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arxiv: 2604.11872 · v2 · submitted 2026-04-13 · 🪐 quant-ph · cond-mat.stat-mech

Eigenstate thermalization

Rohit Patil , Marcos Rigol This is my paper

Pith reviewed 2026-05-10 16:10 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords eigenstate thermalizationquantum thermalizationmany-body systemsrandom matrix theoryentanglement entropyunitary dynamicsisolated quantum systems
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The pith

Eigenstate thermalization explains thermalization in isolated quantum systems under unitary evolution.

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

This paper introduces eigenstate thermalization as the mechanism that lets isolated quantum systems reach thermal equilibrium even though their evolution is strictly unitary. In generic chaotic systems the individual energy eigenstates already look thermal, so time averages of observables coincide with ensemble averages without any external bath. The authors motivate the idea with random matrix theory, review results on the volume-law entanglement entropy of Haar-random states, and show supporting numerical behaviors in quantum many-body models.

Core claim

Eigenstate thermalization is the phenomenon in which the matrix elements of local observables, evaluated in the energy eigenbasis of a generic quantum many-body Hamiltonian, take a specific structure: diagonal elements match the microcanonical thermal expectation value at that energy, while off-diagonal elements are exponentially small in system size. This structure directly accounts for the emergence of thermalization from unitary dynamics in isolated systems.

What carries the argument

The eigenstate thermalization hypothesis (ETH), which encodes the thermal-like structure of observable matrix elements in the energy eigenbasis of chaotic quantum systems.

If this is right

  • Unitary time evolution of local observables relaxes to their thermal values whenever the eigenstates obey ETH.
  • Eigenstates of chaotic Hamiltonians exhibit volume-law entanglement entropy, matching the scaling found in thermal states.
  • Thermalization occurs in closed quantum systems without coupling to an external environment if the eigenstates are thermal.

Where Pith is reading between the lines

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

  • The same matrix-element structure may illuminate how information scrambles in holographic models of black holes.
  • Near-term quantum simulators could test ETH directly by preparing approximate eigenstates and measuring observable expectation values.
  • Systems that violate ETH, such as many-body localized phases, remain non-thermal precisely because they lack the required chaotic eigenstate structure.

Load-bearing premise

The quantum systems under study are generic and sufficiently chaotic that random matrix theory accurately describes their level statistics and eigenstate properties.

What would settle it

An explicit calculation or measurement in a chaotic many-body system showing that the diagonal matrix element of a local observable in an energy eigenstate deviates from the microcanonical thermal value at the corresponding energy density.

read the original abstract

We provide a pedagogical introduction to eigenstate thermalization. This phenomenon, which occurs in generic quantum systems, allows one to understand why thermalization takes place in isolated systems under unitary dynamics. We motivate eigenstate thermalization using random matrix theory and discuss recent complementary results for the volume-law entanglement entropy of Haar-random states. We discuss numerical results that highlight the corresponding behaviors in quantum many-body systems.

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

0 major / 2 minor

Summary. The manuscript provides a pedagogical introduction to the eigenstate thermalization hypothesis (ETH). It claims that this phenomenon occurs in generic quantum systems and explains thermalization in isolated systems evolving under unitary dynamics. The authors motivate ETH via random matrix theory, discuss volume-law entanglement entropy results for Haar-random states, and present numerical results illustrating the corresponding behaviors in quantum many-body systems.

Significance. As a review-style introduction summarizing established results, the paper has moderate pedagogical value for newcomers to quantum many-body physics and thermalization. It consolidates standard random-matrix motivations for ETH and Haar-random entanglement without advancing new theorems or data, so its primary contribution is clarity of exposition and accessibility. The inclusion of numerical illustrations strengthens its utility for teaching.

minor comments (2)
  1. [Abstract] The abstract refers to 'recent complementary results' for Haar-random entanglement without citing specific works; adding explicit references in the main text would improve traceability for readers.
  2. The phrase 'generic quantum systems' is used repeatedly; a short clarification early in the introduction on the precise assumptions (e.g., sufficient chaos for random-matrix applicability) would aid precision without altering the pedagogical tone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript as a pedagogical introduction to the eigenstate thermalization hypothesis (ETH). We appreciate the recognition of its value for newcomers to quantum many-body physics and thermalization, as well as the utility of the numerical illustrations for teaching. The referee's summary accurately reflects the manuscript's scope and intent.

