Pith. sign in

REVIEW 5 major objections 4 minor 2 cited by

A rapid quench can split the supercritical region into two distinct subphases, with the dividing line drawn by the speed of a phase-separation front.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 02:24 UTC pith:TM72OLR3

load-bearing objection A genuinely new nonequilibrium way to carve up the supercritical region, but the central curve is under-evidenced: single protocol, no error bars, no comparison to the static spinodal, and the abstract promises a second line the text never delivers. the 5 major comments →

arxiv 2604.13104 v2 pith:TM72OLR3 submitted 2026-04-10 cond-mat.stat-mech hep-phhep-th

Nonequilibrium crossover in the supercritical region from quench dynamics

Zi-Qiang Zhao , Zhang-Yu Nie , Jing-Fei Zhang , Xin Zhang This is my paper
classification cond-mat.stat-mech hep-phhep-th
keywords supercritical regionquench dynamicsnonequilibrium crossoverinvasion phenomenonphase separationholographic superfluidsymmetry breakingcrossover line
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper argues that the supercritical region beyond a critical point, usually considered featureless, can still be split into distinct subphases by watching how the system evolves after a sudden quench. Using a holographic superfluid model, the authors find that a phase-separation front launched by topological defects sweeps through the system at a constant speed even in the supercritical region. Measuring this invasion velocity as a function of the quench endpoint, they find that the velocity first rises, peaks, and then falls. This peak, mapped across parameter space, defines what they call the nonequilibrium supercritical crossover line — a boundary that carries both thermodynamic and dynamic information. If correct, this provides a new, purely dynamical way to characterize supercritical subphases, applicable in principle to classical fluids, quantum gases, and black-hole systems.

Core claim

Within a holographic superfluid with two higher-order nonlinear terms, a rapid quench from the normal to the superfluid side of the phase diagram triggers spontaneous symmetry breaking. If the initial perturbation contains a single pair of kinks, a uniform phase-separation front invades the region between them at constant velocity — the invasion velocity. The authors find that this velocity is a non-monotonic function of the quench endpoint in the supercritical region: it increases, reaches a clear turning point, and then decreases. This turning point defines a new curve in parameter space, which they term the nonequilibrium supercritical crossover line. Because the curve emerges from time-d

What carries the argument

The central object is the invasion velocity, extracted as the reciprocal of the slope of a linear fit to the front position in the numerical quench simulations. The mechanism is the coupling between symmetry breaking and phase separation: topological-defect kinks act as nucleation sites, and a phase front sweeps inward at constant speed. The enabling step is the claim that during nonequilibrium evolution a globally charge-conserving system behaves like its local subsystems that exchange particles freely, so a locally unstable branch can produce inhomogeneous structures even in the supercritical region. The turning point of the invasion velocity with respect to the quench endpoint is the oper

Load-bearing premise

The load-bearing premise is that a globally charge-conserving system, during nonequilibrium evolution, behaves like its local subsystems that exchange particles freely, so that an unstable branch of those local subsystems creates inhomogeneous structures in the supercritical region; if that equivalence fails, no invasion front forms and the crossover line cannot be defined.

What would settle it

A simulation with strict global charge conservation imposed on the dynamical evolution, rather than allowing local particle exchange, would be decisive: if no phase-separation front appears in the supercritical region, the claimed turning point and crossover line are artifacts of the locality assumption.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The supercritical region is not dynamically featureless: two subphases can be distinguished by how fast a phase front invades after a fast quench.
  • A crossover line defined by a turning point in invasion velocity can be drawn in parameter space, complementing classical static crossover lines with a purely nonequilibrium curve.
  • Because the line encodes both thermodynamic and kinetic information, it may be more sensitive than static response functions to symmetry-breaking structure beyond the critical point.
  • The mechanism is expected to be universal, so the same turning point should appear in any system with an energy barrier and a quench into an unstable region, including classical fluids, quantum gases, and black-hole systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A direct extension is to test whether the turning point shifts with quench rate; if it does, the crossover line is best understood as a kinetic locus rather than a material boundary, setting it even further apart from thermodynamic crossover lines.
  • Because the front speed depends on how fast the symmetry-breaking order parameter relaxes, the turning point might be predicted from a reduced two-well free-energy model without holography, offering a simpler experimental handle.
  • If the locality assumption is dropped — for example, by enforcing strict global charge conservation in the dynamics — the invasion front may fail to form and the entire crossover line would vanish; this is a targeted check that the paper's results invite.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The paper studies the supercritical region of a holographic superfluid model (Einstein–Maxwell–scalar theory with Z2 symmetry and λ, τ, α couplings) by subjecting the system to a rapid quench across the critical point. The authors report that, after a quench from the normal phase to a supercritical endpoint, topological-defect-induced 'invasion' persists and the invasion velocity v_i, extracted from linear fits to the invasion front position, exhibits a maximum as a function of the quench endpoint ρ_f. This maximum is used to define a 'nonequilibrium supercritical crossover line' in the τ–ρ plane. The paper argues that this line encodes both thermodynamic and dynamical information, in contrast to Widom or Frenkel lines.

