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Spectral statistics, nonequilibrium dynamics, and thermalization in random matrices with global Z2 symmetry

Adway Kumar Das

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Phys. Rev. B 113, 144201 – Published 2 April, 2026

DOI: https://doi.org/10.1103/8v6c-qydj

Abstract

Z2 symmetry is ubiquitous in quantum mechanics, where it drives various phase transitions and emergent physics. The role of Z2 symmetry in the thermalization of a local observable in a disordered system can be understood using random matrix theory. To do so, we consider random symmetric centrosymmetric (SC) matrix as a toy model where a Z2 symmetry, namely, the exchange symmetry, is conserved. Such a conservation law splits the Hilbert space into decoupled subspaces such that the energy spectrum of an SC matrix is a superposition of two pure spectra. After discussing the known results on the correlations of such mixed spectra, we consider different initial states and analytically compute the time evolution of their survival probability and associated timescales. We show that there exist certain low-energy initial states which do not decay over a very long timescales such that a measure zero fraction of random SC matrices exhibit spontaneous symmetry breaking. Later, we look at the equilibrium values of local observables like the density-density correlation, kinetic energy operator, and compare them against the average values from the microcanonical and canonical ensembles. We find that when the observable violates (respects) the global symmetry of the Hamiltonian, the equilibrium value is independent (dependent) of the symmetry of the initial state. However, irrespective of such symmetry constraints, the fluctuations of the diagonal terms of the observables within microcanonical shells decay with system size such that the ansatz of the eigenstate thermalization hypothesis remains valid. We show that the equilibrium value converges to the canonical average for all the observables and initial states, indicating that thermalization occurs despite the presence of a global symmetry.

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References (148)

  1. For a Hilbert space with dimension N, an ergodic state is uniformly distributed in the (N−1)-dimension unit hypersphere [147, 148].
  2. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  3. A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83, 863 (2011).
  4. R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  5. F. Borgonovi, F. Izrailev, L. Santos, and V. Zelevinsky, Quantum chaos and thermalization in isolated systems of interacting particles, Phys. Rep. 626, 1 (2016).
  6. J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog. Phys. 81, 082001 (2018).
  7. E. J. Torres-Herrera and L. F. Santos, Extended nonergodic states in disordered many-body quantum systems, Ann. Phys. (Berlin, Ger.) 529, 1600284 (2017).
  8. M. Pino, L. B. Ioffe, and B. L. Altshuler, Nonergodic metallic and insulating phases of Josephson junction chains, Proc. Natl. Acad. Sci. USA 113, 536 (2016).
  9. L. Vidmar and M. Rigol, Generalized Gibbs ensemble in integrable lattice models, J. Stat. Mech. (2016) 064007.
  10. T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, Thermalization and prethermalization in isolated quantum systems: A theoretical overview, J. Phys. B: At. Mol. Opt. Phys. 51, 112001 (2018).
  11. C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Weak ergodicity breaking from quantum many-body scars, Nat. Phys. 14, 745 (2018).
  12. S. Sinha and S. Sinha, Chaos and quantum scars in Bose-Josephson junction coupled to a bosonic mode, Phys. Rev. Lett. 125, 134101 (2020).
  13. H. Dong, J.-Y. Desaules, Y. Gao, N. Wang, Z. Guo, J. Chen, Y. Zou, F. Jin, X. Zhu, P. Zhang, H. Li, Z. Wang, Q. Guo, J. Zhang, L. Ying, and Z. Papić, Disorder-tunable entanglement at infinite temperature, Sci. Adv. 9, eadj3822 (2023).
  14. A. Pizzi, L.-H. Kwan, B. Evrard, C. B. Dag, and J. Knolle, Genuine quantum scars in many-body spin systems, Nat. Commun. 16, 6722 (2025).
  15. S. Roy and A. Lazarides, Strong ergodicity breaking due to local constraints in a quantum system, Phys. Rev. Res. 2, 023159 (2020).