Circularity Check

0 steps flagged

Pedagogical summary of established ETH with no new derivation

full rationale

The manuscript is explicitly a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), an established concept. It motivates ETH via standard random matrix theory applied to generic chaotic systems, discusses known Haar-random entanglement results, and presents illustrative numerics. No new theorem, derivation, or data claim is advanced, so there are no load-bearing steps that reduce by construction to fitted parameters, self-definitions, or self-citation chains. The phrasing relies on prior literature without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard quantum mechanics and random matrix theory as background without introducing new free parameters or entities in the abstract.

axioms (1)
  • domain assumption Random matrix theory accurately models the statistics of generic chaotic quantum many-body Hamiltonians
    Used to motivate eigenstate thermalization.

pith-pipeline@v0.9.0 · 5340 in / 1150 out tokens · 57711 ms · 2026-05-10T16:10:27.458070+00:00 · methodology

discussion (0)

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Forward citations

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Reference graph

Works this paper leans on

99 extracted references · 99 canonical work pages · cited by 3 Pith papers · 1 internal anchor

  1. [1]

    D’Alessio, Y

    L. D’Alessio, Y . Kafri, A. Polkovnikov and M. Rigol,From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics,Adv. Phys.65(2016) 239

    work page 2016

  2. [2]

    Deutsch,Quantum statistical mechanics in a closed system,Phys

    J.M. Deutsch,Quantum statistical mechanics in a closed system,Phys. Rev. A43(1991) 2046

    work page 1991

  3. [3]

    Srednicki,Chaos and quantum thermalization,Phys

    M. Srednicki,Chaos and quantum thermalization,Phys. Rev. E50(1994) 888

    work page 1994

  4. [4]

    Rigol, V

    M. Rigol, V. Dunjko and M. Olshanii,Thermalization and its mechanism for generic isolated quantum systems,Nature452(2008) 854

    work page 2008

  5. [5]

    Zamolodchikov and V.A

    A.B. Zamolodchikov and V.A. Fateev,Model factorized S-matrix and an integrable spin-1 Heisenberg chain,Sov. J. Nucl. Phys.32(1980) 298

    work page 1980

  6. [6]

    Bytsko,On integrable Hamiltonians for higher spin XXZ chain,J

    A.G. Bytsko,On integrable Hamiltonians for higher spin XXZ chain,J. Math. Phys.44(2003) 3698. 22Eigenstate thermalization

    work page 2003

  7. [7]

    Brody, J

    T.A. Brody, J. Flores, J.B. French, P .A. Mello, A. Pandey and S.S.M. Wong,Random-matrix physics: spectrum and strength fluctuations, Rev. Mod. Phys.53(1981) 385

    work page 1981

  8. [8]

    Wigner,Characteristic vectors of bordered matrices with infinite dimensions,Annals of Mathematics62(1955) 548

    E.P . Wigner,Characteristic vectors of bordered matrices with infinite dimensions,Annals of Mathematics62(1955) 548

    work page 1955

  9. [9]

    Wigner,Characteristics vectors of bordered matrices with infinite dimensions II,Annals of Mathematics65(1957) 203

    E.P . Wigner,Characteristics vectors of bordered matrices with infinite dimensions II,Annals of Mathematics65(1957) 203

    work page 1957

  10. [10]

    Wigner,On the distribution of the roots of certain symmetric matrices,Annals of Mathematics67(1958) 325

    E.P . Wigner,On the distribution of the roots of certain symmetric matrices,Annals of Mathematics67(1958) 325

    work page 1958

  11. [11]

    Dyson,Statistical theory of the energy levels of complex systems

    F .J. Dyson,Statistical theory of the energy levels of complex systems. I,Journal of Mathematical Physics3(1962) 140

    work page 1962

  12. [12]

    Alhassid,The statistical theory of quantum dots,Rev

    Y . Alhassid,The statistical theory of quantum dots,Rev. Mod. Phys.72(2000) 895

    work page 2000

  13. [13]

