T (Problem 1). We emphasize that our constructions are more direct
+than in DV19] in the sense that we avoid constructing divisors of certain degree
+and rank, but rather make explicit constructions of tropical morphisms from
+tropical modifications of metric graphs to metric trees.
+Problem 1. Let Γ be a genus 3 metric graph with tree gonality 3 which
+is not hyperelliptic. Construct a tropical modification Γ′ of Γ, a metric tree T
+and a tropical morphism ϕ : Γ′ → T of degree 3.
+2
+
+2
+Preliminaries
+2.1
+Metric graphs.
+A graph G is defined by the following data: a set V called the vertex set, a set
+E called the edge set and a map ∂ : E → P(V ) such that for any e ∈ E we
+have |∂(e)| = 1 or |∂(e)| = 2, where P(V ) is the power set of V . We write
+G = (V, E, ∂). The elements of V (resp. E) are called vertices (resp. edges)
+of G. An edge e ∈ E with |∂(e)| = 1 is called a loop.
+Two or more edges
+e1, e2, . . . , en ∈ E are called multiple edges if there exist v1, v2 ∈ V such that
+∂(ei) = {vi, vj} for all i = 1, 2, . . . , n. The graph G is said to be finite if both
+V and E are finite sets. A length map on G is any function l : E → (0, +∞).
+In this article, unless stated otherwise, a graph is always assumed to be finite
+with multiple edges allowed.
+Let G = (V, E, ∂) be a graph. A path in the graph G is a sequence of edges
+(e1, e2 . . . , en−1) for which there exists a sequence of vertices (v1, v2, . . . , vn) such
+that ∂(ei) = {vi, vi+1} for i = 1, 2, . . . , n − 1. If w = (e1, e2, . . . , en−1) is a path
+in G with vertex sequence (v1, v2, . . . , vn) then w is said to be a path from v1
+to vn. A graph G is said to be connected if for any two vertices v1 and v2 there
+exists a path from v1 to v2. Let e ∈ E with ∂(e) = {v, w}. Subdividing the edge
+e ∈ E with ∂(e) = {v, w} into edges e1, e2 yields the graph G′ = (V ′, E′, ∂′)
+where V ′ = V ∪ {z}, E′ = (E \ {e}) ∪ {e1, e2} and ∂′ is given by ∂′|E\{e} = ∂
+and ∂′(e1) = {v, z} , ∂′(e2) = {z, w}.
+Let G = (V, E, ∂) be a connected graph with no loops. An orientation
+on G is a map ⃗∂ : E → V × V such that if we write ⃗∂(e) = (v1, v2) then
+∂(e) = {v1, v2}. Note that giving an orientation ⃗∂ on G is equivalent to giving
+a map (∂0, ∂1) : E → V × V where ∂0, ∂1 : E → V are endpoint maps.
+Fix an orientation (∂0, ∂1) : E → V ×V on G and choose a length map l on
+G. Let (X, d) be the disjoint union of the real metric spaces [0, l(e)] for e ∈ E
+i.e., the set
+X =
+�
+e∈E
+[0, l(e)] :=
+�
+e∈E
+[0, l(e)] × {e}
+together with the metric d : X × X → [0, ∞] given by
+d((x1, e1), (x2, e2)) =
+�
+|x1 − x2|,
+if e1 = e2
+∞,
+otherwise.
+Consider the relation ∼1 on X defined by x ∼1 y if there exists a vertex v ∈ V
+such that x, y ∈ {(0, e) ∈ X | ∂0(e) = v} ∪ {(l(e), e) ∈ X | ∂1(e) = v} and let ∼
+be the equivalence relation on X generated by ∼1 i.e., x ∼ y if and only if x = y
+or there exists a finite subset {z1, z2, . . . , zn} ⊂ X such that x = z1, zn = y and
+zi ∼1 zi+1 for i = 1, 2, . . ., n − 1. Let ¯X := X/∼ be the quotient space of X
+3
+
+with respect to the equivalence relation ∼ and ¯d : ¯X × ¯X → [0, ∞) be given by
+¯d(¯x, ¯y) := inf
+k
+�
+i=1
+d(xi, yi)
+where the infimum is taken over all k ∈ N and sequences (x1, y1, x2, . . . , xk, yk)
+in X such that x1 ∈ ¯x, xi+1 ∼ yi for i = 1, 2, . . ., k − 1 and yk ∈ ¯y. Then,
+Γ := ( ¯X, ¯d) is a metric space. In this case, we say that the metric space ( ¯X, ¯d)
+is obtained from (G, l) by gluing intervals [0, l(e)], one for each e ∈ E, along
+their endpoints in the manner prescribed by G.
+We often regard each edge
+e ∈ E as a subset of Γ and each vertex v ∈ V as a point in Γ.
+Definition 2.1
+A metric graph is a metric space Γ such that there exists a
+loopless connected graph G with a length map l such that Γ is isometric to
+the metric space obtained from (G, l) by gluing intervals [0, l(e)], one for each
+e ∈ E(G), along their endpoints in the manner prescribed by G.
+The pair (G, l) is called a model of Γ whereas Γ is called a realization of the
+model (G, l). The construction of a metric graph from a graph that may have
+loops will be given in the following way. Let G = (V, E, ∂) be a connected graph
+with loops and l a length function on G. Subdividing all the loops e ∈ E, say
+into e1, e2, . . . , en, yields a graph G′ with no loops. The length map l′ on G′ is
+given by l′ = l on E \ {e ∈ E | ∂(e) = 1} and l′(e1) + l′(e2) + . . . + l′(en) = l(e)
+for edges e1, e2 for which a loop e ∈ E subdivided to e1, e2. Then Γ does not
+depend on the choice of the subdivision (G′, l′). Thus, we define Γ to be the
+realization of (G, l), and we also call (G, l) a model (that may have loops) of Γ.
+The first Betti number of Γ is equal to g(G) := |E(G)| − |V (G)| + 1. It
+is called the genus of Γ and it is denoted by g(Γ). A metric graph Γ of genus
+g(Γ) = 0 is called a metric tree.
+Let Γ be a metric graph. A vertex set of Γ is a finite subset S ⊂ Γ such
+that the subspace Γ \ S is isometric to a disjoint union of finitely many real
+open intervals. Any vertex set S ̸= ∅ of Γ induces a model (GS, lS) of Γ in
+the following way. The graph GS = (V, E, ∂) is given by its vertex set V :=
+S, its edge set E defined to be the set of closures of finitely many connected
+components of Γ \ V and the map ∂ : E → P(V ) given by e �−→ ∂(int(e)),
+where int(e) = e \ S and ∂(int(e)) ⊂ V is its boundary in Γ. Each edge e ∈ E
+is isometric to either a segment or a circle. The length map lS : E → (0, ∞)
+assigns each edge e ∈ E the length of the segment or circle isometric to it.
+We single out a particular model for Γ. A point x ∈ Γ is called an essential
+vertex if for any ε > 0, the open ball B(x, ε) :=
+�
+y ∈ Γ | ¯d(x, y) < ε
+�
+is not
+isometric to (−ε, ε) ⊂ R. If x ∈ Γ is an essential vertex, then for any model
+(G, l) of Γ and any edge e ∈ E(G) we have x /∈ int (e), and so, the set of essential
+vertices of Γ is a subset of E(G) for any model (G, l) of Γ. Since G is a finite
+graph, Γ has only finitely many essential vertices.
+4
+
+Lemma 2.2 Let Γ be a metric graph, E the set of essential vertices of Γ, and
+S a finite nonempty subset of Γ. Then, the set S is a vertex set of Γ if and only
+if E ⊆ S.
+Proof. Suppose that ∅ ̸= S is a vertex set in Γ. Then, S induces a model (G, l)
+of Γ where S = V (G) and, so
+Γ \ S = Γ \ V (G) ≡
+�
+e∈E(G)
+(0, l(e)).
+If E ∩ (Γ \ S) ̸= ∅ then there exists x ∈ E and an edge e ∈ E(G) such that x ∈
+int (e) which contradicts x being an essential vertex. Therefore, E ∩ (Γ \ S) = ∅
+and E ⊆ S. Now, assume that E ⊆ S. If E = ∅ then Γ is isometric to a circle,
+and so, any non-empty finite subset of Γ is a vertex set. Suppose that E ̸= ∅. Let
+(G, l) be a model of Γ, and V , E be the set of vertices, edges of G respectively.
+Then, the set V = �
+e∈E ∂(e), where ∂(e) is the boundary set of e ⊂ Γ, is a
+vertex set of Γ. As E is the set of essential vertices, and V is a vertex set, it
+follows, from what we have shown above, that E ⊆ V . Now, if E = V , then E is
+a vertex set. Assume that E ⊊ V . We know that the set V \ E is always finite.
+If this is a one-element set i.e., V \ E = {x1}, then there exist unique edges
+e1, f1 ∈ E, e1 ̸= f1 such that x1 is a common endpoint of e1 and f1. Then, we
+obtain that
+Γ \ E = (Γ \ V ) ∪ {x1} ≡
+�
+e∈E
+(0, l(e)) ∪ {x1}
+≡
+�
+e∈E
+e̸=e1,f1
+(0, l(e)) ⊔ (0, l(e1) + l(f1))
+which implies that E is a vertex set. If V \ E = {x1, x2}, then there exist unique
+edges ei, fi ∈ E with ei ̸= fi such that xi is a common endpoint of ei and fi
+for i = 1, 2. In the case when one of e1 and e2 is equal to one of f1 and f2, say,
+f1 = e2, we have that
+Γ \ E ≡
+�
+e∈E
+e̸=e1,e2,f2
+(0, l(e)) ⊔ (0, l(e1) + l(e2) + l(f2)).
+If both e1 and e2 are different to both f1 and f2, then
+Γ \ E ≡
+�
+e∈E
+e̸=e1,e2,f1,f2
+(0, l(e)) ⊔ (0, l(e1) + l(e2)) ⊔ (0, l(f1) + l(f2))
+and therefore, E is a vertex set. Similarly we get we get that E is a vertex set if
+V \ E = {x1, x2, . . . , xn}, Thus, Γ \ E is isometric to a disjoint union of finitely
+many open real intervals. Since Γ \ S ⊂ Γ \ E, we have that Γ \ S is also is
+isometric to a disjoint union of finitely many open intervals, and therefore, S is
+a vertex set. □
+5
+
+A metric graph is said to be a metric loop if it is isometric to a circle. If
+Γ is not a metric loop, then E ̸= ∅ is a vertex set of Γ. The model (GE, lE)
+induced by the essential vertex set E is called the essential model of Γ. From
+Lemma 2.1, the essential model (GE, lE) is minimal in the sense that any other
+model of Γ can be obtained by a sequence of edge subdivisions of GE. Thus, all
+models are refinements of the essential model. In addition, this implies that the
+valence of a point x ∈ Γ defined as the valence of x in GS for S a vertex set of
+Γ and x ∈ S, is well-defined notion. The valence of the point x ∈ Γ is denoted
+by val(x).
+2.2
+Harmonic maps and tropical morphisms.
+Definition 2.3
+Let Γ1 and Γ2 be metric graphs with loopless models (G1, l1)
+and (G2, l2) respectively, where E(G1) = {e1} and E(G2) = {e2}.
+A map
+ϕ : Γ1 → Γ2 is said to be linear if there exist isometries ρ1 : Γ1 → [0, l1(e1)] and
+ρ2 : Γ2 → [0, l2(e2)] such that the map ρ2 ◦ ϕ ◦ ρ−1
+1
+: [0, l1(e1)] → [0, l2(e2)] is an
+affine linear map.
+Definition 2.4 Let Γ1 and Γ2 be two metric graphs. A continuous map ϕ :
+Γ1 → Γ2 is said to be piecewise linear if there exist loopless models (G1, l1) and
+(G2, l2) of Γ1 and Γ2 respectively, such that for any edge e1 ∈ E(G1) there exists
+an edge e2 ∈ E(G2) such that ϕ(e1) ⊆ e2 and ϕ|e1 : e1 → e2 is a linear map.
+Let ϕ : Γ1 → Γ2 be a piecewise linear map of metric graphs, v ∈ Γ1 and
+w := ϕ(v). Let (G1, l1) (resp., (G2, l2)) be loopless models of Γ1 (resp., Γ2)
+such that for all e1 ∈ E(G1) there exists e2 ∈ E(G2) such that ϕ(e1) ⊆ e2,
+ϕ|e1 : e1 → e2 is a linear map, and assume that v ∈ V (G1) and w ∈ V (G2).
+Fix a direction ⃗w at w (i.e., a ’unit vector’ starting at w with direction of a
+path emanating from w), and let e2 ∈ E(G2) such that w is an endpoint of e2
+and e2 is in the direction ⃗w. Let {ev1, ev2, . . . , evr} ⊆ E(G1) be the set of edges
+emanating from v. Without loss of generality, assume that
+{ev1, ev2, . . . , evs} = {evj | ϕ(evj) ⊆ e2, j = 1, 2, . . . , r}
+for some s such that 0 ⩽ s ⩽ r. Then, ϕ|evj : evj → e is a linear map for
+j = 1, 2, . . . , s because of the choice of models (G1, l1) and (G2, l2). Denote by
+mϕ, ⃗w(v) the sum of slopes of these linear maps ϕ|evj, j = 1, 2, . . ., s. i.e.,
+mϕ, ⃗w(v) =
+s
+�
+j=1
+slope (ρ ◦ ϕ ◦ ρ−1
+vj )
+where ρ : e2 → [0, l2(e2)] and ρvj : evj → [0, l1(evj)] are the chosen isometries
+with unique parametrizations ρ(w) = ρvj(v) = 0 for i = 1, 2, . . . , s i.e., that map
+initial endpoints of e2, evj, j = 1, 2, . . . , s to 0. This definition of the slope of
+the linear maps ϕ|evj, j = 1, 2, . . . , s, and their sum mϕ, ⃗w(v) is independent of
+the choice of such models (G1, l1) and (G2, l2).
+6
+
+Definition 2.5 A continuous map ϕ : Γ1 → Γ2 is said to be a harmonic map
+of metric graphs if it is piecewise linear with integer slopes and satisfies the
+harmonicity condition: For any point v ∈ Γ and any two directions ⃗w1, ⃗w2
+emanating from w := ϕ(v) we have mϕ, ⃗
+w1(v) = mϕ, ⃗
+w2(v).
+Let ϕ : Γ1 → Γ2 be a harmonic map and v ∈ Γ. Then, mϕ(v) := mϕ, ⃗
+w1(v) =
+mϕ, ⃗
+w2(v) for any two directions ⃗w1, ⃗w2 emanating from ϕ(v) is said to be the
+local degree of ϕ at v. The degree of a non-constant harmonic map ϕ : Γ1 → Γ2
+is defined to be the sum of all local degrees of ϕ at the pre-images under ϕ of
+any point w ∈ Γ′ i.e.,
+deg ϕ :=
+�
+v∈Γ,ϕ(v)=w
+mϕ(w)
+for any w ∈ Γ′. The degree of ϕ is independent of the choice of w. (see Section
+2.4, Kag18]).
+Definition 2.6 A non-constant harmonic map ϕ : Γ → Γ′ of metric graphs is
+said to be a tropical morphism between metric graphs if the slopes of ϕ along
+the edges of linearity are nonzero and the following inequality
+(k − 2) ⩾ mϕ(v) · (l − 2)
+holds for all points v ∈ Γ, where k is the valence of v, and l is the valence of
+w := ϕ(v). The above inequality is known as the Riemann-Hurwitz condition.
+2.3
+Tree gonality.
+Let Γ be a metric graph, T a metric tree, and let v ∈ Γ, w ∈ T be two points
+such that val(w) = 1. Denote by Γ′ the quotient space of Γ ⊔ T with respect
+to the equivalence relation ∼ that identifies v with w. The metric space Γ′ is
+a metric graph, and we say that Γ′ is obtained by grafting the metric tree T
+onto the point v ∈ Γ. In this article, we allow the inverse operation of grafting
+a metric tree onto a point of a metric graph, and we call it deleting a metric
+tree onto a point of the metric graph.
+Definition 2.7
+A tropical modification of a metric graph Γ is another metric
+graph Γ′ that is obtained by grafting or deleting a finite number of metric trees
+onto points of Γ.
+Given a tropical modification Γ′ of Γ and a tropical morphism ϕ : Γ′ → T of
+metric graphs, then there exists a tropical modification Γ′′ (resp. T ′) of Γ′ (resp.
+T ) respectively and a tropical morphism ϕ′ : Γ′′ → T ′ that extends ϕ and has
+the same degree as ϕ (CD18]). The following definition is the key definition in
+this article.
+Definition 2.8 The tree gonality of a metric graph Γ, denoted by tgon(Γ), is
+defined as the minimum of degrees of all tropical morphisms from any tropical
+modification of Γ to any metric tree.
+7
+
+In order to study tree gonality and tropical morphisms of metric graphs, we
+consider the equivalence relation on metric graphs under tropical modification
+called tropical equivalence. Metric graphs under tropical equivalence are said to
+be tropically equivalent.
+First, we recall the notions of contracting and deleting an edge of a graph.
+Let G = (V, E, ∂) be a graph and e ∈ E with ∂(e) = {v, w}. Contracting G
+at the edge e ∈ E yields the graph G1 = (V1, E1, ∂1) where V1 := V/ ∼ where
+∼ identifies v with w, E1 := E \ {e} and ∂1 : E1 → P(V1) given as follows:
+for e′ ∈ E1 such that ∂(e′) = {v′, w′} we define ∂1(e′) = {p(v′), p(w′)}, where
+p : V → V1 is the quotient map. Deleting the edge e ∈ E yields the graph
+G′ := (V, E \ {e} , ∂|E\{e}).
+Next, we work with the notion of dangling edges which is due to DV19].
+Note that we regard a singleton graph (a graph without an edge) as a tree.
+Definition 2.9
+Let G be a connected graph. An edge e ∈ E(G) is said to be
+dangling if deleting e gives a graph with two connected components and one of
+them is a tree.
+Let Γ be a metric graph with model (G, l). Assume that g(Γ) ≥ 2. Denote
+by ˜G the graph obtained by successively contracting the dangling edges of G,
+and let ˜l be a length map on ˜G given as the restriction of l on E( ˜G). Let ˜Γ
+be metric graph which is the realization of ( ˜G, ˜l). Then, the metric graph Γ
+is a tropical modification of ˜Γ, and note that by construction, ˜Γ satisfies the
+following property: ˜Γ is the unique metric graph tropically equivalent to Γ whose
+essential model (E, lE) has valency at least 3 i.e., every vertex point has valence
+at least three.
+2.4
+Hyperelliptic metric graphs.
+We first recall the basic theory of divisors on metric graphs (Cha13], BN07]).
+Let Γ be a metric graph. An element of the free abelian group Div(Γ) generated
+by points of Γ is called a divisor on Γ. If
+D =
+�
+v∈Γ
+D(v) · v
+is a divisor in Γ, then define the degree of D to be
+deg(D) :=
+�
+x∈Γ
+D(v) ∈ Z
+Denote by Div0(Γ) the subgroup of divisors of degree 0. A function f : Γ → R
+is called rational function on Γ if it is continuous, piecewise-linear with integer
+slopes along its domains of linearity. We denote by Rat(Γ) the set of rational
+functions on Γ. For f ∈ Rat(Γ) and a point v in Γ, the sum of the outgoing
+8
+
+slopes of f at v is denoted by ordv(f). This sum is 0 except for all but finitely
+many points of Γ, and therefore,
+div(f) :=
+�
+v∈Γ
+ordv(f)
+is a divisor on Γ. The set of principal divisors on Γ is defined to be Prin(Γ) :=
+{div(f) | f ∈ Rat(Γ)}. Note that Prin(Γ) is a subgroup of Div0(Γ). Two divisors
+D and D′ are said to be linearly equivalent, and we write D ∼ D′, if D − D′ ∈
+Prin(Γ). A divisor D = �
+v∈Γ D(v) · v ∈ Div(Γ) is said to be effective, and we
+write D ⩾ 0, if D(v) ⩾ 0 for all v ∈ Γ. Denote by Divk
++(Γ) the set of all effective
+divisors with degree k. For a divisor D ∈ Div(Γ) a complete linear system |D|
+is defined to be |D| := {D′ ∈ Div(Γ) | D′ ⩾ 0, D′ ∼ D} . The rank of a divisor
+D is defined to be −1 if |D| = ∅, and
+max
+�
+k ∈ Z | ∀D′ ∈ Divk
++(Γ) we have |D − D′| ̸= ∅
+�
+if |D| ̸= ∅.
+The rank of the divisor D is simply denoted by r(D).
+In the
+literature, there exists a notion of a hyperelliptic metric graph. For example
+in Cha13], a metric graph Γ is said to be hyperelliptic if there exists a divisor
+D ∈ Div(Γ) such that deg(D) = 2 and r(D) = 1.
+In this article, we give
+a definiton of hyperelliptic metric graphs in terms of tropical morphisms and
+their tree gonality and which is different to the one given in Cha13].
+Definition 2.10 A metric graph Γ is said to be hyperelliptic if there exists a
+tropical morphism from Γ to a metric tree with degree tgon(Γ).
+One of our goals in this article is to investigate genus 3 nonhyperelliptic metric
+graphs Γ with tree gonality 3. Note that if Γ is hyperelliptic in the sense of
+Kawaguchi-Yamaki (KY15]) that does not imply that Γ is hyperelliptic in our
+sense. For example, the metric graph Γ in Figure 25 is hyperelliptic in the sense
+of Kawaguchi-Yamaki but is not hyperelliptic in our sense. This is because the
+harmonic map coming from the unique hyperelliptic involution ι (Theorem 3.5,
+KY15]) does not satisfy the Riemann-Hurwitz condition.
+9
+
+3
+Construction of tropical morphisms
+The main result in this article is the constructive solution given to the Problem
+1 stated below. Before we do that, we give the following lemma, which will be
+useful to construct tropical morphisms.
+Lemma 3.1
+Let Γ = (G, l), T = (H, m) be two metric graphs where H does
+not have multiple edges and ψ : V (G) → V (H) a map on the set of vertices.
+Suppose that for any v, w ∈ V (G) that are the endpoints of some non-loop edge
+e ∈ E(G), we have ψ(v) = ψ(w), or ψ(v) and ψ(w) are endpoints of some edge
+e′ ∈ H. Then, there exists a unique continuous map ϕ : Γ → T such that
+ϕ|V (G) = ψ and ϕ is linear over each edge e in G.
+Proof.
+If e ∈ E(G) is an edge with endpoints v, w such that ψ(v) = ψ(w), then
+take ϕe : e → T to be the constant map on e with image ψ(v) = ψ(w). In the
+case when e ∈ E(G) is an edge with endpoints v, w such that ψ(v) and ψ(w) are
+endpoints of some edge e′ ∈ H, then choose ϕe : e → e′ to be the linear map
+with slope m(e′)/l(e). Now, we take ϕ : Γ → T to be the unique continuous
+map such that ϕ|e = ϕe for all edges e ∈ E(G). □
+Problem 1. Let Γ be a genus 3 metric graph with tree gonality 3 which is not
+hyperelliptic. Construct a tropical modification Γ′ of Γ, a metric tree T and a
+tropical morphism ϕ : Γ′ → T of degree 3.
+Solution of Problem 1. Consider genus 3 nonhyperelliptic metric graphs with
+tree gonality 3 up to tropical equivalence.
+There is a complete list (up to
+tropical equivalence) of genus 3 metric graphs (Figure 4, Cin15]), and also a
+complete list of genus 3 hyperelliptic metric graphs (the tropical hyperelliptic
+curves of genus 3 with unmarked vertices in Figure 2, Cha13]). Note that there
+is a hyperelliptic metric graph in the latter list, namely the one in Figure 25,
+which is not hyperelliptic in our sense. Based on this, now it is enough to make
+the constructions for the tropically equivalent metric graphs Γ whose essential
+model (G, l) has valency at least 3. They are depicted in Figures 1,5,7,9,.. . ,25.
+We divide the constructions into four cases depending on the number bridges
+(edges of a connected graph whose deletion increases its number of connected
+components) that the essential model (G, l) possesses.
+Case 1. If the metric graph Γ has no bridges, then Γ is one of the metric
+graphs given in Figure 1, 5, 7, 9, 11, or 13.
+Solution of Case 1.
+Case 1.1. Consider the metric graph Γ whose essential model (G, l) is
+given in Figure 1, where the graph G = (V, E, ∂) is given by V = {v1, v2, v3, v4},
+E = {e1, e2, . . . , e6}, and ∂(e1) = {v1, v2}, ∂(e2) = {v1, v3}, ∂(e3) = {v4, v1},
+∂(e5) = {v2, v3}, and ∂(e6) = {v2, v4}. The length map l on E is defined by
+assigning e1 �→ a, e2 �→ b, e3 �→ c, e4 �→ d, e5 �→ e and e6 �→ f, where a, b, c, d, e,
+and f are real positive numbers.
+10
+
+v1
+v2
+v3
+v4
+a
+e
+d
+c
+f
+b
+Curve
+Figure 1. The essential model (G, l) of Γ
+Choose any vertex, say v1 ∈ V (G), and without loss of generality, assume
+that c ⩽ b ⩽ a.
+It is enough to consider the following three subcases: (A)
+c < b ⩽ a, (B) c = b < a, and (C) c = b = a. We give the constructions for each
+subcase separately as follows.
+Case 1.1.A. Let (G1, l1) be another model of Γ given in Figure 1.1, where
+the graph G1 = (V1, E1, ∂1) is obtained by subdividing the edges ei ∈ E into
+e′
+i, e′′
+i , e′′′
+i (i = 1, 2), and ej ∈ E into e′
+j, e′′
+j (j = 4, 5, 6) with orientation given by:
+∂1(e′
+1) = {v1, v6}
+∂1(e′′
+1) = {v6, v4}
+∂1(e′′′
+1 ) = {v5, v2}
+∂1(e′′
+2) = {v8, v7}
+∂1(e′′′
+2 ) = {v7, v3}
+∂1(e′
+4) = {v4, v9}
+∂1(e′′
+4) = {v9, v3}
+∂1(e′
+5) = {v2, v10}
+∂1(e′′
+5) = {v3, v10}
+∂1(e′
+2) = {v1, v8}
+∂1(e′′
+6) = {v4, v11}
+∂1(e′
+6) = {v2, v11}
+and length map l1, which is equal to l on E \ {e1, e2, e4, e5, e6}, whereas on
+{e1, e2, e4, e5, e6} it is equal to
+l1(e′
+1) = l1(e′′
+1) = (a − c)/2
+l1(e′
+4) = l1(e′′
+4) = d/2
+l1(e′
+2) = l1(e′′
+2) = (b − c)/2
+l1(e′
+5) = l1(e′′
+5) = e/2
+l1(e′
+6) = l1(e′′′
+1 ) = l1(e′′′
+2 ) = c
+l1(e′′
+6) = f/2.
+T = (T ′, t′)
+Γ′ = (G′, l′)
+v5
+v2
+c
+v3
+c
+v1
+v4
+c
+w1
+w0
+w6
+w8
+w10
+w11
+w9
+d
+e
+f
+v9
+v11
+v10
+ϕ
+Curve
+v7
+b − c
+a − c
+v8
+v6
+Figure 1.1. The model (G1, l1) of Γ
+11
+
+Let Γ′ be the the tropical modification of Γ with model (G′, l′) in Figure 1.2,
+where the graph G′ is given by its vertex set V (G′) = V1 ∪ {v′
+6, v′
+8, v′
+9, v′
+10, v′
+11},
+and edge set E(G′) = E1 ∪{v2v′
+9, v3v′
+11, v4v′
+10, v5v′
+8, v7v′
+6}. The length map l′ on
+G′ is defined by l′ = l1 on E1, and
+l′(v2v′
+9) = d
+2
+l′(v3v′
+11) = f
+2
+l′(v4v′
+10) = e
+2
+l′(v5v′
+8) = b − c
+2
+l′(v7v′
+6) = a − c
+2
+.
+T = (T ′, t′)
+Γ′ = (G′, l′)
+v5
+v2
+v3
+v1
+v4
+w1
+w0
+w6
+w8
+w10
+w11
+w9
+v9
+v11
+v10
+v′
+11
+v′
+10
+v′
+9
+ϕ
+Curve
+v7
+v8
+v6
+v′
+8
+v′
+6
+Figure 1.2. The model (G′, l′) of Γ′
+Let T be the metric tree with model (T ′, t′) in Figure 1.3, where the tree
+T ′ is given with its vertex set V (T ′) = {w0, w1, w6, w8, w9, w10, w11}, and edge
+set E(T ′) = {w0w1, w0w9, w0w10, w0w11, w1w6, w1w8}, whereas the length map
+t′ on T is defined by
+t′(w0w9) = d
+2
+t′(w0w10) = e
+2
+t′(w0w11) = f
+2
+t′(w0w1) = c
+t′(w1w8) = b − c
+2
+t′(w1w6) = a − c
+2
+.
+12
+
+T = (T ′, t′)
+Γ′ = (G′, l′)
+v5
+v2
+c
+v3
+c
+v1
+v4
+c
+w1
+w0
+w6
+w8
+w10
+w11
+w9
+d
+e
+f
+v′
+9
+v11
+v10
+ϕ
+Curve
+v7
+b − c
+a − c
+v8
+v6
+Figure 1.3. The model (T ′, t′) of T
+Let ψ : V (G′) → V (T ) the map on the set of vertices given by v2, v3, v4 �→
+w0, v1, v7, v5 �→ w1, and vi, v′
+i �→ wi for i = 6, 8, 9, 10, 11. Then, the map ψ
+satisfies the condition in Lemma 3.1, and so, there exist a unique continuous
+map ϕ : Γ′ → T such that ϕ|V (G′) = ψ, and ϕ is linear on each edge e′ ∈ E(G′)
+with slope t′(e)/l′(e′), where e = ϕ(e′) ∈ E(T ) with endpoints ψ(v) and ψ(w).
+The map ϕ given in Figure 2. By construction, the models (G′, l′) and (T ′, t′)
+satisfy the condition in Definition 2.4, and therefore, ϕ is a piecewise linear
+function. From our choice of length maps t′, l′, the slope of ϕ|e′ is equal to 1 for
+all edges e′. Thus, ϕ has non-zero integer slopes along its edges of linearity. It
+is remaining to show that the map ϕ satisfies (i) the harmonicity condition and
+(ii) the Riemann-Hurtswitz condition on every point v ∈ Γ′.
+(i) Assume that v ∈ Γ
+′ is a vertex point, say v = v1 ∈ V (G′). Then, for all
+the directions ⃗w at ϕ(v) = w1, we have that mϕ, ⃗w(v1) = 1. We check that
+the harmonicity condition holds on every other vertex point in a similar
+fashion, and this checking process terminates because the vertex set is
+finite. Whenever v is not a vertex point, say v ∈ int(e) for some edge
+e ∈ E(G′), we have that ϕ(v) ∈ int(e′) where e′ = ϕ(e). Consider the
+new vertex sets on Γ′ and T by adding v and w respectively. There are
+only two directions ⃗w1 and ⃗w2 at ϕ(v) because val(ϕ(v)) = 2. The slopes
+of ϕ at v with directions ⃗w1 and ⃗w2 at ϕ(v) are equal to the slope of the
+same linear map ϕ|e i.e., mϕ, ⃗w1(v) = mϕ, ⃗w2(v), and so, we get that ϕ is
+a harmonic map. Its degree is 3 because for a fixed w ∈ T , say w1, the
+degree of ϕ is given by
+deg(ϕ) =
+�
+v∈Γ′,ϕ(v)=w1
+mϕ(v)
+= mϕ(v1) + mϕ(v7) + mϕ(v5)
+= 3.
+(ii) Assume that v ∈ Γ′ is a vertex point, say v = v8 ∈ V (G′). Then, mϕ(v8) =
+2, val(v8) = 2, and val(ϕ(v8)) = 1. Therefore,
+(val(v8) − 2) − mϕ(v8) ·
+�
+val (ϕ(v8)) − 2
+�
+= 2 > 0.
+Similarly, we check that the Riemann-Hurwitz condition holds on every
+other vertex point. Now, assume that v is not a vertex point. Consider
+13
+
+the new vertex sets on Γ′ and T (just like in part (i)) by adding v and
+w respectively. Then, we have that val(v) = val(ϕ(v)) = 2, and so the
+Riemann-Hurwitz condition holds.
+From (i) and (ii), we obtain that the map ϕ : Γ′ → T is a tropical morphism
+of metric graphs of degree 3, and so, the solution for the Case 1.1.A is finished.
+T
+Γ′
+v2
+c
+v3
+c
+v1
+v4
+c
+c
+a−c
+2
+b−c
+2
+e
+2
+f
+2
+d
+2
+d
+e
+f
+f
+2
+e
+2
+d
+2
+ϕ
+Curve
+b − c
+a − c
+b−c
+2
+a−c
+2
+Figure 2. The tropical morphism ϕ : Γ′ → T
+Remark 3.1 Let ϕ : Γ′ → T be non-constant piecewise linear map with
+nonzero integer slopes (as in Case 1.1.A), where the models (G′, l′), (T ′, t′) of Γ′,
+T respectively, are taken so that the condition in the Definition 2.4 is satisfied.
+In order to show that ϕ satisfies the harmonicity and the Riemann-Hurwitz
+condition on Γ′, it is enough to check those conditions on vertex points. This is
+due to the parts (i) and (ii) above.
