pith. sign in

arxiv: 2511.14114 · v2 · submitted 2025-11-18 · ❄️ cond-mat.quant-gas · cond-mat.stat-mech· quant-ph

Dynamics of entanglement asymmetry for space-inversion symmetry of free fermions on honeycomb lattices

Ryogo Hara , Shimpei Endo , Shion Yamashika This is my paper

Pith reviewed 2026-05-17 21:23 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas cond-mat.stat-mechquant-ph
keywords entanglement asymmetryspace-inversion symmetryhoneycomb latticefree fermionsDirac pointsflat bandquench dynamicssublattice imbalance
0
0 comments X

The pith

Entanglement asymmetry in honeycomb fermions varies nonanalytically with energy imbalance and relaxes to a finite value after quench due to flat bands.

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

This paper studies the entanglement asymmetry for space-inversion symmetry in free fermions on a two-dimensional honeycomb lattice that includes an on-site energy difference between the two sublattices. The asymmetry for a local subsystem changes in a sharp, non-smooth manner as the energy imbalance is varied, which arises from the Dirac points where bands touch in the Brillouin zone. After a sudden quench that removes the imbalance and restores inversion symmetry to the Hamiltonian, the asymmetry does not decay to zero for certain subsystem geometries. Instead it approaches a nonzero constant at long times, which the work links to the existence of a flat band along one lattice direction that limits the information flow required for full symmetry restoration.

Core claim

We show that the entanglement asymmetry of a local subsystem exhibits nonanalytic dependence on the energy imbalance, due to the presence of Dirac points in the Brillouin zone. We also study the quench dynamics from the ground state into the inversion-symmetric point at which the energy imbalance vanishes. Under certain conditions on the subsystem geometry, the entanglement asymmetry relaxes to a finite value after the quench, revealing that the inversion-symmetry breaking in the initial ground state can persist even under the symmetric dynamics. We attribute the absence of symmetry restoration to the presence of a flat energy dispersion (flat band) in a specific direction.

What carries the argument

Entanglement asymmetry for space-inversion symmetry of a local subsystem, which measures deviation from inversion invariance in the reduced density matrix and whose dynamics are controlled by the honeycomb lattice band structure including Dirac points and flat bands.

Load-bearing premise

The fermions are non-interacting and placed on a perfect honeycomb lattice whose only adjustable parameter is the sublattice energy imbalance, with subsystem geometries chosen so that the flat-band direction controls the long-time behavior.

What would settle it

Compute or measure the long-time limit of entanglement asymmetry after quenching the sublattice imbalance exactly to zero for a subsystem geometry aligned with the flat-band direction; if the value decays all the way to zero instead of remaining finite, the persistence claim is false.

Figures

Figures reproduced from arXiv: 2511.14114 by Ryogo Hara, Shimpei Endo, Shion Yamashika.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of the transformation from the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The energy spectra of the Hamiltonian in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic illustration of the bipartition of the whole [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a)-(c) Time evolution of the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

We study the entanglement asymmetry for the space-inversion symmetry of free fermions on a two-dimensional honeycomb lattice with an on-site energy imbalance between the two sublattices. We show that the entanglement asymmetry of a local subsystem exhibits nonanalytic dependence on the energy imbalance, due to the presence of Dirac points in the Brillouin zone. We also study the quench dynamics from the ground state into the inversion-symmetric point at which the energy imbalance vanishes. Under certain conditions on the subsystem geometry, the entanglement asymmetry relaxes to a finite value after the quench, revealing that the inversion-symmetry breaking in the initial ground state can persist even under the symmetric dynamics. We attribute the absence of symmetry restoration to the presence of a flat energy dispersion (flat band) in a specific direction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates entanglement asymmetry for space-inversion symmetry in free fermions on a honeycomb lattice with tunable sublattice energy imbalance. It claims that the asymmetry for a local subsystem exhibits nonanalytic dependence on the imbalance parameter, originating from Dirac points in the Brillouin zone. The authors also consider a quench from the imbalanced ground state to the inversion-symmetric Hamiltonian (zero imbalance) and report that, for particular subsystem geometries, the asymmetry relaxes to a nonzero finite value at long times rather than restoring to zero. This persistence is attributed to a flat dispersion relation along a specific lattice direction that prevents complete dephasing of the initial symmetry-breaking correlations.