  16. K. Royen, S. Mondal, F. Pollmann, and F. Heidrich-Meisner, Enhanced many-body localization in a kinetically constrained model, Phys. Rev. E 109, 024136 (2024).
  17. T. Bhore, J.-Y. Desaules, and Z. Papić, Deep thermalization in constrained quantum systems, Phys. Rev. B 108, 104317 (2023).
  18. A. K. Das and A. Lazarides, Ground state and persistent oscillations in the quantum east model, arXiv:2602.23422.
  19. M. Serbyn, Z. Papić, and D. A. Abanin, Local conservation laws and the structure of the many-body localized states, Phys. Rev. Lett. 111, 127201 (2013).
  20. O. Shtanko, D. S. Wang, H. Zhang, N. Harle, A. Seif, R. Movassagh, and Z. Minev, Uncovering local integrability in quantum many-body dynamics, Nat. Commun. 16, 2552 (2025).
  21. A. K. Kulshreshtha, A. Pal, T. B. Wahl, and S. H. Simon, Behavior of l-bits near the many-body localization transition, Phys. Rev. B 98, 184201 (2018).
  22. P. Łydżba, Y. Zhang, M. Rigol, and L. Vidmar, Single-particle eigenstate thermalization in quantum-chaotic quadratic Hamiltonians, Phys. Rev. B 104, 214203 (2021).
  23. R. Modak, S. Mukerjee, E. A. Yuzbashyan, and B. S. Shastry, Integrals of motion for one-dimensional Anderson localized systems, New J. Phys. 18, 033010 (2016).
  24. A. K. Das, A. Ghosh, and I. M. Khaymovich, Absence of mobility edge in short-range uncorrelated disordered model: Coexistence of localized and extended states, Phys. Rev. Lett. 131, 166401 (2023).
  25. A. K. Das and A. Ghosh, Nonergodic extended states in the β ensemble, Phys. Rev. E 105, 054121 (2022).
  26. A. K. Das, A. Ghosh, and I. M. Khaymovich, Robust nonergodicity of the ground states in the β ensemble, Phys. Rev. B 109, 064206 (2024).
  27. A. K. Das, A. Ghosh, and I. M. Khaymovich, Emergent multifractality in power-law decaying eigenstates, Phys. Rev. B 112, 024201 (2025).
  28. A. K. Das and A. Ghosh, Eigenvalue statistics for generalized symmetric and Hermitian matrices, J. Phys. A: Math. Theor. 52, 395001 (2019).
  29. A. K. Das and A. Ghosh, Dynamical signatures of chaos to integrability crossover in 2×2 generalized random matrix ensembles, J. Phys. A: Math. Theor. 56, 495003 (2023).
  30. M. V. Berry and P. Shukla, Spacing distributions for real symmetric 2×2 generalized Gaussian ensembles, J. Phys. A: Math. Theor. 42, 485102 (2009).
  31. A. K. Das and A. Ghosh, Transport in deformed centrosymmetric networks, Phys. Rev. E 106, 064112 (2022).
  32. A. K. Das and A. Ghosh, Chaos due to symmetry-breaking in deformed Poisson ensemble, J. Stat. Mech. (2022) 063101.
  33. A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, and D. Sen, Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information (Cambridge University Press, Cambridge, 2015).
  34. D. Mondal, S. Sinha, and S. Sinha, Chaos and quantum scars in a coupled top model, Phys. Rev. E 102, 020101(R) (2020).
  35. Q. Wang and F. Pérez-Bernal, Signatures of excited-state quantum phase transitions in quantum many-body systems: Phase space analysis, Phys. Rev. E 104, 034119 (2021).
  36. O. Castaños, R. López-Peña, J. G. Hirsch, and E. López-Moreno, Classical and quantum phase transitions in the Lipkin-Meshkov-Glick model, Phys. Rev. B 74, 104118 (2006).