    Mehta,Random Matrices, vol

    M.L. Mehta,Random Matrices, vol. 142 ofPure and Applied Mathematics, Elsevier/Academic Press, Amsterdam, 3 ed. (2004)

    work page 2004

  14. [14]

    https://doi.org/10.1007/978-3-319-70885-0 Madan Lal Mehta

    G. Livan, M. Novaes and P . Vivo,Introduction to Random Matrices, Springer International Publishing (2018), 10.1007/978-3-319-70885-0

    work page doi:10.1007/978-3-319-70885-0 2018

  15. [15]

    Bohigas, M.J

    O. Bohigas, M.J. Giannoni and C. Schmit,Characterization of chaotic quantum spectra and universality of level fluctuation laws,Phys. Rev. Lett.52(1984) 1

    work page 1984

  16. [16]

    Santos,Integrability of a disordered heisenberg spin-1/2 chain,J

    L.F . Santos,Integrability of a disordered heisenberg spin-1/2 chain,J. Phys. A: Math. Gen.37(2004) 4723

    work page 2004

  17. [17]

    Santos and M

    L.F . Santos and M. Rigol,Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization, Phys. Rev. E81(2010) 036206

    work page 2010

  18. [18]

    Berry and M

    M.V. Berry and M. Tabor,Level clustering in the regular spectrum,Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences356(1977) 375

    work page 1977

  19. [19]

    Y .Y . Atas, E. Bogomolny, O. Giraud and G. Roux,Distribution of the ratio of consecutive level spacings in random matrix ensembles,Phys. Rev. Lett.110(2013) 084101

    work page 2013

  20. [20]

    Oganesyan and D.A

    V. Oganesyan and D.A. Huse,Localization of interacting fermions at high temperature,Phys. Rev. B75(2007) 155111

    work page 2007

  21. [21]

    Santos and M

    L.F . Santos and M. Rigol,Localization and the effects of symmetries in the thermalization properties of one-dimensional quantum systems,Phys. Rev. E82(2010) 031130

    work page 2010

  22. [22]

    Arfken, H.J

    G.B. Arfken, H.J. Weber and F .E. Harris,Chapter 20 - integral transforms, inMathematical Methods for Physicists (Seventh Edition), (Boston), pp. 963–1046, Academic Press (2013), DOI

    work page 2013

  23. [23]

    Porter and R.G

    C.E. Porter and R.G. Thomas,Fluctuations of nuclear reaction widths,Phys. Rev.104(1956) 483

    work page 1956

  24. [24]

    B. Collins,Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability, International Mathematics Research Notices2003(2003) 953

    work page 2003

  25. [25]

    Collins and P

    B. Collins and P . ´Sniady,Integration with respect to the Haar measure on unitary, orthogonal and symplectic group,Communications in Mathematical Physics264(2006) 773

    work page 2006

  26. [26]

    Collins, S

    B. Collins, S. Matsumoto and J. Novak,The Weingarten calculus,Notices of the American Mathematical Society69(2022) 1

    work page 2022

  27. [27]

    Lecture notes titled chaotic dynamics: Eigenstate thermalization hypothesis and beyond

    S. Pappalardi, “Lecture notes titled chaotic dynamics: Eigenstate thermalization hypothesis and beyond.” https://www.qhaos.org/teaching

    work page

  28. [28]

    Amico, R

    L. Amico, R. Fazio, A. Osterloh and V. Vedral,Entanglement in many-body systems,Rev. Mod. Phys.80(2008) 517

    work page 2008

  29. [29]

    Eisert, M

    J. Eisert, M. Cramer and M.B. Plenio,Colloquium: Area laws for the entanglement entropy,Rev. Mod. Phys.82(2010) 277

    work page 2010

  30. [30]

    Bianchi, L

    E. Bianchi, L. Hackl, M. Kieburg, M. Rigol and L. Vidmar,Volume-law entanglement entropy of typical pure quantum states,PRX Quantum 3(2022) 030201

    work page 2022

  31. [31]