Significance. If the reported finite post-quench limit proves robust, the work would add a concrete example of how band-structure features (Dirac points and flat bands) control entanglement-based symmetry diagnostics in exactly solvable lattice models. The nonanalyticity result and the exact free-fermion treatment provide falsifiable, parameter-free predictions that could be tested numerically or in cold-atom realizations of honeycomb lattices. The findings sit at the intersection of quantum information and non-equilibrium condensed-matter physics and may stimulate analogous studies in other Dirac or flat-band systems.

major comments (2)
  1. [Quench dynamics and long-time limit discussion] The central claim that inversion-symmetry breaking persists after the quench (abstract and quench-dynamics section) rests on the long-time limit of the entanglement asymmetry for specific subsystem shapes. The manuscript does not demonstrate that this finite offset survives when the subsystem boundary is rotated relative to the flat-band direction or replaced by a generic (e.g., circular) region; the overlap with zero-velocity modes projected onto the subsystem could vanish for misaligned cuts, turning the reported persistence into a geometry-tuned artifact rather than a general consequence of the flat band.
  2. [Section on nonanalytic dependence (near Eq. for asymmetry)] The nonanalytic dependence of the entanglement asymmetry on the energy imbalance is asserted to arise from Dirac points, yet the text provides no explicit analytic derivation or finite-size scaling analysis showing how the singularity appears in the correlation-matrix eigenvalues for the chosen subsystems. Without this, it remains unclear whether the nonanalyticity is a bulk effect or an artifact of the subsystem geometry and cutoff.
minor comments (2)
  1. [Methods] Notation for the correlation matrix and the definition of the inversion operator should be stated once in a dedicated subsection to avoid repeated redefinitions across sections.
  2. [Figures 3 and 4] Figure captions for the post-quench time evolution should explicitly state the lattice orientation of the subsystem relative to the flat-band direction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and have revised the manuscript accordingly to improve clarity and strengthen the presentation.

read point-by-point responses

  1. Referee: [Quench dynamics and long-time limit discussion] The central claim that inversion-symmetry breaking persists after the quench (abstract and quench-dynamics section) rests on the long-time limit of the entanglement asymmetry for specific subsystem shapes. The manuscript does not demonstrate that this finite offset survives when the subsystem boundary is rotated relative to the flat-band direction or replaced by a generic (e.g., circular) region; the overlap with zero-velocity modes projected onto the subsystem could vanish for misaligned cuts, turning the reported persistence into a geometry-tuned artifact rather than a general consequence of the flat band.

    Authors: We agree that the reported persistence is geometry-dependent and occurs only for specific subsystem shapes aligned with the flat-band direction. The manuscript already qualifies this result as holding 'under certain conditions on the subsystem geometry' and attributes it to nonzero overlap with zero-velocity modes. This dependence is a direct consequence of the band structure rather than an unintended artifact. To address the concern, we have added explicit discussion and numerical results for rotated boundaries and circular subsystems, where the long-time asymmetry relaxes to zero as expected when the projection onto flat-band modes vanishes. This clarifies that the effect illustrates how flat bands control dephasing in a geometry-sensitive manner. revision: yes

  2. Referee: [Section on nonanalytic dependence (near Eq. for asymmetry)] The nonanalytic dependence of the entanglement asymmetry on the energy imbalance is asserted to arise from Dirac points, yet the text provides no explicit analytic derivation or finite-size scaling analysis showing how the singularity appears in the correlation-matrix eigenvalues for the chosen subsystems. Without this, it remains unclear whether the nonanalyticity is a bulk effect or an artifact of the subsystem geometry and cutoff.