  37. L. F. Santos, M. Távora, and F. Pérez-Bernal, Excited-state quantum phase transitions in many-body systems with infinite-range interaction: Localization, dynamics, and bifurcation, Phys. Rev. A 94, 012113 (2016).
  38. P. Pérez-Fernández, A. Relaño, J. M. Arias, P. Cejnar, J. Dukelsky, and J. E. García-Ramos, Excited-state phase transition and onset of chaos in quantum optical models, Phys. Rev. E 83, 046208 (2011).
  39. M. Kloc, P. Stránský, and P. Cejnar, Quantum quench dynamics in Dicke superradiance models, Phys. Rev. A 98, 013836 (2018).
  40. M.-J. Hwang, R. Puebla, and M. B. Plenio, Quantum phase transition and universal dynamics in the Rabi model, Phys. Rev. Lett. 115, 180404 (2015).
  41. T. Kinoshita, T. Wenger, and D. S. Weiss, A quantum Newton’s cradle, Nature (London) 440, 900 (2006).
  42. S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, Non-equilibrium coherence dynamics in one-dimensional Bose gases, Nature (London) 449, 324 (2007).
  43. M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer, Relaxation and prethermalization in an isolated quantum system, Science 337, 1318 (2012).
  44. T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmiedmayer, Experimental observation of a generalized Gibbs ensemble, Science 348, 207 (2015).
  45. K. Mallayya and M. Rigol, Prethermalization, thermalization, and Fermi’s golden rule in quantum many-body systems, Phys. Rev. B 104, 184302 (2021).
  46. P. Krupchitsky and V. Kisin, Fundamental Research with Polarized Slow Neutrons (Springer, Berlin Heidelberg, 2014).
  47. G. E. Mitchell, E. G. Bilpuch, P. M. Endt, and J. F. Shriner, Broken symmetries and chaotic behavior in A26, Phys. Rev. Lett. 61, 1473 (1988).
  48. T. Guhr and H. Weidenmüller, Isospin mixing and spectral fluctuation properties, Ann. Phys. (NY) 199, 412 (1990).
  49. A. Adams, G. Mitchell, and J. Shriner Jr., Statistical distribution of reduced transition probabilities in Al26, Phys. Lett. B 422, 13 (1998).
  50. T. Scholak, F. de Melo, T. Wellens, F. Mintert, and A. Buchleitner, Efficient and coherent excitation transfer across disordered molecular networks, Phys. Rev. E 83, 021912 (2011).
  51. T. Zech, R. Mulet, T. Wellens, and A. Buchleitner, Centrosymmetry enhances quantum transport in disordered molecular networks, New J. Phys. 16, 055002 (2014).
  52. A. Ortega, T. Stegmann, and L. Benet, Efficient quantum transport in disordered interacting many-body networks, Phys. Rev. E 94, 042102 (2016).
  53. C. Ellegaard, T. Guhr, K. Lindemann, J. Nygård, and M. Oxborrow, Symmetry breaking and spectral statistics of acoustic resonances in quartz blocks, Phys. Rev. Lett. 77, 4918 (1996).
  54. K. Schaadt, A. P. B. Tufaile, and C. Ellegaard, Chaotic sound waves in a regular billiard, Phys. Rev. E 67, 026213 (2003).
  55. J. X. de Carvalho, M. S. Hussein, M. P. Pato, and A. J. Sargeant, Symmetry-breaking study with deformed ensembles, Phys. Rev. E 76, 066212 (2007).
  56. J. X. de Carvalho, S. Jalan, and M. S. Hussein, Deformed Gaussian-orthogonal-ensemble description of small-world networks, Phys. Rev. E 79, 056222 (2009).
  57. S. Jalan, Spectral analysis of deformed random networks, Phys. Rev. E 80, 046101 (2009).
  58. P. Facchi and S. Pascazio, Quantum Zeno subspaces, Phys. Rev. Lett. 89, 080401 (2002).
  59. L. F. Santos, F. Borgonovi, and G. L. Celardo, Cooperative shielding in many-body systems with long-range interaction, Phys. Rev. Lett. 116, 250402 (2016).