+Case 1.1.B. Let Γ′
+1 be the tropical modification of Γ with model (G′
+1, l′
+1),
+where the graph G′
+1 is obtained by contracting the edges v1v8, v8v7, and v5v′
+8
+of G′ in Figure 1.2. Let T1 be the metric tree with model (T ′
+1, t′
+1), where the
+tree T ′
+1 is obtained by contracting the edge w1w8 of T ′ in Figure 1.3. Next, let
+ψ1 : V (G′
+1) → V (T1) the map on the set of vertices given by v2, v3, v4 �→ w0,
+v1, v5 �→ w1, and vi, v′
+i �→ wi for i = 6, 9, 10, 11.
+This map ψ satisfies the
+condition in Lemma 3.1, and so, there exist a unique continuous map ϕ1 : Γ′
+1 →
+T1, given in Figure 3, such that ϕ1|V (G′
+1) = ψ1 and ϕ1 is linear on each edge
+e′ ∈ E(G′
+1) with slope t′
+1(e)/l′
+1(e′), where e = ϕ1(e′) ∈ E(T1) with endpoints
+ψ1(v) and ψ1(w). Following the reasoning in (i) and (ii), we get that ϕ1 is a
+tropical map of degree 3, and thus, the solution of Case 1.1.B is done.
+14
+
+T1
+Γ′
+1
+v2
+c
+v3
+v1
+v4
+c
+c
+a−c
+2
+e
+2
+f
+2
+d
+2
+d
+e
+f
+f
+2
+e
+2
+d
+2
+ϕ1
+Curve
+a − c
+a−c
+2
+c
+Figure 3. The tropical map ϕ1 : Γ′
+1 → T1
+T ′
+2
+Γ′
+2
+v2
+v3
+v1
+v4
+c
+c
+e
+2
+f
+2
+d
+2
+d
+e
+f
+f
+2
+e
+2
+d
+2
+ϕ2
+Curve
+c
+c
+Figure 4. The tropical map ϕ2 : Γ′
+2 → T2
+Case 1.1.C. Let Γ′
+2 be the tropical modification of Γ with model (G′
+2, l′
+2),
+where G′
+2 is obtained by contracting the edges v1v6, v6v5, v7v′
+6, v1v8, v8v7 and
+v5v′
+8 of the graph G′ as in Figure 1.2. Let T2 be the metric tree with model
+(T ′
+2, t′
+2), where the tree T ′
+2 is obtained by contracting the edges w1w6, w1w8
+of the tree T ′ as in Figure 1.3. Next, let ψ2 : V (G′
+2) → V (T2) the map on
+the set of vertices given by v2, v3, v4 �→ w0, v1 �→ w1 and vi, v′
+i �→ wi for
+i = 9, 10, 11.
+The function ψ2 satisfies the condition in Lemma 3.1 and so,
+there exist a unique continuous map ϕ2 : Γ′
+2 → T2, given in Figure 4, such that
+ϕ2|V (G′
+2) = ψ2 and ϕ2 is linear on each edge e′ ∈ E(G′
+2) with slope t′
+2(e)/l′
+2(e′),
+15
+
+where e = ϕ2(e′) ∈ E(T2) with endpoints ψ2(v) and ψ2(w).
+Following the
+reasoning in (i) and (ii), we conclude that ϕ2 is a tropical map of degree 3, and
+therefore, the solution of Case 1.1.C is finished.
+Case 1.2.
+Consider the metric graph Γ with essential model (G, l) in
+Figure 5. The graph G is given by its vertex set V (G) = {v1, v2, v3, v4}, and
+edge set E(G) = {v1v2, v3v4, e1, e2, e3, e4}, where e1, e2 (resp., e3, e4) are two
+edges with endpoints v1, v4 (resp., v2, v3). The length map l : E(G) → (0, ∞)
+is defined by assigning v1v2 �→ a, v3v4 �→ b, e1 �→ c, e2 �→ d, e3 �→ e and e4 �→ f
+where a, b, c, d, e, and f are real positive numbers such that a < b. Note that if
+a = b, then Γ is a hyperelliptic metric graph.
+v1
+v2
+v4
+v3
+a
+e
+b
+d
+c
+f
+Curve
+Figure 5. The essential model (G, l) of Γ
+Let (G1, l1) be another model of Γ as in Figure 5.1.
+The graph G1 is
+obtained from G by subdividing the following edges: v3v4 ∈ E(G) into v3v6,
+v6v5, v5v4; e1 ∈ E(G) into v1v7, v7v4; e2 ∈ E(G) into v1v8, v8v4; e3 ∈ E(G) into
+v2v9, v9v3, and e4 ∈ E(G) into v2v10, v10v3, such that
+l1(v3v6) = l1(v6v5) = b − a
+2
+l1(v1v7) = l1(v7v4) = c
+2
+l1(v1v8) = l1(v8v4) = d
+2
+l1(v4v5) = a
+l1(v2v9) = l1(v9v3) = e
+2
+l1(v2v10) = l1(v10v3) = f
+2 .
+16
+
+Γ′
+T
+v1
+v2
+a
+v3
+v4
+v5
+a
+v9
+e
+v10
+f
+a
+v6 b − a
+v8
+d
+v7
+c
+d
+2
+c
+2
+f
+2
+e
+2
+b−a
+2
+ϕ
+Curve
+Figure 5.1. The model (G1, l1) of Γ
+Let Γ′ be the tropical modification of Γ with model (G′, l′) in Figure 5.2,
+where the graph G′ is given with its vertex set V (G′) = V (G1)∪{v′
+6, v′
+7, . . . , v′
+11},
+and edge set E(G′) = {v2v′
+6, v3v′
+11, v′
+11v′
+7, v′
+11v′
+8, v5v′
+9, v5v′
+10}∪E(G1). The length
+map l′ on G′ is given by l′ = l1 on E(G1), and
+l′(v1v7) = l′(v7v4) = l′(v11v′
+7) = c
+2
+l′(v5v′
+10) = l′(v2v10) = l′(v10, v3) = f
+2
+l′(v11v′
+8) = l′(v1v8) = l′(v8v4) = d
+2
+l′(v2v′
+6) = l′(v3v6) = l′(v6v5) = b − a
+2
+l′(v5v′
+9) = l′(v2v9) = l′(v9, v3) = e
+2
+l′(v1v2) = l′(v4v5) = l′(v3v11) = a.
+Γ′
+T
+v1
+v2
+a
+v3
+a
+v4
+v5
+a
+v9
+e
+v10
+f
+a
+v6 b − a
+v8
+d
+v7
+c
+d
+2
+c
+2
+v′
+8
+v′
+7
+f
+2
+e
+2
+b−a
+2
+v��
+6
+b−a
+2
+v′
+10
+f
+2
+v′
+9
+e
+2
+ϕ
+Curve
+Figure 5.2. The model (G′, l′) of Γ′
+Choose T to be the metric tree with model (T ′, t′) in Figure 5.3, where the
+tree T ′ is given by its vertex set V (T ′) = {w1, w2, w6, w7, . . . , w10}, and edge
+set E(T ′) = {w1w2, w1w7, w1w8, w2w6, w2w9, w2w10}. The length map t′ on T ′
+17
+
+is given by
+t′(w2w1) = a
+t′(w2w6) = b − a
+2
+t′(w1w7) = c
+2
+t′(w1w8) = d
+2
+t′(w2w9) = e
+2
+t′(w2w10) = f
+2 .
+Γ′
+T
+v1
+v2
+a
+v3
+v4
+v5
+a
+v9
+e
+v10
+f
+w1
+w2
+a
+v6 b − a
+v8
+d
+v7
+c
+w8
+d
+2
+w7
+c
+2
+w10
+f
+2
+w9
+e
+2
+w6
+b−a
+2
+ϕ
+Curve
+Figure 5.3. The model (T ′, t′) of T
+Let ψ : V (G′) → V (T ) the map on the set of vertices given by v1, v4, v′
+11 �→
+w1, v2, v3, v5 �→ w2, and vi, v′
+i �→ wi for i = 6, 8, 9, 10, 11.
+The function ψ
+satisfies the condition in Lemma 3.1, and so, there exist a unique continuous
+map ϕ : Γ′ → T , shown in Figure 6, such that ϕ|V (G′) = ψ, and ϕ is linear
+on each edge e′ ∈ E(G′) with slope t′(e)/l′(e′), where e = ϕ(e′) ∈ E(T ) with
+endpoints ψ(v) and ψ(w). The tropical morphism ϕ : Γ′ → T is of degree 3
+essentially because of the reasoning in (i) and (ii).
+Remark 3.2 The constructions of tropical morphisms of the remaining metric
+graphs are done similarly as for the metric graph in the Case 1.1.A. In order to
+avoid tedious writing, we give the construction of a model, a tropical modifica-
+tion, a metric tree, and a tropical morphism, using only figures from now on.
+The vertices labeled with a small × are the ’midpoints’ of the edges i.e., when
+subdividing an edge e into e1 and e2 then both lengths of e1 and e2 are equal
+to the half of the length of edge e.
+18
+
+Γ′
+T
+v1
+v2
+a
+v3
+a
+v4
+a
+e
+f
+a
+b − a
+d
+c
+d
+2
+c
+2
+f
+2
+e
+2
+b−a
+2
+b−a
+2
+f
+2
+e
+2
+ϕ
+Curve
+Figure 6. The tropical morphism ϕ : Γ′ → T
+v1
+v4
+v3
+d
+e
+c
+f
+b
+Curve
+Figure 7. The essential model (G, l) of Γ1
+Case 1.3. Consider the metric graph Γ1 with essential model (G, l) in Figure
+7, where b, c, d, e, and f are real positive numbers. The model (G1, l1) which is
+obtained by subdividing (G, l) is shown in Figure 7.1. The tropical modification
+Γ′
+1, the metric tree T1 with models (G′
+1, l′
+1), (T ′
+1, t′
+1) is given in Figure 7.2, 7.3,
+respectively. The construction of the tropical morphism ϕ1 : Γ′
+1 → T1 of degree
+3 is depicted in Figure 8.
+19
+
+v1
+v3
+v4
+v6
+b
+v9
+e
+v10
+f
+v7
+c
+v8
+d
+w1
+w7
+c
+2
+w8
+d
+2
+w9
+e
+2
+w10
+f
+2
+w6
+b
+2
+Curve
+Figure 7.1. The model (G1, l1) of Γ1
+Γ′
+1
+T1
+ϕ1
+v1
+v3
+v4
+v6
+b
+v9
+e
+v10
+f
+v7
+c
+v8
+d
+w1
+w7
+c
+2
+w8
+d
+2
+w9
+e
+2
+w10
+f
+2
+v′
+9
+e
+2
+v′
+10
+f
+2
+w3
+b
+2
+v′
+6
+b
+2
+v′
+8
+d
+2
+v′
+7
+c
+2
+Curve
+Figure 7.2. The model (G′
+1, l′
+1) of Γ′
+1
+v1
+v3
+v4
+v6
+b
+v9
+e
+v10
+f
+v7
+c
+v8
+d
+w1
+w7
+c
+2
+w8
+d
+2
+w9
+e
+2
+w10
+f
+2
+w6
+b
+2
+Curve
+Figure 7.3. The model (T ′
+1, t′
+1) of T1
+20
+
+Γ′
+1
+T1
+ϕ1
+v1
+v3
+v4
+v6
+b
+v9
+e
+v10
+f
+v7
+c
+v8
+d
+w1
+w7
+c
+2
+w8
+d
+2
+w9
+e
+2
+w10
+f
+2
+v′
+9
+e
+2
+v′
+10
+f
+2
+w6
+b
+2
+v′
+6
+b
+2
+v′
+8
+d
+2
+v′
+7
+c
+2
+Curve
+Figure 8. The tropical morphism ϕ1 : Γ′
+1 → T1
+Case 1.4. Consider the metric graph Γ2 with essential model in Figure
+9, where a, b, c, d, and e are real positive numbers such that b > a. Note that
+if a = b, then Γ2 is a hyperelliptic metric graph. The model (G1, l1) that is
+obtained by subdividing (G, l) is shown in Figure 9.1.
+v3
+v1
+v2
+d
+b
+a
+e
+c
+Curve
+Figure 9. The essential model (G, l) of Γ2
+21
+
+Γ′
+2
+T ′
+2
+ϕ2
+w1
+w2
+a
+w6
+b−a
+2
+w9
+e
+2
+w8
+d
+2
+w7
+c
+2
+v4
+v5
+a
+v1
+v2
+a
+v8
+d
+v7
+c
+v6
+b − a
+v9
+e
+Curve
+Figure 9.1. The model (G1, l1) of Γ2
+The tropical modification Γ′
+2, the metric tree T2 with models (G′
+2, l′
+2),
+(T ′
+2, t′
+2) is given in Figure 9.2, 9.3 respectively.
+Γ′
+2
+T ′
+2
+ϕ2
+w1
+w2
+a
+w6
+b−a
+2
+w9
+e
+2
+w8
+d
+2
+w7
+c
+2
+v4
+v5
+a
+v1
+v2
+a
+v8
+d
+v7
+c
+v6
+b − a
+v9
+v11
+a
+v′
+7
+c
+2
+v′
+8
+d
+2
+v′
+9
+e
+2
+v′
+6
+b−a
+2
+e
+Curve
+Figure 9.2. The model (G′
+2, l′
+2) of Γ′
+2
+Γ′
+2
+T ′
+2
+ϕ2
+w1
+w2
+a
+w6
+b−a
+2
+w9
+e
+2
+w8
+d
+2
+w7
+c
+2
+v4
+v5
+a
+v1
+v2
+a
+v8
+d
+v7
+c
+v6
+b − a
+v9
+v11
+a
+v′
+7
+c
+2
+v′
+8
+d
+2
+v′
+9
+e
+2
+v′
+6
+b−a
+2
+e
+Curve
+Figure 9.3. The model (T ′
+2, t′
+2) of T2
+The construction of the tropical morphism ϕ2 : Γ′
+2 → T2 of degree 3 is
+depicted in Figure 10.
+22
+
+Γ′
+2
+T2
+ϕ2
+a
+b−a
+2
+e
+2
+d
+2
+c
+2
+v4
+a
+v1
+v2
+a
+d
+c
+b − a
+a
+c
+2
+d
+2
+e
+2
+b−a
+2
+e
+Curve
+Figure 10. The tropical morphism ϕ2 : Γ′
+2 → T2
+Case 1.5. Consider the metric graph Γ3 with essential model (G, l) in
+Figure 11, where a, b, c, and e are real positive numbers such that b > a. Note
+that if a = b, then Γ2 is a hyperelliptic metric graph. The model (G1, l1) which is
+obtained by subdividing (G, l) is shown in Figure 11.1. The tropical modification
+Γ′
+3, the metric tree T3 with models (G′
+3, l′
+3), (T ′
+3, t′
+3) is given in Figure 11.2, 11.3,
+respectively. The construction of the tropical morphism ϕ3 : Γ′
+3 → T3 of degree
+3 is depicted in Figure 12.
+v1
+c
+v2
+e
+b
+a
+Figure 11. The essential model (G, l) of Γ3
+23
+
+Γ′
+3
+T ′
+3
+ϕ3
+a
+b−a
+2
+e
+2
+c
+2
+v1
+v5
+a
+v2
+v7
+v6
+b − a
+v9
+v11
+a
+v′
+7
+c
+2
+v′
+9
+e
+2
+v′
+6
+b−a
+2
+e
+Curve
+a
+c
+Figure 11.1. The model (G′
+3, l′
+3) of Γ′
+3
+Γ′
+3
+T ′
+3
+ϕ3
+w1
+w2
+a
+w6
+b−a
+2
+w9
+e
+2
+w7
+c
+2
+v1
+v5
+a
+v2
+v7
+v6
+b − a
+v9
+v11
+a
+v′
+7
+c
+2
+v′
+9
+e
+2
+v′
+6
+b−a
+2
+e
+Curve
+a
+c
+Figure 11.2. The model (T ′
+3, t′
+3) of T3
+Γ′
+3
+T3
+ϕ3
+a
+b−a
+2
+e
+2
+c
+2
+v1
+a
+v2
+b − a
+a
+c
+2
+e
+2
+b−a
+2
+e
+Curve
+a
+c
+Figure 12. The tropical morphism ϕ3 : Γ′
+3 → T3
+24
+
+Case 1.6. Consider the metric graph Γ4 with essential model (G, l) in
+Figure 13, where b, c, d, and e are real positive numbers. The model (G1, l1)
+that is obtained by subdividing (G, l) is shown in Figure 13.1. The tropical
+modification Γ′
+4, the metric tree T4 with models (G′
+4, l′
+4), (T ′
+4, t′
+4) is given in
+Figure 13.2, 13.3, respectively. The construction of the tropical morphism ϕ4 :
+Γ′
+4 → T4 of degree 3 is depicted in Figure 14.
+v3
+v1
+d
+e
+c
+b
+Curve
+Figure 13. The essential model (G, l) of Γ4
+Γ′
+4
+T4
+v1
+v′
+3
+d
+v′′
+3
+c
+v′
+2
+b
+v′
+4
+w′
+2
+b
+2
+w′
+4
+w′
+3
+d
+2
+w′′
+3
+c
+2
+v3
+Curve
+e
+2
+e
+Figure 13.1. The model (G1, l1) of Γ4
+Γ′
+4
+T4
+v1
+v′
+3
+d
+v′′
+3
+c
+v′
+2
+b
+x′
+4
+v′
+4
+x′
+2
+w′
+2
+b
+2
+w′
+4
+x′
+3
+d
+2
+x′′
+3
+c
+2
+w′
+3
+d
+2
+w′′
+3
+c
+2
+v3
+Curve
+b
+2
+e
+2
+e
+2
+e
+Figure 13.2. The model (G′
+4, l′
+4) of Γ′
+4
+25
+
+Γ′
+4
+T4
+v1
+d
+c
+b
+w′
+2
+b
+2
+w′
+4
+d
+2
+c
+2
+w′
+3
+d
+2
+w′′
+3
+c
+2
+v3
+Curve
+ϕ4
+b
+2
+e
+2
+e
+2
+e
+Figure 13.3. The model (T ′
+4, t′
+4) of T4
+Γ′
+4
+T4
+v1
+d
+c
+b
+b
+2
+d
+2
+c
+2
+d
+2
+c
+2
+v3
+Curve
+ϕ4
+b
+2
+e
+2
+e
+2
+e
+Figure 14. The tropical morphism ϕ4 : Γ′
+4 → T4
+Case 2. If the metric graph Γ has 1 bridge, then Γ is one of the metric
+graphs given in Figure 15, 17, or 19.
+Solution of Case 2.
+Case 2.1.
+Consider the metric graph Γ with essential model (G, l) in
+Figure 15, where a, b, c, d, e, and f are real positive numbers such that b > a.
+Note that if a = b, then Γ is a hyperelliptic metric graph. The model (G1, l1)
+which is obtained by subdividing (G, l) is shown in Figure 15.1. The tropical
+modification Γ′, the metric tree T with model (G′, l′), (T ′, t′) is given in Figure
+15.2, 15.3, respectively. The construction of the tropical morphism ϕ : Γ′ → T
+of degree 3 is depicted in Figure 16.
+26
+
+v3
+v1
+v2
+v4
+e
+d
+c
+f
+Curve
+b
+a
+Figure 15. The essential model (G, l) of Γ
+Γ′
+T
+v1
+v2
+a
+v3
+v′′
+2
+a
+v′
+3
+d
+v′′
+3
+c
+v′
+2
+b − a
+v4
+f
+v′
+4
+e
+w2
+w′
+2
+b−a
+2
+w4
+f
+w′
+4
+e
+2
+w1
+a
+w′′
+3
+c
+2
+ϕ
+Curve
+Figure 15.1. The model (G1, l1) of Γ
+Γ′
+T
+x1
+v1
+v2
+a
+v3
+v′′
+2
+a
+x2
+a
+v′
+3
+d
+v′′
+3
+c
+v′
+2
+b − a
+x4
+f
+v4
+f
+f
+x′
+4
+e
+2
+v′
+4
+e
+x′
+2
+b−a
+2
+w2
+w′
+2
+b−a
+2
+w4
+f
+w′
+4
+e
+2
+w1
+a
+x′
+3
+d
+2
+x′′
+3
+c
+2
+w′′
+3
+c
+2
+ϕ
+Curve
+Figure 15.2. The model (G′, l′) of Γ′
+27
+
+Γ′
+T
+x1
+v1
+v2
+a
+v3
+v′′
+2
+a
+x2
+a
+v′
+3
+d
+v′′
+3
+c
+v′
+2
+b − a
+x4
+f
+v4
+f
+f
+x′
+4
+e
+2
+v′
+4
+e
+w2
+w′
+2
+b−a
+2
+w4
+f
+w′
+4
+e
+2
+w1
+a
+x′
+3
+d
+2
+c
+2
+w′
+3
+d
+2
+w′′
+3
+c
+2
+ϕ
+Curve
+Figure 15.3. The model (T ′, t′) of T
+Γ′
+T
+v1
+v2
+a
+v3
+a
+a
+d
+c
+b − a
+f
+v4
+f
+f
+e
+2
+e
+b−a
+2
+b−a
+2
+f
+e
+2
+a
+d
+2
+c
+2
+d
+2
+c
+2
+ϕ
+Curve
+Figure 16. The tropical morphism ϕ : Γ′ → T
+Case 2.2. Consider the metric graph Γ1 with essential model (G, l) in
+Figure 17, where a, b, c, d, and e are real positive numbers such that b > a. Note
+that if a = b, then Γ2 is a hyperelliptic metric graph. The model (G1, l1) that
+obtained by subdividing (G, l) is shown in Figure 17.1. The tropical modification
+Γ′
+1, the metric tree T1 with model (G′
+1, l′
+1), (T ′
+1, t′
+1) is given in Figure 17.2, 17.3,
+respectively. The construction of the tropical morphism ϕ : Γ′
+1 → T1 of degree
+3 is depicted in Figure 18.
+v1
+v2
+v4
+d
+e
+Curve
+c
+a
+b
+Figure 17. The essential model (G, l) of Γ1
+28
+
+Γ′
+1
+T1
+v1
+v2
+a
+v′′
+2
+a
+v′′
+3
+v′
+2
+b − a
+v4
+f
+v′
+4
+e
+x′
+2
+w2
+w′
+2
+b−a
+2
+w4
+f
+w′
+4
+e
+2
+w1
+a
+x′′
+3
+w′′
+3
+c
+2
+Curve
+c
+Figure 17.1. The model (G1, l1) of Γ1
+Γ′
+1
+T1
+x1
+v1
+v2
+a
+v′′
+2
+a
+x2
+a
+v′′
+3
+v′
+2
+b − a
+x4
+f
+v4
+f
+f
+x′
+4
+e
+2
+v′
+4
+e
+x′
+2
+b−a
+2
+w2
+w′
+2
+b−a
+2
+w4
+f
+w′
+4
+e
+2
+w1
+a
+x′′
+3
+c
+2
+w′′
+3
+c
+2
+Curve
+c
+Figure 17.2. The model (G′
+1, l′
+1) of Γ′
+1
+Γ′
+1
+T1
+x1
+v1
+v2
+a
+v′′
+2
+a
+x2
+a
+v′′
+3
+v′
+2
+b − a
+x4
+f
+v4
+f
+f
+x′
+4
+e
+2
+v′
+4
+e
+x′
+2
+b−a
+2
+w2
+w′
+2
+b−a
+2
+w4
+f
+w′
+4
+e
+2
+w1
+a
+x′′
+3
+c
+2
+w′′
+3
+c
+2
+Curve
+c
+Figure 17.3. The model (T ′
+1, t′
+1) of T1
+29
+
+Γ′
+1
+T1
+ϕ1
+x1
+v1
+v2
+a
+v′′
+2
+a
+x2
+a
+v′′
+3
+v′
+2
+b − a
+x4
+f
+v4
+f
+f
+x′
+4
+e
+2
+v′
+4
+e
+x′
+2
+b−a
+2
+w2
+w′
+2
+b−a
+2
+w4
+f
+w′
+4
+e
+2
+w1
+a
+x′′
+3
+c
+2
+w′′
+3
+c
+2
+Curve
+c
+Figure 18. The tropical morphism ϕ1 : Γ1 → T1
+Case 2.3. Consider the metric graph Γ2 with essential model in Figure
+19, where a, b, c, d, e, and f are real positive numbers. The model (G1, l1) which
+obtained by subdividing (G, l) is shown in Figure 19.1. The tropical modification
+Γ′
+2, the metric tree T2 with model (G′
+2, l′
+2), (T ′
+2, t′
+2) 7is given in Figure 19.2, 19.3,
+respectively. The construction of the tropical morphism ϕ2 : Γ′
+2 → T2 of degree
+3 is depicted in Figure 20.
+v1
+v3
+b
+d
+c
+v4
+e
+f
+Curve
+Figure 19. The essential model (G, l) of Γ2
+30
+
+Γ′
+2
+T2
+v1
+v′
+3
+d
+v′′
+3
+c
+v′
+2
+b
+v4
+f
+v′
+4
+e
+x′
+2
+b
+2
+w1
+w′
+2
+b
+2
+w4
+f
+w′
+4
+e
+2
+x′′
+3
+w′
+3
+d
+2
+w′′
+3
+c
+2
+v3
+Curve
+ϕ2
+Figure 19.1. The model (G1, l1) of Γ2
+Γ′
+2
+T2
+x1
+v1
+v′
+3
+d
+v′′
+3
+c
+v′
+2
+b
+x4
+f
+v4
+f
+f
+x′
+4
+e
+2
+v′
+4
+e
+x′
+2
+b
+2
+w1
+w′
+2
+b
+2
+w4
+f
+w′
+4
+e
+2
+x′
+3
+d
+2
+x′′
+3
+c
+2
+w′
+3
+d
+2
+w′′
+3
+c
+2
+v3
+Curve
+ϕ2
+Figure 19.2. The model (G′
+2, l′
+2) of Γ′
+2
+Γ′
+2
+T2
+x1
+v1
+v′
+3
+d
+v′′
+3
+c
+v′
+2
+b
+x4
+f
+v4
+f
+f
+x′
+4
+e
+2
+v′
+4
+e
+x′
+2
+b
+2
+w1
+w′
+2
+b
+2
+w4
+f
+w′
+4
+e
+2
+x′
+3
+d
+2
+x′′
+3
+c
+2
+w′
+3
+d
+2
+w′′
+3
+c
+2
+v3
+Curve
+ϕ2
+Figure 19.3. The model (T ′
+2, l′
+2) of T2
+31
+
+Γ′
+2
+T2
+v1
+d
+c
+b
+f
+v4
+f
+f
+e
+2
+e
+b
+2
+b
+2
+f
+e
+2
+d
+2
+c
+2
+d
+2
+c
+2
+v3
+Curve
+ϕ2
+Figure 20. The tropical morphism ϕ2 : Γ′
+2 → T2
+Case 3. If the metric graph Γ has 2 bridges, then Γ is one of the metric
+graphs given in Figure 21 or 23.
+Solution of Case 3.
+Case 3.1.
+Consider the metric graph Γ with essential model (G, l) in
+Figure 21, where a, b, c, d, e, and f are real positive numbers such that b > a.
+Note that if b = a, then Γ is a hyperelliptic metric graph. The model (G1, l1) that
+obtained by subdividing (G, l) is shown in Figure 21.1. The tropical modification
+Γ′, the metric tree T with model (G′, l′), (T ′, t′) is given in Figure 21.2, 21.3,
+respectively. The construction of the tropical morphism ϕ : Γ′ → T of degree 3
+is depicted in Figure 22.
+v1
+v2
+v4
+d
+e
+Curve
+v3
+c
+a
+b
+f
+Figure 21. The essential model (G, l) of Γ
+32
+
+Γ′
+T
+v3
+v1
+c
+v2
+a
+v4
+d
+v′
+3
+f
+v′
+4
+e
+v′′
+2
+a
+w1
+w2
+a
+w′
+2
+b−a
+2
+w4
+d
+w′
+4
+e
+2
+w3
+2c
+w′
+3
+f
+2
+x′
+2
+b − a
+v′
+2
+ϕ
+Curve
+Figure 21.1. The model (G1, l1) of Γ
+Γ′
+T
+v3
+v1
+c
+v2
+a
+v4
+d
+v′
+3
+f
+v′
+4
+e
+v′′
+2
+a
+x4
+d
+x′
+4
+e
+2
+x2
+d
+x1
+a
+x3
+2c
+x′
+3
+f
+2
+w1
+w2
+a
+w′
+2
+b−a
+2
+w4
+d
+w′
+4
+e
+2
+w3
+2c
+w′
+3
+f
+2
+x′
+2
+b−a
+2
+b − a v′
+2
+ϕ
+Curve
+Figure 21.2. The model (G′, l′) of Γ′
+Γ′
+T
+v3
+v1
+c
+v2
+a
+v4
+d
+v′
+3
+f
+v′
+4
+e
+v′′
+2
+a
+x4
+d
+x′
+4
+e
+2
+x2
+d
+x1
+a
+x3
+2c
+x′
+3
+f
+2
+w1
+w2
+a
+w′
+2
+b−a
+2
+w4
+d
+w′
+4
+e
+2
+w3
+2c
+w′
+3
+f
+2
+x′
+2
+b−a
+2
+b − a v′
+2
+ϕ
+Curve
+Figure 21.3. The model (T ′, t′) of T
+33
+
+Γ′
+T
+v3
+v1
+c
+v2
+a
+v4
+d
+f
+e
+a
+d
+e
+2
+d
+a
+2c
+f
+2
+a
+b−a
+2
+d
+e
+2
+2c
+f
+2
+b−a
+2
+b − a
+ϕ
+Figure 22. The tropical morphism ϕ : Γ′ → T
+Case 3.2. Consider the metric graph Γ1 with essential model (G, l) in
+Figure 23, where a, b, c, d, e, and f are real positive numbers. The model (G1, l1)
+that is obtained by subdividing (G, l) is shown in Figure 23.1. The tropical
+modification Γ′
+1, the metric tree T1 with model (G′
+1, l′
+1), (T ′
+1, t′
+1) is given in Figure
+23.2, 23.3, respectively. The construction of the tropical morphism ϕ1 : Γ′
+1 → T1
+of degree 3 is depicted in Figure 24.
+v3
+v1
+v4
+c
+d
+e
+b
+f
+Figure 23. The essential model (G, l) of Γ1
+34
+
+Γ′
+1
+T1
+ϕ1
+v3
+c
+v1
+v4
+d
+v′
+3
+f
+v′
+4
+e
+w1
+w4
+d
+w′
+4
+e
+2
+w3
+2c
+w′
+3
+f
+2
+b
+v′
+2
+Curve
+Figure 23.1. The model (G1, l1) of Γ1
+v3
+c
+v1
+v4
+d
+v′
+3
+f
+v′
+4
+e
+x4
+x′
+4
+e
+2
+d
+x1
+x3
+2c
+x′
+3
+f
+2
+w1
+w4
+d
+w′
+4
+e
+2
+w3
+2c
+w′
+3
+f
+2
+d
+b
+v′
+2
+x′
+2
+b
+2
+w′
+2
+b
+2
+Curve
+Figure 23.2. The model (G′
+1, l′
+1) of Γ′
+1
+v3
+c
+v1
+v4
+d
+v′
+3
+f
+v��
+4
+e
+x4
+x′
+4
+e
+2
+d
+x1
+x3
+2c
+x′
+3
+f
+2
+w1
+w4
+d
+w′
+4
+e
+2
+w3
+2c
+w′
+3
+f
+2
+d
+b
+v′
+2
+x′
+2
+b
+2
+w′
+2
+b
+2
+Curve
+Figure 23.3. The model (T ′
+1, t′
+1) of T1
+35
+
+Γ′
+1
+T1
+ϕ1
+v3
+c
+v1
+v4
+d
+v′
+3
+f
+v′
+4
+e
+x4
+x′
+4
+e
+2
+d
+x1
+x3
+2c
+x′
+3
+f
+2
+w1
+w4
+d
+w′
+4
+e
+2
+w3
+2c
+w′
+3
+f
+2
+d
+b
+v′
+2
+x′
+2
+b
+2
+w′
+2
+b
+2
+Curve
+Figure 24. The tropical morphism ϕ1 : Γ′
+1 → T1
+Case 4. If the metric graph Γ has 3 bridges, then Γ is the metric graph
+given in Figure 25.
+Solution of Case 4.
+Consider the metric graph Γ with essential model (G, l) in Figure 25, where
+a, b, c, d, e, and f are real positive numbers. Note that the metric graph Γ is
+hyperelliptic in the sense of Kawaguchi-Yamaki KY15] i.e., there is a harmonic
+morphism from Γ to a metric tree, but it is not hyperelliptic in our sense be-
+cause the harmonic map coming from the unique hyperelliptic involution ι on
+Γ (see Theorem 3.5, KY15]) is not a tropical morphism in our sense because it
+does not satisfy the Riemann-Hurwitz condition. The model (G1, l1) that is ob-
+tained by subdividing (G, l) is shown in Figure 25.1. The tropical modification
+Γ′, the metric tree T with model (G′, l′), (T ′, t′) is given in Figure 25.2, 25.3,
+respectively. The construction of the tropical morphism ϕ : Γ′ → T of degree 3
+is depicted in Figure 26. This ends our constructive solution of Problem 1.