    Miao and T

    Q. Miao and T. Barthel,Eigenstate entanglement: Crossover from the ground state to volume laws,Phys. Rev. Lett.127(2021) 040603

    work page 2021

  32. [32]

    Barthel and Q

    T. Barthel and Q. Miao,Scaling functions for eigenstate entanglement crossovers in harmonic lattices,Phys. Rev. A104(2021) 022414

    work page 2021

  33. [33]

    Miao and T

    Q. Miao and T. Barthel,Eigenstate entanglement scaling for critical interacting spin chains,Quantum6(2022) 642

    work page 2022

  34. [34]

    Vidmar and M

    L. Vidmar and M. Rigol,Entanglement entropy of eigenstates of quantum chaotic Hamiltonians,Phys. Rev. Lett.119(2017) 220603

    work page 2017

  35. [35]

    Garrison and T

    J.R. Garrison and T. Grover,Does a single eigenstate encode the full Hamiltonian?,Phys. Rev. X8(2018) 021026

    work page 2018

  36. [36]

    Storms and R.R.P

    M. Storms and R.R.P . Singh,Entanglement in ground and excited states of gapped free-fermion systems and their relationship with Fermi surface and thermodynamic equilibrium properties,Phys. Rev. E89(2014) 012125

    work page 2014

  37. [37]

    Vidmar, L

    L. Vidmar, L. Hackl, E. Bianchi and M. Rigol,Entanglement entropy of eigenstates of quadratic fermionic Hamiltonians,Phys. Rev. Lett. 119(2017) 020601

    work page 2017

  38. [38]

    Hackl, L

    L. Hackl, L. Vidmar, M. Rigol and E. Bianchi,Average eigenstate entanglement entropy of the XY chain in a transverse field and its universality for translationally invariant quadratic fermionic models,Phys. Rev. B99(2019) 075123

    work page 2019

  39. [39]

    LeBlond, K

    T. LeBlond, K. Mallayya, L. Vidmar and M. Rigol,Entanglement and matrix elements of observables in interacting integrable systems, Phys. Rev. E100(2019) 062134

    work page 2019

  40. [40]

    Page,Average entropy of a subsystem,Phys

    D.N. Page,Average entropy of a subsystem,Phys. Rev. Lett.71(1993) 1291

    work page 1993

  41. [41]

    Y auk, R

    Y . Y auk, R. Patil, Y . Zhang, M. Rigol and L. Hackl,Typical entanglement entropy in systems with particle-number conservation,Phys. Rev. B110(2024) 235154

    work page 2024

  42. [42]

    Patil, L

    R. Patil, L. Hackl, G.R. Fagan and M. Rigol,Average pure-state entanglement entropy in spin systems with SU(2) symmetry,Phys. Rev. B 108(2023) 245101

    work page 2023

  43. [43]

    Bianchi, P

    E. Bianchi, P . Dona and R. Kumar,Non-Abelian symmetry-resolved entanglement entropy,SciPost Phys.17(2024) 127

    work page 2024

  44. [44]

    Chakraborty, L

    A. Chakraborty, L. Hackl and M. Kieburg, “Random matrix prediction of average entanglement entropy in non-Abelian symmetry sectors.” arXiv:2512.22942, 2025

    work page arXiv 2025

  45. [45]

    Łyd˙zba, M

    P . Łyd˙zba, M. Rigol and L. Vidmar,Eigenstate entanglement entropy in random quadratic Hamiltonians,Phys. Rev. Lett.125(2020) 180604

    work page 2020

  46. [46]

    Bianchi, L

    E. Bianchi, L. Hackl and M. Kieburg,Page curve for fermionic Gaussian states,Phys. Rev. B103(2021) L241118

    work page 2021

  47. [47]

    Huang,Universal entanglement of mid-spectrum eigenstates of chaotic local Hamiltonians,Nuclear Physics B966(2021) 115373

    Y . Huang,Universal entanglement of mid-spectrum eigenstates of chaotic local Hamiltonians,Nuclear Physics B966(2021) 115373

    work page 2021

  48. [48]