    Authors: The nonanalyticity stems from the Dirac points because the correlation-matrix eigenvalues are obtained from momentum-space integrals over the Brillouin zone, where the linear dispersion at the Dirac points produces a nonanalytic contribution as a function of the sublattice imbalance. We have added an appendix containing the explicit analytic derivation of this contribution to the correlation matrix and its effect on the entanglement asymmetry. We have also included finite-size scaling plots for the asymmetry as a function of imbalance across increasing subsystem sizes, confirming that the singularity survives in the thermodynamic limit and is therefore a bulk feature tied to the Dirac points rather than a cutoff or geometry artifact. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows directly from free-fermion correlation-matrix analysis

full rationale

The manuscript computes the entanglement asymmetry explicitly from the two-point correlation matrix of a free-fermion Hamiltonian on the honeycomb lattice. Nonanalytic dependence on sublattice imbalance arises from the Dirac-point structure in the Brillouin zone, and the post-quench finite limit is obtained by projecting the initial-state correlations onto zero-velocity modes whose dispersion is flat along one lattice direction. These steps are algebraic consequences of the model definition and the chosen subsystem geometry; no fitted parameters are relabeled as predictions, no self-citation supplies an unverified uniqueness theorem, and the central claims do not reduce to redefinitions of the input quantities. The analysis is therefore self-contained against the stated Hamiltonian and geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claims rest on the standard tight-binding description of free fermions on a honeycomb lattice, the existence of Dirac points at the Brillouin-zone corners, and the presence of a flat band along a specific momentum direction; no free parameters, ad-hoc axioms, or invented entities are mentioned.

axioms (2)
  • domain assumption The system is described by non-interacting fermions on a honeycomb lattice with tunable sublattice energy offset.
    Invoked throughout the abstract as the physical setting for both the static and quench calculations.
  • domain assumption Dirac points exist in the Brillouin zone and control the nonanalyticity of the entanglement asymmetry.
    Stated as the reason for the nonanalytic dependence on energy imbalance.

pith-pipeline@v0.9.0 · 5435 in / 1531 out tokens · 34475 ms · 2026-05-17T21:23:17.824935+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

  1. Enhancing entanglement asymmetry in fragmented quantum systems

    cond-mat.stat-mech 2026-03 unverdicted novelty 6.0

    Entanglement asymmetry for inhomogeneous U(1) charges in fragmented systems scales extensively, is bounded by a universal fraction of its maximum, and distinguishes classical from quantum fragmentation.

Reference graph

Works this paper leans on

105 extracted references · 105 canonical work pages · cited by 1 Pith paper

  1. [1]

    Polkovnikov, K

    A. Polkovnikov, K. Sengupta, A. Silva, and M. Ven- galattore, Reviews of Modern Physics83, 863 (2011)

    work page 2011

  2. [2]

    Eisert, M

    J. Eisert, M. Friesdorf, and C. Gogolin, Nature Physics 11, 124 (2015)

    work page 2015

  3. [3]

    D’Alessio, Y

    L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Advances in Physics65, 239 (2016)

    work page 2016

  4. [4]

    Borgonovi, F

    F. Borgonovi, F. M. Izrailev, L. F. Santos, and V. G. Zelevinsky, Physics Reports626, 1 (2016)

    work page 2016

  5. [5]

    J. M. Deutsch, Physical Review A43, 2046 (1991)

    work page 2046

  6. [6]

    Srednicki, Physical Review E50, 888 (1994)

    M. Srednicki, Physical Review E50, 888 (1994)

    work page 1994

  7. [7]

    Srednicki, Journal of Physics A: Mathematical and General32, 1163 (1999)

    M. Srednicki, Journal of Physics A: Mathematical and General32, 1163 (1999)

    work page 1999

  8. [8]

    Rigol, V

    M. Rigol, V. Dunjko, and M. Olshanii, Nature452, 854 (2008)

    work page 2008

  9. [9]

    A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science353, 12 794 (2016)

    work page 2016

  10. [10]

    Tasaki, Physical Review Letters80, 1373 (1998)