+v1
+v2
+a
+v3
+b
+v4
+c
+f
+e
+d
+Figure 25. The essential model (G, l) of Γ
+36
+
+Γ′
+T
+v′
+2
+d
+v2
+v1
+a
+w2
+w′
+2
+v3
+b
+e
+v′
+3
+x3
+x′
+3
+e
+2
+w3
+w′
+3
+e
+2
+v4
+c
+v′
+4
+w′
+4
+ϕ
+f
+Curve
+Figure 25.1. The model (G1, l1) of Γ
+Γ′
+T
+v′
+2
+d
+v2
+v1
+a
+x2
+2a
+x′
+2
+d
+2
+w2
+w′
+2
+v3
+b
+e
+v′
+3
+x3
+2b
+x′
+3
+e
+2
+w3
+w′
+3
+e
+2
+v4
+c
+v′
+4
+x4
+2c
+x′
+4
+f
+2
+w′
+4
+ϕ
+f
+Curve
+Figure 25.2. The model (G′, l′) of Γ′
+Γ′
+T
+d
+v2
+v1
+a
+2a
+d
+2
+w1
+w2
+2a
+w′
+2
+d
+2
+v3
+b
+e
+2b
+w3
+2b
+w′
+3
+e
+2
+v4
+c
+2c
+f
+2
+w4
+2c
+w′
+4
+f
+2
+ϕ
+f
+Curve
+Figure 25.3. The model (T ′, t′) of T
+37
+
+Γ′
+T
+d
+v2
+v1
+a
+2a
+d
+2
+2a
+d
+2
+v3
+b
+e
+2b
+e
+2
+2b
+e
+2
+v4
+c
+2c
+f
+2
+2c
+f
+2
+ϕ
+f
+Figure 26. The tropical morphism ϕ : Γ′ → T
+References
+[KY15] Shu Kawaguchi and Kazuhiko Yamaki, Rank of Divisors on Hyperelliptic Curves
+and Graphs Under Specialization, Vol. 12, 2015.
+[Cha13] Melody Chan, Tropical hyperelliptic curves, Vol. 37, 2013.
+[BN07] Matthew Baker and Serguei Norine, Riemann-Roch and Abel - Jacobi theory on a
+finite graph, Vol. 215, 2007.
+[Cap14] Lucia Caporaso, Gonality of Algebraic Curves and Graphs, Vol. 71, 2014.
+[Mik17] Grigory Mikhalkin, Tropical Geometry and Its Applications, 2017.
+[BBM11] Benoˆıt Bertrand, Erwan Brugall´e, and Grigory Mikhalkin, Tropical Open Hurwitz
+Numbers, Vol. 125, 2011.
+[Bak08] Matthew Baker, Specialization of linear systems from curves to graphs, Vol. 2, 2008.
+[BN09] Matthew Baker and Serguei Norine, Harmonic morphisms and hyperelliptic graphs,
+Vol. 2009, 2009.
+[CKK15] Gunther Cornelissen, Fumiharo Kato, and Janne Kool, A combinatorial Li-Yau
+inequality and rational points on curves, Vol. 1-2, 2015.
+[BN19] Matthew Baker and Serguei Norine, Harmonic morphisms and hyperelliptic graphs,
+Vol. 2009, 2019.
+[CD18] Filip Cools and Jan Draisma, On Metric Graphs with Prescribed Gonality, Vol. 156,
+2018.
+[DV19] Jan Draisma and Alejandro Vargas, Catalan-many tropical morphisms to trees; Part
+I: Constructions, https: // arxiv. org/ abs/ 1909. 12924 , 2019.
+[Cin15] Zubeyir Cinkir, Admissible invariants of genus 3 curves, Manuscripta math 148
+(2015), 317-339.
+[Kag18] Yuki Kageyama, Divisorial condition for the stable gonality of tropical curves,
+https: // arxiv. org/ abs/ 1801. 07405 , 2018.
+38
+

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+1
+Marine IoT Systems with Space-Air-Sea Integrated
+Networks: Hybrid LEO and UAV Edge Computing
+Sooyeob Jung, Seongah Jeong, Jinkyu Kang, and Joonhyuk Kang
+Abstract—Marine Internet of Things (IoT) systems have grown
+substantially with the development of non-terrestrial networks
+(NTN) via aerial and space vehicles in the upcoming sixth-
+generation (6G), thereby assisting environment protection, mili-
+tary reconnaissance, and sea transportation. Due to unpredictable
+climate changes and the extreme channel conditions of maritime
+networks, however, it is challenging to efficiently and reliably
+collect and compute a huge amount of maritime data. In
+this paper, we propose a hybrid low-Earth orbit (LEO) and
+unmanned aerial vehicle (UAV) edge computing method in space-
+air-sea integrated networks for marine IoT systems. Specifically,
+two types of edge servers mounted on UAVs and LEO satellites
+are endowed with computational capabilities for the real-time
+utilization of a sizable data collected from ocean IoT sensors.
+Our system aims at minimizing the total energy consumption
+of the battery-constrained UAV by jointly optimizing the bit
+allocation of communication and computation along with the
+UAV path planning under latency, energy budget and opera-
+tional constraints. For availability and practicality, the proposed
+methods were developed for three different cases according to
+the accessibility of the LEO satellite, “Always On,” “Always
+Off” and “Intermediate Disconnected”, by leveraging successive
+convex approximation (SCA) strategies. Via numerical results,
+we verify that significant energy savings can be accrued for
+all cases of LEO accessibility by means of joint optimization
+of bit allocation and UAV path planning compared to partial
+optimization schemes that design for only the bit allocation or
+trajectory of the UAV.
+Index terms — Marine networks, Internet of Things (IoT), edge
+computing, low-Earth orbit (LEO) satellite, unmanned aerial
+vehicles (UAVs), successive convex approximation (SCA).
+I. INTRODUCTION
+M
+ARINE Internet of Things (IoT) systems have evolved
+significantly with the rapid development of non-
+terrestrial network (NTN) technologies composed of space
+and airborne platforms to collect and process a variety of
+ocean data. The vast amount of ocean data plays an important
+This work was supported by the Institute of Information & communica-
+tions Technology Planning & Evaluation (IITP) grant funded by the Korea
+government (MSIT) (No.2021-0-00847, Development of 3D Spatial Satellite
+Communications Technology).
+This research was supported by the Ministry of Science and ICT (MSIT),
+Korea, under the Information Technology Research Center (ITRC) support
+program (IITP-2020-0-01787) supervised by the IITP.
+Sooyeob Jung is with the Department of Electrical Engineering, Korea
+Advanced Institute of Science and Technology (KAIST), and with the Satellite
+Wide-Area Infra Research Section, Electronics and Telecommunications Re-
+search Institute (ETRI), Daejeon, South Korea (Email: [email protected]).
+Seongah Jeong is with the School of Electronics Engineering, Kyungpook
+National University, Daegu 14566, South Korea (Email: [email protected]).
+Jinkyu Kang is with the Department of Information and Communications
+Engineering, Myongji University, Gyeonggi-do 17058, South Korea (Email:
+[email protected]).
+Joonhyuk Kang is with the Department of Electrical Engineering, Korea
+Advanced Institute of Science and Technology (KAIST), Daejeon, South
+Korea (Email: [email protected]).
+role in marine monitoring, which contributes to environ-
+mental protection, natural disaster prevention, oceanographic
+research, mineral exploration, military surveillance, etc. [1]-
+[3]. In particular, continuous monitoring of various physical
+phenomena of marine networks, such as sounds, vibrations and
+images, requires high-precision and wide-range measurements.
+Currently, three types of marine monitoring platforms are
+being investigated according to the relay node: shore-based
+radar, survey vessels and satellites [1], most of which have the
+following procedures. By using existing information communi-
+cation technologies, the marine data collected from ocean IoT
+sensors is transferred to a ground cloud server with sufficient
+computation storage capacity. The ground cloud server stores
+and analyzes the collected data, thereby managing various ap-
+plications based on ocean utilization and exploration. In shore-
+based radar systems installed on offshore buoys and automatic
+weather stations located on the coast or islands, there are
+difficulties in installation and maintenance due to their spatial
+constraints. Meanwhile, survey vessel-based platforms have
+temporal constraints, which limit the time for data collection.
+In addition, unexpected loss and defects of collected data may
+occur in point measurements attained by platforms with shore-
+based radar or survey vessel platforms due to extreme channel
+environments and unpredictable climate changes in the ocean
+[2].
+To address these spatial and temporal limitations, satellite-
+based monitoring can be an alternative that provides full
+coverage of the area of interest with one or multiple satellites.
+With the participation of global companies in the satellite
+business such as SpaceX, Amazon, and Telesat [4], low-
+Earth orbit (LEO) satellites are gaining more attention than
+ever before, and cost-effective easy-to-deploy large-scale satel-
+lite networks are being established. In addition, conventional
+satellite operators such as Spire, Kepler, Fleet, Lacuna space
+and Eutelsat, are preparing to provide satellite IoT services
+with global coverage [5], [6]. Until recently, satellites have
+mostly been adopted as a relay with terrestrial networks;
+however, for future 6G IoT services, they can operate as
+functional network components, e.g., computing servers [7]-
+[12]. Traditionally, the critical drawback of satellite-assisted
+networks is the latency resulting from round-trip delays due
+to the IoT sensor-satellite-terrestrial station link as well as
+the rapidly increasing volume of transmitted data. Therefore,
+it is beneficial to bring computing functions in the satellite
+to handle processing capabilities of the collected data, rather
+than sending it to the ground cloud server. In the following
+section, we briefly summarize the recent research activities
+that focus on hierarchical integrated networks using satellites
+as computing servers.
+arXiv:2301.03815v1  [eess.SY]  10 Jan 2023
+
+2
+A. Related Works
+Satellite-assisted edge computing systems have been ac-
+tively studied in space-ground integrated networks [7]-[13],
+space-air-ground integrated networks (SAGIN) [14]-[21] and
+space-air-sea-based non-terrestrial networks (SAS-NTN) [22],
+[23]. In particular, the authors in [7] propose a three-tier
+computation architecture consisting of ground users, LEO
+satellites and ground servers to minimize the total energy
+consumption of the system. In [8], network slice scheduling
+for satellite-assisted computing architecture is studied, where
+satellite servers and ground servers are considered for IoT
+applications. Although satellite-assisted edge computing can
+provide real-time offloading services to large areas, such as
+the ocean, it still faces several practical problems. For long-
+distance communication with a satellite, more transmit power
+and larger antenna size are preferred at ground user terminals,
+which is costly and spatially-limited in real applications.
+Moreover, the transceiver for satellite communications must
+be robustly designed against severe fading due to atmospheric
+turbulence.
+Unmanned aerial vehicles (UAVs) can be adopted to provide
+enhanced coverage for overcoming path loss and fading issues
+of satellite-assisted edge computing. UAVs can receive and
+compute data in close proximity to ocean IoT sensors, or can
+relay the data to the cloud server for computing. Recently,
+UAV-assisted satellite IoT networks have been suggested in
+several studies [14], [15]. Cheng et al. [14] propose offloading
+systems of remote IoT applications in the space-air-ground
+scenario, where UAVs provide computational capability to
+nearby users as edge servers, while satellites relay the of-
+floaded data to the ground cloud server. In [15], LEO satellite-
+assisted UAV data collection for IoT sensors is proposed,
+where the delay-tolerant data and delay-sensitive data are
+transferred to the ground cloud server via UAV and LEO
+satellite, respectively.
+As briefly reviewed above, most of existing works on
+hierarchical offloading systems in the integrated space and
+air networks assume terrestrial infrastructures, which may
+result in latency caused by the extreme channel variation of
+marine IoT systems. Furthermore, even though space or aerial
+computing platforms are considered, most studies assume full
+accessibility of the LEO satellite during mission time, which
+may not be guaranteed according to the orbit of revolution of
+the LEO satellite under insufficient deployments. To perform
+real-time data mining and analysis of ocean data in marine
+IoT systems, the use of aerial/space moving cloudlets play an
+important role considering their availability.
+B. Main Contributions
+In this paper, we focus on a marine IoT system with
+space-air-sea integrated networks, as illustrated in Fig. 1,
+where both UAV and LEO satellite-mounted cloudlets are
+deployed to offer computing opportunities. In the proposed
+system, a number of ocean IoT sensors are distributed only to
+collect abundant marine information with limited battery, and
+transmit the collected data to a designated computing server
+among UAV or LEO-mounted cloudlets so as to satisfy the
+LEO satellite
+(Cloud server)
+UAV
+(Edge server)
+IoT 1
+End user
+Frame 1
+Frame n
+Frame N
+IoT k
+IoT K
+(
+)
+,
+,0
+I
+I
+I
+k
+k
+k
+x
+y
+=
+p
+(
+)
+1
+,
+,
+E
+E
+E
+n
+n
+n
+x
+y
+h
+=
+p
+(
+)
+1
+2
+,
+,
+C
+C
+C
+x
+y
+h
+h
+=
++
+p
+1
+I
+K +
+p
+1
+E
+N +
+p
+IoT → UAV (for UAV computing)
+UAV → LEO
+LEO → UAV
+UAV → User
+IoT → UAV → LEO → UAV (for LEO computing) 
+IoT → UAV (for edge computing)
+IoT → UAV (for cloud computing) 
+UAV → LEO (offloading)
+LEO → UAV
+UAV → User
+IoT → UAV (for edge computing)
+IoT → UAV (for cloud computing) 
+UAV → LEO (offloading)
+LEO → UAV
+UAV → User
+LEO satellite-mounted cloudlet
+End user
+Ocean 
+IoT sensor k
+(
+)
+,
+,0
+I
+I
+I
+k
+k
+k
+x
+y
+=
+p
+(
+)
+,
+,
+U
+U
+U
+n
+n
+n
+U
+x
+y
+h
+=
+p
+UAV-mounted cloudlet
+LEO → Ground
+Orbit
+Ocean 
+IoT sensor 1
+Ocean 
+IoT sensor K
+IoT → UAV (for UAV computing)
+IoT → UAV → LEO → UAV (for LEO computing) 
+Space
+Air
+Sea
+(
+)
+,
+,
+L
+L
+L
+n
+n
+n
+U
+L
+x
+y
+h
+h
+=
++
+p
+LEO satellite-mounted cloudlet
+End user
+Ocean 
+IoT sensor k
+UAV-mounted 
+cloudlet
+Orbit
+Space
+Air
+Sea
+(
+)
+,
+,
+L
+L
+L
+n
+n
+n
+U
+L
+x
+y
+h
+h
+=
++
+p
+(
+)
+,
+,
+U
+U
+U
+n
+n
+n
+U
+x
+y
+h
+=
+p
+(
+)
+,
+,0
+I
+I
+I
+k
+k
+k
+x
+y
+=
+p
+1
+U
+p
+U
+N
+p
+1
+Ip
+I
+K
+p
+UAV computing: Sensor → UAV →
+LEO computing: Sensor → UAV →
+UAV computing
+LEO computing
+LEO satellite-mounted cloudlet
+End user
+Ocean 
+IoT sensor k
+UAV-mounted cloudlet
+Orbit
+Space
+Air
+Sea
+(
+)
+,
+,
+L
+L
+L
+n
+n
+n
+U
+L
+x
+y
+h
+h
+=
++
+p
+(
+)
+,
+,
+U
+U
+U
+n
+n
+n
+U
+x
+y
+h
+=
+p
+(
+)
+,
+,0
+I
+I
+I
+k
+k
+k
+x
+y
+=
+p
+1
+U
+p
+U
+N
+p
+1
+I
+p
+I
+K
+p
+End user
+: 
+: Sensor → UAV →
+LEO → UAV → End user
+UAV computing
+LEO computing
+UAV computing: Sensor → UAV → End user
+LEO computing: Sensor → UAV →LEO → UAV → End user
+Fig. 1: Marine IoT system model with a space-air-sea inte-
+grated network using hybrid LEO and UAV edge computing
+for real-time data utilization.
+system design criterion. Here, the LEO satellite is assumed
+to have a higher computational capability to process the task
+than that of the UAV. When the IoT data size exceeds the
+computation capacity of the UAV, the computational task is
+totally offloaded to the LEO satellite. The computation results
+executed at LEO are retransmitted to the UAV, are stored
+until it arrives over the end user, and is finally sent to the
+end user. To this end, we tackle the key design problem of
+jointly optimizing the bit allocation for communication and
+computing and the trajectory of the UAV, with the aim of
+minimizing its energy consumption. The main contributions
+of this paper are summarized as follows:
+• For marine IoT systems with extreme channel environ-
+ments and unpredictable climate changes, we propose
+a hybrid LEO and UAV edge computing method. The
+scheduling between UAV and LEO satellite-mounted
+cloudlets depends on the size of the offloaded ocean data
+and the LEO connection status.
+• For practicality and usability, we consider three different
+scenarios according to LEO availability such as “Always
+On,” “Always Off” and “Intermediate Disconnected”.
+For each case, we develop the joint optimization of bit
+allocation required for offloading and UAV path planning.
+• The non-convex optimization problems formulated for
+three different cases depending on the availability of the
+LEO satellite are tackled by means of a successive convex
+approximation (SCA) algorithm [24], [25], which can
+guarantee the local minimum of the original non-convex
+problems by using an efficient iterative algorithm.
+The rest of this paper is organized as follows. The system
+model is presented in Section II. Section III, IV and V provide
+problem formulations and proposed methods for the LEO
+access status of “Always On,” “Always Off” and “Intermediate
+Disconnected”, respectively. Simulation results are given in
+Section VI, and conclusions are summarized in Section VII.
+
+C3
+Frame n-1
+Frame n
+IoT sensor 1
+〮〮〮 
+IoT sensor k
+〮〮〮 
+IoT sensor K
+
+…
+…
+K
+
+1
+U
+n−
+p
+U
+np
+1
+U
+n+
+p
+2
+U
+n+
+p
+Frame n+1
+Fig. 2: Frame structure of orthogonal access for multiple ocean
+IoT sensors.
+II. SYSTEM MODEL
+A. Set-up
+Fig. 1 illustrates a marine IoT system with a space-air-
+sea integrated network using hybrid LEO and UAV edge
+computing, where 𝐾 ocean IoT sensors collect marine data
+to be entirely transferred to available cloudlets for computing.
+The computed results are then designated to an end user. For
+real-time data utilization, two types of cloudlets mounted on
+the UAV and LEO satellite are considered, between which
+the scheduling depends on the UAV computing capability and
+LEO accessibility. Specifically, when the collected data size
+exceeds the computation capacity of the UAV, the data should
+be entirely offloaded to the LEO. The computing capability
+of the LEO satellite is assumed to be higher than that of
+the UAV. Another major factor for scheduling is whether
+the LEO satellite is available or not since its beam coverage
+varies according to the orbit of revolution. Here, we consider
+three different cases according to the availability of the LEO
+satellite during mission time: “Always On,” “Always Off” and
+“Intermediate Disconnected”. For each scenario, we developed
+the joint optimization of the bit allocation for communication
+and computation and the trajectory of the UAV. Depending on
+the types of cloudlets, we refer to UAV computing and LEO
+computing, where computing of the IoT sensor task is executed
+at the UAV and LEO, respectively. In UAV computing, the task
+of the IoT sensor 𝑘 is offloaded to the UAV-mounted cloudlet
+until the UAV arrives over the end user and the output results
+are conveyed to them. In LEO computing, the UAV receives
+and relays the offloaded data of the IoT sensor to LEO for the
+LEO execution. The computed results at LEO are then sent to
+the end user via the UAV when the UAV arrives above them.
+For communication links between IoT sensors and the UAV,
+and between the UAV and LEO satellite, a frequency division
+duplex (FDD) scheme is assumed with equal bandwidth 𝐵 for
+the uplink and downlink. Each IoT sensor 𝑘 has the number
+𝐼𝑘 of input information bits to be processed. The results for
+LEO computing and UAV computing are characterized as the
+number 𝑂𝐿
+𝑘 and 𝑂𝑈
+𝑘 of bits produced per input bit of the IoT
+sensor 𝑘, and the number 𝐶𝐿
+𝑘 and 𝐶𝑈
+𝑘 of CPU cycles per input
+bit for computing, respectively. We assume that all tasks must
+be computed within the total mission time 𝑇. Here, a three-
+dimensional Cartesian coordinate system is adopted based on
+the metric unit. We assume that the IoT sensor 𝑘 is deployed
+at position 𝒑𝐼
+𝑘 = (𝑥𝐼
+𝑘, 𝑦𝐼
+𝑘, 𝑎𝑘), for 𝑘 ∈ {1, · · ·, 𝐾 + 1}, with
+𝑎𝑘 being the average sea surface level, where the position
+TABLE I: List of Symbols
+Symbol
+Definition
+𝐾
+Number of ocean IoT sensors
+𝑇
+Total mission time
+Δ
+Frame duration
+𝑁
+Number of frames within 𝑇
+ℎ𝑈 , ℎ𝐿
+Altitudes of UAV and LEO satellite with respect to
+average sea surface level and UAV, respectively
+𝑔𝑘,𝑛, ℎ𝑛
+Path loss between the IoT sensor 𝑘 and UAV and
+between the UAV and LEO at the 𝑛th frame
+𝑔0
+Channel gain at reference distance 1 m
+𝑇𝑣
+Visible time of an LEO satellite
+𝑣𝑠
+Speed of an LEO satellite
+ℎ
+Height of an LEO satellite orbit
+𝜃, 𝜑
+Elevation angle and beamwidth of the LEO satellite
+𝑀
+the gross mass of the UAV
+𝒗𝑈
+𝑛
+velocity vector of the UAV at the 𝑛th frame
+𝜀
+Energy budget of the IoT sensor 𝑘 at each frame
+𝐼𝑘
+Number of input bits of the IoT sensor 𝑘
+𝐸𝐼,𝑈
+𝑘,𝑛
+Energy consumption for uplink communication at the
+IoT sensor 𝑘 at the 𝑛th frame
+𝐸𝑈
+𝑘,𝑛, 𝐸𝑈,𝐿
+𝑘,𝑛
+Energy consumption for computing and uplink com-
+munication at the UAV-mounted cloudlet for the IoT
+sensor 𝑘 at the 𝑛th frame
+𝐸𝑈,𝐸
+Energy consumption for downlink communication at
+the UAV-mounted cloudlet
+𝐸 𝐿
+𝑘,𝑛, 𝐸 𝐿,𝑈
+𝑘,𝑛
+Energy consumption for computing and downlink com-
+munication at the LEO-mounted cloudlet for the IoT
+sensor 𝑘 at the 𝑛th frame
+𝐸𝐹
+𝑛
+Energy consumption for a UAV flying at the 𝑛th frame
+𝐿𝐼,𝑈
+𝑘,𝑛
+Number of bits for uplink communication at the IoT
+sensor 𝑘 at the 𝑛th frame
+𝑙𝑈
+𝑘,𝑛, 𝐿𝑈,𝐿
+𝑘,𝑛
+Number of bits for computing and uplink communica-
+tion at a UAV-mounted cloudlet for the IoT sensor 𝑘 at
+the 𝑛th frame
+𝐿𝑈,𝐸
+Number of bits for downlink communication at the
+UAV-mounted cloudlet
+𝑙𝐿
+𝑘,𝑛, 𝐿𝐿,𝑈
+𝑘,𝑛
+Number of bits for computing and downlink communi-
+cation at the LEO-mounted cloudlet for the IoT sensor
+𝑘 at the 𝑛th frame
+𝑂𝐿
+𝑘 , 𝑂𝑈
+𝑘
+Number of output bits produced per input bit of the IoT
+sensor 𝑘
+𝑓 𝐿
+𝑛 , 𝑓 𝑈
+𝑛
+CPU frequency at the LEO and UAV-mounted cloudlets
+for the 𝑛th frame
+𝐶𝐿
+𝑘 , 𝐶𝑈
+𝑘
+CPU cycles per input bit at the LEO and UAV-mounted
+cloudlets for the task of the IoT sensor 𝑘
+𝛾𝐿, 𝛾𝑈
+Effective switched capacitances of the LEO and UAV,
+respectively
+𝒑𝐼
+𝑘, 𝒑𝑈
+𝑛 , 𝒑𝐿𝑛
+Positions of the IoT sensor 𝑘, UAV and LEO for the
+𝑛th frame
+𝛼𝑘,𝑛, 𝛽𝑘,𝑛
+Variables to indicate LEO connection and offloading
+scheduling of the IoT sensor 𝑘 at the 𝑛th frame
+𝑁𝑡
+Frame number during LEO disconnection
+of the end user is considered with an index of 𝐾 + 1. The
+UAV flies along a trajectory 𝒑𝑈 (𝑡) = (𝑥𝑈 (𝑡), 𝑦𝑈 (𝑡), ℎ𝑈)
+with a fixed altitude ℎ𝑈 assumed for system stability, for
+0 ≤ 𝑡 ≤ 𝑇, and the position of the LEO satellite is defined as
+𝒑𝐿(𝑡) = (𝑥𝐿(𝑡), 𝑦𝐿(𝑡), ℎ𝑈 + ℎ𝐿) with a fixed altitude ℎ𝑈 + ℎ𝐿,
+for 0 ≤ 𝑡 ≤ 𝑇, all the altitudes are measured with respect to
+the average sea surface level 𝑎𝑘. For the multiple access of 𝐾
+ocean IoT sensors, orthogonal access is assumed, as shown in
+Fig. 2. For tractability, in this paper, the total time duration 𝑇
+is divided into 𝑁 frames of duration Δ seconds, each of which
+is equally divided as Δ/𝐾 seconds, and is preallocated to
+IoT sensors for uplink and downlink communication required
+
+4
+Er
+h
+L
+LEO
+Satellite
+s
+
+
+Earth
+Orbit
+IoT Sensor
+
+sv
+Er
+h
+L
+LEO
+Satellite
+s
+
+
+Earth
+Orbit
+IoT Sensor
+
+sv
+Fig. 3: Geometric relationship between the ground user and
+the LEO satellite.
+for offloading. Accordingly, the IoT sensors do not interfere
+with each other in the offloading procedure. Moreover, the
+information data collected from the IoT sensor 𝑘 at the 𝑛 th
+frame is assumed to be entirely computed and transferred to
+the designated node within the corresponding frame during
+Δ/𝐾 seconds, for 𝑛 ∈ {1, · · ·, 𝑁}, so that the computational
+task cannot be partitioned. According to the discretized time
+unit, the trajectory of the UAV 𝒑𝑈 (𝑡) and the position of the
+LEO satellite 𝒑𝐿(𝑡) is expressed as 𝒑𝑈
+𝑛 = (𝑥𝑈
+𝑛 , 𝑦𝑈
+𝑛 , ℎ𝑈) and
+𝒑𝐿
+𝑛 = (𝑥𝐿
+𝑛 , 𝑦𝐿
+𝑛, ℎ𝑈 + ℎ𝐿), for 𝑛 ∈ N, respectively. The LEO
+satellite generally flies at a constant speed along its orbit and
+the relative positional coordinates of the LEO and UAV should
+vary constantly. For the task mission of marine IoT systems,
+the initial location 𝒑𝑈
+𝐼 and the final location 𝒑𝑈
+𝐹 of the UAV
+are assigned to 𝒑𝑈
+1 and 𝒑𝑈
+𝑁 +1, respectively, and its maximum
+speed constraint is given as
+��𝒗𝑈
+𝑛
+�� =
+�� 𝒑𝑈
+𝑛+1 − 𝒑𝑈
+𝑛
+��
+Δ
+≤ 𝑣max,
+(1)
+where the velocity vector 𝒗𝑈
+𝑛
+of the UAV is defined as
+( 𝒑𝑈
+𝑛+1 − 𝒑𝑈
+𝑛 )/Δ, and 𝑣max is its maximum velocity. The overall
+system variables and parameters are summarized in Table I.
+We assume that communication channels between the IoT
+sensors and UAV [16], [26], and between the UAV and LEO
+satellite [15], [16] are dominated by line-of-sight (LoS) links.
+At the 𝑛th frame, the channel gains for the IoT sensor 𝑘-UAV
+link and UAV-LEO link are written as
+𝑔𝑘,𝑛( 𝒑𝑈
+𝑛 ) =
+𝑔0
+(𝑥𝑈𝑛 − 𝑥𝐼
+𝑘)2 + (𝑦𝑈𝑛 − 𝑦𝐼
+𝑘)2 + ℎ𝑈 2
+(2)
+and
+ℎ𝑛( 𝒑𝑈
+𝑛 ) =
+𝑔0𝐺
+(𝑥𝐿𝑛 − 𝑥𝑈𝑛 )2 + (𝑦𝐿𝑛 − 𝑦𝑈𝑛 )2 + ℎ𝐿2 ,
+(3)
+respectively, where 𝑔0 represents the channel gain at the
+reference distance 1 m, and 𝐺 is an antenna gain for the long-
+distance satellite communication consisting of the transmission
+antenna gain of the UAV and the receiver antenna gain of
+the LEO satellite [15], [27]. In real applications, note that
+ℎ𝑛( 𝒑𝑈
+𝑛 ) ≫ 𝑔𝑘,𝑛( 𝒑𝑈
+𝑛 ) is guaranteed. For communication links,
+an additive white Gaussian noise is considered with zero mean
+and power spectral density 𝑁0 [dBm/Hz].
+B. Coverage Model of the LEO Satellite
+In this section, we explore the beam coverage model [7],
+[28] of an LEO satellite that accounts for the effect of the
+orbit of revolution. As shown in Fig. 3, when the LEO satellite
+makes an orbit round, the available communication time with
+the UAV can be limited, which is referred to as the LEO visible
+time window. The length of the visible time window is defined
+as
+𝑇𝑣 = 𝐿
+𝑣𝑠
+= 2 (𝑟𝐸 + ℎ) 𝛾
+𝑣𝑠
+,
+(4)
+where 𝑣𝑠 is the speed of the LEO satellite. 𝐿 is the arc length to
+define the coverage where IoT sensors can communicate with
+the LEO satellite, and is calculated by 𝐿 = 2 (𝑟𝐸 + ℎ) 𝛾 with
+𝑟𝐸 being the radius of Earth, ℎ being the height of the LEO
+satellite orbit, and 𝛾 being the angle of the satellite coverage.
+In general, due to the very low altitude of a UAV in comparison
+to the orbit height, the same visible time window is applied to
+the UAV and IoT sensors. The maximum length of the LEO
+visible time window can be achieved when 𝛾 = 𝜋. The angle
+𝛾 of the satellite coverage is calculated by
+𝛾 = cos−1
+�
+𝑟𝐸
+𝑟𝐸 + ℎ · cos 𝜃
+�
+− 𝜃,
+(5)
+where 𝜃 and 𝜑 are the elevation angle and the beamwidth
+of the satellite, respectively, and are derived as 𝜃
+=
+cos−1 �
+𝑟𝐸+ℎ
+𝑠
+· cos (𝜃 + 𝜑)
+�
+and 𝜑 = 𝜋/2 − (𝜃 + 𝛾) with 𝑠
+indicating the distance between the IoT sensor and LEO
+satellite. We assume that the UAV can fully access the LEO
+satellite within the visible time window of 𝑇𝑣. According to
+the availability of LEO communication based on the coverage
+model, three different cases can be considered: “Always On,”
+“Always Off” and “Intermediate Disconnected”, the details for
+which are described below.
+1) “Always On” scenario (𝑇 ≤ 𝑇𝑣): The first scenario is
+when the UAV can communicate with the LEO satellite during
+the entire mission time since the total mission time is within
+the LEO visible time, i.e., ���� ≤ 𝑇𝑣. In this scenario, we have
+𝛼𝑘,𝑛 = 1 for all 𝑛 ∈ N; therefore, the computation capability
+of the UAV determines whether the UAV or LEO will be used
+for computing.
+2) “Always Off” scenario (𝑇𝑣 = 0): The second scenario
+is when LEO communication is not available during the entire
+mission time since the UAV flies outside the beam coverage
+of the LEO satellite, i.e., 𝑇𝑣 = 0. In this scenario, we have
+𝛼𝑘,𝑛 = 0 for all 𝑛 ∈ N, and only the UAV computing can be
+performed. Furthermore, when the offloaded data size exceeds
+the UAV computation capability, it is transferred to the end
+user via the UAV without computing.
+3) “Intermediate Disconnected” scenario (𝑇 > 𝑇𝑣): The
+final scenario is when LEO connection is lost during the
+mission time, since the total mission time is larger than the
+LEO visible time, i.e., 𝑇 > 𝑇𝑣. In this scenario, when 𝑡 ≤ 𝑇𝑣,
+we have 𝛼𝑘,𝑛 = 1 for 𝑛 ∈ {1, · · ·, 𝑁𝑡}, with 𝑁𝑡 being the
+last frame within 𝑇𝑣, where both LEO computing and UAV
+computing can be performed: that is, 𝛽𝑘,𝑛 ∈ {0, 1}. When
+𝑡 > 𝑇𝑣, 𝛼𝑘,𝑛 = 0 for 𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁}, where only UAV
+computing is available: that is, 𝛽𝑘,𝑛 = 0. For example, if the
+
+5
+TABLE II: Three different scenarios according to LEO availability.
+Scenario
+𝛼𝑘,𝑛
+𝛽𝑘,𝑛
+Available types of computing
+“Always On” (𝑇 ≤ 𝑇𝑣)
+1, for all 𝑛 ∈ N
+0, for all 𝑛 ∈ N
+UAV Computing
+1, for all 𝑛 ∈ N
+LEO Computing
+“Always Off” (𝑇𝑣 = 0)
+0, for all 𝑛 ∈ N
+0, for all 𝑛 ∈ N
+UAV Computing
+“Intermediate Disconnected” (𝑇 > 𝑇𝑣)
+1, for 𝑛 ∈ {1, · · ·, 𝑁𝑡 },
+0, for 𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁 }
+0, for 𝑛 ∈ {1, · · ·, 𝑁𝑡 },
+0, for 𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁 }
+UAV Computing
+1, for 𝑛 ∈ {1, · · ·, 𝑁𝑡 },
+0, for 𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁 }
+LEO Computing →
+UAV Computing
+LEO connection is lost at 𝑇𝑣 = 𝑇/2, 𝑁𝑡 is defined as 𝑁/2. The
+frame data of 𝑛 ∈ {1, · · ·, 𝑁𝑡} is computed by the LEO or UAV,
+while the frame data of 𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁} is computed by
+the UAV. The details for these three scenarios are summarized
+in Table II.