Significance. If the reported maximum in v_i(ρ_f) is robust and does not simply coincide with the static grand-canonical spinodal, the proposed crossover line would be a genuinely nonequilibrium probe of supercritical subphase structure, combining phase-separation and symmetry-breaking dynamics. The idea is interesting and potentially relevant beyond holography, as the authors relate it to Ginzburg–Landau mechanisms. However, the paper currently lacks several checks that are essential for establishing the claim: no protocol-sensitivity study, no comparison with the static stability boundary, no uncertainty or convergence analysis, and an unproved locality assumption in the theoretical justification. The abstract also promises two crossover lines while the body delivers only one.

major comments (5)
  1. [Abstract / Sections IV–V] The abstract states that the paper defines a supercritical crossover curve based on 'the time at which inhomogeneous structures appear most rapidly' and, in addition, a second crossover line from the invasion-velocity turning point. The main text never defines or analyzes the former quantity; Sections IV and V only discuss the invasion-velocity line. This discrepancy affects the paper's central claims and must be resolved: either remove the first line from the abstract or present the corresponding analysis.
  2. [Section IV, Fig. 3] The proposed crossover line is the red dashed curve in the lower panel, defined as the maximum of v_i as a function of ρ_f for fixed τ. The text itself notes in Section III that the grand-canonical ensemble retains an unstable branch in the supercritical region, and the lower panel shows a static spinodal region (black dashed line). The red dashed line is never compared with that static boundary. If the v_i maximum simply tracks the spinodal or the normal/superfluid boundary, the claimed novelty is substantially weakened. The authors should overlay the static spinodal and the v_i maxima for all τ values shown, and quantify their separation or coincidence.
  3. [Section IV, Fig. 3] All simulations use a single quench protocol: τ_Q=0.1, ρ_s=1.3, L_x=600, and a fixed step initial perturbation ±1e-5. In holographic quenches, front velocities and unstable-mode growth depend on quench rate, so v_i(ρ_f) and the location of its maximum could shift or smear with τ_Q. The paper must provide v_i(ρ_f) for at least a few quench rates (e.g., τ_Q = 0.05, 0.1, 0.2) and, ideally, for different initial perturbation amplitudes and system sizes, to show that the turning point is a robust dynamical feature rather than a protocol artifact.
  4. [Section IV, Fig. 3] The v_i values are extracted from linear fits (Fig. 2) but no error bars, fit residuals, or convergence tests are reported. The black circles in Fig. 3 therefore cannot be used to assess whether the maximum is statistically significant or whether the observed nonmonotonicity is within numerical uncertainty. The authors should report uncertainties from the fits and test sensitivity to time step, spatial resolution, and number of grid points.
  5. [Section III, last paragraph] The argument that the globally charge-conserving canonical system can be described by local grand-canonical subsystems, and that dynamical stability coincides with the stability of local subsystems, is asserted without proof. This assumption is load-bearing because it is what allows inhomogeneous structures and invasion to exist in the supercritical region. The authors should either provide a derivation or a quantitative numerical test (e.g., comparing canonical and grand-canonical response or directly computing local stability eigenvalues) to justify this step.
minor comments (4)
  1. [Section IV, Fig. 3 caption/text] The text says the invasion velocity exhibits a 'distinct inflection point' but describes a maximum (turning point). Please use consistent terminology.
  2. [Figure 2 caption] The quench rate is written as τ_Q = 0.1s, but the holographic model uses dimensionless boundary time; the unit 's' is either a typo or requires clarification of the conversion to physical units.
  3. [General] The paper relies heavily on the companion preprint Ref. [81] for the phase diagram and the invasion mechanism. The present manuscript should state more explicitly which results are new here and which are inherited from Ref. [81].
  4. [Section IV, paragraph 1] The description of the initial condition for the step perturbation could be clearer: it is introduced only in the caption of Fig. 2, and the main text should define it at first use.

