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Anomalous energy correlations and spectral form factor in the nonergodic phase of the β ensemble

Basudha Roy*, Adway Kumar Das†, and Anandamohan Ghosh‡

Ivan M. Khaymovich§

  • *Contact author: br23rs008@iiserkol.ac.in
  • †Contact author: akd19rs062@iiserkol.ac.in
  • ‡Contact author: anandamohan@iiserkol.ac.in
  • §Contact author: ivan.khaymovich@gmail.com
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Phys. Rev. B 112, 094206 – Published 23 September, 2025

DOI: https://doi.org/10.1103/s2d6-wkz9

Abstract

The β ensemble is a prototypical model of a single-particle system on a one-dimensional disordered lattice with inhomogeneous nearest-neighbor hopping. The corresponding nonergodic phase has an anomalous critical energy scale, Ec: Correlations are present above and absent below Ec, as reflected in the number variance. We study the dynamical properties of the β ensemble where the critical energy controls the characteristic timescales. In particular, the spectral form factor equilibrates at a relaxation time tR≡Ec−1, which is parametrically smaller than the Heisenberg time, tH, given by the inverse of the mean level spacing. Incidentally, the dimensionless relaxation time, τR≡tR/tH≪1 is equal to the Dyson index, β. We show that the energy correlations are absent within a temporal window tR<t<tH, which we term as the correlation void. This is in contrast to the mechanism of equilibration in a typical many-body system. We analytically explain the qualitative behavior of the number variance and the spectral form factor of the β ensemble by a spatially local mapping to the Anderson model.

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References (146)