+C. Energy Consumption Model for Offloading
+In the proposed hierarchical architecture, IoT sensors and
+the UAV are battery-limited, while the available energy of the
+LEO satellite is much more sufficient due to its larger size
+and mass, which is therefore negligible for the system design.
+With the aim of minimizing the total energy consumption of
+the UAV, we cover the energy consumption model for compu-
+tation, communication and flying required for offloading. Here,
+the LEO satellite is assumed to have sufficient battery capacity
+compared to the UAV and IoT sensors [7], [13], which is not
+reflected in the system design.
+1) Computation energy model: First, we define the
+amount of computation energy consumption at the LEO and
+UAV-mounted cloudlets at the 𝑛th frame for IoT sensor 𝑘 as
+[29], [30]
+𝐸𝑑
+𝑘,𝑛(𝑙𝑑
+𝑘,𝑛) =
+𝛾𝑑𝐶𝑑
+𝑘 𝑙𝑑
+𝑘,𝑛
+Δ2
+� 𝐾
+∑︁
+𝑘′=1
+𝐶𝑑
+𝑘′𝑙𝑑
+𝑘′,𝑛
+�2
+,
+(6)
+where 𝑑 ∈ {𝐿,𝑈} with 𝐿 indicating the LEO satellite and 𝑈
+indicating the UAV; 𝑙𝑑
+𝑘,𝑛 is the number of bits to be computed
+at the cloudlet and 𝛾𝑑 is the effective switched capacitance of
+the cloudlet.
+2) Communication energy model: In the proposed system,
+the transmit energy consumption from the UAV to LEO at the
+𝑛th frame for offloading the task of the IoT sensor 𝑘 is defined
+as [26], [31]
+𝐸𝑈,𝐿
+𝑘,𝑛 (𝐿𝑈,𝐿
+𝑘,𝑛 , 𝒑𝑈
+𝑛 ) = 𝑁0𝐵Δ/𝐾
+ℎ𝑛( 𝒑𝑈𝑛 )
+�
+2
+𝐿𝑈,𝐿
+𝑘,𝑛
+𝐵Δ/𝐾 − 1
+�
+,
+(7)
+where 𝐿𝑈,𝐿
+𝑘,𝑛
+is the number of uplink bits. At the final
+destination of the UAV above the end user, the downlink
+communication energy consumption is required so that the
+UAV can transmit the computing results accumulated during
+flying, which is given as
+𝐸𝑈,𝐸 (𝐿𝑈,𝐸, 𝒑𝑈
+𝑁 +1) =
+𝑁0𝐵Δ/𝐾
+𝑔𝐾+1,𝑁 +1( 𝒑𝑈
+𝑁 +1)
+�
+2
+𝐿𝑈,𝐸
+𝐵Δ/𝐾 − 1
+�
+,
+(8)
+where 𝐿𝑈,𝐸 is the number of downlink bits and is the same as
+the sum of output bits of the UAV and LEO-mounted cloudlets
+as follows:
+𝐿𝑈,𝐸 = 𝑂𝑈
+𝑘
+𝑁 −2
+∑︁
+𝑛=1
+𝑙𝑈
+𝑘,𝑛+1 + 𝑂𝐿
+𝑘
+𝑁 −4
+∑︁
+𝑛=1
+𝑙𝐿
+𝑘,𝑛+2.
+(9)
+In addition, the transmit energy consumption from the LEO
+and IoT sensor 𝑘 to the UAV at the 𝑛th frame is defined as
+𝐸 𝐿,𝑈
+𝑘,𝑛 (𝐿𝐿,𝑈
+𝑘,𝑛 , 𝒑𝑈
+𝑛 ) = 𝑁0𝐵Δ/𝐾
+ℎ𝑛( 𝒑𝑈𝑛 )
+�
+2
+𝐿𝐿,𝑈
+𝑘,𝑛
+𝐵Δ/𝐾 − 1
+�
+(10)
+and
+𝐸 𝐼 ,𝑈
+𝑘,𝑛 (𝐿𝐼 ,𝑈
+𝑘,𝑛 , 𝒑𝑈
+𝑛 ) = 𝑁0𝐵Δ/𝐾
+𝑔𝑘,𝑛( 𝒑𝑈𝑛 )
+�
+2
+𝐿𝐼,𝑈
+𝑘,𝑛
+𝐵Δ/𝐾 − 1
+�
+,
+(11)
+where 𝐿𝐿,𝑈
+𝑘,𝑛
+is the number of downlink bits transmitted at
+the LEO and 𝐿𝐼 ,𝑈
+𝑘,𝑛 is the number of uplink bits transmitted
+at the IoT sensor 𝑘. The energy consumption for reception is
+excluded since it is much smaller than the transmission energy
+consumption.
+3) Flying energy model: Following [32], [33], the flying
+energy consumption of the UAV at the 𝑛th frame is written as
+𝐸𝐹
+𝑛 (𝒗𝑈
+𝑛 ) = 𝜅∥𝒗𝑈
+𝑛 ∥2,
+(12)
+where 𝜅 = 0.5𝑀Δ and 𝑀 is the mass of the UAV. The flying
+energy consumption depends only on the velocity vector 𝒗𝑈
+𝑛
+of the UAV, and the level flight entails no change in the
+gravitational potential energy.
+Our purpose is to minimize the total energy consumption of
+the UAV, which must be calculated as the sum of the energy
+consumption of computation, communication and flying:
+𝐸𝑡𝑜𝑡𝑎𝑙
+𝑘,𝑛
+= 𝛼𝑘,𝑛
+�
+𝛽𝑘,𝑛𝐸𝑈,𝐿
+𝑘,𝑛 (𝐿𝑈,𝐿
+𝑘,𝑛 , 𝒑𝑈
+𝑛 ) + (1 − 𝛽𝑘,𝑛)𝐸𝑈
+𝑘,𝑛(𝑙𝑈
+𝑘,𝑛)
+�
++ (1 − 𝛼𝑘,𝑛)(1 − 𝛽𝑘,𝑛)𝐸𝑈
+𝑘,𝑛(𝑙𝑈
+𝑘,𝑛) + 𝐸𝐹
+𝑛 (𝒗𝑈
+𝑛 ),
+(13)
+where 𝛼𝑘,𝑛 and 𝛽𝑘,𝑛 are variables for the LEO availability
+and scheduling between LEO computing and UAV computing,
+respectively, which are given as
+𝛼𝑘,𝑛 =
+�
+1,
+if LEO communication is available,
+0,
+otherwise,
+(14)
+𝛽𝑘,𝑛 =
+�
+1,
+if LEO computing is performed,
+0,
+if UAV computing is performed.
+(15)
+Note that the energy consumption 𝐸𝑈,𝐸 for downlink com-
+munication with the end user in (8) is excluded from (13)
+
+6
+since it is constant regardless of optimization. In addition,
+LEO computing is considered by 𝛽𝑘,𝑛 = 1 when the input
+bits of the IoT sensor 𝑘 exceeds the computation capability of
+the UAV: that is,
+𝑁
+∑︁
+𝑛=1
+𝐿𝐼 ,𝑈
+𝑘,𝑛 >
+𝑁
+∑︁
+𝑛=1
+�
+𝑓 𝑈
+𝑛 · Δ
+𝐾
+�
+1
+𝐶𝑈
+𝑘
+,
+(16)
+where 𝑓 𝑈
+𝑛
+[CPU cycles/s] is the CPU frequency at the UAV
+edge server.
+III. OPTIMAL ENERGY CONSUMPTION FOR THE
+“ALWAYS ON” SCENARIO
+In this section, we formulate an optimization problem and
+the proposed algorithm to obtain a solution for the “Always
+On” scenario. Depending on the size of the offloaded data,
+either LEO computing or UAV computing is selected. As
+mentioned above, the total UAV energy consumption 𝐸𝑡𝑜𝑡𝑎𝑙
+𝑘,𝑛
+in (13) is rewritten with 𝛼𝑘,𝑛 = 1, for all 𝑛 ∈ N, as
+𝐸𝑡𝑜𝑡𝑎𝑙
+𝑘,𝑛
+= 𝛽𝑘,𝑛𝐸𝑈,𝐿
+𝑘,𝑛 (𝐿𝑈,𝐿
+𝑘,𝑛 , 𝒑𝑈
+𝑛 ) + (1 − 𝛽𝑘,𝑛)𝐸𝑈
+𝑘,𝑛(𝑙𝑈
+𝑘,𝑛)
++ 𝐸𝐹
+𝑛 (𝒗𝑈
+𝑛 ).
+(17)
+When LEO computing is considered, i.e., 𝛽𝑘,𝑛
+=
+1,
+we
+need
+to
+jointly
+optimize
+the
+bit
+allocation
+of
+{𝐿𝐼 ,𝑈
+𝑘,𝑛 }𝑛∈{1,···,𝑁 −4},𝑘 ∈K,
+{𝐿𝑈,𝐿
+𝑘,𝑛 }𝑛∈{2,···,𝑁 −3},𝑘 ∈K,
+{𝑙𝐿
+𝑘,𝑛}𝑛∈{3,···,𝑁 −2},𝑘 ∈K
+and
+{𝐿𝐿,𝑈
+𝑘,𝑛 }𝑛∈{4,···,𝑁 −1},𝑘 ∈K
+along
+with
+the
+UAV
+trajectory
+{ 𝒑𝑈
+𝑛 }𝑛∈{2,···,𝑁 }.
+When
+UAV
+computing is performed, that is, 𝛽𝑘,𝑛 = 0, we must jointly
+optimize
+the
+bit
+allocation
+of
+{𝐿𝐼 ,𝑈
+𝑘,𝑛 }𝑛∈{1,···,𝑁 −2},𝑘 ∈K
+and {𝑙𝑈
+𝑘,𝑛}𝑛∈{2,···,𝑁 −1},𝑘 ∈K along with the UAV trajectory
+{ 𝒑𝑈
+𝑛 }𝑛∈{2,···,𝑁 }. This problem is formulated with (17) as
+follows:
+min
+𝐿𝐼,𝑈
+𝑘,𝑛 ,𝐿𝑈,𝐿
+𝑘,𝑛 ,𝐿𝐿,𝑈
+𝑘,𝑛
+𝑙𝑈
+𝑘,𝑛,𝑙𝐿
+𝑘,𝑛,𝒑𝑈
+𝑛
+𝐾
+∑︁
+𝑘=1
+�𝑁 −4
+∑︁
+𝑛=1
+𝛽𝑘,𝑛𝐸𝑈,𝐿
+𝑘,𝑛+1(𝐿𝑈,𝐿
+𝑘,𝑛+1, 𝒑𝑈
+𝑛+1)
++
+𝑁 −2
+∑︁
+𝑛=1
+(1 − 𝛽𝑘,𝑛)𝐸𝑈
+𝑘,𝑛+1(𝑙𝑈
+𝑘,𝑛+1)
+�
++
+𝑁
+∑︁
+𝑛=1
+𝐸𝐹
+𝑛 (𝒗𝑈
+𝑛 )
+(18a)
+s.t. 𝐸 𝐼 ,𝑈
+𝑘,𝑛 (𝐿𝐼 ,𝑈
+𝑘,𝑛 , 𝒑𝑈
+𝑛 ) ≤ 𝜀, ∀𝑘 ∈ K, 𝑛 ∈ N
+(18b)
+𝑛
+∑︁
+𝑖=1
+𝑙𝑈
+𝑘,𝑖+1 ≤
+𝑛
+∑︁
+𝑖=1
+𝐿𝐼 ,𝑈
+𝑘,𝑖 , ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁 − 2
+(18c)
+𝑛
+∑︁
+𝑖=1
+𝐿𝑈,𝐿
+𝑘,𝑖+1 ≤
+𝑛
+∑︁
+𝑖=1
+𝐿𝐼 ,𝑈
+𝑘,𝑖 , ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁 − 4
+(18d)
+𝑛
+∑︁
+𝑖=1
+𝑙𝐿
+𝑘,𝑖+2 ≤
+𝑛
+∑︁
+𝑖=1
+𝐿𝑈,𝐿
+𝑘,𝑖+1, ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁 − 4
+(18e)
+𝑛
+∑︁
+𝑖=1
+𝐿𝐿,𝑈
+𝑘,𝑖+3 ≤ 𝑂𝐿
+𝑘
+𝑛
+∑︁
+𝑖=1
+𝑙𝐿
+𝑘,𝑖+2, ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁 − 4 (18f)
+𝑁 −4
+∑︁
+𝑛=1
+𝛽𝑘,𝑛𝐿𝐼 ,𝑈
+𝑘,𝑛 +
+𝑁 −2
+∑︁
+𝑛=1
+(1 − 𝛽𝑘,𝑛)𝐿𝐼 ,𝑈
+𝑘,𝑛 = 𝐼𝑘, ∀𝑘 ∈ K
+(18g)
+𝑁 −4
+∑︁
+𝑛=1
+𝛽𝑘,𝑛𝑙𝐿
+𝑘,𝑛+2 +
+𝑁 −2
+∑︁
+𝑛=1
+(1 − 𝛽𝑘,𝑛)𝑙𝑈
+𝑘,𝑛+1 = 𝐼𝑘, ∀𝑘 ∈ K
+(18h)
+𝑁 −4
+∑︁
+𝑛=1
+𝑙𝐿
+𝑘,𝑛+2 =
+𝑁 −4
+∑︁
+𝑛=1
+𝐿𝑈,𝐿
+𝑘,𝑛+1, ∀𝑘 ∈ K
+(18i)
+𝑁 −4
+∑︁
+𝑛=1
+𝐿𝐿,𝑈
+𝑘,𝑛+3 = 𝑂𝐿
+𝑘
+𝑁 −4
+∑︁
+𝑛=1
+𝐿𝑈,𝐿
+𝑘,𝑛+1, ∀𝑘 ∈ K
+(18j)
+𝐿𝐼 ,𝑈
+𝑘,𝑛 , 𝐿𝑈,𝐿
+𝑘,𝑛 , 𝐿𝐿,𝑈
+𝑘,𝑛 , 𝑙𝑈
+𝑘,𝑛, 𝑙𝐿
+𝑘,𝑛 ≥ 0, ∀𝑘 ∈ K, 𝑛 ∈ N
+(18k)
+𝒑𝑈
+1 = 𝒑𝑈
+𝐼 , 𝒑𝑈
+𝑁 +1 = 𝒑𝑈
+𝐹 ,
+(18l)
+��𝒗𝑈
+𝑛
+�� ≤ 𝑣max, ∀𝑛 ∈ N,
+(18m)
+where 𝜀 in (18b) represents the energy budget constraint per
+frame for the IoT sensors. The inequality constraint (18c)
+and (18e) ensures that the number of bits computed at the
+UAV and LEO-mounted cloudlet is less than or equal to the
+number of uplink bits transmitted from the IoT sensor and
+UAV, respectively. The inequality constraints (18d) and (18f)
+ensure that the number of uplink bits from the UAV is less than
+or equal to the number of uplink bits from the IoT sensor, and
+the number of downlink bits from the LEO is limited by the
+number of output bits from the LEO. The equality constraints
+(18g) and (18h) enforce that the sum of the uplink bits of
+the IoT sensor and the sum of the computation bits for the
+LEO and UAV computing are equal to the input bits of the
+IoT sensor. The equality constraints (18i) and (18j) enforce the
+completion of LEO computing, while (18k) is imposed for the
+non-negative bit allocations. The constraints (18l) and (18m)
+represent the flying UAV’s initial and final position constraint
+and the maximum speed constraint, respectively.
+Problem (18) is non-convex because the objective function
+and the energy budget constraint are non-convex. To address
+this non-convexity, we apply the SCA-based strategy [24], [25]
+which builds on the inner convex approximation framework.
+In particular, we develop proposed algorithm 1 by using the
+following lemmas.
+Lemma 1: Given that a non-convex objective function
+𝑈(𝒙) = 𝑓1(𝒙) 𝑓2(𝒙) is the product of 𝑓1 and 𝑓2 convex and
+non-negative for any 𝒚 in the domain of 𝑈(𝒙), a convex
+approximation that satisfies the conditions required by the
+SCA algorithm is given as
+¯𝑈 (𝒙; 𝒚) = 𝑓1(𝒙) 𝑓2(𝒚) + 𝑓1(𝒚) 𝑓2(𝒙)
++ 𝜏𝑖
+2 (𝒙 − 𝒚)T𝑯(𝒚)(𝒙 − 𝒚),
+(19)
+where 𝜏𝑖 > 0 is a positive constant, 𝑯(𝒚) is a positive definite
+matrix, and (·)T indicates the transpose.
+Lemma 2: Given a non-convex constraint 𝑔(𝒙1, 𝒙2) ≤ 0,
+where 𝑔(𝒙1, 𝒙2) = ℎ1(𝒙1)ℎ2(𝒙2) is the product of the ℎ1 and
+ℎ2 convex and non-negative, for any (𝒚1, 𝒚2) in the domain of
+𝑔(𝒙1, 𝒙2), a convex approximation that satisfies the conditions
+required by the SCA algorithm is given as
+¯𝑔 (𝒙1, 𝒙2; 𝒚1, 𝒚2)
+Δ= 1
+2 (ℎ1(𝒙1) + ℎ2(𝒙2))2 − 1
+2 (ℎ12(𝒚1) + ℎ22(𝒚2))
+− ℎ1(𝒚1)ℎ1
+′(𝒚1)(𝒙1 − 𝒚1) − ℎ2(𝒚2)ℎ2
+′(𝒚2)(𝒙2 − 𝒚2),
+(20)
+where the partial derivative of 𝑓 (·) is 𝑓
+′ (·).
+
+7
+We set the primal variables for the formulated Problem
+(18) as 𝒛 = {𝒛𝑛}𝑛∈N with 𝒛𝑛 = ({𝐿𝐼 ,𝑈
+𝑘,𝑛 }𝑘 ∈K, {𝐿𝑈,𝐿
+𝑘,𝑛 }𝑘 ∈K,
+{𝐿𝐿,𝑈
+𝑘,𝑛 }𝑘 ∈K, {𝑙𝑈
+𝑘,𝑛}𝑘 ∈K, {𝑙𝐿
+𝑘,𝑛}𝑘 ∈K, 𝒑𝑈
+𝑛 ). We observe that the
+function 𝐸𝑈,𝐿
+𝑘,𝑛 (𝒛𝑛)
+Δ= 𝐸𝑈,𝐿
+𝑘,𝑛 (𝐿𝑈,𝐿
+𝑘,𝑛 , 𝒑𝑈
+𝑛 ) in (18a) is the product
+of two convex and non-negative functions, namely
+𝑓1(𝐿𝑈,𝐿
+𝑘,𝑛 ) = 𝑁0𝐵Δ/𝐾
+𝑔0𝐺
+�
+2
+𝐿𝑈,𝐿
+𝑘,𝑛
+𝐵Δ/𝐾 − 1
+�
+(21)
+and
+𝑓2( 𝒑𝑈
+𝑛 ) = (𝑥𝐿
+𝑛 − 𝑥𝑈
+𝑛 )2 + (𝑦𝐿
+𝑛 − 𝑦𝑈
+𝑛 )2 + ℎ𝐿2.
+(22)
+Then,
+by
+using
+Lemma
+1
+and
+defining
+𝒛𝑛(𝑣)
+=
+({𝐿𝐼 ,𝑈
+𝑘,𝑛 (𝑣)}𝑘 ∈K,
+{𝐿𝑈,𝐿
+𝑘,𝑛 (𝑣)}𝑘 ∈K,
+{𝐿𝐿,𝑈
+𝑘,𝑛 (𝑣)}𝑘 ∈K,
+{𝑙𝑈
+𝑘,𝑛(𝑣)}𝑘 ∈K, {𝑙𝐿
+𝑘,𝑛(𝑣)}𝑘 ∈K, 𝒑𝑈
+𝑛 (𝑣))∈ X for the 𝑣th iterate
+within the feasible set X of (18), we obtain a strongly convex
+surrogate function ¯𝐸𝑈,𝐿
+𝑘,𝑛 (𝒛𝑛; 𝒛𝑛(𝑣)) of 𝐸𝑈,𝐿
+𝑘,𝑛 (𝒛𝑛) as
+¯𝐸𝑈,𝐿
+𝑘,𝑛 (𝒛𝑛; 𝒛𝑛(𝑣))
+Δ= ¯𝐸𝑈,𝐿
+𝑘,𝑛 (𝐿𝑈,𝐿
+𝑘,𝑛 , 𝒑𝑈
+𝑛 ; 𝐿𝑈,𝐿
+𝑘,𝑛 (𝑣), 𝒑𝑈
+𝑛 (𝑣))
+= 𝑓1(𝐿𝑈,𝐿
+𝑘,𝑛 ) 𝑓2( 𝒑𝑈
+𝑛 (𝑣)) + 𝑓1(𝐿𝑈,𝐿
+𝑘,𝑛 (𝑣)) 𝑓2( 𝒑𝑈
+𝑛 )
++
+𝜏𝐿𝑈,𝐿
+𝑘,��
+2
+(𝐿𝑈,𝐿
+𝑘,𝑛 − 𝐿𝑈,𝐿
+𝑘,𝑛 (𝑣))2 +
+𝜏𝑥𝑈
+𝑛
+2 (𝑥𝑈
+𝑛 − 𝑥𝑈
+𝑛 (𝑣))2
++
+𝜏𝑦𝑈
+𝑛
+2 (𝑦𝑈
+𝑛 − 𝑦𝑈
+𝑛 (𝑣))2,
+(23)
+where 𝜏𝐿𝑈,𝐿
+𝑘,𝑛 , 𝜏𝑥𝑈
+𝑛 , 𝜏𝑦𝑈
+𝑛
+> 0. Also, the function 𝐸𝑈
+𝑘,𝑛(𝒛𝑛)
+Δ=
+𝐸𝑈
+𝑘,𝑛(𝑙𝑈
+𝑘,𝑛) in (18a) is the product of two convex and non-
+negative functions, namely
+𝑓1(𝑙𝑈
+𝑘,𝑛) =
+𝛾𝑈𝐶𝑈
+𝑘 𝑙𝑈
+𝑘,𝑛
+Δ2
+(24)
+and
+𝑓2(𝑙𝑈
+𝑘′,𝑛) =
+� 𝐾
+∑︁
+𝑘′=1
+𝐶𝑈
+𝑘′𝑙𝑈
+𝑘′,𝑛
+�2
+.
+(25)
+As in (23), we obtain a strongly convex surrogate function
+¯𝐸𝑈
+𝑘,𝑛(𝒛𝑛; 𝒛𝑛(𝑣)) of 𝐸𝑈
+𝑘,𝑛(𝒛𝑛) as
+¯𝐸𝑈
+𝑘,𝑛(𝒛𝑛; 𝒛𝑛(𝑣))
+Δ= ¯𝐸𝑈
+𝑘,𝑛(𝑙𝑈
+𝑘,𝑛, 𝑙𝑈
+𝑘′,𝑛; 𝑙𝑈
+𝑘,𝑛(𝑣), 𝑙𝑈
+𝑘′,𝑛(𝑣))
+= 𝑓1(𝑙𝑈
+𝑘,𝑛) 𝑓2(𝑙𝑈
+𝑘′,𝑛(𝑣)) + 𝑓1(𝑙𝑈
+𝑘,𝑛(𝑣)) 𝑓2(𝑙𝑈
+𝑘′,𝑛)
++
+𝜏𝑙𝑈
+𝑘,𝑛
+2 (𝑙𝑈
+𝑘,𝑛 − 𝑙𝑈
+𝑘,𝑛(𝑣))2 +
+𝜏𝑙𝑈
+𝑘′,𝑛
+2
+(𝑙𝑈
+𝑘′,𝑛 − 𝑙𝑈
+𝑘′,𝑛(𝑣))2,
+(26)
+where 𝜏𝑙𝑈
+𝑘,𝑛, 𝜏𝑙𝑈
+𝑘′,𝑛 > 0.
+For the non-convex energy budget constraint (18b), we
+derive a convex upper bound by using Lemma 2. The function
+𝐸 𝐼 ,𝑈
+𝑘,𝑛 (𝒛𝑛)
+Δ= 𝐸 𝐼 ,𝑈
+𝑘,𝑛 (𝐿𝐼 ,𝑈
+𝑘,𝑛 , 𝒑𝑈
+𝑛 ) is the product of two convex and
+non-negative functions, namely
+ℎ1(𝐿𝐼 ,𝑈
+𝑘,𝑛 ) = 2
+𝐿𝐼,𝑈
+𝑘,𝑛
+𝐵Δ/𝐾 − 1
+(27)
+and
+ℎ2( 𝒑𝑈
+𝑛 ) = (𝑥𝑈
+𝑛 − 𝑥𝐼
+𝑘)2 + (𝑦𝑈
+𝑛 − 𝑦𝐼
+𝑘)2 + ℎ𝑈 2.
+(28)
+Algorithm 1 Proposed algorithm for the “Always On” scenario
+Input: 𝛾(𝑣) ∈ (0, 1], 𝒛(0) = {𝒛𝑛(0)}𝑛∈N ∈ X; Set 𝑣 = 0.
+Output: {𝐿𝐼 ,𝑈
+𝑘,𝑛 }, {𝐿𝑈,𝐿
+𝑘,𝑛 }, {𝐿𝐿,𝑈
+𝑘,𝑛 }, {𝑙𝑈
+𝑘,𝑛}, {𝑙𝐿
+𝑘,𝑛}, { 𝒑𝑈
+𝑛 }.
+1: If 𝒛(𝑣) is a stationary solution of (18): STOP.
+2: Compute ˆ𝒛 (𝒛(𝑣)) of (30) using dual decomposition or
+CVX.
+3: Set 𝒛(𝑣 + 1) = 𝒛(𝑣) + 𝛾(𝑣) (ˆ𝒛 (𝒛(𝑣)) − 𝒛(𝑣)).
+4: 𝑣 ← 𝑣 + 1 and go to step 1.
+Then, by using Lemma 2 and defining 𝒛𝑛(𝑣) for the 𝑣th
+iterate, we obtain a strongly convex surrogate function
+¯𝐸 𝐼 ,𝑈
+𝑘,𝑛 (𝒛𝑛; 𝒛𝑛(𝑣)) of 𝐸 𝐼 ,𝑈
+𝑘,𝑛 (𝒛𝑛) as
+¯𝐸 𝐼 ,𝑈
+𝑘,𝑛 (𝒛𝑛; 𝒛𝑛(𝑣))
+Δ= 𝐸 𝐼 ,𝑈
+𝑘,𝑛 (𝐿𝐼 ,𝑈
+𝑘,𝑛 , 𝒑𝑈
+𝑛 ; 𝐿𝐼 ,𝑈
+𝑘,𝑛 (𝑣), 𝒑𝑈
+𝑛 (𝑣))
+= 𝑁0𝐵Δ/𝐾
+2𝑔0
+������
+�
+2
+𝐿𝐼,𝑈
+𝑘,𝑛
+𝐵Δ/𝐾 − 1 + (𝑥𝑈
+𝑛 − 𝑥𝐼
+𝑘)
+2 + (𝑦𝑈
+𝑛 − 𝑦𝐼
+𝑘)
+2 + ℎ𝑈 2
+�2
+−
+�
+2
+𝐿𝐼,𝑈
+𝑘,𝑛 (𝑣)
+𝐵Δ/𝐾
+− 1
+�2
+−
+�
+(𝑥𝑈
+𝑛 (𝑣) − 𝑥𝐼
+𝑘)
+2 + (𝑦𝑈
+𝑛 (𝑣) − 𝑦𝐼
+𝑘)
+2 + ℎ𝑈 2�2������
+− 𝑁0 ln 2
+𝑔0
+2
+𝐿𝐼,𝑈
+𝑘,𝑛 (𝑣)
+𝐵Δ/𝐾
+�
+2
+𝐿𝐼,𝑈
+𝑘,𝑛 (𝑣)
+𝐵Δ/𝐾
+− 1
+� �
+𝐿𝐼 ,𝑈
+𝑘,𝑛 − 𝐿𝐼 ,𝑈
+𝑘,𝑛 (𝑣)
+�
+− 2𝑁0𝐵Δ/𝐾
+𝑔0
+�
+(𝑥𝑈
+𝑛 (𝑣) − 𝑥𝐼
+𝑘)
+2 + (𝑦𝑈
+𝑛 (𝑣) − 𝑦𝐼
+𝑘)
+2 + ℎ𝑈 2�
+�
+(𝑥𝑈
+𝑛 (𝑣) − 𝑥𝐼
+𝑘)(𝑥𝑈
+𝑛 − 𝑥𝑈
+𝑛 (𝑣)) + (𝑦𝑈
+𝑛 (𝑣) − 𝑦𝐼
+𝑘)(𝑦𝑈
+𝑛 − 𝑦𝑈
+𝑛 (𝑣))
+�
+.
+(29)
+Finally, the problem in Equation (18) can be transformed
+into the strongly convex inner approximation for a given
+feasible 𝒛(𝑣) = {𝒛𝑛(𝑣)}𝑛∈N, as
+min
+𝒛
+𝐾
+∑︁
+𝑘=1
+�𝑁 −4
+∑︁
+𝑛=1
+𝛽𝑘,𝑛 ¯𝐸𝑈,𝐿
+𝑘,𝑛+1(𝒛𝑛+1; 𝒛𝑛+1(𝑣))
++
+𝑁 −2
+∑︁
+𝑛=1
+(1 − 𝛽𝑘,𝑛) ¯𝐸𝑈
+𝑘,𝑛+1(𝒛𝑛+1; 𝒛𝑛+1(𝑣))
+�
++
+𝑁
+∑︁
+𝑛=1
+𝐸𝐹
+𝑛 (𝒗𝑈
+𝑛 )
+(30a)
+s.t. ¯𝐸 𝐼 ,𝑈
+𝑘,𝑛 (𝒛𝑛; 𝒛𝑛(𝑣)) ≤ 𝜀, ∀𝑘 ∈ K, 𝑛 ∈ N
+(30b)
+(18c) − (18m),
+(30c)
+which has a unique solution denoted by ˆ𝒛 (𝒛(𝑣)). Since Prob-
+lem (30) is convex, we can obtain the closed-form solutions
+via dual decomposition [34] or a standard convex optimization
+solver such as CVX [35]. The proposed algorithm based
+on the SCA method is summarized as Algorithm 1. The
+sequence {𝒛(𝑣)} generated by Algorithm 1 converges if the
+step size 𝛾(𝑣) is chosen so that 𝛾(𝑣) ∈ (0, 1], 𝛾(𝑣) → 0, and
+�
+𝑣 𝛾(𝑣) = ∞. Also, {𝒛(𝑣)} is bounded and every limit point
+of {𝒛(𝑣)} is stationary. Furthermore, if Algorithm 1 does not
+stop after a finite number of steps, none of the stationary points
+are a local minimum of Problem (18).
+
+8
+Algorithm 2 Proposed algorithm for the “Always Off” sce-
+nario
+Input: 𝛾(𝑣) ∈ (0, 1], 𝒛(0) = {𝒛𝑛(0)}𝑛∈N ∈ X; Set 𝑣 = 0.
+Output: {𝐿𝐼 ,𝑈
+𝑘,𝑛 }, {𝑙𝑈
+𝑘,𝑛}, { 𝒑𝑈
+𝑛 }.
+1: If 𝒛(𝑣) is a stationary solution of (32): STOP.
+2: Compute ˆ𝒛 (𝒛(𝑣)) of (33) using dual decomposition or
+CVX.
+3: Set 𝒛(𝑣 + 1) = 𝒛(𝑣) + 𝛾(𝑣) (ˆ𝒛 (𝒛(𝑣)) − 𝒛(𝑣)).
+4: 𝑣 ← 𝑣 + 1 and go to step 1.
+IV. OPTIMAL ENERGY CONSUMPTION FOR THE
+“ALWAYS OFF” SCENARIO
+In this section, we find the optimal bit allocation and UAV
+path planning when the LEO communication is not available
+during the entire mission time. Therefore, the total UAV
+energy consumption 𝐸𝑡𝑜𝑡𝑎𝑙
+𝑘,𝑛
+in (13) is rewritten with 𝛼𝑘,𝑛 = 0
+for all 𝑛 ∈ N, as
+𝐸𝑡𝑜𝑡𝑎𝑙
+𝑘,𝑛
+= (1 − 𝛽𝑘,𝑛)𝐸𝑈
+𝑘,𝑛(𝑙𝑈
+𝑘,𝑛) + 𝐸𝐹
+𝑛 (𝒗𝑈
+𝑛 ).
+(31)
+For UAV computing with 𝛽𝑘,𝑛 = 0, the problem is given with
+(31) by
+min
+𝐿𝐼,𝑈
+𝑘,𝑛 ,𝑙𝑈
+𝑘,𝑛,𝒑𝑈
+𝑛
+𝐾
+∑︁
+𝑘=1
+𝑁 −2
+∑︁
+𝑛=1
+(1 − 𝛽𝑘,𝑛)𝐸𝑈
+𝑘,𝑛+1(𝑙𝑈
+𝑘,𝑛+1) +
+𝑁
+∑︁
+𝑛=1
+𝐸𝐹
+𝑛 (𝒗𝑈
+𝑛 )
+(32a)
+s.t.