Circularity Check

1 steps flagged

Main circularity: the supercritical-invasion premise is imported from overlapping-author preprint [81], while the v_i maximum itself is new data; the 'crossover line' is definitional rather than independently validated.

specific steps
  1. self citation load bearing [Sec. I (Introduction); Sec. III (static solutions/phase diagram); Sec. IV (quench dynamics); Refs. [81]]
    "The full phase diagram in terms of the coefficients of the higher-order terms has been presented in previous work [81]. ... In Ref. [81], the authors demonstrated that when symmetry breaking is present and the quench endpoint resides within the unstable region of the first-order phase transition, topological defects can serve as nucleation sites for phase separation, thereby inducing a uniform invasion phenomenon."

    The paper's central measurable, the invasion velocity v_i in the supercritical region, is justified by phenomena (topological-defect invasion, supercritical phase-separation mechanism, phase-diagram regime) attributed to Ref. [81], a companion preprint by the same authors (Zhao, Nie, Zhang, Zhang). No independent check of [81] is given. The new v_i(ρ_f) maximum is measured here, so the central claim has independent numerical content, but its physical premise is a load-bearing self-citation.

full rationale

The paper does not derive the new crossover line from first principles; it operationally defines it as the turning point of the invasion velocity v_i(ρ_f) measured in its own simulations. That is a stipulative definition, not a circular derivation: the line is, by construction, the locus of the maximum of v_i, and the paper says so explicitly ('This turning point defines a new curve in parameter space'). The absence of a comparison with the static grand-canonical spinodal, error bars, or τ_Q-variation is a validation/correctness gap, not a circularity. The unproved locality assumption in Sec. III ('the dynamical stability of the system coincides with that of the local subsystems') is likewise a missing proof rather than a self-reduction and is weighed as a correctness risk, not as circularity. The clearest circularity concern is the load-bearing reliance on Ref. [81], an overlapping-author preprint, for the invasion mechanism, the phase diagram, and scale-independence; this is what raises the score. Because the v_i turning-point data are new and self-contained, the paper retains independent content, consistent with a moderate circularity score of 4.

Axiom & Free-Parameter Ledger

5 free parameters · 4 axioms · 0 invented entities

The central contribution is a numerical observable and a curve drawn through its extremum, not a new physical entity. The model is carried over from companion work [81], and the main added modeling assumptions are the local grand-canonical stability criterion and the identification of the v_i maximum with a supercritical subphase boundary.

free parameters (5)
  • λ (quartic scalar coupling) = -4
    Chosen by hand to induce a first-order phase transition; Section III states λ=-4.
  • τ (sextic scalar coupling) = 2.93 at critical point; varied to generate crossover line
    Tuned to place the system near the critical point with λ=-4, then scanned to define the red dashed crossover line in Fig. 3.
  • α (coupling in h(Ψ)=e^{αΨ²}) = 5
    Given in Section III as a fixed model parameter; chosen without sensitivity analysis.
  • m² (scalar mass squared) = not stated in text
    Appears explicitly in Eq. (12); no numerical value or standard quantization value is provided, yet the equations of motion depend on it.
  • Quench protocol parameters = ρ_s=1.3, τ_Q=0.1, L_x=600, n_x=1800, δt=0.05, initial perturbation 10^-5
    All hand-picked and fixed; no scan over quench rate, initial amplitude, or system size is shown, so the inferred crossover locus may depend on this protocol.
axioms (4)
  • domain assumption AdS/CFT duality maps the boundary superfluid to a classical Einstein–Maxwell–scalar bulk theory.
    The holographic dictionary is used throughout Section II without proof; accepted framework but not established in this paper.
  • domain assumption The metric is taken in a fixed probe/fixed-background form f(z)=1-(z/z_h)^3 with no backreaction evolved.
    Section II adopts the ingoing Eddington metric with fixed f(z); the dynamical equations (5)-(8) do not include metric fluctuations, yet this restriction is not stated.
  • ad hoc to paper Dynamical stability of the globally charge-conserving system coincides with the grand-canonical stability of local subsystems.
    Section III's final paragraph uses this to explain why supercritical phase separation and invasion can occur even when the canonical ensemble is in the supercritical region; it is load-bearing and unproved.
  • ad hoc to paper A genuine supercritical subphase boundary can be identified with the maximum of the invasion velocity v_i(ρ_f) at fixed quench protocol.
    Section IV introduces the turning point as the definition of the crossover line; no independent criterion selects this extremum as physically meaningful.