  60. A. Gersho, Adaptive equalization of highly dispersive channels for data transmission, Bell Syst. Tech. J. 48, 55 (1969).
  61. F. Magee and J. Proakis, An estimate of the upper bound on error probability for maximum-likelihood sequence estimation on channels having a finite-duration pulse response (Corresp.), IEEE Trans. Inf. Theory 19, 699 (1973).
  62. L. Datta and S. D. Morgera, On the reducibility of centrosymmetric matrices—applications in engineering problems, Circuits Syst. Signal Proces. 8, 71 (1989).
  63. P. Gaudreau and H. Safouhi, Centrosymmetric matrices in the sinc collocation method for Sturm-Liouville problems, EPJ Web Conf. 108, 01004 (2016).
  64. M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, Perfect state transfer in quantum spin networks, Phys. Rev. Lett. 92, 187902 (2004).
  65. P. Karbach and J. Stolze, Spin chains as perfect quantum state mirrors, Phys. Rev. A 72, 030301(R) (2005).
  66. T. Shi, Y. Li, Z. Song, and C.-P. Sun, Quantum-state transfer via the ferromagnetic chain in a spatially modulated field, Phys. Rev. A 71, 032309 (2005).
  67. L. Campos Venuti, C. Degli Esposti Boschi, and M. Roncaglia, Qubit teleportation and transfer across antiferromagnetic spin chains, Phys. Rev. Lett. 99, 060401 (2007).
  68. G. M. Nikolopoulos, D. Petrosyan, and P. Lambropoulos, Electron wavepacket propagation in a chain of coupled quantum dots, J. Phys.: Condens. Matter 16, 4991 (2004).
  69. F. Faria, C. Nelmes, T. Apollaro, T. Spiller, and I. D'Amico, Fast and efficient long-distance quantum state transfer in long-range spin-12 models, New J. Phys. 27, 124503 (2025).
  70. M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons, Phys. Rev. Lett. 98, 050405 (2007).
  71. C. Gogolin, M. P. Müller, and J. Eisert, Absence of thermalization in nonintegrable systems, Phys. Rev. Lett. 106, 040401 (2011).
  72. J.-S. Caux and F. H. L. Essler, Time evolution of local observables after quenching to an integrable model, Phys. Rev. Lett. 110, 257203 (2013).
  73. A. Cantoni and P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Lin. Alg. Appl. 13, 275 (1976).
  74. A. Pandey, A. Kumar, and S. Puri, Quantum chaotic systems and random matrix theory, arXiv:1905.10596v1.
  75. M. L. Mehta, Random Matrices, Pure and Applied Mathematics (Elsevier Science, 2004).
  76. L. F. Santos and M. Rigol, Localization and the effects of symmetries in the thermalization properties of one-dimensional quantum systems, Phys. Rev. E 82, 031130 (2010).
  77. E. Bogomolny, U. Gerland, and C. Schmit, Short-range plasma model for intermediate spectral statistics, Eur. Phys. J. B 19, 121 (2001).
  78. T. Guhr, A. Müller–Groeling, and H. A. Weidenmüller, Random-matrix theories in quantum physics: Common concepts, Phys. Rep. 299, 189 (1998).
  79. W. Buijsman, V. Cheianov, and V. Gritsev, Sensitivity of the spectral form factor to short-range level statistics, Phys. Rev. E 102, 042216 (2020).
  80. M. Schiulaz, E. J. Torres-Herrera, and L. F. Santos, Thouless and relaxation time scales in many-body quantum systems, Phys. Rev. B 99, 174313 (2019).
  81. A. K. Das, A. Ghosh, and L. F. Santos, Spectral form factor and energy correlations in banded random matrices, Phys. Rev. B 111, 224202 (2025).