+𝑁 −2
+∑︁
+𝑛=1
+(1 − 𝛽𝑘,𝑛)𝐿𝐼 ,𝑈
+𝑘,𝑛 = 𝐼𝑘, ∀𝑘 ∈ K
+(32b)
+𝑁 −2
+∑︁
+𝑛=1
+(1 − 𝛽𝑘,𝑛)𝑙𝑈
+𝑘,𝑛+1 = 𝐼𝑘, ∀𝑘 ∈ K
+(32c)
+(18b), (18c), (18k) − (18m),
+(32d)
+where the equality constraints (32b) and (32c) guarantee that
+the total number of uplink bits from the IoT sensor and the
+total number of computation bits at the UAV must be equal to
+the input bits of the IoT sensor for complete offloading.
+In the “Always Off” case, the primal variables are defined as
+𝒛 = {𝒛𝑛}𝑛∈N with 𝒛𝑛 = ({𝐿𝐼 ,𝑈
+𝑘,𝑛 }𝑘 ∈K, {𝑙𝑈
+𝑘,𝑛}𝑘 ∈K, 𝒑𝑈
+𝑛 ). Since
+Problem (32) is non-convex, it can be transformed into the
+strongly convex inner approximation, for a given a feasible
+𝒛(𝑣) = {𝒛𝑛(𝑣)}𝑛∈N, as
+min
+𝒛
+𝐾
+∑︁
+𝑘=1
+𝑁 −2
+∑︁
+𝑛=1
+(1 − 𝛽𝑘,𝑛) ¯𝐸𝑈
+𝑘,𝑛+1(𝒛𝑛+1; 𝒛𝑛+1(𝑣)) +
+𝑁
+∑︁
+𝑛=1
+𝐸𝐹
+𝑛 (𝒗𝑈
+𝑛 )
+(33a)
+s.t. (32b), (32c), (30b), (18c), (18k) − (18m),
+(33b)
+where ¯𝐸𝑈
+𝑘,𝑛 of the objective function is defined equally in (26).
+Problem (33) has a unique solution denoted by ˆ𝒛 (𝒛(𝑣)) due to
+its convexity. As in Problem (30), the locally optimal solution
+can be obtained by dual decomposition or a standard convex
+optimization solver. The proposed SCA-based algorithm is
+summarized in Algorithm 2.
+Algorithm 3 Proposed algorithm for the “Intermediate Dis-
+connected” scenario
+Input: 𝛾(𝑣) ∈ (0, 1], 𝒛(0) = {𝒛𝑛(0)}𝑛∈N ∈ X; Set 𝑣 = 0.
+Output: {𝐿𝐼 ,𝑈
+𝑘,𝑛 }, {𝐿𝑈,𝐿
+𝑘,𝑛 }, {𝐿𝐿,𝑈
+𝑘,𝑛 }, {𝑙𝑈
+𝑘,𝑛}, {𝑙𝐿
+𝑘,𝑛}, { 𝒑𝑈
+𝑛 }.
+1: If 𝒛(𝑣) is a stationary solution of (34): STOP.
+2: Compute ˆ𝒛 (𝒛(𝑣)) of (35) using dual decomposition or
+CVX.
+3: Set 𝒛(𝑣 + 1) = 𝒛(𝑣) + 𝛾(𝑣) (ˆ𝒛 (𝒛(𝑣)) − 𝒛(𝑣)).
+4: 𝑣 ← 𝑣 + 1 and go to step 1.
+V. OPTIMAL ENERGY CONSUMPTION FOR THE
+“INTERMEDIATE DISCONNECTED” SCENARIO
+For the “Intermediate Disconnected” case, we provide joint
+path planning and resource allocation when the LEO commu-
+nication is intermediately disconnected. The total UAV energy
+consumption in this case follows (13).
+During the LEO computing for 𝑛
+∈
+{1, · · ·, 𝑁𝑡} with
+𝛼𝑘,𝑛
+=
+1
+and
+𝛽𝑘,𝑛
+=
+1,
+we
+jointly
+optimize
+the
+bit allocation {𝐿𝐼 ,𝑈
+𝑘,𝑛 }𝑛∈{1,···,𝑁𝑡 },𝑘 ∈K, {𝐿𝑈,𝐿
+𝑘,𝑛 }𝑛∈{2,···,𝑁𝑡+1},𝑘 ∈K,
+{𝑙𝐿
+𝑘,𝑛}𝑛∈{3,···,𝑁𝑡+2},𝑘 ∈K
+and
+{𝐿𝐿,𝑈
+𝑘,𝑛 }𝑛∈{4,···,𝑁𝑡+3},𝑘 ∈K
+along
+with the UAV trajectory { 𝒑𝑈
+𝑛 }𝑛∈{2,···,𝑁𝑡+4}. During UAV com-
+puting for 𝑛 ∈ {1, · · ·, 𝑁𝑡} with 𝛼𝑘,𝑛 = 1 and 𝛽𝑘,𝑛 = 0 and
+𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁} with 𝛼𝑘,𝑛 = 0 and 𝛽𝑘,𝑛 = 0, the bit
+allocation and the UAV path planning are jointly designed
+as in the UAV computing process of the “Always On” case.
+Accordingly, we can formulate the problem as
+min
+𝐿𝐼,𝑈
+𝑘,𝑛 ,𝐿𝑈,𝐿
+𝑘,𝑛 ,𝐿𝐿,𝑈
+𝑘,𝑛
+𝑙𝑈
+𝑘,𝑛,𝑙𝐿
+𝑘,𝑛,𝒑𝑈
+𝑛
+𝐾
+∑︁
+𝑘=1
+𝑁𝑡
+∑︁
+𝑛=1
+𝛼𝑘,𝑛
+�
+𝛽𝑘,𝑛𝐸𝑈,𝐿
+𝑘,𝑛+1(𝐿𝑈,𝐿
+𝑘,𝑛+1, 𝒑𝑈
+𝑛+1)
++ �1 − 𝛽𝑘,𝑛
+� 𝐸𝑈
+𝑘,𝑛+1(𝑙𝑈
+𝑘,𝑛+1)
+�
++
+𝐾
+∑︁
+𝑘=1
+𝑁 −2
+∑︁
+𝑛=𝑁𝑡+1
+�1 − 𝛼���,𝑛
+� (1 − 𝛽𝑘,𝑛)𝐸𝑈
+𝑘,𝑛+1(𝑙𝑈
+𝑘,𝑛+1)
++
+𝑁
+∑︁
+𝑛=1
+𝐸𝐹
+𝑛 (𝒗𝑈
+𝑛 )
+(34a)
+s.t.
+𝑛
+∑︁
+𝑖=1
+𝐿𝑈,𝐿
+𝑘,𝑖+1 ≤
+𝑛
+∑︁
+𝑖=1
+𝐿𝐼 ,𝑈
+𝑘,𝑖 , ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁𝑡
+(34b)
+𝑛
+∑︁
+𝑖=1
+𝑙𝐿
+𝑘,𝑖+2 ≤
+𝑛
+∑︁
+𝑖=1
+𝐿𝑈,𝐿
+𝑘,𝑖+1, ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁𝑡
+(34c)
+𝑛
+∑︁
+𝑖=1
+𝐿𝐿,𝑈
+𝑘,𝑖+3 ≤ 𝑂𝐿
+𝑘
+𝑛
+∑︁
+𝑖=1
+𝑙𝐿
+𝑘,𝑖+2, ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁𝑡
+(34d)
+𝑁𝑡
+∑︁
+𝑛=1
+𝛽𝑘,𝑛𝑙𝐿
+𝑘,𝑛+2 +
+𝑁 −2
+∑︁
+𝑛=1
+(1 − 𝛽𝑘,𝑛)𝑙𝑈
+𝑘,𝑛+1 = 𝐼𝑘, ∀𝑘 ∈ K
+(34e)
+𝑁𝑡
+∑︁
+𝑛=1
+𝑙𝐿
+𝑘,𝑛+2 =
+𝑁𝑡
+∑︁
+𝑛=1
+𝐿𝑈,𝐿
+𝑘,𝑛+1, ∀𝑘 ∈ K
+(34f)
+𝑁𝑡
+∑︁
+𝑛=1
+𝐿𝐿,𝑈
+𝑘,𝑛+3 = 𝑂𝐿
+𝑘
+𝑁𝑡
+∑︁
+𝑛=1
+𝐿𝑈,𝐿
+𝑘,𝑛+1, ∀𝑘 ∈ K
+(34g)
+(18b), (18c), (32b), (18k) − (18m),
+(34h)
+
+9
+TABLE III: Simulation Parameters
+Parameter
+Value
+Parameter
+Value
+𝑣𝑠
+7.5 km/s
+𝑟𝐸
+6371 km
+𝜃
+10 ◦
+𝑇𝑣
+830 s
+ℎ𝑈
+1 km
+ℎ𝐿
+600 km
+𝐾
+10
+𝑣max
+50 m/s
+𝑀
+9.65 kg
+𝑂𝐿
+𝑘 , 𝑂𝑈
+𝑘
+0.5
+𝑓 𝑈
+𝑛
+19.5 × 109 cycles/s [7]
+𝐺
+10 dB
+𝛾𝐿, 𝛾𝑈
+10−28 [29], [30]
+𝐶𝐿
+𝑘 , 𝐶𝑈
+𝑘
+1550.7 [29], [30]
+𝐵
+40 MHz
+𝑁0
+-174 dBm/Hz
+𝜀
+0.11 J
+ref. SNR
+80 dB
+where the inequality constraints (34b)-(34d) and equality con-
+straints (34e)-(34g) limit the number of frames to 𝑛 = 1, ···, 𝑁𝑡
+instead of 𝑛 = 1, ···, 𝑁 −4 in constraints (18d)-(18f) and (18h)-
+(18j), respectively.
+In the“Intermediate Disconnected” case, the primal vari-
+ables are defined the same as in the“Always On” case. By
+applying the SCA method,the non-convex Problem (34) can
+be transformed into the strongly convex inner approximation
+for a given a feasible 𝒛(𝑣) = {𝒛𝑛(𝑣)}𝑛∈N, as
+min
+𝒛
+𝐾
+∑︁
+𝑘=1
+𝑁𝑡
+∑︁
+𝑛=1
+𝛼𝑘,𝑛
+�
+𝛽𝑘,𝑛 ¯𝐸𝑈,𝐿
+𝑘,𝑛+1(𝒛𝑛+1; 𝒛𝑛+1(𝑣))
++ �1 − 𝛽𝑘,𝑛
+� ¯𝐸𝑈
+𝑘,𝑛+1(𝒛𝑛+1; 𝒛𝑛+1(𝑣))
+�
++
+𝐾
+∑︁
+𝑘=1
+𝑁 −2
+∑︁
+𝑛=𝑁𝑡+1
+�1 − 𝛼𝑘,𝑛
+� (1 − 𝛽𝑘,𝑛) ¯𝐸𝑈
+𝑘,𝑛+1(𝒛𝑛+1; 𝒛𝑛+1(𝑣))
++
+𝑁
+∑︁
+𝑛=1
+𝐸𝐹
+𝑛 (𝒗𝑈
+𝑛 )
+(35a)
+s.t. (34b) − (34g), (30b), (18c), (32b), (18k) − (18m), (35b)
+which has a unique solution denoted by ˆ𝒛 (𝒛(𝑣)) to be obtained
+by dual decomposition or a standard convex optimization
+solver. Algorithm 3 describes the proposed method for the
+“Intermediate Disconnected” scenario.
+VI. SIMULATION RESULTS
+In this section, we evaluate the performance of the proposed
+algorithms to jointly optimize the bit allocation and the UAV
+trajectory for marine IoT systems in various LEO accessible
+statuses. For reference, we consider the following schemes:
+(i) No optimization: The equal bit allocation is considered for
+communication and computation per frame, while the UAV
+flies at constant velocity between the initial and final positions
+as 𝒑𝑈
+𝑛 = 𝒑𝑈
+𝐼 + (𝑛 − 1) � 𝒑𝑈
+𝐹 − 𝒑𝑈
+𝐼
+��𝑁, for 𝑛 ∈ N; (ii) Opti-
+mized bit allocation: The communication and computation bits
+are optimized by the proposed algorithms while considering
+the UAV trajectory with the constant-velocity as in (i); (iii)
+Optimized UAV trajectory: The path planning of the UAV
+is obtained by the proposed algorithms with fixed equal bit
+allocation per frame. The simulation parameters are provided
+in Table III. Particularly, the space segment considers Iridium-
+like LEO satellite networks that provide global coverage with
+66 satellites distributed in 6 polar orbits [15], where the orbit
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+x [km]
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+y [km]
+IoT2
+IoT5
+IoT3
+IoT4
+IoT7
+IoT8
+IoT9
+IoT10
+IoT6
+IoT1
+LEO
+UAV trajectory
+("Always On")
+UAV trajectory
+("Always Off")
+UAV trajectory
+("Intermediate Disconnected")
+Fig. 4: Optimal UAV trajectories according to the different
+LEO access scenarios.
+height is ℎ = 601 km with the elevation angle 𝜃 = 10 ◦, and
+satellites in the orbit travel at a speed of around 𝑣𝑠 = 7.5 km/s.
+To better understand the proposed algorithms, Figs. 4 and 5
+consider the partial optimization of UAV path planning or bit
+allocation. As shown in Fig. 4, there are 𝐾 = 10 IoT sensors
+distributed randomly in a 10 km × 10 km area within the
+beam coverage of the central LEO satellite, i.e., 𝛼𝑘,𝑛 = 1,
+for all 𝑛 ∈ N and 𝑘 ∈ K. With the LEO visible time of
+𝑇𝑣 = 830 s obtained from (4) and a bandwidth of 𝐵 = 40
+MHz [15], the data size collected from each IoT sensor is
+randomly determined based on the computation capability of
+the UAV in (16). In our simulation, the scheduling variable
+𝛽𝑘,𝑛 is defined from (16), i.e., 𝛽𝑘,𝑛 = [0 0 0 1 1 1 0 1 0 0],
+for 𝑘 ∈ K and 𝑛 ∈ N, as shown in Fig. 4. The IoT sensors
+with 𝛽𝑘,𝑛 = 0 for UAV computing and with 𝛽𝑘,𝑛 = 1 for LEO
+computing are indicated by black-colored circles and green-
+colored circles, respectively, while the LEO satellite, indicated
+by a red-colored hexagram, travels along the red dotted line.
+The initial and final positions of the UAV are 𝒑𝑈
+𝐼 = (5, 0, 0)
+to 𝒑𝑈
+𝐹 = (10, 5, 0).
+Fig. 4 shows the optimized UAV trajectories with the fixed
+equal bit allocation according to the different LEO satellite
+access scenarios. For this experiment, the latency constraint is
+𝑇 = 360 s with 𝑁 = 60 and Δ = 6 s. In the “Always On” case,
+the optimized UAV trajectory, represented by a blue asterisk
+line, is designed to fly close to the IoT sensors with LEO
+computing until its final destination. This can significantly
+reduce the large amount of uplink communication energy
+consumption induced by the long distance between the LEO
+satellite and UAV. In the “Always Off” case, where only UAV
+computing is considered, the UAV flies along a straight path
+to a destination, which is represented by a yellow crossed
+line. In this case, the flying energy consumption must be
+reduced to to minimize the total UAV energy due to the fixed
+computation bit allocation. In the “Intermediate Disconnected”
+case, where the LEO communication is lost at 𝑁𝑡 = 𝑁/2, the
+
+10
+10
+20
+30
+40
+50
+60
+Frame number
+0
+1
+2
+IoT6's number of bits
+107
+ LI,U
+6,n
+ lU
+6,n
+ LU,L
+6,n
+ lL
+6,n
+ LL,U
+6,n
+(a) “Always On” scenario
+10
+20
+30
+40
+50
+60
+Frame number
+0
+5
+10
+IoT6's number of bits
+106
+(b) “Always Off” scenario
+10
+20
+30
+40
+50
+60
+Frame number
+0
+0.5
+1
+1.5
+2
+IoT6's number of bits
+107
+(c) “Intermediate Disconnected” scenario
+Fig. 5: Optimal bit allocations for IoT sensor 6 in Fig. 4
+according to the different LEO access scenarios.
+optimized UAV trajectory, represented by a purple square line,
+tends to fly close to the IoT sensors with LEO computing for
+𝑛 = 1, · · ·, 𝑁𝑡. Then, in the frame period of 𝑛 = 𝑁𝑡 + 1, · · ·, 𝑁
+where LEO communication is disconnected, the UAV flies
+straight to the final destination because it performs only UAV
+computing.
+Fig. 5 illustrates the optimized bit allocations for IoT sensor
+6 shown in Fig. 4 with the fixed constant-velocity UAV
+trajectory according to different LEO access scenarios. Except
+for the UAV trajectory, the simulation environment is the same
+as in Fig. 4. In Fig. 5(a), the optimal bit allocations 𝐿𝐼 ,𝑈
+𝑘,𝑛 ,
+𝐿𝑈,𝐿
+𝑘,𝑛 , 𝑙𝐿
+𝑘,𝑛, 𝐿𝐿,𝑈
+𝑘,𝑛
+by proposed Algorithm 1 are shown for
+LEO computing in the “Always On” case. First, most of the
+uplink bits 𝐿𝐼 ,𝑈
+𝑘,𝑛 are allocated between frames 20 to 35, which
+corresponds to the period where the UAV flies closest to IoT
+sensor 6. The offloading bits 𝐿𝑈,𝐿
+𝑘,𝑛
+are allocated equally in
+the entire frame because the equal bit allocation can achieve
+the minimal communication energy from (7). Finally, the LEO
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+x [km]
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+y [km]
+IoT3
+IoT4
+IoT10
+IoT6
+IoT2
+IoT8
+IoT1
+IoT7
+IoT9
+IoT5
+LEO
+2nd 
+orbit
+1st 
+orbit
+3rd 
+orbit
+UAV trajectory
+(1st orbit)
+UAV trajectory
+(2nd orbit)
+UAV trajectory
+(3rd orbit)
+Fig. 6: Optimal UAV trajectories according to different LEO
+satellite orbits, where the IoT sensors with LEO computing
+are deployed at the corner.
+computing bits 𝑙𝐿
+𝑘,𝑛 and LEO downlink bits 𝐿𝐿,𝑈
+𝑘,𝑛 are mostly
+allocated in the latter parts between frames 50 to 60 to satisfy
+the inequality constraints of (18e) and (18f). In Fig. 5(b), the
+optimized bit allocations 𝐿𝐼 ,𝑈
+𝑘,𝑛 and 𝑙𝑈
+𝑘,𝑛 obtained by proposed
+Algorithm 2 are shown for UAV computing of the“Always
+Off” case. Since the UAV cannot communicate with the LEO
+satellite, the computing process is entirely at the UAV-mounted
+cloudlet. The uplink bits 𝐿𝐼 ,𝑈
+𝑘,𝑛 and the computing bits 𝑙𝑈
+𝑘,𝑛 are
+assigned the same as 𝐿𝐼 ,𝑈
+𝑘,𝑛 and 𝐿𝑈,𝐿
+𝑘,𝑛 in Fig. 5(a), respectively.
+However, 𝑙𝑈
+𝑘,𝑛 is dramatically reduced to 8 × 106 per frame
+compared to 10 × 106, as illustrated in Fig. 5(a). This is
+because the amount of data exceeding the UAV computation
+capability is excluded from the UAV computing. Fig. 5(c)
+shows the optimization result of bit allocation attained by
+proposed Algorithm 3 in the “Intermediate Disconnected”
+case. LEO computing is performed during the first half of
+frames, i.e., 𝑛 = 1, ···, 𝑁𝑡, while UAV computing is performed
+during the second half of frames, i.e., 𝑛 = 𝑁𝑡 + 1, · · ·, 𝑁. The
+computing bits 𝑙𝐿
+𝑘,𝑛 at LEO and the downlink bits 𝐿𝐿,𝑈
+𝑘,𝑛 are
+reduced in proportion to the reduced frame duration of LEO
+computing compared to those shown in Fig. 5(a). For UAV
+computing, there are more computing bits 𝑙𝑈
+𝑘,𝑛 allocated at
+the UAV than those from the case in Fig. 5(b). This means
+that less data exceeds the computational capability of the UAV
+thanks to the LEO computing.
+Fig. 6 shows the optimal UAV trajectories according to the
+different LEO satellite orbits in the “Always On” scenario,
+where the IoT sensors that need LEO computing are clustered
+at the corner, i.e., 𝛽𝑘,𝑛 = [0 1 1 0 1 1 0 0 0 0], for 𝑘 ∈ K and
+𝑛 ∈ N. In this deployment, the three different movements of
+the LEO satellite in different orbital directions are considered.
+In the first orbit moving from the upper right corner to the
+lower left corner, the UAV flies near the corner area with IoT
+sensors with LEO computing to its final destination. In the
+
+11
+400
+600
+800
+1000
+1200
+1400
+1600
+Total time T (s)
+0
+1
+2
+3
+4
+5
+6
+7
+Total UAV energy consumption (J)
+106
+No opt. - "Always On"
+No opt. - "Always Off"
+No opt. - "Intermediate Disconnected"
+Opt. bit allocation - "Always On"
+Opt. UAV trajectory - "Always On"
+Joint opt. - "Always On"
+Joint opt. - "Always Off"
+Joint opt. - "Intermediate Disconnected"
+Fig. 7: Comparison of the total UAV energy consumption
+for different optimization schemes in the three LEO satellite
+access scenarios.
+second orbit moving from the upper left corner to the lower
+right corner, the UAV flies in a diagonally downward direction
+along its own orbit rather than the optimized UAV trajectory
+for the first orbit. In the third orbit moving upwards from
+below the midpoint, the UAV flies in an upward direction along
+its own orbit rather than the optimal UAV trajectory for the first
+orbit. From these results, we can see that the LEO movements
+resulting from the orbit influences the optimal UAV path so
+as to reduce the communication energy consumption between
+the UAV and the LEO satellite.
+Fig. 7 compares the total UAV energy consumption of the
+joint optimization scheme with reference schemes in three
+LEO satellite access scenarios. For this experiment, the latency
+constraint is 𝑇 = [360:90:1620] s with 𝑁 = [60:15:270] and
+Δ = 6 s, while the remaining simulation parameters are
+the same as in Figs. 4 and 5. First, the no optimization
+scheme consumes the highest energy in the three scenarios,
+among which the largest energy consumption takes place in
+the “Always Off” case, where only the UAV computing is
+performed. This is natural since the UAV-mounted cloudlet has
+a slightly larger burden in terms of the energy consumption
+with no support of the LEO. In the “Always On” case, for
+𝑇 = 360 s, the total UAV energy consumption for the joint
+optimization scheme is the lowest at 4.6 × 106 J, whereas
+the optimized UAV trajectory scheme with fixed equal bit
+allocation requires 5.5×106 J, and the optimized bit allocation
+with the constant-velocity UAV and no optimization schemes
+requires 6.1 × 106 J. This implies that the UAV path planning
+is more effective in terms of UAV energy consumption than
+bit allocation. Moreover, the total energy consumption in all
+schemes decreases as the total time increases. This is because
+the same amount of data is processed over a longer period
+of time. Compared to the total UAV energy consumption of
+the joint optimization scheme in the “Always Off” scenario,
+those of the joint optimization scheme in other scenarios
+0
+1/8
+2/8
+4/8
+6/8
+7/8
+1
+LEO satellite access time rate
+2.3
+2.4
+2.5
+2.6
+2.7
+2.8
+2.9
+3
+3.1
+3.2
+3.3
+3.4
+Total UAV energy consumption (J)
+106
+55
+60
+65
+70
+75
+80
+85
+90
+95
+100
+Collected data usage rate (%)
+Total energy - "Always On"
+Total energy - "Always Off"
+Total energy - "Intermediate Disconnected"
+Data usage rate - "Always On"
+Data usage rate - "Always Off"
+Data usage rate - "Intermediate Disconnected"
+Fig. 8: Relationship between the total UAV energy consump-
+tion and the collected data usage rate in three LEO satellite
+access scenarios according to the LEO satellite access time
+rate.
+are much higher since the UAV flies straight to its final
+destination when the LEO satellite connection is lost, as in Fig.
+4. However, there is a trade-off between the total UAV energy
+consumption and the collected data usage rate for computing,
+which determines the amount of data executed at cloudlet,
+which is analyzed in the following figure.
+Fig. 8 shows the relationship between the total UAV energy
+consumption and the collected data usage rate for computing in
+the different LEO accessibility scenarios. Any amount of data
+exceeding the UAV computation capability is excluded from
+UAV computing. For this experiment, the scheduling variables
+are defined as 𝛽𝑘,𝑛 = [0 0 1 1 1 1 0 1 0 0], for 𝑘 ∈ K
+and 𝑛 ∈ N. The UAV computation capability is applied to
+226 Mbits by using the CPU frequency at the UAV server
+𝑓 𝑈
+𝑛
+= 9.75 × 109 cycles/s. In the “Always On” scenario, the
+LEO satellite access time rate is 1. At this time, the total
+UAV energy consumption is 3.3×106 J and the collected data
+usage rate is 100%. In the “Always Off” case, where the LEO
+satellite access time rate is 0, the total UAV energy consump-
+tion is 2.24 × 106 J and the collected data usage rate is 54%.
+Although the energy consumption in the “Always Off” case is
+dramatically reduced, the utilization rate of the collected data
+is also cut in half. In the “Intermediate Disconnected” case,
+as the LEO satellite access time rate increases, the total UAV
+energy consumption and the collected data usage rate increase
+differently. When the LEO satellite access time rate is above
+6/8, the total UAV energy consumption is saturated with the
+total UAV energy consumption of the “Always On” case. This
+is because the straight flight segment of the UAV to the final
+destination after disconnecting with the LEO satellite matches
+that of the “Always On” case. Also, when the LEO satellite
+access time rate is more than 7/8, the collected data usage
+rate is more than about 95%. In this simulation environment,
+adequate data usage and energy consumption is achieved with
+more than a 7/8 LEO satellite access time rate.
+
+12
+VII. CONCLUSIONS
+In this paper, a marine IoT system using hybrid LEO
+and UAV computing for real-time utilization of marine data
+has been analyzed according to the different LEO satellite
+access scenarios: “Always On,” “Always Off” and “Inter-
+mediate Disconnected”. For each scenario, we proposed the
+joint optimization problem of bit allocation for computing
+and communication in offloading and UAV path planning to
+minimize the total UAV energy consumption under latency,
+energy budget, and UAV operational constraints. To solve the
+optimization problem, we developed an SCA-based algorithm
+whose performance in terms of energy efficiency was validated
+via numerical results compared to conventional approaches
+with partial optimization that design only the bit allocation or
+UAV trajectory. According to LEO satellite access time and
+its orbit direction, the path planning of the UAV is optimized
+differently for energy saving, whose impact is pronounced for
+the case when the LEO connectivity is unstable or discon-
+nected. In future works, different existing LEO deployments
+should be further considered with various heights of multiple
+satellites and UAVs.
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+

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+Compact stars in Quantum Field Theory
+Ignacio A. Reyes1 and Giovanni Maria Tomaselli2
+1Institute for Theoretical Physics and
+2GRAPPA,
+University of Amsterdam, Amsterdam, 1098 XH, The Netherlands
+Very compact stars seem to be forbidden in General Relativity. While Buchdahl’s theorem sets
+an upper bound on compactness, further no-go results rely on the existence of two light rings, the
+inner of which has been associated to gravitational instabilities. However, little is known about
+the role of quantum fields in these strong gravity regimes. Working in the probe approximation
+where the backreaction is ignored, we show that the trapping of modes around the inner light ring
+leads the renormalized stress tensor of Conformal Field Theories to diverge faster than the classical
+source in the Buchdahl limit. This leads to the violation of the Null Energy Condition, as well as
+the isotropy assumption used in Buchdahl’s theorem. The backreaction of quantum fields in this
+regime therefore cannot be ignored. This happens as the star’s surface approaches the Buchdahl
+radius 9GM/4 rather than the Schwarzschild radius, with the quantum fields having support in
+a small region around the center, becoming negligible at the surface. These are generic quantum
+features and do not depend on the details of the interactions. Our findings open a way for further
+investigation into the role of QFT in astrophysics.
+COMPACT RELATIVISTIC STARS
+The General Relativistic prediction of the existence of
+compact objects, such as white dwarfs and neutron stars,
+has been confirmed by many observations. Their macro-
+scopic properties follow from the Tolman-Oppenheimer-
+Volkoff equation. However, quantum theory is essential
+in understanding the physics of these stars, as it provides
+the ultimate reason for their existence, namely, Fermi’s
+exclusion principle.
+The question regarding the maximum mass of such
+compact object is crucial: it is the main criterion used to
+discriminate between what we suspect is a neutron star or
+a black hole. Well-known upper limits were set by Chan-
+drasekhar [1] and Rhoades-Ruffini [2]. A more generic re-
+sult, that is independent of the equation of state of the
+matter, was established by Buchdahl [3] and gives an up-
+per bound on compactness in GR. Consider an isotropic
+perfect fluid star, with stress tensor
+T µ
+ν = diag(−ρ, p, p, p) ,
+(1)
+on a static, spherically symmetric metric
+ds2 = −f(r) dt2 + h(r) dr2 + r2(dθ2 + sin2 θ dφ2) . (2)
+Assuming in addition that ρ > 0, ∂rρ ≤ 0 and that
+Einstein’s equations hold, the requirement that the met-
+ric is everywhere regular leads to
+R ≥ 9GM/4 ,
+(3)
+where R is the radius of the star and M is its mass. The
+saturation of the bound is known as the Buchdahl limit.
+Notice one can also formulate this bound in a coordinate-
+independent way.
+A particularly simple solution that manifestly sat-
+urates Buchdahl’s second assumption is the constant-
+density star or ‘Schwarzschild interior metric’.
+These
+configurations have uniform density ρ
+=
+3M/4πR3
+throughout the star, and as is well known they can sat-
+urate the Buchdahl limit (3). Although they are unre-
+alistic models of an astrophysical object, they are the
+standard example when studying the TOV equations.
+The metric for this equation of state takes the form
+(2), with
+f(r) =
+�
+3
+2
+�
+1 − 2GM
+R
+− 1
+2
+�
+1 − 2GMr2
+R3
+�2
+,
+(4)
+h(r) =
+�
+1 − 2GMr2
+R3
+�−1
+,
+(5)
+and is matched to the usual exterior Schwarzschild vac-
+uum solution at the sphere’s surface.
+The simplicity of these solutions make them an excel-
+lent setup to test Quantum Field Theory (QFT) in the
+strong gravity regime.
+A final motivation to consider
+this metric is that it is conformally (Weyl) flat. In fact,
+the uniform density metric above is the unique solution
+to Einstein’s equations coupled to a static perfect fluid
+that is conformally flat [4, 5]. This will allow us to ob-
+tain explicit analytic results.
+We will thus work with
+this spacetime, and comment about the generality of our
+results later on.
+WAVE EQUATION
+In order to understand the behaviour of quantum fields
+in this spacetime, let us begin by first considering the
+propagation of classical waves in it. The wave equation
+for the uniform density background was first discussed
+by Chandrasekhar and Ferrari [6]. For simplicity we take
+a massless scalar Φ using the usual decomposition
+Φ =
+�
+fωℓm ,
+fωℓm(x) = u(r)
+r
+Yℓm(θ, φ)e−iωt . (6)
+arXiv:2301.00826v1  [gr-qc]  2 Jan 2023
+
+2
+−30
+−20
+−10
+0
+10
+0
+0.05
+0.1
+0.15
+r∗/(GM)
+V/(GM)2
+FIG. 1. Potential (for ℓ = 1) given in (9), for R/(GM) =
+9/4 (blue), 2.3 (orange) and 2.4 (green). The dashed lines
+mark the the values of r∗ corresponding to the centre of the
+star in the two latter cases.
+A discontinuous jump at the
+star’s surface matches it to the exterior vacuum Schwarzschild
+solution.
+The wave equation □Φ
+=
+0 can be recast in a
+Schr¨odinger-like form:
+−∂2
+r∗u + V (r∗)u = ω2u ,
+(7)
+where we defined the tortoise coordinate r∗ via
+dr∗
+dr =
+�
+h(r)/f(r) .
+(8)
+The potential V (r∗) takes the form
+V = 1
+r ∂2
+r∗r + ℓ(ℓ + 1)
+r2
+f
+(9)
+and is plotted in Fig. 1 for ℓ = 1 and various values of
+R/(GM). The potential at r > R corresponds to the
+Schwarzschild vacuum metric, and vanishes at infinity. It
+connects to the interior of the star with a discontinuous
+step.
+As we can see from Fig. 1, when R > 9GM/4, the
+tortoise coordinate has a finite minimum possible value
+(dashed lines) corresponding to the centre of the star,
+because the factor h/f is always regular around r = 0.
+Moreover, V (r∗) reaches a local minimum greater than
+zero and then increases towards the surface.
+When R → 9GM/4, however, one has h/f ∼ r−2
+and therefore the domain of r∗ becomes infinite on both
+sides, while V (r∗) vanishes at the centre of the star. This
+closely resembles the situation for black holes, but in that
+case it is the horizon that is mapped to r∗ → −∞. The
+field modes can thus be trapped inside the star, leading
+to a spectrum of quasi-bound states whose magnitudes
+are amplified close to the origin.
+These properties of the effective potential, together
+with the behavior of the tortoise coordinate, suggests
+that upon quantization the renormalized stress tensor
+can become important in the Buchdahl limit. The rest
+of our analysis will be done in a more generic way that
+depends less on the specific theory considered.