pith-pipeline@v1.3.0-alltime-deepseek · 9789 in / 17206 out tokens · 170429 ms · 2026-08-03T02:24:22.898687+00:00 · methodology

0 comments
read the original abstract

Distinguishing different subphases in the supercritical region is an important issue in statistical physics and condensed matter physics. Traditional approaches rely mainly on static thermodynamic response functions or equilibrium correlation functions, which are essentially limited to quasistatic processes. In this paper, we investigate the evolution behavior of a system after a rapid quench from the perspective of nonequilibrium dynamics within a holographic model. We find that, using the time at which inhomogeneous structures appear most rapidly, we can define a supercritical crossover curve based on the pure phase separation process. In addition, the uniform invasion phenomenon induced by topological defects persists in the supercritical region, and the invasion velocity exhibits a clear turning point as a function of the quench endpoint. This turning point can define another new nonequilibrium supercritical crossover line that simultaneously incorporates the effects of both symmetry breaking and phase separation. Unlike the classical Widom line or Frenkel line, these two new crossover lines contain both thermodynamic information and dynamical information, reflecting the dynamical nature of the supercritical region under nonequilibrium conditions. This work provides a novel nonequilibrium dynamical approach for characterizing supercritical subphases.

Figures

Figures reproduced from arXiv: 2604.13104 by Jing-Fei Zhang, Xin Zhang, Zhang-Yu Nie, Zi-Qiang Zhao.

Figure 2
Figure 2. Figure 2: FIG. 2. The process of invasion phenomenon during a quench [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. The condensate and free energy curve for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The relationship between the invasion velocity in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Phase separation seeded by Z2 and U(1) topological defects from holography

    hep-th 2026-05 unverdicted novelty 6.0

    Holographic simulations demonstrate that Z2 and U(1) topological defects universally seed phase separation, with cores expanding into domains under a double-quench protocol.

  2. Boson star-black hole binaries: initial data and head-on collisions

    gr-qc 2026-04 unverdicted novelty 6.0

    A one-body conformal-factor correction stabilizes boson star-black hole initial data, enabling gravitational-wave analysis that shows higher multipoles can discriminate mixed mergers from pure black-hole binaries.

Reference graph

Works this paper leans on

85 extracted references · 55 linked inside Pith · cited by 2 Pith papers

  1. [1]

    Burke, J

    T. Burke, J. L. Lebowitz, and E. Lieb, Phys. Rev.149, 118 (1966)

    1966

  2. [2]

    D. T. Son, M. Stephanov, and H.-U. Yee, J. Stat. Mech. 2101, 013105 (2021), arXiv:2006.01156 [cond-mat.quant- gas]

    Pith/arXiv arXiv 2021

  3. [3]

    J. L. Jim´ enez, S. P. G. Crone, E. Fogh, M. E. Za- yed, R. Lortz, E. Pomjakushina, K. Conder, A. M. L¨ auchli, L. Weber, S. Wessel, A. Honecker, B. Normand, C. R¨ uegg, P. Corboz, H. M. Rønnow, and F. Mila, Na- ture592, 370–375 (2021)

    2021

  4. [4]

    V. V. Brazhkin and K. Trachenko, Physics Today65, 68 (2012)

    2012

  5. [5]

    Kubiznak and R

    D. Kubiznak and R. B. Mann, JHEP07, 033 (2012), arXiv:1205.0559 [hep-th]

    Pith/arXiv arXiv 2012

  6. [6]

    Kubiznak, R

    D. Kubiznak, R. B. Mann, and M. Teo, Class. Quant. Grav.34, 063001 (2017), arXiv:1608.06147 [hep-th]

    Pith/arXiv arXiv 2017

  7. [7]