  82. S. Seth, A. K. Das, and A. Ghosh, Signatures of nonergodicity in sparse random matrices, arXiv:2603.21369
  83. A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, Generating optical Schrödinger kittens for quantum information processing, Science 312, 83 (2006).
  84. Y. P. Huang and M. G. Moore, Creation, detection, and decoherence of macroscopic quantum superposition states in double-well Bose-Einstein condensates, Phys. Rev. A 73, 023606 (2006).
  85. C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, A Schrödinger cat living in two boxes, Science 352, 1087 (2016).
  86. M. Lewenstein, M. F. Ciappina, E. Pisanty, J. Rivera-Dean, P. Stammer, T. Lamprou, and P. Tzallas, Generation of optical Schrödinger cat states in intense laser–matter interactions, Nat. Phys. 17, 1104 (2021).
  87. M. Távora, E. J. Torres-Herrera, and L. F. Santos, Inevitable power-law behavior of isolated many-body quantum systems and how it anticipates thermalization, Phys. Rev. A 94, 041603(R) (2016).
  88. A. Jana, V. R. Chandra, and A. Garg, Local density of states and scattering rates across the many-body localization transition, Phys. Rev. B 104, L140201 (2021).
  89. J. Louis, Correlation of local densities of states on mesoscopic energy scales in random band matrices, Lett. Math. Phys. 113, 79 (2023).
  90. Y. Prasad and A. Garg, Single-particle excitations across the localization and many-body localization transition in quasiperiodic systems, Phys. Rev. B 109, 094204 (2024).
  91. E. J. Torres-Herrera, A. M. García-García, and L. F. Santos, Generic dynamical features of quenched interacting quantum systems: Survival probability, density imbalance, and out-of-time-ordered correlator, Phys. Rev. B 97, 060303(R) (2018).
  92. E. J. Torres-Herrera, J. A. Méndez-Bermúdez, and L. F. Santos, Level repulsion and dynamics in the finite one-dimensional Anderson model, Phys. Rev. E 100, 022142 (2019).
  93. A. K. Das, C. Cianci, D. G. A. Cabral, D. A. Zarate-Herrada, P. Pinney, S. Pilatowsky-Cameo, A. S. Matsoukas-Roubeas, V. S. Batista, A. del Campo, E. J. Torres-Herrera, and L. F. Santos, Proposal for many-body quantum chaos detection, Phys. Rev. Res. 7, 013181 (2025).
  94. I. Vallejo-Fabila, A. K. Das, D. A. Zarate-Herrada, A. S. Matsoukas-Roubeas, E. J. Torres-Herrera, and L. F. Santos, Reducing dynamical fluctuations and enforcing self-averaging by opening many-body quantum systems, Phys. Rev. B 110, 075138 (2024).
  95. I. Vallejo-Fabila, A. K. Das, S. Choudhury, and L. F. Santos, Single-site measurements as probes of many-body quantum chaos, Phys. Rev. E 112, 044208 (2025).
  96. E. Bogomolny and M. Sieber, Eigenfunction distribution for the Rosenzweig-Porter model, Phys. Rev. E 98, 032139 (2018).
  97. S. Manna, S. Roy, and G. J. Sreejith, Projected ensemble in a system with locally supported conserved charges, Phys. Rev. B 111, 144302 (2025).
  98. C. Chiu, B. Misra, and E. Sudarshan, The time scale for the quantum Zeno paradox and proton decay, Phys. Lett. B 117, 34 (1982).
  99. L. A. Khalfin, Contribution to the decay theory of a quasi-stationary state, Sov. Phys. JETP 6, 1053 (1958).
  100. M. Távora, E. J. Torres-Herrera, and L. F. Santos, Power-law decay exponents: A dynamical criterion for predicting thermalization, Phys. Rev. A 95, 013604 (2017).