+QFT IN CURVED SPACETIME
+QFT in curved spacetime has seen significant progress
+in the last half century. In the semi-classical approxima-
+tion, gravity is still treated classically and one considers
+some quantum fields as another dynamical source to Ein-
+stein’s equations,
+Rµν − 1
+2gµνR = 8πG
+�
+Tµν + ⟨ ˆTµν⟩
+�
+.
+(10)
+We shall denote by ˆTµν the operator of the QFT to dis-
+tinguish it from the classical source (1).
+However, most work in this field has focused on either
+cosmology or black holes. Here, we will study the role
+it plays for astrophysical compact stars. The question
+we will address in this work is whether there exists some
+generic feature of QFT, independent of the details of
+the nuclear interactions and the quantum states involved,
+that becomes important for very compact stars, in the
+regime of strong gravitational fields. We will show that
+there is indeed such an effect.
+We shall focus on the effects of conformally coupled
+fields where the computation is easier, taking it as a toy
+model for more generic scenarios. We work in 3+1 dimen-
+sions, but the generalization to even higher dimensions is
+straightforward. The non-conformal case will be treated
+elsewhere.
+As is well known, conformally coupled classical mat-
+ter has a vanishing trace of its stress tensor. However,
+its quantum counterpart develops a trace anomaly. In
+3 + 1 dimensions, the vacuum expectation value of the
+trace of the renormalized stress tensor for quantum fields
+propagating in a curved spacetime is
+⟨ ˆT µ
+µ ⟩ =
+1
+(4π)2 [cF − aG − d□R] ,
+(11)
+where R is the Ricci scalar, F is the square of the Weyl
+tensor and G is the Gauss-Bonnet invariant. Amongst the
+three real coefficients, c > 0 and a > 0 are well under-
+stood and characterize the particular theory in question.
+On the other hand, d is not determined by the bare La-
+grangian as it depends on the renormalization scheme,
+and is closely related to the quadratic corrections to the
+gravity action as we review below. As such, it should
+be fixed by experiments. For now we will leave d as a
+fixed but undetermined constant and proceed with the
+calculation.
+If additionally the metric is conformally flat – as is the
+case for the constant-density star – then all components
+
+3
+of the renormalized stress tensor are fixed [7]:
+⟨ ˆT µν⟩ =
+−
+a
+(4π)2
+�
+gµν
+�R2
+2 − RαβRαβ
+�
++ 2RµλRν
+λ − 4
+3RRµν
+�
++
+d
+(4π)2
+� 1
+12gµν(R2 − 4R,λ
+;λ) − 1
+3(RRµν − R,µ;ν)
+�
+.
+(12)
+The quantum state chosen for (12) is the vacuum, but
+this will not play an important role. It could be a state
+at finite temperature or with a large number of fermions:
+this would only add an extra contribution independent of
+the curvature. The vacuum stress tensor for the interior
+of the uniform density star is therefore given by (12). We
+now proceed to evaluate it and examine its properties.
+QUANTUM FIELDS IN THE BUCHDAHL LIMIT
+In this section, we describe the main features of
+the
+quantum
+stress
+tensor
+(12)
+evaluated
+on
+the
+Schwarzschild interior metric. In particular, we wish to
+understand its behavior as we approach the Buchdahl
+limit
+R = (9/4 + ϵ)GM ,
+ϵ → 0 .
+(13)
+We will report the results to leading orders in ϵ.
+The Buchdahl limit (13) is a finite distance above the
+black hole compactness corresponding to R = 2GM.
+Nevertheless, this regime is no less extreme: the Ricci
+scalar R of the background metric at the center diverges
+in this limit as
+R(0) = − 3
+R2ϵ + O(1) .
+(14)
+Correspondingly, the central density and pressure of the
+classical uniform density star solution behave as
+ρ(0) =
+1
+3πGR2 + O(ϵ) ,
+(15)
+p(0) =
+1
+8πGR2ϵ + O(1) .
+(16)
+Let us contrast this behavior with its quantum coun-
+terpart (12). For generic ϵ, this takes the form
+⟨ ˆT µ
+ν ⟩ = diag(−⟨ˆρ⟩, ⟨ˆpr⟩, ⟨ˆpθ⟩, ⟨ˆpθ⟩) ,
+(17)
+with ⟨ˆpr⟩ ̸= ⟨ˆpθ⟩. The radial dependence of the compo-
+nents are illustrated in Fig. 2. In the limit ϵ → 0, their
+central values scale as
+⟨ˆρ(0)⟩ =
+9d
+(8πR2ϵ)2 +
+d
+6(πR2)2ϵ + O(1) ,
+(18)
+⟨ˆpr(0)⟩ = ⟨ˆpθ(0)⟩ = −
+3d
+(8πR2ϵ)2 +
+2a − d
+(3πR2)2ϵ + O(1) . (19)
+0
+0.5
+1
+1.5
+2
+0
+0.1
+0.2
+r/(GM)
+⟨ˆρ⟩
+⟨ˆpr⟩
+⟨ˆpθ⟩
+FIG. 2.
+Radial profile of the three components of ⟨ ˆT µ
+ν ⟩,
+for a = d = 1/360, ϵ = 0.003, in units where GM = 1. The
+location of the inner light ring is depicted by the dashed line.
+We emphasize that the pressures match only at the cen-
+ter, and not elsewhere. Moreover, notice that the leading
+order of ⟨ˆρ⟩ and ⟨ˆp⟩ have opposite signs. There is no con-
+tribution from c because the Weyl tensor vanishes.
+By comparing (15) and (16) with (18) and (19), we see
+that the components of the renormalized stress tensor
+scale with higher powers of ϵ than the classical contribu-
+tions, and therefore cannot be ignored in the Buchdahl
+limit. Furthermore, notice that the leading divergence of
+the quantum terms depends only on d: in this regime the
+quantum effects are dominated by the scheme-dependent
+terms proportional to d, and not by c or a.
+The backreaction of quantum effects cannot be ne-
+glected if they become of the same order as the clas-
+sical ones.
+By comparing the classical and quantum
+central pressures (16) and (19), this crossover happens
+at ϵ ∼ |d| (ℓP /R)2, which corresponds to a pressure
+p ∼ |d|−1ℓ−4
+P
+and central curvature R ∼ −|d|−1ℓ−2
+P ,
+where ℓP is the Planck length. If d ≪ 1, this corresponds
+to sub-Planckain lengths and therefore we cannot trust
+our semi-classical analysis. Instead, if d ≫ 1, the QFT ef-
+fects cannot be neglected in this regime. For this specific
+equation of state, a different, earlier crossover is found
+if the energy densities are compared instead. However,
+the impact of the constant energy density on the metric
+is negligible compared to that of the diverging central
+pressure, in the Buchdahl limit.
+We will not address the problem of full backreaction in
+this work. Nevertheless, some general features of a lin-
+earized approximation provide useful insight. Consider
+the trace of the semi-classical equations (10),
+−R = 8πG (−ρ + 3p − ⟨ˆρ⟩ + ⟨ˆpr⟩ + 2⟨ˆpθ⟩) ,
+(20)
+evaluated at the origin, as we approach the Buchdahl
+limit.
+In the absence of the quantum corrections, the
+right-hand side of (20) diverges as ϵ−1 as shown in (14).
+However, as we see from (18) and (19), the quantum
+contributions of the last three terms scale as −dϵ−2.
+
+4
+If d < 0, the quantum terms on the right side of (20)
+grow without bound with the same sign as the classi-
+cal ones. This suggests a runaway: as the curvature in-
+creases, so do quantum effects, which increase the curva-
+ture further and so on. Conversely, if d > 0, the quantum
+contributions to the trace have the opposite sign, which
+decreases the curvature. This suggests the possible exis-
+tence of a backreacted solution, but only for d > 0. Such
+an equilibrium would require a small but finite ϵ of the
+order discussed above, so the surface of such an object
+would lie very close the Buchdahl radius, and far from
+the Schwarzschild radius.
+ROLE OF THE LIGHT RING
+Light rings (photon spheres) play a key role in our anal-
+ysis. These are defined as regions where null geodesics
+form circles, and they always come in pairs due to topo-
+logical arguments [8].
+For 9GM/4 < R ≤ 3GM, the
+above metric develops two light rings located at:
+rext = 3GM ,
+rint = 1
+3
+�
+R3
+GM
+4R − 9GM
+R − 2GM .
+(21)
+The outer ring rext, also present for black holes, corre-
+sponds to the usual photon sphere outside the surface of
+the star and is unstable: photons crossing it either es-
+cape to infinity or spiral inwards. It has been probed by
+recent observations [9–11].
+The inner ring rint lies in the interior and is a stable
+attractor of null geodesics, meaning that massless fields
+remain trapped around it. Notice that it shrinks to the
+origin in the Buchdahl limit. As illustrated in Fig. 2, the
+quantum stress tensor (12) is maximum at the center and
+falls steeply around the inner light ring. Indeed, in the
+Buchdahl limit the inner light ring sets the location at
+which the field values have dropped roughly by one order
+of magnitude, i.e.
+⟨ˆρ(rint)⟩
+⟨ˆρ(0)⟩
+∼ 0.1
+(22)
+and similarly for the pressures. This shows that the re-
+gion inside the inner photon sphere is where the quantum
+fields have most support, which is the quantum analogue
+to the classical trapping of modes discussed above us-
+ing the wave equation. The crossover when the classical
+and quantum pressures become comparable corresponds
+to an inner light ring of radius r ∼
+�
+|d| ℓP .
+The inner photon sphere plays yet another important
+role: it is the location where the Null Energy Condition
+(NEC) is violated. Given a null vector kµ, one defines an
+operator by contracting the (total) stress tensor with it
+NEC =
+�
+Tµν + ⟨ ˆTµν⟩
+�
+kµkν ,
+(23)
+where we have included both the classical and quantum
+contributions. For classical matter, one expects NEC ≥
+0, while it is well known that quantum fields can violate
+this.
+In the star’s interior, but far from the inner light ring,
+the NEC will be positive, since quantum effects there are
+negligible.
+In order to investigate the behavior of the
+NEC in the vicinity of the inner photon sphere as we
+approach the Buchdahl bound, we choose the null vector
+as kµ = (1, kr, 0, 0). We then compute (23) inside the
+star, in the limit ϵ → 0, keeping fixed the ratio r/rint.
+This yields
+NEC(r) =
+2d
+27π2G4R4
+r2
+int − r2
+r2
+int + r2 + O(ϵ) .
+(24)
+This is effectively ‘tracking’ the NEC in the region
+around the inner photon sphere as the configuration ap-
+proaches the Buchdahl bound, since rint → 0 in this limit.
+The NEC clearly changes sign at the light ring and is
+thus violated. Notice that the classical contribution is
+subdominant in this limit and is contained in the sub-
+leading orders. On the other hand, choosing kµ along
+the (t, φ) plane does not lead to a violation.
+The analysis above posits an interesting question. Sta-
+ble light rings have been recently associated with gravita-
+tional instabilities due to the existence of slowly decaying
+modes around it [12, 13], which would rule out ultra com-
+pact objects [8]. However, we have shown here that it is
+precisely this feature that enhances the quantum effects
+there, leading to the violation of energy conditions and
+to significant backreaction. Exploring this interaction at
+the non-linear level is an interesting direction.
+COMMENTS ON BUCHDAHL’S THEOREM.
+Buchdahl’s theorem relies on several assumptions as
+stated in the introduction. Our results show that QFT
+in curved spacetime violates two of these assumptions,
+namely isotropy of the matter and the effective equations
+of motion.
+As we have seen in (12) and is illustrated in Fig. 2, the
+renormalized vacuum stress tensor of the quantum fields
+is not isotropic, thus violating one of the assumptions of
+Buchdahl’s theorem. Anisotropic versions of Buchdahl’s
+bound exist but they require extra assumptions [14–18].
+These typically take the form of energy conditions, with
+the strength of the bound depending on the strength
+of the conditions. Here, we have shown that quantum
+fields violate energy conditions in the probe approxima-
+tion. We leave it for future work to examine whether the
+equations including backreaction violate the assumptions
+leading to these generalized theorems.
+Second, close to the compactness bound the relevant
+equations of motion to solve are (10), rather than the
+
+5
+classical Einstein equations.
+These differ by the pres-
+ence of the quantum source which, as we have shown,
+becomes the dominant term in the Buchdahl limit. This
+contribution depends explicitly on the curvature tensors,
+and therefore the differential equations to solve are of a
+different nature than the purely classical ones.
+This last feature has an alternative description in terms
+of quadratic gravity.
+For our specific background, we
+have shown that among the terms that determine ⟨ ˆTµν⟩
+in (11) and (12), only those controlled by d diverge faster
+than the classical Tµν as ϵ → 0. The ones associated with
+a diverge with the same power as the classical terms, but
+come with a coefficient that is very small for astrophysi-
+cal objects. Now as anticipated, d is a scheme-dependent
+parameter that can be generated by adding the countert-
+erm −
+d
+12(4π)2 R2 to the Lagrangian. This means that our
+results can also be interpreted as coming from quadratic
+corrections to Einstein’s gravity.
+The Weyl-flatness of
+the background, then, is not essential to find the leading
+terms of ⟨ ˆTµν⟩.
+This two-faced interpretation is akin to Starobinsky’s
+inflation [19], initially formulated in terms of the back-
+reaction of quantum fields, then as R2 gravity (in the
+Jordan frame) or Einstein gravity coupled to a scalar
+field (in the Einstein frame). In the latter picture, the
+stability of the scalar field requires the condition d > 0,
+the same we found and discussed earlier.
+It is worth noticing that Buchdahl’s theorem holds in
+a local form as
+r
+Gm(r) ≥ 9
+4, where the radius and mass
+of the star are replaced by an arbitrary coordinate ra-
+dius r and the Misner-Sharp mass m(r) = 4π
+� r
+0 dr r2ρ
+contained within it, provided the assumptions are met
+inside that sphere. For example, the star could consist of
+an incompressible dense core surrounded by an external
+crust obeying a softer equation of state. Our results also
+apply to this generalized scenario.
+Interesting recent work has also considered quantum
+fields in the Buchdahl limit [20–22] in the approxima-
+tion of a two-dimensional reduction. This corresponds
+to the s-wave (ℓ = 0) sector, and leaves the stress tensor
+undetermined up to an arbitrary function. Our results
+differ from theirs in that (12) fully captures the 3 + 1-
+dimensional features, leaving no functional freedom. For
+other applications of similar techniques see [23–25].
+SUMMARY
+We have investigated the universal behavior of QFT
+in the interior of very compact stars. A useful arena to
+probe this is the strong gravity regime close to Buch-
+dahl’s limit that, classically, sets an upper bound on the
+compactness of static, spherically symmetric spheres in
+General Relativity. As a proxy for this, we have worked
+with the constant-density Schwarzschild interior solution.
+Motivated by the trapping of classical waves in this
+metric close to Buchdahl’s limit, we have studied quan-
+tum fields propagating on this background in the approx-
+imation of no backreaction.
+Exploiting the conformal
+flatness of this solution, we have evaluated the full renor-
+malized stress tensor (12) for Conformal Field Theories.
+This depends on two coefficients a and d, the latter of
+which is not fixed by the theory in question.
+The vacuum renormalized stress tensor (17) is not
+isotropic, since the radial and angular pressures are dif-
+ferent. The sign of the energy density is opposite to that
+of the pressures. Its components acquire their maximum
+magnitude at the origin, and fall steeply around the inner
+light ring, as shown in (22).
+As we approach the Buchdahl limit, the d term of
+the renormalized stress tensor (18)-(19) diverges faster
+than the classical source (15)-(16), meaning that quan-
+tum fields respond stronger to changes in compactness
+than their classical counterpart.
+The crossover when
+classical and quantum contributions are of the same or-
+der happens when the proper radius of the inner light
+ring is rint ∼
+�
+|d|ℓP . The radial Null Energy Condition
+– including both classical and quantum contributions –
+changes sign at the inner photon sphere as shown in (24),
+and is thus violated inside the star. Whether the scales
+involved are Planckian or not depends on the value of d.
+If d ≪ 1, we cannot trust our semi-classical analysis. On
+the other hand, if d ≫ 1, the effects of the QFT cannot
+be ignored in this regime.
+We emphasize that the enhancement of quantum ef-
+fects discussed here happens as the surface of the star
+approaches the Buchdahl radius 9GM/4 instead of 2GM.
+Moreover, the effect of the quantum fields is localized in
+a small region around the center – the inner light ring
+– and not the surface. This is different from ultra com-
+pact objects close to the Schwarzschild radius. There,
+the renormalized stress tensor in the Boulware vacuum
+is well known to diverge at the surface as the star ap-
+proaches the black hole limit [26].
+The isotropy assumption used in Buchdahl’s theorem
+is violated by vacuum quantum fields. Whether the con-
+ditions leading to the anisotropic generalizations of this
+bound hold or not requires further investigation.
+We have not attempted to solve the semi-classical
+equations (10) here. Nevertheless, our results suggests
+that if d > 0, quantum fields act by decreasing the cur-
+vature, suggesting that a self-consistent solution to these
+equations might exist that avoids curvature singularities.
+It is intriguing to wonder whether quantum physics
+may play yet another, unexpected, role in the determi-
+nation of the maximum mass of compact stars.
+Acknowledgments.
+We thank Max Ba˜nados, Pablo
+Bosch, Alejandra Castro, Jan de Boer and Erik Verlinde
+for insightful discussions.
+We also thank Daniel Bau-
+mann and Vitor Cardoso for feedback on the manuscript.
+We are particularly grateful to Ben Freivogel for exten-
+sive discussions.
+
+6
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+

MNAyT4oBgHgl3EQf6vpX/content/tmp_files/load_file.txt

@@ -0,0 +1,382 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf,len=381
+page_content='Compact stars in Quantum Field Theory  Ignacio A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Reyes1 and Giovanni Maria Tomaselli2  1Institute for Theoretical Physics and  2GRAPPA,  University of Amsterdam, Amsterdam, 1098 XH, The Netherlands  Very compact stars seem to be forbidden in General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' While Buchdahl’s theorem sets  an upper bound on compactness, further no-go results rely on the existence of two light rings, the  inner of which has been associated to gravitational instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' However, little is known about  the role of quantum fields in these strong gravity regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Working in the probe approximation  where the backreaction is ignored, we show that the trapping of modes around the inner light ring  leads the renormalized stress tensor of Conformal Field Theories to diverge faster than the classical  source in the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' This leads to the violation of the Null Energy Condition, as well as  the isotropy assumption used in Buchdahl’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The backreaction of quantum fields in this  regime therefore cannot be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' This happens as the star’s surface approaches the Buchdahl  radius 9GM/4 rather than the Schwarzschild radius, with the quantum fields having support in  a small region around the center, becoming negligible at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' These are generic quantum  features and do not depend on the details of the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Our findings open a way for further  investigation into the role of QFT in astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  COMPACT RELATIVISTIC STARS  The General Relativistic prediction of the existence of  compact objects, such as white dwarfs and neutron stars,  has been confirmed by many observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Their macro-  scopic properties follow from the Tolman-Oppenheimer-  Volkoff equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' However, quantum theory is essential  in understanding the physics of these stars, as it provides  the ultimate reason for their existence, namely, Fermi’s  exclusion principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The question regarding the maximum mass of such  compact object is crucial: it is the main criterion used to  discriminate between what we suspect is a neutron star or  a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Well-known upper limits were set by Chan-  drasekhar [1] and Rhoades-Ruffini [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' A more generic re-  sult, that is independent of the equation of state of the  matter, was established by Buchdahl [3] and gives an up-  per bound on compactness in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Consider an isotropic  perfect fluid star, with stress tensor  T µ  ν = diag(−ρ, p, p, p) ,  (1)  on a static, spherically symmetric metric  ds2 = −f(r) dt2 + h(r) dr2 + r2(dθ2 + sin2 θ dφ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' (2)  Assuming in addition that ρ > 0, ∂rρ ≤ 0 and that  Einstein’s equations hold, the requirement that the met-  ric is everywhere regular leads to  R ≥ 9GM/4 ,  (3)  where R is the radius of the star and M is its mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The  saturation of the bound is known as the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  Notice one can also formulate this bound in a coordinate-  independent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  A particularly simple solution that manifestly sat-  urates Buchdahl’s second assumption is the constant-  density star or ‘Schwarzschild interior metric’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  These  configurations have uniform density ρ  =  3M/4πR3  throughout the star, and as is well known they can sat-  urate the Buchdahl limit (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Although they are unre-  alistic models of an astrophysical object, they are the  standard example when studying the TOV equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The metric for this equation of state takes the form  (2), with  f(r) =  �  3  2  �  1 − 2GM  R  − 1  2  �  1 − 2GMr2  R3  �2  ,  (4)  h(r) =  �  1 − 2GMr2  R3  �−1  ,  (5)  and is matched to the usual exterior Schwarzschild vac-  uum solution at the sphere’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The simplicity of these solutions make them an excel-  lent setup to test Quantum Field Theory (QFT) in the  strong gravity regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  A final motivation to consider  this metric is that it is conformally (Weyl) flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' In fact,  the uniform density metric above is the unique solution  to Einstein’s equations coupled to a static perfect fluid  that is conformally flat [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' This will allow us to ob-  tain explicit analytic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  We will thus work with  this spacetime, and comment about the generality of our  results later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  WAVE EQUATION  In order to understand the behaviour of quantum fields  in this spacetime, let us begin by first considering the  propagation of classical waves in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The wave equation  for the uniform density background was first discussed  by Chandrasekhar and Ferrari [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' For simplicity we take  a massless scalar Φ using the usual decomposition  Φ =  �  fωℓm ,  fωℓm(x) = u(r)  r  Yℓm(θ, φ)e−iωt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' (6)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='00826v1  [gr-qc]  2 Jan 2023  2  −30  −20  −10  0  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='15  r∗/(GM)  V/(GM)2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Potential (for ℓ = 1) given in (9), for R/(GM) =  9/4 (blue), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='3 (orange) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='4 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The dashed lines  mark the the values of r∗ corresponding to the centre of the  star in the two latter cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  A discontinuous jump at the  star’s surface matches it to the exterior vacuum Schwarzschild  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The wave equation □Φ  =  0 can be recast in a  Schr¨odinger-like form:  −∂2  r∗u + V (r∗)u = ω2u ,  (7)  where we defined the tortoise coordinate r∗ via  dr∗  dr =  �  h(r)/f(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  (8)  The potential V (r∗) takes the form  V = 1  r ∂2  r∗r + ℓ(ℓ + 1)  r2  f  (9)  and is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' 1 for ℓ = 1 and various values of  R/(GM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The potential at r > R corresponds to the  Schwarzschild vacuum metric, and vanishes at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' It  connects to the interior of the star with a discontinuous  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  As we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' 1, when R > 9GM/4, the  tortoise coordinate has a finite minimum possible value  (dashed lines) corresponding to the centre of the star,  because the factor h/f is always regular around r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  Moreover, V (r∗) reaches a local minimum greater than  zero and then increases towards the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  When R → 9GM/4, however, one has h/f ∼ r−2  and therefore the domain of r∗ becomes infinite on both  sides, while V (r∗) vanishes at the centre of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' This  closely resembles the situation for black holes, but in that  case it is the horizon that is mapped to r∗ → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The  field modes can thus be trapped inside the star, leading  to a spectrum of quasi-bound states whose magnitudes  are amplified close to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  These properties of the effective potential, together  with the behavior of the tortoise coordinate, suggests  that upon quantization the renormalized stress tensor  can become important in the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The rest  of our analysis will be done in a more generic way that  depends less on the specific theory considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  QFT IN CURVED SPACETIME  QFT in curved spacetime has seen significant progress  in the last half century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' In the semi-classical approxima-  tion, gravity is still treated classically and one considers  some quantum fields as another dynamical source to Ein-  stein’s equations,  Rµν − 1  2gµνR = 8πG  �  Tµν + ⟨ ˆTµν⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  (10)  We shall denote by ˆTµν the operator of the QFT to dis-  tinguish it from the classical source (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  However, most work in this field has focused on either  cosmology or black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Here, we will study the role  it plays for astrophysical compact stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The question  we will address in this work is whether there exists some  generic feature of QFT, independent of the details of  the nuclear interactions and the quantum states involved,  that becomes important for very compact stars, in the  regime of strong gravitational fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' We will show that  there is indeed such an effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  We shall focus on the effects of conformally coupled  fields where the computation is easier, taking it as a toy  model for more generic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' We work in 3+1 dimen-  sions, but the generalization to even higher dimensions is  straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The non-conformal case will be treated  elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  As is well known, conformally coupled classical mat-  ter has a vanishing trace of its stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' However,  its quantum counterpart develops a trace anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' In  3 + 1 dimensions, the vacuum expectation value of the  trace of the renormalized stress tensor for quantum fields  propagating in a curved spacetime is  ⟨ ˆT µ  µ ⟩ =  1  (4π)2 [cF − aG − d□R] ,  (11)  where R is the Ricci scalar, F is the square of the Weyl  tensor and G is the Gauss-Bonnet invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Amongst the  three real coefficients, c > 0 and a > 0 are well under-  stood and characterize the particular theory in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  On the other hand, d is not determined by the bare La-  grangian as it depends on the renormalization scheme,  and is closely related to the quadratic corrections to the  gravity action as we review below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' As such, it should  be fixed by experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' For now we will leave d as a  fixed but undetermined constant and proceed with the  calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  If additionally the metric is conformally flat – as is the  case for the constant-density star – then all components  3  of the renormalized stress tensor are fixed [7]:  ⟨ ˆT µν⟩ =  −  a  (4π)2  �  gµν  �R2  2 − RαβRαβ  �  + 2RµλRν  λ − 4  3RRµν  �  +  d  (4π)2  � 1  12gµν(R2 − 4R,λ  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='λ) − 1  3(RRµν − R,µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='ν)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  (12)  The quantum state chosen for (12) is the vacuum, but  this will not play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' It could be a state  at finite temperature or with a large number of fermions:  this would only add an extra contribution independent of  the curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The vacuum stress tensor for the interior  of the uniform density star is therefore given by (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' We  now proceed to evaluate it and examine its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  QUANTUM FIELDS IN THE BUCHDAHL LIMIT  In this section, we describe the main features of  the  quantum  stress  tensor  (12)  evaluated  on  the  Schwarzschild interior metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' In particular, we wish to  understand its behavior as we approach the Buchdahl  limit  R = (9/4 + ϵ)GM ,  ϵ → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  (13)  We will report the results to leading orders in ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The Buchdahl limit (13) is a finite distance above the  black hole compactness corresponding to R = 2GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  Nevertheless, this regime is no less extreme: the Ricci  scalar R of the background metric at the center diverges  in this limit as  R(0) = − 3  R2ϵ + O(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  (14)  Correspondingly, the central density and pressure of the  classical uniform density star solution behave as  ρ(0) =  1  3πGR2 + O(ϵ) ,  (15)  p(0) =  1  8πGR2ϵ + O(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  (16)  Let us contrast this behavior with its quantum coun-  terpart (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' For generic ϵ, this takes the form  ⟨ ˆT µ  ν ⟩ = diag(−⟨ˆρ⟩, ⟨ˆpr⟩, ⟨ˆpθ⟩, ⟨ˆpθ⟩) ,  (17)  with ⟨ˆpr⟩ ̸= ⟨ˆpθ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The radial dependence of the compo-  nents are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' In the limit ϵ → 0, their  central values scale as  ⟨ˆρ(0)⟩ =  9d  (8πR2ϵ)2 +  d  6(πR2)2ϵ + O(1) ,  (18)  ⟨ˆpr(0)⟩ = ⟨ˆpθ(0)⟩ = −  3d  (8πR2ϵ)2 +  2a − d  (3πR2)2ϵ + O(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' (19)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='5  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='2  r/(GM)  ⟨ˆρ⟩  ⟨ˆpr⟩  ⟨ˆpθ⟩  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  Radial profile of the three components of ⟨ ˆT µ  ν ⟩,  for a = d = 1/360, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='003, in units where GM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The  location of the inner light ring is depicted by the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  We emphasize that the pressures match only at the cen-  ter, and not elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Moreover, notice that the leading  order of ⟨ˆρ⟩ and ⟨ˆp⟩ have opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' There is no con-  tribution from c because the Weyl tensor vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  By comparing (15) and (16) with (18) and (19), we see  that the components of the renormalized stress tensor  scale with higher powers of ϵ than the classical contribu-  tions, and therefore cannot be ignored in the Buchdahl  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Furthermore, notice that the leading divergence of  the quantum terms depends only on d: in this regime the  quantum effects are dominated by the scheme-dependent  terms proportional to d, and not by c or a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The backreaction of quantum effects cannot be ne-  glected if they become of the same order as the clas-  sical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  By comparing the classical and quantum  central pressures (16) and (19), this crossover happens  at ϵ ∼ |d| (ℓP /R)2, which corresponds to a pressure  p ∼ |d|−1ℓ−4  P  and central curvature R ∼ −|d|−1ℓ−2  P ,  where ℓP is the Planck length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' If d ≪ 1, this corresponds  to sub-Planckain lengths and therefore we cannot trust  our semi-classical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Instead, if d ≫ 1, the QFT ef-  fects cannot be neglected in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' For this specific  equation of state, a different, earlier crossover is found  if the energy densities are compared instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' However,  the impact of the constant energy density on the metric  is negligible compared to that of the diverging central  pressure, in the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  We will not address the problem of full backreaction in  this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Nevertheless, some general features of a lin-  earized approximation provide useful insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Consider  the trace of the semi-classical equations (10),  −R = 8πG (−ρ + 3p − ⟨ˆρ⟩ + ⟨ˆpr⟩ + 2⟨ˆpθ⟩) ,  (20)  evaluated at the origin, as we approach the Buchdahl  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  In the absence of the quantum corrections, the  right-hand side of (20) diverges as ϵ−1 as shown in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  However, as we see from (18) and (19), the quantum  contributions of the last three terms scale as −dϵ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  4  If d < 0, the quantum terms on the right side of (20)  grow without bound with the same sign as the classi-  cal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' This suggests a runaway: as the curvature in-  creases, so do quantum effects, which increase the curva-  ture further and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Conversely, if d > 0, the quantum  contributions to the trace have the opposite sign, which  decreases the curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' This suggests the possible exis-  tence of a backreacted solution, but only for d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Such  an equilibrium would require a small but finite ϵ of the  order discussed above, so the surface of such an object  would lie very close the Buchdahl radius, and far from  the Schwarzschild radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  ROLE OF THE LIGHT RING  Light rings (photon spheres) play a key role in our anal-  ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' These are defined as regions where null geodesics  form circles, and they always come in pairs due to topo-  logical arguments [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  For 9GM/4 < R ≤ 3GM, the  above metric develops two light rings located at:  rext = 3GM ,  rint = 1  3  �  R3  GM  4R − 9GM  R − 2GM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  (21)  The outer ring rext, also present for black holes, corre-  sponds to the usual photon sphere outside the surface of  the star and is unstable: photons crossing it either es-  cape to infinity or spiral inwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' It has been probed by  recent observations [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The inner ring rint lies in the interior and is a stable  attractor of null geodesics, meaning that massless fields  remain trapped around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Notice that it shrinks to the  origin in the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' 2, the  quantum stress tensor (12) is maximum at the center and  falls steeply around the inner light ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Indeed, in the  Buchdahl limit the inner light ring sets the location at  which the field values have dropped roughly by one order  of magnitude, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  ⟨ˆρ(rint)⟩  ⟨ˆρ(0)⟩  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='1  (22)  and similarly for the pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' This shows that the re-  gion inside the inner photon sphere is where the quantum  fields have most support, which is the quantum analogue  to the classical trapping of modes discussed above us-  ing the wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The crossover when the classical  and quantum pressures become comparable corresponds  to an inner light ring of radius r ∼  �  |d| ℓP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The inner photon sphere plays yet another important  role: it is the location where the Null Energy Condition  (NEC) is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Given a null vector kµ, one defines an  operator by contracting the (total) stress tensor with it  NEC =  �  Tµν + ⟨ ˆTµν⟩  �  kµkν ,  (23)  where we have included both the classical and quantum  contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' For classical matter, one expects NEC ≥  0, while it is well known that quantum fields can violate  this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  In the star’s interior, but far from the inner light ring,  the NEC will be positive, since quantum effects there are  negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  In order to investigate the behavior of the  NEC in the vicinity of the inner photon sphere as we  approach the Buchdahl bound, we choose the null vector  as kµ = (1, kr, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' We then compute (23) inside the  star, in the limit ϵ → 0, keeping fixed the ratio r/rint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  This yields  NEC(r) =  2d  27π2G4R4  r2  int − r2  r2  int + r2 + O(ϵ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  (24)  This is effectively ‘tracking’ the NEC in the region  around the inner photon sphere as the configuration ap-  proaches the Buchdahl bound, since rint → 0 in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The NEC clearly changes sign at the light ring and is  thus violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Notice that the classical contribution is  subdominant in this limit and is contained in the sub-  leading orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' On the other hand, choosing kµ along  the (t, φ) plane does not lead to a violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The analysis above posits an interesting question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Sta-  ble light rings have been recently associated with gravita-  tional instabilities due to the existence of slowly decaying  modes around it [12, 13], which would rule out ultra com-  pact objects [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' However, we have shown here that it is  precisely this feature that enhances the quantum effects  there, leading to the violation of energy conditions and  to significant backreaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Exploring this interaction at  the non-linear level is an interesting direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  COMMENTS ON BUCHDAHL’S THEOREM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  Buchdahl’s theorem relies on several assumptions as  stated in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Our results show that QFT  in curved spacetime violates two of these assumptions,  namely isotropy of the matter and the effective equations  of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  As we have seen in (12) and is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' 2, the  renormalized vacuum stress tensor of the quantum fields  is not isotropic, thus violating one of the assumptions of  Buchdahl’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Anisotropic versions of Buchdahl’s  bound exist but they require extra assumptions [14–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  These typically take the form of energy conditions, with  the strength of the bound depending on the strength  of the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Here, we have shown that quantum  fields violate energy conditions in the probe approxima-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' We leave it for future work to examine whether the  equations including backreaction violate the assumptions  leading to these generalized theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  Second, close to the compactness bound the relevant  equations of motion to solve are (10), rather than the  5  classical Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  These differ by the pres-  ence of the quantum source which, as we have shown,  becomes the dominant term in the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' This  contribution depends explicitly on the curvature tensors,  and therefore the differential equations to solve are of a  different nature than the purely classical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  This last feature has an alternative description in terms  of quadratic gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  For our specific background, we  have shown that among the terms that determine ⟨ ˆTµν⟩  in (11) and (12), only those controlled by d diverge faster  than the classical Tµν as ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The ones associated with  a diverge with the same power as the classical terms, but  come with a coefficient that is very small for astrophysi-  cal objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Now as anticipated, d is a scheme-dependent  parameter that can be generated by adding the countert-  erm −  d  12(4π)2 R2 to the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' This means that our  results can also be interpreted as coming from quadratic  corrections to Einstein’s gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The Weyl-flatness of  the background, then, is not essential to find the leading  terms of ⟨ ˆTµν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  This two-faced interpretation is akin to Starobinsky’s  inflation [19], initially formulated in terms of the back-  reaction of quantum fields, then as R2 gravity (in the  Jordan frame) or Einstein gravity coupled to a scalar  field (in the Einstein frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' In the latter picture, the  stability of the scalar field requires the condition d > 0,  the same we found and discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  It is worth noticing that Buchdahl’s theorem holds in  a local form as  r  Gm(r) ≥ 9  4, where the radius and mass  of the star are replaced by an arbitrary coordinate ra-  dius r and the Misner-Sharp mass m(r) = 4π  � r  0 dr r2ρ  contained within it, provided the assumptions are met  inside that sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' For example, the star could consist of  an incompressible dense core surrounded by an external  crust obeying a softer equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Our results also  apply to this generalized scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  Interesting recent work has also considered quantum  fields in the Buchdahl limit [20–22] in the approxima-  tion of a two-dimensional reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' This corresponds  to the s-wave (ℓ = 0) sector, and leaves the stress tensor  undetermined up to an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Our results  differ from theirs in that (12) fully captures the 3 + 1-  dimensional features, leaving no functional freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' For  other applications of similar techniques see [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  SUMMARY  We have investigated the universal behavior of QFT  in the interior of very compact stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' A useful arena to  probe this is the strong gravity regime close to Buch-  dahl’s limit that, classically, sets an upper bound on the  compactness of static, spherically symmetric spheres in  General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' As a proxy for this, we have worked  with the constant-density Schwarzschild interior solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  Motivated by the trapping of classical waves in this  metric close to Buchdahl’s limit, we have studied quan-  tum fields propagating on this background in the approx-  imation of no backreaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  Exploiting the conformal  flatness of this solution, we have evaluated the full renor-  malized stress tensor (12) for Conformal Field Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  This depends on two coefficients a and d, the latter of  which is not fixed by the theory in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The vacuum renormalized stress tensor (17) is not  isotropic, since the radial and angular pressures are dif-  ferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The sign of the energy density is opposite to that  of the pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Its components acquire their maximum  magnitude at the origin, and fall steeply around the inner  light ring, as shown in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  As we approach the Buchdahl limit, the d term of  the renormalized stress tensor (18)-(19) diverges faster  than the classical source (15)-(16), meaning that quan-  tum fields respond stronger to changes in compactness  than their classical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The crossover when  classical and quantum contributions are of the same or-  der happens when the proper radius of the inner light  ring is rint ∼  �  |d|ℓP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' The radial Null Energy Condition  – including both classical and quantum contributions –  changes sign at the inner photon sphere as shown in (24),  and is thus violated inside the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Whether the scales  involved are Planckian or not depends on the value of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  If d ≪ 1, we cannot trust our semi-classical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' On  the other hand, if d ≫ 1, the effects of the QFT cannot  be ignored in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  We emphasize that the enhancement of quantum ef-  fects discussed here happens as the surface of the star  approaches the Buchdahl radius 9GM/4 instead of 2GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  Moreover, the effect of the quantum fields is localized in  a small region around the center – the inner light ring  – and not the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' This is different from ultra com-  pact objects close to the Schwarzschild radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' There,  the renormalized stress tensor in the Boulware vacuum  is well known to diverge at the surface as the star ap-  proaches the black hole limit [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  The isotropy assumption used in Buchdahl’s theorem  is violated by vacuum quantum fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Whether the con-  ditions leading to the anisotropic generalizations of this  bound hold or not requires further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  We have not attempted to solve the semi-classical  equations (10) here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content=' Nevertheless, our results suggests  that if d > 0, quantum fields act by decreasing the cur-  vature, suggesting that a self-consistent solution to these  equations might exist that avoids curvature singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  It is intriguing to wonder whether quantum physics  may play yet another, unexpected, role in the determi-  nation of the maximum mass of compact stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  We thank Max Ba˜nados, Pablo  Bosch, Alejandra Castro, Jan de Boer and Erik Verlinde  for insightful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  We also thank Daniel Bau-  mann and Vitor Cardoso for feedback on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
+page_content='  We are particularly grateful to Ben Freivogel for exten-  sive discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'}
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N9AzT4oBgHgl3EQfk_2S/content/tmp_files/2301.01541v1.pdf.txt

@@ -0,0 +1,573 @@
+Condensed Matter Physics, 2022, Vol. 25, No. 4, 43710: 1–9
+DOI: 10.5488/CMP.25.43710
+http://www.icmp.lviv.ua/journal
+Electric field induced polarization rotation in squaric
+acid crystals revisited
+A. P. Moina
+Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
+1 Svientsitskii St., 79011 Lviv, Ukraine
+Received July 10, 2022, in final form July 26, 2022
+Using the previously developed model we revisit the problem of the electric field induced polarization rotation
+in antiferroelectric crystals of squaric acid. We test an alternative set of the model parameters, according to
+which the dipole moments associated with the H2C4O4 groups are assumed to be parallel to the diagonals of
+the 𝑎𝑐 plane. The 𝑇-𝐸 phase diagrams and the polarization curves 𝑃(𝐸) for the fields directed along the 𝑎 axis
+and along one of the diagonals are considered. Comparison of the theoretical results with the newly published
+experimental data confirm the validity of the model. The calculations reveal no apparent advantage of the new
+set of the parameters over the previously used set.