    Cai, L.-M

    R.-G. Cai, L.-M. Cao, L. Li, and R.-Q. Yang, JHEP09, 005 (2013), arXiv:1306.6233 [gr-qc]

    Pith/arXiv arXiv 2013

  8. [8]

    Wei and Y.-X

    S.-W. Wei and Y.-X. Liu, Phys. Rev. D90, 044057 (2014), arXiv:1402.2837 [hep-th]

    Pith/arXiv arXiv 2014

  9. [9]

    L. Xu, P. Kumar, S. V. Buldyrev, S.-H. Chen, P. H. Poole, F. Sciortino, and H. E. Stanley, Proceedings of the National Academy of Sciences102, 16558–16562 (2005)

    2005

  10. [10]

    Ruppeiner, A

    G. Ruppeiner, A. Sahay, T. Sarkar, and G. Sen- gupta, Physical Review E86(2012), 10.1103/phys- reve.86.052103

    doi:10.1103/phys- 2012

  11. [11]

    J. Luo, L. Xu, E. Lascaris, H. E. Stanley, and S. V. Buldyrev, Phys. Rev. Lett.112, 135701 (2014)

    2014

  12. [12]

    D. T. Banuti, M. Raju, and M. Ihme, Phys. Rev. E95, 052120 (2017)

    2017

  13. [13]

    Gallo, D

    P. Gallo, D. Corradini, and M. Rovere, Nature Commu- nications5, 5806 (2014)

    2014

  14. [14]

    T. J. Yoon, M. Y. Ha, W. B. Lee, and Y.-W. Lee, The Journal of Physical Chemistry Letters9, 4550–4554 (2018). 6

    2018

  15. [15]

    V. V. Brazhkin, Y. D. Fomin, A. G. Lyapin, V. N. Ryzhov, E. N. Tsiok, and K. Trachenko, Phys. Rev. Lett.111, 145901 (2013)

    2013

  16. [16]

    Bolmatov, V

    D. Bolmatov, V. V. Brazhkin, and K. Trachenko, Nature Communications4, 2331 (2013)

    2013

  17. [17]

    Prescher, Y

    C. Prescher, Y. D. Fomin, V. B. Prakapenka, J. Stefanski, K. Trachenko, and V. V. Brazhkin, Physical Review B 95(2017), 10.1103/physrevb.95.134114

    doi:10.1103/physrevb.95.134114 2017

  18. [18]

    Bolmatov, M

    D. Bolmatov, M. Zhernenkov, D. Zav’yalov, S. N. Tkachev, A. Cunsolo, and Y. Q. Cai, Scientific Reports 5(2015), 10.1038/srep15850

    doi:10.1038/srep15850 2015

  19. [19]

    Y. D. Fomin, V. N. Ryzhov, E. N. Tsiok, J. E. Proc- tor, C. Prescher, V. B. Prakapenka, K. Trachenko, and V. V. Brazhkin, Journal of Physics: Condensed Matter 30, 134003 (2018)

    2018

  20. [20]

    Y. D. Fomin, V. N. Ryzhov, E. N. Tsiok, and V. V. Brazhkin, Scientific Reports5, 14234 (2015)

    2015

  21. [21]

    V. V. Brazhkin, Y. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and K. Trachenko, Phys. Rev. E85, 031203 (2012)

    2012

  22. [22]

    Huang, M

    D. Huang, M. Baggioli, S. Lu, Z. Ma, and Y. Feng, Phys- ical Review Research5, 013149 (2023), arXiv:2301.08449 [physics.plasm-ph]

    Pith/arXiv arXiv 2023

  23. [23]

    Jiang, Z

    C. Jiang, Z. Zheng, Y. Chen, M. Baggioli, and J. Zhang, arXiv preprint arXiv:2403.08285 (2024)

    Pith/arXiv arXiv 2024

  24. [24]

    Zhao, Z.-Y

    Z.-Q. Zhao, Z.-Y. Nie, J.-F. Zhang, X. Zhang, and M. Baggioli, Eur. Phys. J. C85, 464 (2025), arXiv:2406.05345 [hep-th]

    Pith/arXiv arXiv 2025

  25. [25]

    Xu and R

    Z.-M. Xu and R. B. Mann, Phys. Rev. Lett.136, 041402 (2026), arXiv:2504.05708 [gr-qc]

    arXiv 2026

  26. [26]