    Haque, P .A

    M. Haque, P .A. McClarty and I.M. Khaymovich,Entanglement of midspectrum eigenstates of chaotic many-body systems: Reasons for deviation from random ensembles,Phys. Rev. E105(2022) 014109

    work page 2022

  49. [49]

    Kliczkowski, R

    M. Kliczkowski, R. ´Swi ˛ etek, L. Vidmar and M. Rigol,Average entanglement entropy of midspectrum eigenstates of quantum-chaotic interacting Hamiltonians,Phys. Rev. E107(2023) 064119

    work page 2023

  50. [50]

    Rodriguez-Nieva, C

    J.F . Rodriguez-Nieva, C. Jonay and V. Khemani,Quantifying quantum chaos through microcanonical distributions of entanglement,Phys. Rev. X14(2024) 031014

    work page 2024

  51. [51]

    Langlett and J.F

    C.M. Langlett and J.F . Rodriguez-Nieva,Entanglement patterns of quantum chaotic hamiltonians with a scalar U(1) charge,Phys. Rev. Lett.134(2025) 230402

    work page 2025

  52. [52]

    Bianchi and P

    E. Bianchi and P . Donà,Typical entanglement entropy in the presence of a center: Page curve and its variance,Phys. Rev. D100(2019) 105010

    work page 2019

  53. [53]

    ´Swie ¸tek, M

    R. ´Swie ¸tek, M. Kliczkowski, L. Vidmar and M. Rigol,Eigenstate entanglement entropy in the integrable spin-1 2 XYZ model,Phys. Rev. E 109(2024) 024117

    work page 2024

  54. [54]

    Srednicki,The approach to thermal equilibrium in quantized chaotic systems,J

    M. Srednicki,The approach to thermal equilibrium in quantized chaotic systems,J. Phys. A.32(1999) 1163. Eigenstate thermalization23

    work page 1999

  55. [55]

    Foini and J

    L. Foini and J. Kurchan,Eigenstate thermalization hypothesis and out of time order correlators,Phys. Rev. E99(2019) 042139

    work page 2019

  56. [56]

    Pappalardi, L

    S. Pappalardi, L. Foini and J. Kurchan,Eigenstate thermalization hypothesis and free probability,Phys. Rev. Lett.129(2022) 170603

    work page 2022

  57. [57]

    Dymarsky,Bound on eigenstate thermalization from transport,Phys

    A. Dymarsky,Bound on eigenstate thermalization from transport,Phys. Rev. Lett.128(2022) 190601

    work page 2022

  58. [58]

    Wang, M.H

    J. Wang, M.H. Lamann, J. Richter, R. Steinigeweg, A. Dymarsky and J. Gemmer,Eigenstate thermalization hypothesis and its deviations from random-matrix theory beyond the thermalization time,Phys. Rev. Lett.128(2022) 180601

    work page 2022

  59. [59]

    Capizzi, J

    L. Capizzi, J. Wang, X. Xu, L. Mazza and D. Poletti,Hydrodynamics and the eigenstate thermalization hypothesis,Phys. Rev. X15(2025) 011059

    work page 2025

  60. [60]

    Mierzejewski and L

    M. Mierzejewski and L. Vidmar,Quantitative impact of integrals of motion on the eigenstate thermalization hypothesis,Phys. Rev. Lett. 124(2020) 040603

    work page 2020

  61. [61]

    Łyd˙zba, R

    P . Łyd˙zba, R. ´Swi ˛ etek, M. Mierzejewski, M. Rigol and L. Vidmar,Normal weak eigenstate thermalization,Phys. Rev. B110(2024) 104202

    work page 2024

  62. [62]

    Beugeling, R

    W. Beugeling, R. Moessner and M. Haque,Finite-size scaling of eigenstate thermalization,Phys. Rev. E89(2014) 042112

    work page 2014

  63. [63]

    Biroli, C

    G. Biroli, C. Kollath and A.M. Läuchli,Effect of rare fluctuations on the thermalization of isolated quantum systems,Phys. Rev. Lett.105 (2010) 250401

    work page 2010

  64. [64]

    Alba,Eigenstate thermalization hypothesis and integrability in quantum spin chains,Phys