+Key words: polarization, electric field, phase transition, phase diagram, squaric acid
+1. Introduction
+The squaric acid H2C4O4 is a classical two-dimensional antiferroelectric. The crystal is tetrago-
+nal, 𝐼4/𝑚, in the paraelectric phase and monoclinic, 𝑃21/𝑚, in the antiferroelectric phase. The hydrogen
+bonded C4O4 groups form sheets, parallel to the 𝑎𝑐 plane and stacked along the 𝑏-axis. Below the tran-
+sition at 373 K, a spontaneous polarization arises in these sheets, with the neighboring sheets polarized
+in the opposite directions [1–3].
+External electric fields applied to a uniaxial antiferroelectric can switch a sublattice polarization by
+180◦ and induce thereby the transition from antiferroelectric (AFE) to ferroelectric (FE) phase. The
+(pseudo)tetragonal symmetry of the squaric acid crystal lattice and of its hydrogen bond networks allows
+the sublattice polarizations to be directed along two perpendicular axes in the fully ordered system.
+As a result, here the external field can rotate one of the sublattice polarizations by 90◦, whereupon
+a noncollinear ferrielectric phase with perpendicular sublattice polarizations (NC90 [4]) is induced.
+The possibility of such a rotation has been suggested by Horiuchi et al [5], and their hysteresis loop
+measurements and Berry phase calculations gave evidence for it. Further calculations [6] indicated that
+the 90◦ rotation is possible at different orientations of the field within the 𝑎𝑐 plane. It is also predicted [5, 6]
+that application of higher fields along the diagonals of the 𝑎𝑐 plane can lead to the second rotation of the
+negative sublattice polarization by 90◦ and induction of the collinear ferroelectric phase.
+Recently [4, 7] we developed a deformable [8] two-sublattice proton-ordering model for a description
+of squaric acid behaviour in external electric fields, applied arbitrarily within the plane of hydrogen
+bonds. The model calculations confirm the two-step process of polarization reorientation [5, 6] at low
+temperatures, with the negative sublattice polarization being switched twice by 90◦ at each transition,
+for any orientation of the field within the 𝑎𝑐 plane, but a few exceptional directions. The exceptional
+directions are those, when the field is either i) collinear to the axes of the sublattice polarization in the AFE
+phase, or ii) directed at 45◦ to these axes. In the case i), the crystal behaves like a uniaxial antiferroelectric,
+undergoing a single-step polarization switching to the FE phase without the intermediate noncollinear
+phase, while in the case ii), the transition field from the NC90 to the FE phase goes to infinity, i.e., the
+transition never occurs [7].
+This work is licensed under a Creative Commons Attribution 4.0 International License. Further distribution
+of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
+43710-1
+arXiv:2301.01541v1  [cond-mat.mtrl-sci]  4 Jan 2023
+
+A. P. Moina
+The temperature-electric field phase diagrams of squaric acid were constructed [4, 7] for the field
+𝐸1(𝐸3) directed along the 𝑎(𝑐) tetragonal axis, for the fields denoted for brevity as 𝐸1 ± 𝐸3 and directed
+along the diagonals of the 𝑎𝑐 plane, as well as for the fields of the two above mentioned exceptional
+directions i) and ii). Note that the 𝑇-𝐸 diagrams are identical for the fields rotated by 90◦ around the 𝑏
+axis, because of the pseudotetragonal symmetry of the model [7].
+Experimentally, the low-temperature transition between the NC90 and FE phases has not been detected
+yet due to the dielectric breakdown of the samples. As follows from the model calculations [7], the field
+of this transition is the lowest when its direction is close to the axis of the sublattice polarization, so it is
+most likely to be experimentally observed at this field orientation.
+On the other hand, for the AFE-NC90 switching, the experimental data by Horiuchi et al [5] had
+been available, when our calculations were carried out. The polarization hysteresis curves at different
+temperatures for the field 𝐸1 had been measured, and the temperature dependence of the switching
+field had been deduced from those; for the field 𝐸1 + 𝐸3, the measurements had been performed for
+one temperature only [5]. With the fitting procedure for the model being based on the data [5] for the
+static dielectric permittivity, the obtained agreement between the theory and the experiment for the
+switching fields and for the 𝑃(𝐸) curves was only qualitative [4, 7]. Quantitatively, the agreement was
+conspicuously unsatisfactory, which led us to believe that the model used was not completely appropriate,
+and that essential modifications were required [7]. Quite recently, however, the same group of Horiuchi
+et al reported [9] the results of their new measurements of the polarization loops for the squaric acid
+crystals of an improved dielectric strength. This permitted to increase the maximum electric field that
+could be applied to the samples in the hysteresis experiments. Our preliminary calculations showed that
+the new experimental data were much closer to the predictions of the model [4, 7] than the previous data
+of [5], and that the doubts concerning the model validity were premature.
+It was extensively discussed in [4] that the accepted set of the values of the model parameters, in
+particular of the dipole moments assigned to the ground state configurations of the H2C4O4 groups,
+is not unique. While the magnitude of the dipole moment vector is constrained by the fitting to the
+permittivity [5], its orientation (and thereby the orientation of the ground state sublattice polarizations)
+can be varied within the 𝑎𝑐 plane. With the set of the model parameters adopted in [4, 7] these vectors
+are oriented at about 56◦ to the 𝑎(𝑐) axes. On the other hand, the Berry phase calculations [5, 9] indicate
+that the axes of the spontaneous sublattice polarization, in fact, are very close or even coincide with
+the diagonals of the 𝑎𝑐 plane. In terms of our model, this means that the crystallographic axes and the
+diagonals are the above mentioned exceptional directions: the axis 𝑎(𝑐) is the direction ii), while the
+diagonals of the 𝑎𝑐 plane are the direction i). The topology of the 𝑇-𝐸 diagrams and the shape of the
+𝑃(𝐸) curves for the fields 𝐸1(𝐸3) and 𝐸1 ± 𝐸3 will change accordingly. The availability of the new, more
+reliable experimental data [9] makes a quantitative comparison of theoretical and experimental 𝑃(𝐸)
+curves meaningful and could help to ascertain the orientation of the model dipole moment vectors.
+Thus, it seems worthwhile to revisit the problem of polarization rotation in squaric acid, to perform
+calculations with an alternative set of the model parameters, where the sublattice polarizations are
+oriented along the diagonals of the 𝑎𝑐 plane, and to compare the theoretical results with the most
+recent [9] experimental data. The model [4, 7], briefly described in section 2, is used without any further
+modification of the formulae. In section 3 the results of the theoretical calculations with the old and new
+sets of the model parameters are compared with the experimental data.
+2. The model
+The model has been introduced and explicated in [4], and a concise outline is given in [7]. Below
+we present a brief qualitative description of the model; all the formulae and other relevant details and
+discussions can be found in the mentioned papers.
+Protons on the hydrogen bonds in squaric acid move in double-well potentials, so each of the protons
+can occupy one of the two sites on the bond: closer to the given C4O4 group or to the neighboring
+group. The motion of protons is described by Ising pseudospins, whose two eigenvalues are assigned to
+two equilibrium positions of each proton. Two interpenetrating sublattices (layers) of pseudospins are
+considered.
+43710-2
+
+Electric field induced polarization rotation in squaric acid crystals revisited
+a)
+b)
+a
+Figure 1. (Colour online) a) The crystal structure of squaric acid as viewed along the 𝑏 axis. Two adjacent
+layers are shown, with black and open circles each. The A and B type C4O4 groups are indicated (see [4, 8]
+for explanation), and the hydrogen bonds are numbered, 𝑓 = 1, 2, 3, 4. b) The dipole moments assigned
+to one of the four lateral proton configurations (the configuration 1 in tables 1 in [4, 7]). Directions of
+the dipole moments associated with protons 𝝁𝐻
+1 = (2𝜇𝐻 , 0, 0) and with electrons 𝝁𝜋
+1 = (2𝜇𝜋
+∥ , 0, −2𝜇𝜋
+⊥)
+are shown with blue and red arrows, respectively; the green arrow is the total dipole moment of the
+configuration; the vector lengths are nominal. 𝜑0 = arctan(𝜇𝐻 + 𝜇𝜋
+∥ )/𝜇𝜋
+⊥ is the angle between the total
+dipole moment of configuration 1 and the 𝑐 axis. Figures are taken from [4, 7, 8, 10].
+The total system Hamiltonian [4, 7] includes ferroelectric intralayer long-range interactions between
+pseudospins, ensuring ferroelectric ordering within each separate layer, antiferroelectric interlayer inter-
+actions responsible for alternation of polarizations in the stacked layers, and the short-range interactions,
+which include also the coupling with external electric fields 𝐸1 and 𝐸3 directed along the tetragonal
+(paraelectric) 𝑎 and 𝑐 axes of the crystal.
+The short-range Hamiltonian describes the four-particle configurational correlations between protons
+placed around each C4O4 group. The usual Slater-Takagi type scheme [4, 8, 11, 12] of 16 degenerate
+levels of lateral/diagonal/single-ionized/double-ionized proton configurations is assumed. The lateral and
+single-ionized configurations have dipole moments in the 𝑎𝑐 plane; the degeneracy of their energy levels
+is removed by the electric fields 𝐸1 and 𝐸3, which break the equivalence of the hydrogen bonds that link
+the C4O4 groups along the 𝑎 and 𝑐 axes (see tables 1 in [4, 7]).
+Assignment of the dipole moments to the ground-state lateral configurations is the crucial point of the
+model. We rely on the results of the Berry phase calculations [5], which have shown that the ground-state
+sublattice polarization in this crystal is formed directly by displacements of protons along the hydrogen
+bonds and, mostly, by the electronic contributions of switchable 𝜋-bond dipoles.
+Positions of the 𝜋-bonds are determined by the proton arrangement around the given C4O4 group: in
+the lateral configurations the 𝜋-bond is formed between the two neighboring carbons, near which protons
+sit on the hydrogen bonds (see fig. 1b), and also between the carbons and adjacent to them oxygens, next
+to which there is no proton (meaning that the protons on these H-bonds sit in the minima close to the
+neighboring C4O4 groups). The field-induced polarization rotation by 90◦ or 180◦ occurs via flipping
+of one or two protons in each molecule to the other sites along the same hydrogen bonds and via a
+simultaneous switching of the 𝜋-bonds. For the depicted in figure 1b lateral proton configuration, the
+vector of the proton contribution to the dipole moment is oriented along the 𝑎 axis, while the electronic
+contribution is at the angle to this axis. The dipole moments of the three remaining lateral configurations
+can be obtained from the scheme of figure 1b by rotation by a multiple of 90◦.
+After going from the representation of proton configuration energies to the pseudospin representation,
+the four-particle cluster approximation for the obtained short-range Hamiltonian is employed. The mean
+field approximation is used for the long-range interlayer and intralayer interactions [4, 8]. The dependence
+of all proton-proton interaction parameters on the diagonal components of the lattice strain tensor and
+on the H-site distance, which are changed by the thermal expansion and potentially by an external stress
+if such is applied, is taken into account [8]. The expression for the thermodynamic potential has been
+obtained [4]; the order parameters and lattice strains are found by numerical minimization thereof.
+43710-3
+
+H
+c
+02Po,H
+μiA. P. Moina
+The values of all model parameters were chosen earlier [4, 7, 8]. In particular, they were required [8]
+to provide the best fit to the experimental temperature curves of the order parameter at ambient pressure,
+to the temperature and hydrostatic pressure dependences of the diagonal lattice strains, and to the pressure
+dependence of the transition temperature 𝑇N in squaric acid.
+The dielectric characteristics and other electric field effects in our model are mostly governed by
+values of the dipole moments, which enter the final expressions only via the sum 𝜇𝐻 + 𝜇𝜋
+∥ and via 𝜇𝜋
+⊥.
+These values are found by fitting the calculated curve of the static dielectric permittivity 𝜀11 at zero
+external bias field to the experimental points of [5], while trying to get the best possible agreement
+with the experiment for the values of the switching fields, corresponding to the first 90◦ rotation of the
+sublattice polarization by the field 𝐸1. It can be shown that in the paraelectric phase 𝜀11 ∼ ¯𝜇2, where
+¯𝜇 =
+√︃
+(𝜇𝐻 + 𝜇𝜋
+∥ )2 + (𝜇𝜋
+⊥)2
+is half the magnitude of the dipole moment, assigned to the H2C4O4 groups. It means that above 𝑇N the
+permittivity 𝜀11 at zero field is determined by the magnitude of the dipole moment vector only, whereas
+the orientation of the vector within the 𝑎𝑐 plane can be varied. For the set, adopted in [4, 7] and presented
+in table 1 as the set A, the dipole moment and the ground state sublattice polarization are oriented at the
+angle 𝜑0 = arctan(𝜇𝐻 + 𝜇𝜋
+∥ )/𝜇𝜋
+⊥ ≈ 56◦ to the crystallographic axes. However, the results of the Berry
+phase calculations [9] indicate that the angle should be closer to 45◦. Thus, we find an alternative set of
+the dipole moment values with 𝜇𝐻 + 𝜇𝜋
+∥ = 𝜇𝜋
+⊥ and with the same ¯𝜇 as in the set A, which yields the same
+fit to the permittivity in the paraelectric phase; this is the set B in table 1. In the next section, using the
+set B, we construct the 𝑇-𝐸 phase diagrams and explore the 𝑃(𝐸) curves for the electric fields 𝐸1 and
+𝐸1 + 𝐸3. The results are compared with the previous calculations [7] performed with the set A, as well
+with the experimental data of [5, 9].
+Table 1. The adopted values of the model dipole moments. The set A is taken from [7]. The values of all
+other model parameters are the same as in [4, 7, 8].
+𝜇𝐻 + 𝜇𝜋
+∥
+𝜇𝜋
+⊥
+¯𝜇
+(10−29 C m)
+set A
+3.16
+2.12
+3.8
+set B
+2.66
+2.66
+3.8
+3. Calculations
+3.1. Phase diagrams
+In figure 2 we redraw the 𝑇-𝐸 diagrams of squaric acid for the fields 𝐸1 and 𝐸1 + 𝐸3, obtained earlier
+in [4, 7] along with the newly available experimental points of [9]. Here, the set A of the dipole moment
+values was used in the calculations. The diagrams overlap the color gradient plots of the introduced in [4]
+noncollinearity angle 𝜃, which is the angle between the vectors of the sublattice polarizations.
+Different phases in the diagrams are separated by the lines of the first order phase transitions I, II,
+and III, and of the second order phase transitions IV. All these lines terminate at various critical points
+(bicritical end points BCE, tricritical point TCP, critical end points CEP). Some of the critical points can
+be artifacts of the mean field approximation, used for the long-range interactions. This was discussed
+extensively in [4, 7]; we shall not dwell on this here. The phase denoted as AFE* (the red region) is
+non-collinear antiferrielectric, very close to the initial AFE phase with 𝜃 ∼ 180◦. The purple region is the
+collinear field-induced ferroelectric phase (FE) with 𝜃 = 0. The phase between the transition lines II, III,
+and IV (green and blue) is the noncollinear ferrielectric phase NC90, where 𝜃 mostly remains close to 90◦,
+only rapidly decreasing to zero near the second-order phase transition line IV. In the region NC135*
+43710-4
+
+Electric field induced polarization rotation in squaric acid crystals revisited
+250
+300
+350
+400
+450
+0
+200
+400
+600
+800
+1000
+1200
+V
+NC135*
+I
+BCE1
+CEP
+TCP
+BCE2
+IV
+III
+ 
+ 
+T  (K)
+E1 (kV/cm)
+FE
+NC90
+0
+1
+11
+22
+33
+45
+56
+67
+78
+90
+107
+115
+120
+125
+130
+146
+157
+169
+180
+AFE*
+θ  (deg)
+II
+300
+350
+400
+200
+400
+V
+NC135*
+ 
+ 
+FE
+NC90
+I
+IV
+III
+CEP2
+CEP1
+BCE3
+BCE1
+T  (K)
+E1+E3 (kV/cm)
+BCE2
+II
+AFE*
+Figure 2. (Colour online) The 𝑇-𝐸 phase diagrams of the squaric acid, overlapping the 𝑇-𝐸 color
+contour plots of the noncollinearity angle 𝜃. The set A is used in calculations. Solid and dashed lines
+indicate the first and second order phase transitions, respectively; dotted lines are the supercritical lines,
+corresponding to the loci of maxima in the field dependences of d𝑃(𝐸)/d𝐸. The open squares □, star �,
+and full circles • indicate the critical end points (CEP), tricritical point (TCP), and bicritical end points
+(BCE), respectively. Blue full triangles ▲, ▶, and ▼ are the experimental points of [5], the electronic
+supplementary material thereto, and [9], respectively.
+(orange to yellow), a crossover between the AFE* and NC90 phases occurs. Here, 𝜃 changes gradually
+from ∼ 180◦ to ∼ 90◦: the negative sublattice polarization rotates continuously with increasing field and
+becomes perpendicular to the positive sublattice polarization. As discussed in [4, 7], this continuous
+rotation is a statistically averaged effect, possible only in presence of thermal fluctuations.
+Crossovers are often marked by the lines formed by the loci of the extrema of the response functions —
+second derivatives of the thermodynamic potentials. Those supercritical lines are continuations of the
+first order transition lines beyond the critical points terminating them. The major drawback of this method
+is that the extrema of different response functions yield different supercritical lines; moreover, the super-
+critical lines formed by the extrema of the same response function taken along different thermodynamic
+paths (e.g., isotherms or isofields) are different as well (see [13]). In order to compare the theory and the
+experimental data derived from the field dependence of polarization, we mark the crossover between the
+AFE* and NC90 phases using the lines formed by the maxima of the d𝑃(𝐸)/d𝐸 isotherms (the inflection
+points of the 𝑃(𝐸) isotherms), where 𝑃 is the projection of the net polarization vector on the field axis.
+These are the dotted lines V in the phase diagrams.
+As one can see in the left-hand panel of figure 2, for the field 𝐸1, the most recent data obtained in [9]
+for the sample with the improved dielectric strength appear to be in a much better agreement with the
+theory than the earlier experimental data of [5]. The theoretical switching fields, calculated with the set
+A, are much higher than the experimental values of [5] with the relative error 𝜂 = 1 − 𝐸exp/𝐸theor ≈ 0.42
+at room temperature (295 K). The error decreases down to ≈ 0.23 for the experimental points of [9],
+which is still not quite satisfactory, but evidently much better.
+In the case of 𝐸1 + 𝐸3, the switching fields calculated with the set A are higher than predicted for
+the field 𝐸1. This is in a qualitative agreement with all available experimental observations [5, 9]. The
+relative errors are about 0.3 for [5] at 324 K and 0.22 for [9] at 295 K, that is, the improvement here is
+not so striking.
+Now let us see how the situation changes, when the set B is used in calculations. For this set, the
+ground state spontaneous polarization axis is oriented along the diagonal of the 𝑎𝑐 plane. It means that
+the fields 𝐸1 + 𝐸3 and 𝐸1 are directed along this axis and at 45◦ to it, respectively, that is, along the
+exceptional directions i) and ii), discussed in Introduction. It is then expected that the 𝑇-𝐸 diagrams will
+be topologically different from those, depicted in figure 2. For the field 𝐸1 + 𝐸3, the crystal of squaric
+acid should behave like a uniaxial antiferroelectric, exhibiting a one-step polarization rotation by 180◦
+43710-5
+
+A. P. Moina
+without the intermediate noncollinear phase. For the field 𝐸1, the field of switching to the ferroelectric
+phase is expected to tend to infinity, and only the AFE*-NC90 transition can be observed.
+300
+400
+100
+200
+300
+400
+T  (K)
+I
+IV
+CEP
+BCE1
+FE
+NC90
+ 
+ 
+0
+1
+11
+22
+33
+45
+56
+67
+78
+90
+107
+115
+120
+135
+146
+157
+169
+180
+E1 (kV/cm)
+AFE*
+BCE2
+II
+θ (deg)
+250
+300
+350
+100
+200
+300
+400
+FI
+E1+E3 (kV/cm)
+IV
+I
+ 
+FE
+TP
+BCE
+TCP2
+T (K)
+TCP1
+AFE*
+III
+V
+VII
+238
+240
+296
+300
+304
+VI
+VII
+III
+BCE
+AFE*
+ 
+ 
+ 
+FE
+TP
+III
+V
+FI
+Figure 3. (Colour online) The same as in figure 2. The set B is used in calculations. The open triangle
+△ indicates the triple point TP. The dash-dotted line VII corresponds to the loci of minima in the field
+dependences of d𝑃(𝐸)/d𝐸. The other notations are the same as in figure 2.
+The 𝑇-𝐸 phase diagrams, calculated with the set B and presented in figure 3, are indeed in a total
+agreement with the above described picture. As the magnitude of the dipole moment 2 ¯𝜇 is the same
+for both sets, these diagrams are numerically identical to those obtained in [7] with the set A for
+the exceptional directions ii) and i), respectively (see figures 8, 9 in [7]). This identity can be proved
+algebraically, using the expression for the thermodynamic potential of the system [4, 7].
+For the field 𝐸1, the positions of the lines II of the AFE*-NC90 phase transitions, calculated with
+the sets A and B, are very close but not the same (c.f. the left-hand panels in figures 2 and 3). The
+closeness can be explained by the found in [7] dependence of this switching field at low temperatures on
+the orientation of the spontaneous sublattice polarization axis 𝐸 𝐼 𝐼 ∼ 1/cos(𝛿𝜑 − π/4), where 𝛿𝜑 is the
+angle between this axis and the external field 𝐸. Since 𝛿𝜑 for the sets A and B differ by about 11◦ only,
+the difference between the corresponding switching fields is small as well. It is then trivial to say that for
+𝐸1, the sets A and B yield about the same agreement with the experimental data for the switching field.
+For the field 𝐸1 + 𝐸3, the intermediate phase NC90 is absent, and 𝜃 is always either 180◦ or zero (see
+the right-hand panel of figure 3), i.e., all phases are collinear. The polarization switching occurs either as
+a first order phase transition across lines III directly to the FE phase and across line VI to an intermediate
+collinear ferrielectric phase FI, or gradually. In the latter case, the magnitude of one of the sublattice
+polarizations decreases down to zero with increasing field, changes its sign continuously at line VII, and
+then increases until the second order transition to the FE phase occurs at line IV. Interestingly, line VII,
+where the angle 𝜃 changes from 180◦ to 0, is formed by the loci of the minima of the d𝑃(𝐸)/d𝐸 isotherms,
+as opposed to line V, formed by the loci of the maxima. Line VI, emanating from the critical point BCE
+(see the inset in figure 3), marks the crossover between AFE* and FI phases. It is to be compared with
+the experimental data for the switching fields, and it yields nearly the same agreement as the set A, with
+the relative errors about 0.3 for [5] at 324 K and 0.23 for [9].
+3.2. Polarization
+In figure 4 we plot the field dependences of the projections of the net polarization vector on the field
+direction for the fields 𝐸1 and 𝐸1 + 𝐸3. The experimental points of [5] and [9] are also presented. The
+drastic changes in the experimental hysteresis curves, brought by the improvement of the sample quality
+and by the increase of the maximum value of the applied field in [9], are very well seen. It is obvious that
+the comparison of the theoretical polarization curves with the earlier data of [5] could be only qualitative.
+As one can see, for 𝐸1, the sets A and B predict three and two plateaus of polarization, respectively.
+43710-6
+
+Electric field induced polarization rotation in squaric acid crystals revisited
+100
+1000
+0
+10
+20
+30
+40
+200
+400
+600
+0
+10
+20
+30
+40
+III
+  set A
+  set B
+E1 (kV/cm)
+P1 (µC/cm2)
+II
+V
+IV
+E1+E3 (kV/cm)
+P (µC/cm2)
+Figure 4. (Colour online) The field dependences of polarizations at 295 K. Full triangles: experimental
+points taken from [5] (▲) and [9] (▼). The arrows indicate phase transitions of the first order across lines
+II, III (left-hand) and of the second order across lines IV (right-hand). The arrow and full circles (•)
+indicate the crossovers at lines V (right-hand). Lines II-V are from the 𝑇-𝐸 phase diagrams, figures 2, 3.
+In the physically reasonable field range, which includes the AFE*-NC90 first order phase transition (at
+lines II from the phase diagrams 2, 3), the two sets yield very similar polarizations. The calculated
+polarization jumps are 17.9 µC/cm2 for the set A and 20.4 µC/cm2 for the set B, in a fair agreement
+with the experimental 17.2 µC/cm2 [9]. The set A also predicts the second step of polarization at a much
+higher field, at the transition to the FE phase (across line III from the phase diagram, figure 2). It seems
+unlikely, however, that the field of such high a magnitude could ever be applied in an experiment without
+the squaric acid samples suffering the dielectric breakdown.
+For the field 𝐸1 + 𝐸3, the two sets of the model parameters yield different behaviour of polarization
+even at experimentally accessible fields. The 𝑃(𝐸) curve, calculated with the set A, has three smeared
+plateaus, with a clear rounded step, corresponding to the AFE*- NC90 crossover across line V, and then
+a cusp at the NC90-FE second order transition across line IV. The lower part of this curve, albeit being
+shifted to higher fields, is in a good qualitative and quantitative agreement with the experimental points.
+The polarization, calculated with the set B, on the other hand, has only two smeared plateaus: at low
+fields and above the cusp at line IV. No pronounced intermediate plateau is seen. The change of concavity
+at the inflection point, marked by a full circle in the figure, is hardly discernible. The agreement with
+the experiment is visibly worse than for the set A. However, the switching field magnitude for 𝐸1 + 𝐸3 is
+predicted [5, 7, 9] to be higher than for the field 𝐸1. It means that, despite the increased dielectric strength
+of the samples, the maximum applied fields 𝐸1 +𝐸3 [9] could still be insufficient to obtain correct data for
+polarization. A potential further improvement of the crystal quality (if such is still possible) may change
+the measured values of polarization and switching field for the diagonally directed field in the same way,
+as such an improvement did in the case of the field 𝐸1 in [9] as compared to [5], which is well illustrated
+in the left-hand panel of figure 4. Then, the agreement with the theoretical curves can be reexamined.
+4. Concluding remarks
+Using the previously developed [4] deformable two-sublattice proton ordering model, we revisit
+the problem of polarization rotation in antiferroelectric crystals of squaric acid under the influence
+of external electric fields. The unique structure of the two-dimensional hydrogen bond networks in
+squaric acid permits 90◦ rotation of the sublattice polarization. The model predicts [4, 7] that except
+for some particular directions of the field, the polarization reorientation at low temperatures is a two-
+step process: first, to the noncollinear phase with perpendicular sublattice polarizations and then to the
+collinear ferroelectric phase. However, when the field is directed along the axis of spontaneous sublattice
+polarizations, the intermediate noncollinear phase is absent; when the field is at 45◦ to this axis, the field
+of the transition to the ferroelectric phase tends to infinity.