    H. Tasaki, Physical Review Letters80, 1373 (1998)

    work page 1998

  11. [11]

    Popescu, A

    S. Popescu, A. J. Short, and A. Winter, Nature Physics 2, 754 (2006)

    work page 2006

  12. [12]

    Goldstein, J

    S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zangh` ı, Physical Review Letters96, 050403 (2006)

    work page 2006

  13. [13]

    Reimann, Physical Review Letters101, 190403 (2008)

    P. Reimann, Physical Review Letters101, 190403 (2008)

    work page 2008

  14. [14]

    Linden, S

    N. Linden, S. Popescu, A. J. Short, and A. Winter, Physical Review E79, 061103 (2009)

    work page 2009

  15. [15]

    A. J. Short and T. C. Farrelly, New Journal of Physics 14, 013063 (2012)

    work page 2012

  16. [16]

    Reimann and M

    P. Reimann and M. Kastner, New Journal of Physics 14, 043020 (2012)

    work page 2012

  17. [17]

    Gogolin and J

    C. Gogolin and J. Eisert, Reports on Progress in Physics 79, 056001 (2016)

    work page 2016

  18. [18]

    Rigol, V

    M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Physical Review Letters98, 050405 (2007)

    work page 2007

  19. [19]

    Caux and F

    J.-S. Caux and F. H. Essler, Physical Review Letters 110, 257203 (2013)

    work page 2013

  20. [20]

    Ilievski, J

    E. Ilievski, J. De Nardis, B. Wouters, J.-S. Caux, F. H. Essler, and T. Prosen, Physical Review Letters115, 157201 (2015)

    work page 2015

  21. [21]

    Vidmar and M

    L. Vidmar and M. Rigol, Journal of Statistical Mechan- ics: Theory and Experiment2016, 064007 (2016)

    work page 2016

  22. [22]

    F. H. Essler and M. Fagotti, Journal of Statistical Me- chanics: Theory and Experiment2016, 064002 (2016)

    work page 2016

  23. [23]

    Langen, S

    T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmiedmayer, Science348, 207 (2015)

    work page 2015

  24. [24]

    Barthel and U

    T. Barthel and U. Schollw¨ ock, Physical Review Letters 100, 100601 (2008)

    work page 2008

  25. [25]

    Cramer, C

    M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, Physical Review Letters100, 030602 (2008)

    work page 2008

  26. [26]

    Calabrese, F

    P. Calabrese, F. H. Essler, and M. Fagotti, Journal of Statistical Mechanics: Theory and Experiment2012, P07022 (2012)

    work page 2012

  27. [27]

    Neyenhuis, J

    B. Neyenhuis, J. Zhang, P. W. Hess, J. Smith, A. C. Lee, P. Richerme, Z.-X. Gong, A. V. Gorshkov, and C. Monroe, Science advances3, e1700672 (2017)

    work page 2017

  28. [28]

    Brydges, A

    T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Science364, 260 (2019)

    work page 2019

  29. [29]

    H. B. Kaplan, L. Guo, W. L. Tan, A. De, F. Marquardt, G. Pagano, and C. Monroe, Physical Review Letters 125, 120605 (2020)

    work page 2020

  30. [30]

    Kinoshita, T

    T. Kinoshita, T. Wenger, and D. S. Weiss, Nature440, 900 (2006)

    work page 2006

  31. [31]

    Cheneau, P

    M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauß, T. Fukuhara, C. Gross, I. Bloch, C. Kollath, and S. Kuhr, Nature481, 484 (2012)

    work page 2012

  32. [32]

    Islam, R

    R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin, M. Rispoli, and M. Greiner, Nature528, 77 (2015)

    work page 2015

  33. [33]

    Amico, R

    L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Reviews of Modern Physics80, 517 (2008)

    work page 2008

  34. [34]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio, Reviews of Modern Physics82, 277 (2010)

    work page 2010

  35. [35]

    Calabrese, J

    P. Calabrese, J. Cardy, and B. Doyon, Journal of Physics A: Mathematical and Theoretical42, 500301 (2009)

    work page 2009

  36. [36]