  1. J. von Neumann, Proof of the ergodic theorem and the H-theorem in quantum mechanics, Eur. Phys. J. H 35, 201 (2010).
  2. 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. 466, 3203 (2010).
  3. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  4. M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature (London) 452, 854 (2008).
  5. 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).
  6. J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog. Phys. 81, 082001 (2018).
  7. O. Bohigas, M. J. Giannoni, and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52, 1 (1984).
  8. F. Borgonovi, F. Izrailev, L. Santos, and V. Zelevinsky, Quantum chaos and thermalization in isolated systems of interacting particles, Phys. Rep. 626, 1 (2016).
  9. F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
  10. N. Macé, F. Alet, and N. Laflorencie, Multifractal scalings across the many-body localization transition, Phys. Rev. Lett. 123, 180601 (2019).
  11. A. Solórzano, L. F. Santos, and E. J. Torres-Herrera, Multifractality and self-averaging at the many-body localization transition, Phys. Rev. Res. 3, L032030 (2021).
  12. M. A. Bastarrachea-Magnani, D. Villaseñor, J. Chávez-Carlos, S. Lerma-Hernández, L. F. Santos, and J. G. Hirsch, Quantum multifractality as a probe of phase space in the Dicke model, Phys. Rev. E 109, 034202 (2024).
  13. 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).
  14. A. K. Das, A. Ghosh, and I. M. Khaymovich, Emergent multifractality in power-law decaying eigenstates, Phys. Rev. B 112, 024201 (2025).
  15. A. K. Das and A. Ghosh, Transport in deformed centrosymmetric networks, Phys. Rev. E 106, 064112 (2022).
  16. T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, Thermalization and prethermalization in isolated quantum systems: A theoretical overview, J. Phys. B 51, 112001 (2018).
  17. L. Vidmar and M. Rigol, Generalized Gibbs ensemble in integrable lattice models, J. Stat. Mech. (2016) 064007.
  18. S. Sinha and S. Sinha, Chaos and quantum scars in Bose-Josephson junction coupled to a bosonic mode, Phys. Rev. Lett. 125, 134101 (2020).
  19. 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).
  20. 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).
  21. 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).
  22. 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).
  23. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  24. 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).
  25. 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).
  26. E. Rosenberg, T. Andersen, R. Samajdar, A. Petukhov, J. Hoke, D. Abanin, A. Bengtsson, I. Drozdov, C. Erickson, P. Klimov et al., Dynamics of magnetization at infinite temperature in a Heisenberg spin chain, Science 384, 48 (2024).
  27. Y. Bar Lev, G. Cohen, and D. R. Reichman, Absence of diffusion in an interacting system of spinless fermions on a one-dimensional disordered lattice, Phys. Rev. Lett. 114, 100601 (2015).
  28. D. J. Luitz and Y. Bar Lev, Anomalous thermalization in ergodic systems, Phys. Rev. Lett. 117, 170404 (2016).
  29. C. M. Soukoulis and E. N. Economou, Fractal character of eigenstates in disordered systems, Phys. Rev. Lett. 52, 565 (1984).
  30. L. Erdős and A. Knowles, The Altshuler–Shklovskii formulas for random band matrices I: The unimodular case, Commun. Math. Phys. 333, 1365 (2015).
  31. A. Jagannathan and M. Tarzia, Electronic states of a disordered two-dimensional quasiperiodic tiling: From critical states to Anderson localization, Phys. Rev. B 107, 054206 (2023).
  32. L. Leviandier, M. Lombardi, R. Jost, and J. P. Pique, Fourier transform: A tool to measure statistical level properties in very complex spectra, Phys. Rev. Lett. 56, 2449 (1986).
  33. J. P. Pique, Y. Chen, R. W. Field, and J. L. Kinsey, Chaos and dynamics on 0.5–300 ps time scales in vibrationally excited acetylene: Fourier transform of stimulated-emission pumping spectrum, Phys. Rev. Lett. 58, 475 (1987).
  34. Y. Alhassid and R. D. Levine, Spectral autocorrelation function in the statistical theory of energy levels, Phys. Rev. A 46, 4650 (1992).
  35. L. Michaille and J.-P. Pique, Influence of experimental resolution on the spectral statistics used to show quantum chaos: The case of molecular vibrational chaos, Phys. Rev. Lett. 82, 2083 (1999).
  36. E. J. Torres-Herrera and L. F. Santos, Dynamical manifestations of quantum chaos: Correlation hole and bulge, Philos. Trans. R. Soc. A 375, 20160434 (2017).
  37. 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).
  38. 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).
  39. J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, Many-body localization in a quantum simulator with programmable random disorder, Nat. Phys. 12, 907 (2016).