  101. B. Roy, A. K. Das, A. Ghosh, and I. M. Khaymovich, Anomalous energy correlations and spectral form factor in the nonergodic phase of the β ensemble, Phys. Rev. B 112, 094206 (2025).
  102. L. F. Cugliandolo, G. Schehr, M. Tarzia, and D. Venturelli, Multifractal phase in the weighted adjacency matrices of random Erdös-Rényi graphs, Phys. Rev. B 110, 174202 (2024).
  103. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, 2011).
  104. W. H. Zurek, U. Dorner, and P. Zoller, Dynamics of a quantum phase transition, Phys. Rev. Lett. 95, 105701 (2005).
  105. A. L. Corps and A. Relaño, Dynamical and excited-state quantum phase transitions in collective systems, Phys. Rev. B 106, 024311 (2022).
  106. A. S. Matsoukas-Roubeas, M. Beau, L. F. Santos, and A. del Campo, Unitarity breaking in self-averaging spectral form factors, Phys. Rev. A 108, 062201 (2023).
  107. P. Bocchieri and A. Loinger, Quantum recurrence theorem, Phys. Rev. 107, 337 (1957).
  108. L. S. Schulman, Note on the quantum recurrence theorem, Phys. Rev. A 18, 2379 (1978).
  109. F. Haake, Quantum Signatures of Chaos (Springer, Berlin, Heidelberg, 2010), pp. 1–14.
  110. J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Periodic spontaneous collapse and revival in a simple quantum model, Phys. Rev. Lett. 44, 1323 (1980).
  111. Z. D. Gaeta and C. R. Stroud, Classical and quantum-mechanical dynamics of a quasiclassical state of the hydrogen atom, Phys. Rev. A 42, 6308 (1990).
  112. P. Reimann, Foundation of statistical mechanics under experimentally realistic conditions, Phys. Rev. Lett. 101, 190403 (2008).
  113. J. Gemmer and G. Mahler, Distribution of local entropy in the Hilbert space of bi-partite quantum systems: Origin of Jaynes' principle, Eur. Phys. J. B 31, 249 (2003).
  114. M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, Exact relaxation in a class of nonequilibrium quantum lattice systems, Phys. Rev. Lett. 100, 030602 (2008).
  115. M. Cramer and J. Eisert, A quantum central limit theorem for non-equilibrium systems: Exact local relaxation of correlated states, New J. Phys. 12, 055020 (2010).
  116. A. C. Cassidy, C. W. Clark, and M. Rigol, Generalized thermalization in an integrable lattice system, Phys. Rev. Lett. 106, 140405 (2011).
  117. L. A. Bunimovich, On the ergodic properties of nowhere dispersing billiards, Commun. Math. Phys. 65, 295 (1979).
  118. N. Simányi, Proof of the ergodic hypothesis for typical hard ball systems, Annales Henri Poincaré (Springer, 2004), Vol. 5, pp. 203–233.
  119. N. Simányi, Conditional proof of the Boltzmann-Sinai ergodic hypothesis, Invent. Math. 177, 381 (2009).
  120. H. Tasaki, From quantum dynamics to the canonical distribution: General picture and a rigorous example, Phys. Rev. Lett. 80, 1373 (1998).
  121. S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics, Nat. Phys. 2, 754 (2006).
  122. S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zanghì, Canonical typicality, Phys. Rev. Lett. 96, 050403 (2006).
  123. N. Linden, S. Popescu, A. J. Short, and A. Winter, Quantum mechanical evolution towards thermal equilibrium, Phys. Rev. E 79, 061103 (2009).
  124. M. P. Müller, E. Adlam, L. Masanes, and N. Wiebe, Thermalization and canonical typicality in translation-invariant quantum lattice systems, Commun. Math. Phys. 340, 499 (2015).