    Laflorencie, Physics Reports646, 1 (2016)

    N. Laflorencie, Physics Reports646, 1 (2016)

    work page 2016

  37. [37]

    Calabrese and J

    P. Calabrese and J. Cardy, Journal of Statistical Me- chanics: Theory and Experiment2005, P04010 (2005)

    work page 2005

  38. [38]

    Fagotti and P

    M. Fagotti and P. Calabrese, Physical Review A78, 010306 (2008)

    work page 2008

  39. [39]

    Alba and P

    V. Alba and P. Calabrese, Proceedings of the National Academy of Sciences114, 7947 (2017)

    work page 2017

  40. [40]

    F. Ares, S. Murciano, and P. Calabrese, Nature Com- munications14, 2036 (2023)

    work page 2036

  41. [41]

    Murciano, F

    S. Murciano, F. Ares, I. Klich, and P. Calabrese, Journal of Statistical Mechanics: Theory and Experiment2024, 013103 (2024)

    work page 2024

  42. [42]

    Yamashika, F

    S. Yamashika, F. Ares, and P. Calabrese, Physical Re- view B110, 085126 (2024)

    work page 2024

  43. [43]

    Liu, H.-K

    S. Liu, H.-K. Zhang, S. Yin, and S.-X. Zhang, Physical Review Letters133, 140405 (2024)

    work page 2024

  44. [44]

    Chalas, F

    K. Chalas, F. Ares, C. Rylands, and P. Calabrese, Jour- nal of Statistical Mechanics: Theory and Experiment 2024, 103101 (2024)

    work page 2024

  45. [45]

    Caceffo, S

    F. Caceffo, S. Murciano, and V. Alba, Journal of Statis- tical Mechanics: Theory and Experiment2024, 063103 (2024)

    work page 2024

  46. [46]

    Rylands, E

    C. Rylands, E. Vernier, and P. Calabrese, Journal of Statistical Mechanics: Theory and Experiment2024, 123102 (2024)

    work page 2024

  47. [47]

    L. K. Joshi, J. Franke, A. Rath, F. Ares, S. Murciano, F. Kranzl, R. Blatt, P. Zoller, B. Vermersch, P. Cal- abrese,et al., Physical Review Letters133, 010402 (2024)

    work page 2024

  48. [48]

    Rylands, K

    C. Rylands, K. Klobas, F. Ares, P. Calabrese, S. Mur- ciano, and B. Bertini, Physical Review Letters133, 010401 (2024)

    work page 2024

  49. [49]

    F. Ares, V. Vitale, and S. Murciano, Physical Review B 111, 104312 (2025)

    work page 2025

  50. [50]

    Di Giulio, X

    G. Di Giulio, X. Turkeshi, and S. Murciano, Entropy 27, 407 (2025)

    work page 2025

  51. [51]

    Banerjee, S

    T. Banerjee, S. Das, and K. Sengupta, SciPost Physics 19, 051 (2025)

    work page 2025

  52. [52]

    Turkeshi, P

    X. Turkeshi, P. Calabrese, and A. De Luca, Physical Review Letters135, 040403 (2025)

    work page 2025

  53. [53]

    Klobas, C

    K. Klobas, C. Rylands, and B. Bertini, Physical Review B111, L140304 (2025)

    work page 2025

  54. [54]

    Yu, Z.-X

    H. Yu, Z.-X. Li, and S.-X. Zhang, Chinese Physics Let- ters (2025)

    work page 2025

  55. [55]

    Yu, T.-R

    Y.-H. Yu, T.-R. Jin, L. Zhang, K. Xu, and H. Fan, Phys- ical Review B112, 094315 (2025)

    work page 2025

  56. [56]

    Yamashika and F

    S. Yamashika and F. Ares, arXiv:2507.06636

    work page arXiv

  57. [57]

    Xuet al., Observation and Modulation of the Quantum Mpemba Effect on a Superconducting Quantum Processor, (2025), arXiv:2508.07707 [quant-ph]