    V. Alba,Eigenstate thermalization hypothesis and integrability in quantum spin chains,Phys. Rev. B91(2015) 155123

    work page 2015

  65. [65]

    Beugeling, R

    W. Beugeling, R. Moessner and M. Haque,Off-diagonal matrix elements of local operators in many-body quantum systems,Phys. Rev. E 91(2015) 012144

    work page 2015

  66. [66]

    Brenes, J

    M. Brenes, J. Goold and M. Rigol,Low-frequency behavior of off-diagonal matrix elements in the integrable XXZ chain and in a locally perturbed quantum-chaotic XXZ chain,Phys. Rev. B102(2020) 075127

    work page 2020

  67. [67]

    LeBlond and M

    T. LeBlond and M. Rigol,Eigenstate thermalization for observables that break Hamiltonian symmetries and its counterpart in interacting integrable systems,Phys. Rev. E102(2020) 062113

    work page 2020

  68. [68]

    Zhang, L

    Y . Zhang, L. Vidmar and M. Rigol,Statistical properties of the off-diagonal matrix elements of observables in eigenstates of integrable systems,Phys. Rev. E106(2022) 014132

    work page 2022

  69. [69]

    Essler and A.J.J.M

    F .H.L. Essler and A.J.J.M. de Klerk,Statistics of matrix elements of local operators in integrable models,Phys. Rev. X14(2024) 031048

    work page 2024

  70. [70]

    Rottoli and V

    F . Rottoli and V. Alba,Eigenstate thermalization hypothesis for off-diagonal matrix elements in integrable spin chains,Phys. Rev. B113 (2026) 054308

    work page 2026

  71. [71]

    Jansen, J

    D. Jansen, J. Stolpp, L. Vidmar and F . Heidrich-Meisner,Eigenstate thermalization and quantum chaos in the Holstein polaron model, Phys. Rev. B99(2019) 155130

    work page 2019

  72. [72]

    Brenes, S

    M. Brenes, S. Pappalardi, M.T. Mitchison, J. Goold and A. Silva,Out-of-time-order correlations and the fine structure of eigenstate thermalization,Phys. Rev. E104(2021) 034120

    work page 2021

  73. [73]

    Kubo,Statistical-mechanical theory of irreversible processes

    R. Kubo,Statistical-mechanical theory of irreversible processes. I. general theory and simple applications to magnetic and conduction problems,Journal of the Physical Society of Japan12(1957) 570

    work page 1957

  74. [74]

    Martin and J

    P .C. Martin and J. Schwinger,Theory of many-particle systems. I,Phys. Rev.115(1959) 1342

    work page 1959

  75. [75]

    Schönle, D

    C. Schönle, D. Jansen, F . Heidrich-Meisner and L. Vidmar,Eigenstate thermalization hypothesis through the lens of autocorrelation functions,Phys. Rev. B103(2021) 235137

    work page 2021

  76. [76]

    Abdelshafy, R

    M. Abdelshafy, R. Mondaini and M. Rigol,Onset of quantum chaos and ergodicity in spin systems with highly degenerate Hilbert spaces, Phys. Rev. B112(2025) L161108

    work page 2025

  77. [77]

    Brenes, T

    M. Brenes, T. LeBlond, J. Goold and M. Rigol,Eigenstate thermalization in a locally perturbed integrable system,Phys. Rev. Lett.125 (2020) 070605

    work page 2020

  78. [78]

    Richter, A

    J. Richter, A. Dymarsky, R. Steinigeweg and J. Gemmer,Eigenstate thermalization hypothesis beyond standard indicators: Emergence of random-matrix behavior at small frequencies,Phys. Rev. E102(2020) 042127

    work page 2020

  79. [79]

    Kliczkowski, R

    M. Kliczkowski, R. ´Swi ˛ etek, M. Hopjan and L. Vidmar,Fading ergodicity,Phys. Rev. B110(2024) 134206

    work page 2024

  80. [80]

    ´Swie ¸tek, P

    R. ´Swie ¸tek, P . Łyd˙zba and L. Vidmar,Fading ergodicity meets maximal chaos,Phys. Rev. B111(2025) 184203

    work page 2025

Showing first 80 references.