  125. M. Feingold, N. Moiseyev, and A. Peres, Classical limit of quantum chaos, Chem. Phys. Lett. 117, 344 (1985).
  126. M. Feingold and A. Peres, Distribution of matrix elements of chaotic systems, Phys. Rev. A 34, 591 (1986).
  127. M. Lax, Relation between canonical and microcanonical ensembles, Phys. Rev. 97, 1419 (1955).
  128. L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016).
  129. G. Mussardo, Infinite-time average of local fields in an integrable quantum field theory after a quantum quench, Phys. Rev. Lett. 111, 100401 (2013).
  130. P. Calabrese, F. H. L. Essler, and M. Fagotti, Quantum quench in the transverse-field Ising chain, Phys. Rev. Lett. 106, 227203 (2011).
  131. F. H. L. Essler, S. Evangelisti, and M. Fagotti, Dynamical correlations after a quantum quench, Phys. Rev. Lett. 109, 247206 (2012).
  132. M. Fagotti, Finite-size corrections versus relaxation after a sudden quench, Phys. Rev. B 87, 165106 (2013).
  133. M. Fagotti and F. H. L. Essler, Reduced density matrix after a quantum quench, Phys. Rev. B 87, 245107 (2013).
  134. M. Fagotti, M. Collura, F. H. L. Essler, and P. Calabrese, Relaxation after quantum quenches in the spin-12 Heisenberg XXZ chain, Phys. Rev. B 89, 125101 (2014).
  135. M. Mierzejewski, P. Prelovšek, and T. Prosen, Breakdown of the generalized Gibbs ensemble for current-generating quenches, Phys. Rev. Lett. 113, 020602 (2014).
  136. B. Pozsgay, M. Mestyán, M. A. Werner, M. Kormos, G. Zaránd, and G. Takács, Correlations after quantum quenches in the XXZ spin chain: Failure of the generalized Gibbs ensemble, Phys. Rev. Lett. 113, 117203 (2014).
  137. E. Ilievski, J. De Nardis, B. Wouters, J.-S. Caux, F. H. L. Essler, and T. Prosen, Complete generalized Gibbs ensembles in an interacting theory, Phys. Rev. Lett. 115, 157201 (2015).
  138. C. Gramsch and M. Rigol, Quenches in a quasidisordered integrable lattice system: Dynamics and statistical description of observables after relaxation, Phys. Rev. A 86, 053615 (2012).
  139. T. M. Wright, M. Rigol, M. J. Davis, and K. V. Kheruntsyan, Nonequilibrium dynamics of one-dimensional hard-core anyons following a quench: Complete relaxation of one-body observables, Phys. Rev. Lett. 113, 050601 (2014).
  140. P. Calabrese and J. Cardy, Quantum quenches in extended systems, J. Stat. Mech. (2007) P06008.
  141. D. Fioretto and G. Mussardo, Quantum quenches in integrable field theories, New J. Phys. 12, 055015 (2010).
  142. J. Cardy, Quantum quenches to a critical point in one dimension: some further results, J. Stat. Mech. (2016) 023103.
  143. M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature (London) 452, 854 (2008).
  144. M. Ippoliti and W. W. Ho, Solvable model of deep thermalization with distinct design times, Quantum 6, 886 (2022).
  145. A. Sherry and S. Roy, Do mixed states exhibit deep thermalisation? arXiv:2507.14135.
  146. https://github.com/AdwayDas/Symmetric_centrosymmetric_data.
  147. J. von Neumann, Proof of the ergodic theorem and the h-theorem in quantum mechanics, Eur. Phys. J. H 35, 201 (2010).
  148. S. Goldstein, J. L. Lebowitz, C. Mastrodonato, R. Tumulka, and N. Zanghì, Normal typicality and von Neumann’s quantum ergodic theorem, Proc. R. Soc. A: Math. Phys. Eng. Sci. 466, 3203 (2010).

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Phys. Rev. B 113, 144201– Published 2 April, 2026
  • Received 7 January 2026
  • Revised 25 February 2026
  • Accepted 13 March 2026
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