+The previously obtained 𝑇-𝐸 phase diagrams and newly calculated polarization curves are compared
+43710-7
+
+A. P. Moina
+with the most recent experimental data [9], measured using the crystal samples of the increased dielectric
+strength. We also test an alternative set of the model parameters, for which the dipole moments assigned
+to the H2C4O4 groups are of the same magnitude as in the previous calculations, but oriented along the
+diagonals of the 𝑎𝑐 plane.
+The new experimental data [9] are in a drastically better agreement with the theory than the earlier
+results [5], especially for the polarization curves, as well as for the switching fields. It shows that the
+simplicity of the model was not the major reason of the earlier [4, 7] disagreement between theory and
+experiment and gives a strong evidence for the model validity.
+Results of testing the new set of the model parameters are inconclusive. Overall, the comparison of the
+theoretical polarization curves with the experimental data seems to slightly favor the previous set [4, 7],
+according to which the axes of the spontaneous sublattice polarization are close, but not exactly parallel
+to the diagonals of the 𝑎𝑐 plane. Further experimental studies may shed some light on this problem.
+As far as a further verification of the model is concerned, the appropriateness of the mean field
+approximation, used for the long-range interactions, may be addressed. This approximation is, most
+likely, the origin of the artifact splitting [4, 7] of some tricritical points in the 𝑃-𝐸 phase diagrams into
+the systems of bicritical and critical endpoints and also of the appearance of the intermediate FI phase,
+seen in the right-hand panels of figures 2, 3. Monte Carlo calculations may be used to construct more
+accurate diagrams.
+References
+1. Semmingsen D., Tun Z., Nelmes R. J., McMullan R. K., Koetzle T. F., Z. Kristallogr. Cryst. Mater., 1995, 210,
+934–947, doi:10.1524/zkri.1995.210.12.934.
+2. Semmingsen D., Hollander F. J., Koetzle T. F., J. Chem. Phys., 1977, 66, 4405–4412, doi:10.1063/1.433745
+3. Hollander F. J., Semmingsen D., Koetzle T. F., J. Chem. Phys., 1977, 67, 4825–4831, doi:10.1063/1.434686.
+4. Moina A. P., Phys. Rev. B, 2021, 103, 214104, doi:10.1103/PhysRevB.103.214104.
+5. Horiuchi S., Kumai R., Ishibashi S., Chem. Sci., 2018, 9, 425–432, doi:10.1039/C7SC03859C.
+6. Ishibashi S., Horiuchi S., Kumai R., Phys. Rev. B, 2018, 97, 184102, doi:10.1103/PhysRevB.97.184102.
+7. Moina A. P., Condens. Matter Phys., 2021, 24, No. 4, 43703, doi:10.5488/CMP.24.43703.
+8. Moina A. P., Condens. Matter Phys., 2020, 23, No. 3, 33704, doi:10.5488/CMP.23.33704.
+9. Horiuchi S., Ishibashi S., Chem. Phys., 2021, 12, 14198–14206, doi:10.1039/d1sc02729h.
+10. Semmingsen D., Feder J., Solid State Commun., 1974, 15, 1369–1372, doi:10.1016/0038-1098(74)91382-9.
+11. Matsushita E., Yoshimitsu K., Matsubara T., Progr. Theor. Phys., 1980, 64, No. 4, 1176–1192,
+doi:10.1143/PTP.64.1176.
+12. Matsushita E., Matsubara T., Progr. Theor. Phys., 1982, 68, No. 6, 1811–1826, doi:10.1143/PTP.68.1811.
+13. Schienbein P., Marx D., Phys. Rev. E, 2018, 98, 022104, doi:10.1103/PhysRevE.98.022104.
+43710-8
+
+Electric field induced polarization rotation in squaric acid crystals revisited
+Ще раз про обертання поляризацiї електричним полем в
+кристалах квадратної кислоти
+А. П. Моїна
+Iнститут фiзики конденсованих систем Нацiональної академiї наук України
+79011, м. Львiв, вул. Свєнцiцького, 1
+З використанням запропонованої ранiше моделi розглядаються процеси обертання поляризацiї зовнiшнi-
+ми електричними полями в антисегнетоелектричних кристалах квадратної кислоти. Обчислення також
+проводяться з альтернативним набором параметрiв теорiї, в якому дипольнi моменти, якi приписуються
+групам H2C4O4, паралельнi до дiагоналей площини 𝑎𝑐. Дослiджено фазовi дiаграми 𝑇-𝐸 та кривi поля-
+ризацiї 𝑃(𝐸) для полiв, прикладених уздовж осi 𝑎 та уздовж дiагоналi площини 𝑎𝑐. Порiвняння теорети-
+чних результатiв з нещодавно опублiкованими експериментальними даними пiдтверджує правильнiсть
+запропонованої моделi. Не виявлено суттєвої переваги нового набору параметрiв моделi перед тим, що
+використовувався в попереднiх розрахунках.
+Ключовi слова: поляризацiя, електричне поле, фазовий перехiд, антисегнетоелектрик, фазова дiаграма,
+квадратна кислота
+43710-9
+
+

N9AzT4oBgHgl3EQfk_2S/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf,len=388
+page_content='Condensed Matter Physics, 2022, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' 25, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' 4, 43710: 1–9  DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='5488/CMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='43710  http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='icmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='lviv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='ua/journal  Electric field induced polarization rotation in squaric  acid crystals revisited  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Moina  Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,  1 Svientsitskii St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=', 79011 Lviv, Ukraine  Received July 10, 2022, in final form July 26, 2022  Using the previously developed model we revisit the problem of the electric field induced polarization rotation  in antiferroelectric crystals of squaric acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' We test an alternative set of the model parameters, according to  which the dipole moments associated with the H2C4O4 groups are assumed to be parallel to the diagonals of  the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The 𝑇-𝐸 phase diagrams and the polarization curves 𝑃(𝐸) for the fields directed along the 𝑎 axis  and along one of the diagonals are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Comparison of the theoretical results with the newly published  experimental data confirm the validity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The calculations reveal no apparent advantage of the new  set of the parameters over the previously used set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  Key words: polarization, electric field, phase transition, phase diagram, squaric acid  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Introduction  The squaric acid H2C4O4 is a classical two-dimensional antiferroelectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The crystal is tetrago-  nal, 𝐼4/𝑚, in the paraelectric phase and monoclinic, 𝑃21/𝑚, in the antiferroelectric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The hydrogen  bonded C4O4 groups form sheets, parallel to the 𝑎𝑐 plane and stacked along the 𝑏-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Below the tran-  sition at 373 K, a spontaneous polarization arises in these sheets, with the neighboring sheets polarized  in the opposite directions [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  External electric fields applied to a uniaxial antiferroelectric can switch a sublattice polarization by  180◦ and induce thereby the transition from antiferroelectric (AFE) to ferroelectric (FE) phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The  (pseudo)tetragonal symmetry of the squaric acid crystal lattice and of its hydrogen bond networks allows  the sublattice polarizations to be directed along two perpendicular axes in the fully ordered system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  As a result, here the external field can rotate one of the sublattice polarizations by 90◦, whereupon  a noncollinear ferrielectric phase with perpendicular sublattice polarizations (NC90 [4]) is induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  The possibility of such a rotation has been suggested by Horiuchi et al [5], and their hysteresis loop  measurements and Berry phase calculations gave evidence for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Further calculations [6] indicated that  the 90◦ rotation is possible at different orientations of the field within the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' It is also predicted [5, 6]  that application of higher fields along the diagonals of the 𝑎𝑐 plane can lead to the second rotation of the  negative sublattice polarization by 90◦ and induction of the collinear ferroelectric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  Recently [4, 7] we developed a deformable [8] two-sublattice proton-ordering model for a description  of squaric acid behaviour in external electric fields, applied arbitrarily within the plane of hydrogen  bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The model calculations confirm the two-step process of polarization reorientation [5, 6] at low  temperatures, with the negative sublattice polarization being switched twice by 90◦ at each transition,  for any orientation of the field within the 𝑎𝑐 plane, but a few exceptional directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The exceptional  directions are those, when the field is either i) collinear to the axes of the sublattice polarization in the AFE  phase, or ii) directed at 45◦ to these axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' In the case i), the crystal behaves like a uniaxial antiferroelectric,  undergoing a single-step polarization switching to the FE phase without the intermediate noncollinear  phase, while in the case ii), the transition field from the NC90 to the FE phase goes to infinity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=', the  transition never occurs [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Further distribution  of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  43710-1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='01541v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='mtrl-sci]  4 Jan 2023  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Moina  The temperature-electric field phase diagrams of squaric acid were constructed [4, 7] for the field  𝐸1(𝐸3) directed along the 𝑎(𝑐) tetragonal axis, for the fields denoted for brevity as 𝐸1 ± 𝐸3 and directed  along the diagonals of the 𝑎𝑐 plane, as well as for the fields of the two above mentioned exceptional  directions i) and ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Note that the ��-𝐸 diagrams are identical for the fields rotated by 90◦ around the 𝑏  axis, because of the pseudotetragonal symmetry of the model [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  Experimentally, the low-temperature transition between the NC90 and FE phases has not been detected  yet due to the dielectric breakdown of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' As follows from the model calculations [7], the field  of this transition is the lowest when its direction is close to the axis of the sublattice polarization, so it is  most likely to be experimentally observed at this field orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  On the other hand, for the AFE-NC90 switching, the experimental data by Horiuchi et al [5] had  been available, when our calculations were carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The polarization hysteresis curves at different  temperatures for the field 𝐸1 had been measured, and the temperature dependence of the switching  field had been deduced from those;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' for the field 𝐸1 + 𝐸3, the measurements had been performed for  one temperature only [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' With the fitting procedure for the model being based on the data [5] for the  static dielectric permittivity, the obtained agreement between the theory and the experiment for the  switching fields and for the 𝑃(𝐸) curves was only qualitative [4, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Quantitatively, the agreement was  conspicuously unsatisfactory, which led us to believe that the model used was not completely appropriate,  and that essential modifications were required [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Quite recently, however, the same group of Horiuchi  et al reported [9] the results of their new measurements of the polarization loops for the squaric acid  crystals of an improved dielectric strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' This permitted to increase the maximum electric field that  could be applied to the samples in the hysteresis experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Our preliminary calculations showed that  the new experimental data were much closer to the predictions of the model [4, 7] than the previous data  of [5], and that the doubts concerning the model validity were premature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  It was extensively discussed in [4] that the accepted set of the values of the model parameters, in  particular of the dipole moments assigned to the ground state configurations of the H2C4O4 groups,  is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' While the magnitude of the dipole moment vector is constrained by the fitting to the  permittivity [5], its orientation (and thereby the orientation of the ground state sublattice polarizations)  can be varied within the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' With the set of the model parameters adopted in [4, 7] these vectors  are oriented at about 56◦ to the 𝑎(𝑐) axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' On the other hand, the Berry phase calculations [5, 9] indicate  that the axes of the spontaneous sublattice polarization, in fact, are very close or even coincide with  the diagonals of the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' In terms of our model, this means that the crystallographic axes and the  diagonals are the above mentioned exceptional directions: the axis 𝑎(𝑐) is the direction ii), while the  diagonals of the 𝑎𝑐 plane are the direction i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The topology of the 𝑇-𝐸 diagrams and the shape of the  𝑃(𝐸) curves for the fields 𝐸1(𝐸3) and 𝐸1 ± 𝐸3 will change accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The availability of the new, more  reliable experimental data [9] makes a quantitative comparison of theoretical and experimental 𝑃(𝐸)  curves meaningful and could help to ascertain the orientation of the model dipole moment vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  Thus, it seems worthwhile to revisit the problem of polarization rotation in squaric acid, to perform  calculations with an alternative set of the model parameters, where the sublattice polarizations are  oriented along the diagonals of the 𝑎𝑐 plane, and to compare the theoretical results with the most  recent [9] experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The model [4, 7], briefly described in section 2, is used without any further  modification of the formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' In section 3 the results of the theoretical calculations with the old and new  sets of the model parameters are compared with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The model  The model has been introduced and explicated in [4], and a concise outline is given in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Below  we present a brief qualitative description of the model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' all the formulae and other relevant details and  discussions can be found in the mentioned papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  Protons on the hydrogen bonds in squaric acid move in double-well potentials, so each of the protons  can occupy one of the two sites on the bond: closer to the given C4O4 group or to the neighboring  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The motion of protons is described by Ising pseudospins, whose two eigenvalues are assigned to  two equilibrium positions of each proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Two interpenetrating sublattices (layers) of pseudospins are  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  43710-2  Electric field induced polarization rotation in squaric acid crystals revisited  a)  b)  a  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' (Colour online) a) The crystal structure of squaric acid as viewed along the 𝑏 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Two adjacent  layers are shown, with black and open circles each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The A and B type C4O4 groups are indicated (see [4, 8]  for explanation), and the hydrogen bonds are numbered, 𝑓 = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' b) The dipole moments assigned  to one of the four lateral proton configurations (the configuration 1 in tables 1 in [4, 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Directions of  the dipole moments associated with protons 𝝁𝐻  1 = (2𝜇𝐻 , 0, 0) and with electrons 𝝁𝜋  1 = (2𝜇𝜋  ∥ , 0, −2𝜇𝜋  ⊥)  are shown with blue and red arrows, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' the green arrow is the total dipole moment of the  configuration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' the vector lengths are nominal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' 𝜑0 = arctan(𝜇𝐻 + 𝜇𝜋  ∥ )/𝜇𝜋  ⊥ is the angle between the total  dipole moment of configuration 1 and the 𝑐 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Figures are taken from [4, 7, 8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  The total system Hamiltonian [4, 7] includes ferroelectric intralayer long-range interactions between  pseudospins, ensuring ferroelectric ordering within each separate layer, antiferroelectric interlayer inter-  actions responsible for alternation of polarizations in the stacked layers, and the short-range interactions,  which include also the coupling with external electric fields 𝐸1 and 𝐸3 directed along the tetragonal  (paraelectric) 𝑎 and 𝑐 axes of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  The short-range Hamiltonian describes the four-particle configurational correlations between protons  placed around each C4O4 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The usual Slater-Takagi type scheme [4, 8, 11, 12] of 16 degenerate  levels of lateral/diagonal/single-ionized/double-ionized proton configurations is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The lateral and  single-ionized configurations have dipole moments in the 𝑎𝑐 plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' the degeneracy of their energy levels  is removed by the electric fields 𝐸1 and 𝐸3, which break the equivalence of the hydrogen bonds that link  the C4O4 groups along the 𝑎 and 𝑐 axes (see tables 1 in [4, 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  Assignment of the dipole moments to the ground-state lateral configurations is the crucial point of the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' We rely on the results of the Berry phase calculations [5], which have shown that the ground-state  sublattice polarization in this crystal is formed directly by displacements of protons along the hydrogen  bonds and, mostly, by the electronic contributions of switchable 𝜋-bond dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  Positions of the 𝜋-bonds are determined by the proton arrangement around the given C4O4 group: in  the lateral configurations the 𝜋-bond is formed between the two neighboring carbons, near which protons  sit on the hydrogen bonds (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' 1b), and also between the carbons and adjacent to them oxygens, next  to which there is no proton (meaning that the protons on these H-bonds sit in the minima close to the  neighboring C4O4 groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The field-induced polarization rotation by 90◦ or 180◦ occurs via flipping  of one or two protons in each molecule to the other sites along the same hydrogen bonds and via a  simultaneous switching of the 𝜋-bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' For the depicted in figure 1b lateral proton configuration, the  vector of the proton contribution to the dipole moment is oriented along the 𝑎 axis, while the electronic  contribution is at the angle to this axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The dipole moments of the three remaining lateral configurations  can be obtained from the scheme of figure 1b by rotation by a multiple of 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  After going from the representation of proton configuration energies to the pseudospin representation,  the four-particle cluster approximation for the obtained short-range Hamiltonian is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The mean  field approximation is used for the long-range interlayer and intralayer interactions [4, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The dependence  of all proton-proton interaction parameters on the diagonal components of the lattice strain tensor and  on the H-site distance, which are changed by the thermal expansion and potentially by an external stress  if such is applied, is taken into account [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The expression for the thermodynamic potential has been  obtained [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' the order parameters and lattice strains are found by numerical minimization thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  43710-3  H  c  02Po,H  μiA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Moina  The values of all model parameters were chosen earlier [4, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' In particular, they were required [8]  to provide the best fit to the experimental temperature curves of the order parameter at ambient pressure,  to the temperature and hydrostatic pressure dependences of the diagonal lattice strains, and to the pressure  dependence of the transition temperature 𝑇N in squaric acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  The dielectric characteristics and other electric field effects in our model are mostly governed by  values of the dipole moments, which enter the final expressions only via the sum 𝜇𝐻 + 𝜇𝜋  ∥ and via 𝜇𝜋  ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  These values are found by fitting the calculated curve of the static dielectric permittivity 𝜀11 at zero  external bias field to the experimental points of [5], while trying to get the best possible agreement  with the experiment for the values of the switching fields, corresponding to the first 90◦ rotation of the  sublattice polarization by the field 𝐸1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' It can be shown that in the paraelectric phase 𝜀11 ∼ ¯𝜇2, where  ¯𝜇 =  √︃  (𝜇𝐻 + 𝜇𝜋  ∥ )2 + (𝜇𝜋  ⊥)2  is half the magnitude of the dipole moment, assigned to the H2C4O4 groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' It means that above 𝑇N the  permittivity 𝜀11 at zero field is determined by the magnitude of the dipole moment vector only, whereas  the orientation of the vector within the 𝑎𝑐 plane can be varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' For the set, adopted in [4, 7] and presented  in table 1 as the set A, the dipole moment and the ground state sublattice polarization are oriented at the  angle 𝜑0 = arctan(𝜇𝐻 + 𝜇𝜋  ∥ )/𝜇𝜋  ⊥ ≈ 56◦ to the crystallographic axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' However, the results of the Berry  phase calculations [9] indicate that the angle should be closer to 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Thus, we find an alternative set of  the dipole moment values with 𝜇𝐻 + 𝜇𝜋  ∥ = 𝜇𝜋  ⊥ and with the same ¯𝜇 as in the set A, which yields the same  fit to the permittivity in the paraelectric phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' this is the set B in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' In the next section, using the  set B, we construct the 𝑇-𝐸 phase diagrams and explore the 𝑃(𝐸) curves for the electric fields 𝐸1 and  𝐸1 + 𝐸3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The results are compared with the previous calculations [7] performed with the set A, as well  with the experimental data of [5, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The adopted values of the model dipole moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The set A is taken from [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The values of all  other model parameters are the same as in [4, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  𝜇𝐻 + 𝜇𝜋  ∥  𝜇𝜋  ⊥  ¯𝜇  (10−29 C m)  set A  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='16  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='8  set B  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='66  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='66  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Calculations  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Phase diagrams  In figure 2 we redraw the 𝑇-𝐸 diagrams of squaric acid for the fields 𝐸1 and 𝐸1 + 𝐸3, obtained earlier  in [4, 7] along with the newly available experimental points of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Here, the set A of the dipole moment  values was used in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The diagrams overlap the color gradient plots of the introduced in [4]  noncollinearity angle 𝜃, which is the angle between the vectors of the sublattice polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  Different phases in the diagrams are separated by the lines of the first order phase transitions I, II,  and III, and of the second order phase transitions IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' All these lines terminate at various critical points  (bicritical end points BCE, tricritical point TCP, critical end points CEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Some of the critical points can  be artifacts of the mean field approximation, used for the long-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' This was discussed  extensively in [4, 7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' we shall not dwell on this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The phase denoted as AFE* (the red region) is  non-collinear antiferrielectric, very close to the initial AFE phase with 𝜃 ∼ 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The purple region is the  collinear field-induced ferroelectric phase (FE) with 𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The phase between the transition lines II, III,  and IV (green and blue) is the noncollinear ferrielectric phase NC90, where 𝜃 mostly remains close to 90◦,  only rapidly decreasing to zero near the second-order phase transition line IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' In the region NC135*  43710-4  Electric field induced polarization rotation in squaric acid crystals revisited  250  300  350  400  450  0  200  400  600  800  1000  1200  V  NC135*  I  BCE1  CEP  TCP  BCE2  IV  III  T  (K)  E1 (kV/cm)  FE  NC90  0  1  11  22  33  45  56  67  78  90  107  115  120  125  130  146  157  169  180  AFE*  θ  (deg)  II  300  350  400  200  400  V  NC135*  FE  NC90  I  IV  III  CEP2  CEP1  BCE3  BCE1  T  (K)  E1+E3 (kV/cm)  BCE2  II  AFE*  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' (Colour online) The 𝑇-𝐸 phase diagrams of the squaric acid, overlapping the 𝑇-𝐸 color  contour plots of the noncollinearity angle 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The set A is used in calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Solid and dashed lines  indicate the first and second order phase transitions, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' dotted lines are the supercritical lines,  corresponding to the loci of maxima in the field dependences of d𝑃(𝐸)/d𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The open squares □, star �,  and full circles • indicate the critical end points (CEP), tricritical point (TCP), and bicritical end points  (BCE), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Blue full triangles ▲, ▶, and ▼ are the experimental points of [5], the electronic  supplementary material thereto, and [9], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  (orange to yellow), a crossover between the AFE* and NC90 phases occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Here, 𝜃 changes gradually  from ∼ 180◦ to ∼ 90◦: the negative sublattice polarization rotates continuously with increasing field and  becomes perpendicular to the positive sublattice polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' As discussed in [4, 7], this continuous  rotation is a statistically averaged effect, possible only in presence of thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  Crossovers are often marked by the lines formed by the loci of the extrema of the response functions —  second derivatives of the thermodynamic potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Those supercritical lines are continuations of the  first order transition lines beyond the critical points terminating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The major drawback of this method  is that the extrema of different response functions yield different supercritical lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' moreover, the super-  critical lines formed by the extrema of the same response function taken along different thermodynamic  paths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=', isotherms or isofields) are different as well (see [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' In order to compare the theory and the  experimental data derived from the field dependence of polarization, we mark the crossover between the  AFE* and NC90 phases using the lines formed by the maxima of the d𝑃(𝐸)/d𝐸 isotherms (the inflection  points of the 𝑃(𝐸) isotherms), where 𝑃 is the projection of the net polarization vector on the field axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  These are the dotted lines V in the phase diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  As one can see in the left-hand panel of figure 2, for the field 𝐸1, the most recent data obtained in [9]  for the sample with the improved dielectric strength appear to be in a much better agreement with the  theory than the earlier experimental data of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The theoretical switching fields, calculated with the set  A, are much higher than the experimental values of [5] with the relative error 𝜂 = 1 − 𝐸exp/𝐸theor ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='42  at room temperature (295 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The error decreases down to ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='23 for the experimental points of [9],  which is still not quite satisfactory, but evidently much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  In the case of 𝐸1 + 𝐸3, the switching fields calculated with the set A are higher than predicted for  the field 𝐸1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' This is in a qualitative agreement with all available experimental observations [5, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The  relative errors are about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='3 for [5] at 324 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='22 for [9] at 295 K, that is, the improvement here is  not so striking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  Now let us see how the situation changes, when the set B is used in calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' For this set, the  ground state spontaneous polarization axis is oriented along the diagonal of the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' It means that  the fields 𝐸1 + 𝐸3 and 𝐸1 are directed along this axis and at 45◦ to it, respectively, that is, along the  exceptional directions i) and ii), discussed in Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' It is then expected that the 𝑇-𝐸 diagrams will  be topologically different from those, depicted in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' For the field 𝐸1 + 𝐸3, the crystal of squaric  acid should behave like a uniaxial antiferroelectric, exhibiting a one-step polarization rotation by 180◦  43710-5  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Moina  without the intermediate noncollinear phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' For the field 𝐸1, the field of switching to the ferroelectric  phase is expected to tend to infinity, and only the AFE*-NC90 transition can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  300  400  100  200  300  400  T  (K)  I  IV  CEP  BCE1  FE  NC90  0  1  11  22  33  45  56  67  78  90  107  115  120  135  146  157  169  180  E1 (kV/cm)  AFE*  BCE2  II  θ (deg)  250  300  350  100  200  300  400  FI  E1+E3 (kV/cm)  IV  I  FE  TP  BCE  TCP2  T (K)  TCP1  AFE*  III  V  VII  238  240  296  300  304  VI  VII  III  BCE  AFE*  FE  TP  III  V  FI  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' (Colour online) The same as in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The set B is used in calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The open triangle  △ indicates the triple point TP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The dash-dotted line VII corresponds to the loci of minima in the field  dependences of d𝑃(𝐸)/d𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The other notations are the same as in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  The 𝑇-𝐸 phase diagrams, calculated with the set B and presented in figure 3, are indeed in a total  agreement with the above described picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' As the magnitude of the dipole moment 2 ¯𝜇 is the same  for both sets, these diagrams are numerically identical to those obtained in [7] with the set A for  the exceptional directions ii) and i), respectively (see figures 8, 9 in [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' This identity can be proved  algebraically, using the expression for the thermodynamic potential of the system [4, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  For the field 𝐸1, the positions of the lines II of the AFE*-NC90 phase transitions, calculated with  the sets A and B, are very close but not the same (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' the left-hand panels in figures 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The  closeness can be explained by the found in [7] dependence of this switching field at low temperatures on  the orientation of the spontaneous sublattice polarization axis 𝐸 𝐼 𝐼 ∼ 1/cos(𝛿𝜑 − π/4), where 𝛿𝜑 is the  angle between this axis and the external field 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Since 𝛿𝜑 for the sets A and B differ by about 11◦ only,  the difference between the corresponding switching fields is small as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' It is then trivial to say that for  𝐸1, the sets A and B yield about the same agreement with the experimental data for the switching field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  For the field 𝐸1 + 𝐸3, the intermediate phase NC90 is absent, and 𝜃 is always either 180◦ or zero (see  the right-hand panel of figure 3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=', all phases are collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The polarization switching occurs either as  a first order phase transition across lines III directly to the FE phase and across line VI to an intermediate  collinear ferrielectric phase FI, or gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' In the latter case, the magnitude of one of the sublattice  polarizations decreases down to zero with increasing field, changes its sign continuously at line VII, and  then increases until the second order transition to the FE phase occurs at line IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Interestingly, line VII,  where the angle 𝜃 changes from 180◦ to 0, is formed by the loci of the minima of the d𝑃(𝐸)/d𝐸 isotherms,  as opposed to line V, formed by the loci of the maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Line VI, emanating from the critical point BCE  (see the inset in figure 3), marks the crossover between AFE* and FI phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' It is to be compared with  the experimental data for the switching fields, and it yields nearly the same agreement as the set A, with  the relative errors about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='3 for [5] at 324 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='23 for [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Polarization  In figure 4 we plot the field dependences of the projections of the net polarization vector on the field  direction for the fields 𝐸1 and 𝐸1 + 𝐸3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The experimental points of [5] and [9] are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The  drastic changes in the experimental hysteresis curves, brought by the improvement of the sample quality  and by the increase of the maximum value of the applied field in [9], are very well seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' It is obvious that  the comparison of the theoretical polarization curves with the earlier data of [5] could be only qualitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  As one can see, for 𝐸1, the sets A and B predict three and two plateaus of polarization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  43710-6  Electric field induced polarization rotation in squaric acid crystals revisited  100  1000  0  10  20  30  40  200  400  600  0  10  20  30  40  III  set A  set B  E1 (kV/cm)  P1 (µC/cm2)  II  V  IV  E1+E3 (kV/cm)  P (µC/cm2)  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' (Colour online) The field dependences of polarizations at 295 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Full triangles: experimental  points taken from [5] (▲) and [9] (▼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The arrows indicate phase transitions of the first order across lines  II, III (left-hand) and of the second order across lines IV (right-hand).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The arrow and full circles (•)  indicate the crossovers at lines V (right-hand).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Lines II-V are from the 𝑇-𝐸 phase diagrams, figures 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  In the physically reasonable field range, which includes the AFE*-NC90 first order phase transition (at  lines II from the phase diagrams 2, 3), the two sets yield very similar polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The calculated  polarization jumps are 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='9 µC/cm2 for the set A and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='4 µC/cm2 for the set B, in a fair agreement  with the experimental 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='2 µC/cm2 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The set A also predicts the second step of polarization at a much  higher field, at the transition to the FE phase (across line III from the phase diagram, figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' It seems  unlikely, however, that the field of such high a magnitude could ever be applied in an experiment without  the squaric acid samples suffering the dielectric breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  For the field 𝐸1 + 𝐸3, the two sets of the model parameters yield different behaviour of polarization  even at experimentally accessible fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The 𝑃(𝐸) curve, calculated with the set A, has three smeared  plateaus, with a clear rounded step, corresponding to the AFE*- NC90 crossover across line V, and then  a cusp at the NC90-FE second order transition across line IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The lower part of this curve, albeit being  shifted to higher fields, is in a good qualitative and quantitative agreement with the experimental points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  The polarization, calculated with the set B, on the other hand, has only two smeared plateaus: at low  fields and above the cusp at line IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' No pronounced intermediate plateau is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The change of concavity  at the inflection point, marked by a full circle in the figure, is hardly discernible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The agreement with  the experiment is visibly worse than for the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' However, the switching field magnitude for 𝐸1 + 𝐸3 is  predicted [5, 7, 9] to be higher than for the field 𝐸1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' It means that, despite the increased dielectric strength  of the samples, the maximum applied fields 𝐸1 +𝐸3 [9] could still be insufficient to obtain correct data for  polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' A potential further improvement of the crystal quality (if such is still possible) may change  the measured values of polarization and switching field for the diagonally directed field in the same way,  as such an improvement did in the case of the field 𝐸1 in [9] as compared to [5], which is well illustrated  in the left-hand panel of figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Then, the agreement with the theoretical curves can be reexamined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Concluding remarks  Using the previously developed [4] deformable two-sublattice proton ordering model, we revisit  the problem of polarization rotation in antiferroelectric crystals of squaric acid under the influence  of external electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The unique structure of the two-dimensional hydrogen bond networks in  squaric acid permits 90◦ rotation of the sublattice polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' The model predicts [4, 7] that except  for some particular directions of the field, the polarization reorientation at low temperatures is a two-  step process: first, to the noncollinear phase with perpendicular sublattice polarizations and then to the  collinear ferroelectric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' However, when the field is directed along the axis of spontaneous sublattice  polarizations, the intermediate noncollinear phase is absent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' when the field is at 45◦ to this axis, the field  of the transition to the ferroelectric phase tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  The previously obtained 𝑇-𝐸 phase diagrams and newly calculated polarization curves are compared  43710-7  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Moina  with the most recent experimental data [9], measured using the crystal samples of the increased dielectric  strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' We also test an alternative set of the model parameters, for which the dipole moments assigned  to the H2C4O4 groups are of the same magnitude as in the previous calculations, but oriented along the  diagonals of the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  The new experimental data [9] are in a drastically better agreement with the theory than the earlier  results [5], especially for the polarization curves, as well as for the switching fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' It shows that the  simplicity of the model was not the major reason of the earlier [4, 7] disagreement between theory and  experiment and gives a strong evidence for the model validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  Results of testing the new set of the model parameters are inconclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Overall, the comparison of the  theoretical polarization curves with the experimental data seems to slightly favor the previous set [4, 7],  according to which the axes of the spontaneous sublattice polarization are close, but not exactly parallel  to the diagonals of the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Further experimental studies may shed some light on this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  As far as a further verification of the model is concerned, the appropriateness of the mean field  approximation, used for the long-range interactions, may be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' This approximation is, most  likely, the origin of the artifact splitting [4, 7] of some tricritical points in the 𝑃-𝐸 phase diagrams into  the systems of bicritical and critical endpoints and also of the appearance of the intermediate FI phase,  seen in the right-hand panels of figures 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Monte Carlo calculations may be used to construct more  accurate diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Semmingsen D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
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+page_content='1143/PTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Schienbein P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=', Marx D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' E, 2018, 98, 022104, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='1103/PhysRevE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='022104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  43710-8  Electric field induced polarization rotation in squaric acid crystals revisited  Ще раз про обертання поляризацiї електричним полем в  кристалах квадратної кислоти  А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' П.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Моїна  Iнститут фiзики конденсованих систем Нацiональної академiї наук України  79011, м.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Львiв, вул.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Свєнцiцького, 1  З використанням запропонованої ранiше моделi розглядаються процеси обертання поляризацiї зовнiшнi-  ми електричними полями в антисегнетоелектричних кристалах квадратної кислоти.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Обчислення також  проводяться з альтернативним набором параметрiв теорiї, в якому дипольнi моменти, якi приписуються  групам H2C4O4, паралельнi до дiагоналей площини 𝑎𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Дослiджено фазовi дiаграми 𝑇-𝐸 та кривi поля-  ризацiї 𝑃(𝐸) для полiв, прикладених уздовж осi 𝑎 та уздовж дiагоналi площини 𝑎𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Порiвняння теорети-  чних результатiв з нещодавно опублiкованими експериментальними даними пiдтверджує правильнiсть  запропонованої моделi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content=' Не виявлено суттєвої переваги нового набору параметрiв моделi перед тим, що  використовувався в попереднiх розрахунках.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}
+page_content='  Ключовi слова: поляризацiя, електричне поле, фазовий перехiд, антисегнетоелектрик, фазова дiаграма,  квадратна кислота  43710-9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'}

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