    Zhao, Z.-Y

    Z.-Q. Zhao, Z.-Y. Nie, J.-F. Zhang, and X. Zhang, Chin. Phys. Lett.42, 101102 (2025), arXiv:2504.04995 [gr-qc]

    arXiv 2025

  27. [27]

    S. Wang, X. Li, Y. Jin, and L. Li, (2025), arXiv:2506.10808 [gr-qc]

    Pith/arXiv arXiv 2025

  28. [28]

    Li, X.-R

    Z.-Y. Li, X.-R. Chen, B. Wu, and Z.-M. Xu, (2025), arXiv:2511.10357 [hep-th]

    arXiv 2025

  29. [29]

    Guo and Z.-M

    F. Guo and Z.-M. Xu, (2026), arXiv:2602.09770 [hep-th]

    Pith/arXiv arXiv 2026

  30. [30]

    R. Li, K. Zhang, J. Yang, R. B. Mann, and J. Wang, Phys. Rev. D112, 064004 (2025), arXiv:2505.24148 [gr- qc]

    Pith/arXiv arXiv 2025

  31. [31]

    J. M. Maldacena, Adv. Theor. Math. Phys.2, 231 (1998), arXiv:hep-th/9711200

    Pith/arXiv arXiv 1998

  32. [32]

    S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, Phys. Rev. Lett.101, 031601 (2008), arXiv:0803.3295 [hep-th]

    Pith/arXiv arXiv 2008

  33. [33]

    S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, JHEP 12, 015 (2008), arXiv:0810.1563 [hep-th]

    Pith/arXiv arXiv 2008

  34. [34]

    C. P. Herzog, Phys. Rev. D81, 126009 (2010), arXiv:1003.3278 [hep-th]

    Pith/arXiv arXiv 2010

  35. [35]

    Q. Chen, Y. Liu, Y. Tian, B. Wang, C.-Y. Zhang, and H. Zhang, JHEP01, 056 (2023), arXiv:2209.12789 [hep- th]

    Pith/arXiv arXiv 2023

  36. [36]

    Li, Z.-Y

    X. Li, Z.-Y. Nie, and Y. Tian, JHEP09, 063 (2020), arXiv:2003.12987 [hep-th]

    Pith/arXiv arXiv 2020

  37. [37]

    Q. Chen, Y. Liu, Y. Tian, X. Wu, and H. Zhang, Phys. Rev. D108, 106017 (2023), arXiv:2211.11291 [hep-th]

    Pith/arXiv arXiv 2023

  38. [38]

    Xia and H.-B

    C.-Y. Xia and H.-B. Zeng, Phys. Rev. D102, 126005 (2020), arXiv:2009.00435 [hep-th]

    Pith/arXiv arXiv 2020

  39. [39]

    del Campo, F

    A. del Campo, F. J. G´ omez-Ruiz, Z.-H. Li, C.-Y. Xia, H.-B. Zeng, and H.-Q. Zhang, JHEP06, 061 (2021), arXiv:2101.02171 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2021

  40. [40]

    Li, H.-B

    Z.-H. Li, H.-B. Zeng, and H.-Q. Zhang, JHEP04, 295 (2021), arXiv:2101.08405 [hep-th]

    Pith/arXiv arXiv 2021

  41. [41]

    Li, C.-Y

    Z.-H. Li, C.-Y. Xia, H.-B. Zeng, and H.-Q. Zhang, JHEP 10, 124 (2021), arXiv:2103.01485 [hep-th]

    Pith/arXiv arXiv 2021

  42. [42]

    Xia and H.-B

    C.-Y. Xia and H.-B. Zeng, (2021), arXiv:2110.07969 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2021

  43. [43]

    Zeng, C.-Y

    H.-B. Zeng, C.-Y. Xia, and A. del Campo, Phys. Rev. Lett.130, 060402 (2023), arXiv:2204.13529 [cond- mat.stat-mech]

    Pith/arXiv arXiv 2023

  44. [44]

    Yang, C.-Y

    P. Yang, C.-Y. Xia, S. Grieninger, H.-B. Zeng, and M. Baggioli, Phys. Rev. Lett.136, 051602 (2026), arXiv:2508.05964 [cond-mat.stat-mech]

    arXiv 2026

  45. [45]