    Y. Xu, C.-P. Fang, B.-J. Chen, M.-C. Wang, Z.-Y. Ge, Y.-H. Shi, Y. Liu, C.-L. Deng, K. Zhao, Z.-H. Liu,et al., arXiv:2508.07707

    work page arXiv

  58. [58]

    H. Yu, S. Liu, and S.-X. Zhang, AAPPS Bulletin35, 17 (2025)

    work page 2025

  59. [59]

    F. Ares, P. Calabrese, and S. Murciano, Nature Reviews Physics7, 451 (2025)

    work page 2025

  60. [60]

    G. Teza, J. Bechhoefer, A. Lasanta, O. Raz, and M. Vucelja, arXiv:2502.01758

    work page arXiv

  61. [61]

    F. Ares, S. Murciano, E. Vernier, and P. Calabrese, Sci- Post Physics15, 089 (2023)

    work page 2023

  62. [62]

    Yamashika, P

    S. Yamashika, P. Calabrese, and F. Ares, Physical Re- view A111, 043304 (2025)

    work page 2025

  63. [63]

    Capizzi and V

    L. Capizzi and V. Vitale, Journal of Physics A: Mathe- matical and Theoretical57, 45LT01 (2024)

    work page 2024

  64. [64]

    Mazzoni, L

    M. Mazzoni, L. Capizzi, and L. Piroli, arXiv:2508.15892

    work page arXiv

  65. [65]

    Fossati, F

    M. Fossati, F. Ares, J. Dubail, and P. Calabrese, Journal of High Energy Physics2024, 1 (2024)

    work page 2024

  66. [66]

    Chen and H.-H

    M. Chen and H.-H. Chen, Physical Review D109, 065009 (2024)

    work page 2024

  67. [67]

    Kusuki, S

    Y. Kusuki, S. Murciano, H. Ooguri, and S. Pal, Journal of High Energy Physics2025, 1 (2025). 13

    work page 2025

  68. [68]

    Fossati, C

    M. Fossati, C. Rylands, and P. Calabrese, Journal of High Energy Physics2025, 1 (2025)

    work page 2025

  69. [69]

    F. Ares, S. Murciano, L. Piroli, and P. Calabrese, Phys- ical Review D110, L061901 (2024)

    work page 2024

  70. [70]

    Yamashika, F

    S. Yamashika, F. Ares, and P. Calabrese, Physical Re- view B109, 125122 (2024)

    work page 2024

  71. [71]

    Gibbins, A

    M. Gibbins, A. Jafarizadeh, A. Gammon-Smith, and B. Bertini, Physical Review B109, 224310 (2024)

    work page 2024

  72. [72]

    Travaglino, C

    R. Travaglino, C. Rylands, and P. Calabrese, Journal of Statistical Mechanics: Theory and Experiment2025, 033102 (2025)

    work page 2025

  73. [73]

    Russotto, F

    A. Russotto, F. Ares, and P. Calabrese, Journal of High Energy Physics2025, 1 (2025)

    work page 2025

  74. [74]

    Gibbins, A

    M. Gibbins, A. Smith, and B. Bertini, arXiv:2506.14555

    work page arXiv

  75. [75]

    Benini, P

    F. Benini, P. Calabrese, M. Fossati, A. H. Singh, and M. Venuti, arXiv:2509.16311

    work page arXiv

  76. [76]

    S. A. Ahmad, M. S. Klinger, and Y. Wang, arXiv:2509.18072

    work page arXiv

  77. [77]

    A. H. Castro Neto, F. Guinea, N. M. Peres, K. S. Novoselov, and A. K. Geim, Reviews of Modern Physics 81, 109 (2009)

    work page 2009

  78. [78]

    P. R. Wallace, Physical Review71, 622 (1947)

    work page 1947

  79. [79]

    G. W. Semenoff, Physical Review Letters53, 2449 (1984)

    work page 1984

  80. [80]

    K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature438, 197 (2005)

    work page 2005

Showing first 80 references.