    C.-Y. Xia, A. Grabarits, H.-B. Zeng, and A. del Campo, (2026), arXiv:2601.14328 [hep-th]

    arXiv 2026

  46. [46]

    Xia, H.-B

    C.-Y. Xia, H.-B. Zeng, C.-M. Chen, and A. del Campo, Phys. Rev. D108, 026017 (2023), arXiv:2302.11597 [hep- th]

    Pith/arXiv arXiv 2023

  47. [47]

    Su, C.-Y

    J.-H. Su, C.-Y. Xia, W.-C. Yang, and H.-B. Zeng, Phys. Rev. D109, 046019 (2024), arXiv:2311.05856 [hep-th]

    Pith/arXiv arXiv 2024

  48. [48]

    Zhao, Z.-Y

    X. Zhao, Z.-Y. Nie, Z.-Q. Zhao, H.-B. Zeng, Y. Tian, and M. Baggioli, JHEP02, 184 (2024), arXiv:2311.08277 [hep-th]

    Pith/arXiv arXiv 2024

  49. [49]

    Y.-P. An, L. Li, C.-Y. Xia, and H.-B. Zeng, Phys. Rev. D 109, 106022 (2024), arXiv:2401.09189 [cond-mat.quant- gas]

    Pith/arXiv arXiv 2024

  50. [50]

    Yang, C.-Y

    W.-C. Yang, C.-Y. Xia, Y. Tian, M. Tsubota, and H.-B. Zeng, (2024), arXiv:2402.17980 [hep-th]

    Pith/arXiv arXiv 2024

  51. [51]

    Xia and H.-B

    C.-Y. Xia and H.-B. Zeng, Phys. Rev. D111, 026004 (2025), arXiv:2404.04274 [hep-th]

    Pith/arXiv arXiv 2025

  52. [52]

    Zeng, C.-Y

    H.-B. Zeng, C.-Y. Xia, W.-C. Yang, Y. Tian, and M. Tsubota, Phys. Rev. Lett.134, 091603 (2025), arXiv:2408.13620 [hep-th]

    Pith/arXiv arXiv 2025

  53. [53]

    S. A. Hartnoll, G. T. Horowitz, J. Kruthoff, and J. E. Santos, SciPost Phys.10, 009 (2021), arXiv:2008.12786 [hep-th]

    Pith/arXiv arXiv 2021

  54. [54]

    R.-G. Cai, L. Li, and R.-Q. Yang, JHEP03, 263 (2021), arXiv:2009.05520 [gr-qc]

    Pith/arXiv arXiv 2021

  55. [55]

    Y.-Q. Wang, Y. Song, Q. Xiang, S.-W. Wei, T. Zhu, and Y.-X. Liu, (2020), arXiv:2009.06277 [hep-th]

    Pith/arXiv arXiv 2020

  56. [56]

    Liu, H.-D

    Y. Liu, H.-D. Lyu, and A. Raju, JHEP10, 140 (2021), arXiv:2108.04554 [hep-th]

    Pith/arXiv arXiv 2021

  57. [57]

    Y.-S. An, L. Li, and F.-G. Yang, Phys. Rev. D104, 024040 (2021), arXiv:2106.01069 [gr-qc]

    Pith/arXiv arXiv 2021

  58. [58]

    S. A. H. Mansoori, L. Li, M. Rafiee, and M. Baggioli, JHEP10, 098 (2021), arXiv:2108.01471 [hep-th]

    Pith/arXiv arXiv 2021

  59. [59]

    R.-G. Cai, C. Ge, L. Li, and R.-Q. Yang, JHEP02, 139 (2022), arXiv:2112.04206 [gr-qc]

    Pith/arXiv arXiv 2022

  60. [60]

    Y.-S. An, L. Li, F.-G. Yang, and R.-Q. Yang, JHEP08, 133 (2022), arXiv:2205.02442 [hep-th]

    Pith/arXiv arXiv 2022

  61. [61]

    Sword and D

    L. Sword and D. Vegh, JHEP12, 045 (2022), arXiv:2210.01046 [hep-th]

    Pith/arXiv arXiv 2022

  62. [62]

    Liu and H.-D

    Y. Liu and H.-D. Lyu, JHEP09, 071 (2022), arXiv:2205.14803 [hep-th]

    Pith/arXiv arXiv 2022

  63. [63]

    Zhao, X.-K

    Z.-Q. Zhao, X.-K. Zhang, and Z.-Y. Nie, JHEP02, 023 (2023), arXiv:2211.14762 [hep-th]

    Pith/arXiv arXiv 2023

  64. [64]

    Y. Xu, D. Wang, and Q. Pan, Phys. Rev. D110, 046003 (2024), arXiv:2311.13145 [hep-th]

    Pith/arXiv arXiv 2024

  65. [65]

    L.-L. Gao, Y. Liu, and H.-D. Lyu, JHEP03, 034 (2023), arXiv:2301.01468 [hep-th]

    Pith/arXiv arXiv 2023

  66. [66]

    L.-L. Gao, Y. Liu, and H.-D. Lyu, JHEP01, 063 (2024), arXiv:2310.15781 [hep-th]

    Pith/arXiv arXiv 2024

  67. [67]

    Zhang, X

    X.-K. Zhang, X. Zhao, Z.-Y. Nie, Y.-P. Hu, and Y.-S. An, Phys. Lett. B868, 139684 (2025), arXiv:2411.07693 [hep-th]

    Pith/arXiv arXiv 2025

  68. [68]

    Zhang, X

    X.-K. Zhang, X. Zhao, Z.-Y. Nie, Y.-P. Hu, and Y.-S. An, Phys. Lett. B872, 140110 (2026), arXiv:2506.19419 [hep-th]. 7

    arXiv 2026

  69. [69]

    Y. Xu, L. Li, and W.-J. Li, (2025), arXiv:2511.00877 [hep-th]

    arXiv 2025

  70. [70]

    L.-L. Gao, Y. Liu, and R.-L. Zhao, (2025), arXiv:2511.19079 [hep-th]

    Pith/arXiv arXiv 2025

  71. [71]

    Xiong and H

    Z.-X. Xiong and H. Lu, (2026), arXiv:2602.22314 [gr-qc]

    arXiv 2026

  72. [72]

    Zhao, Z.-Y

    Z.-Q. Zhao, Z.-Y. Nie, S.-W. Wei, J.-F. Zhang, and X. Zhang, (2026), arXiv:2604.01818 [gr-qc]

    arXiv 2026

  73. [73]

    R. A. Janik, J. Jankowski, and H. Soltanpanahi, Phys. Rev. Lett.117, 091603 (2016), arXiv:1512.06871 [hep- th]

    Pith/arXiv arXiv 2016

  74. [74]

    R. A. Janik, J. Jankowski, and H. Soltanpanahi, JHEP 06, 047 (2016), arXiv:1603.05950 [hep-th]

    Pith/arXiv arXiv 2016

  75. [75]

    R. A. Janik, J. Jankowski, and H. Soltanpanahi, Phys. Rev. Lett.119, 261601 (2017), arXiv:1704.05387 [hep- th]

    Pith/arXiv arXiv 2017

  76. [76]

    Attems, Y

    M. Attems, Y. Bea, J. Casalderrey-Solana, D. Mateos, M. Triana, and M. Zilhao, JHEP06, 129 (2017), arXiv:1703.02948 [hep-th]

    Pith/arXiv arXiv 2017

  77. [77]

    Attems, Y

    M. Attems, Y. Bea, J. Casalderrey-Solana, D. Mateos, and M. Zilh˜ ao, JHEP01, 106 (2020), arXiv:1905.12544 [hep-th]

    Pith/arXiv arXiv 2020

  78. [78]

    Bellantuono, R

    L. Bellantuono, R. A. Janik, J. Jankowski, and H. Soltanpanahi, JHEP10, 146 (2019), arXiv:1906.00061 [hep-th]

    Pith/arXiv arXiv 2019

  79. [79]

    Attems, JHEP08, 155 (2021), arXiv:2012.15687 [hep- th]

    M. Attems, JHEP08, 155 (2021), arXiv:2012.15687 [hep- th]

    Pith/arXiv arXiv 2021

  80. [80]

    Z. Ning, Q. Chen, Y. Tian, X. Wu, and H. Zhang, Phys. Rev. D109, 064082 (2024), arXiv:2307.14156 [gr-qc]

    Pith/arXiv arXiv 2024

Showing first 80 references.