  40. H. Dong, P. Zhang, C. B. Dağ, Y. Gao, N. Wang, J. Deng, X. Zhang, J. Chen, S. Xu, K. Wang, Y. Wu, C. Zhang, F. Jin, X. Zhu, A. Zhang, Y. Zou, Z. Tan, Z. Cui, Z. Zhu, F. Shen et al., Measuring the spectral form factor in many-body chaotic and localized phases of quantum processors, Phys. Rev. Lett. 134, 010402 (2025).
  41. 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).
  42. 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).
  43. I. Vallejo-Fabila, A. K. Das, S. Choudhury, and L. F. Santos, Proposal for many-body quantum chaos detection with single-site measurements, arXiv:2505.05572.
  44. D. A. Zarate-Herrada, I. Vallejo-Fabila, L. F. Santos, and E. J. Torres-Herrera, Dynamical detection of extended nonergodic states in many-body quantum systems, arXiv:2505.23910.
  45. I. Dumitriu and A. Edelman, Matrix models for beta ensembles, J. Math. Phys. 43, 5830 (2002).
  46. P. J. Forrester, Log-gases and Random Matrices (Princeton University Press, Princeton, NJ, 2010).
  47. M. Mehta, Random Matrices, Pure and Applied Mathematics (Elsevier Science, Amsterdam, 2004).
  48. A. K. Das and A. Ghosh, Nonergodic extended states in the beta ensemble, Phys. Rev. E 105, 054121 (2022).
  49. 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).
  50. A. K. Das, A. Ghosh, and I. M. Khaymovich, Robust nonergodicity of the ground states in the beta ensemble, Phys. Rev. B 109, 064206 (2024).
  51. L. Colmenarez, D. J. Luitz, I. M. Khaymovich, and G. De Tomasi, Subdiffusive Thouless time scaling in the Anderson model on random regular graphs, Phys. Rev. B 105, 174207 (2022).
  52. D. J. Luitz, N. Laflorencie, and F. Alet, Many-body localization edge in the random-field Heisenberg chain, Phys. Rev. B 91, 081103(R) (2015).
  53. P. Sierant, D. Delande, and J. Zakrzewski, Thouless time analysis of Anderson and many-body localization transitions, Phys. Rev. Lett. 124, 186601 (2020).
  54. A. K. Das and A. Ghosh, Chaos due to symmetry-breaking in deformed poisson ensemble, J. Stat. Mech. (2022) 063101.
  55. A. L. Corps, R. A. Molina, and A. Relaño, Thouless energy challenges thermalization on the ergodic side of the many-body localization transition, Phys. Rev. B 102, 014201 (2020).
  56. U. Sivan and Y. Imry, Energy-level correlation function and ac conductivity of a finite disordered system, Phys. Rev. B 35, 6074 (1987).
  57. A. G. Aronov and A. D. Mirlin, Fluctuations of the number of energy levels at the mobility edge, Phys. Rev. B 51, 6131 (1995).
  58. M. Hopjan and L. Vidmar, Scale-invariant survival probability at eigenstate transitions, Phys. Rev. Lett. 131, 060404 (2023).
  59. N. Rosenzweig and C. E. Porter, “Repulsion of energy levels” in complex atomic spectra, Phys. Rev. B 120, 1698 (1960).
  60. V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, and M. Amini, A random matrix model with localization and ergodic transitions, New J. Phys. 17, 122002 (2015).
  61. C. Monthus, Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models: Wigner-Weisskopf approach, J. Phys. A: Math. Theor. 50, 295101 (2017).
  62. D. Venturelli, L. F. Cugliandolo, G. Schehr, and M. Tarzia, Replica approach to the generalized Rosenzweig-Porter model, SciPost Phys. 14, 110 (2023).
  63. G. de Tomasi, M. Amini, S. Bera, I. M. Khaymovich, and V. E. Kravtsov, Survival probability in generalized Rosenzweig-Porter random matrix ensemble, SciPost Phys. 6, 014 (2019).
  64. M. Amini, Spread of wave packets in disordered hierarchical lattices, Europhys. Lett. 117, 30003 (2017).
  65. T. Guhr, A. Müller–Groeling, and H. A. Weidenmüller, Random-matrix theories in quantum physics: Common concepts, Phys. Rep. 299, 189 (1998).
  66. C. B. Dağ, S. I. Mistakidis, A. Chan, and H. R. Sadeghpour, Many-body quantum chaos in stroboscopically-driven cold atoms, Commun. Phys. 6, 136 (2023).
  67. F. Fritzsch, M. F. I. Kieler, and A. Bäcker, Eigenstate correlations in dual-unitary quantum circuits: Partial spectral form factor, Quantum 9, 1709 (2025).
  68. T. Kalsi, A. Romito, and H. Schomerus, Hierarchical analytical approach to universal spectral correlations in Brownian quantum chaos, Phys. Rev. B 111, 094211 (2025).
  69. A. Vikram and V. Galitski, Dynamical quantum ergodicity from energy level statistics, Phys. Rev. Res. 5, 033126 (2023).
  70. A. K. Das, Validity of generalized Gibbs ensemble in a random matrix model with a global Z2-symmetry, arXiv:2506.16176.
  71. J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, Quantum chaos challenges many-body localization, Phys. Rev. E 102, 062144 (2020).
  72. C. L. Bertrand and A. M. García-García, Anomalous Thouless energy and critical statistics on the metallic side of the many-body localization transition, Phys. Rev. B 94, 144201 (2016).
  73. P. Sierant and J. Zakrzewski, Level statistics across the many-body localization transition, Phys. Rev. B 99, 104205 (2019).
  74. 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).
  75. T. Guhr and H. Weidenmuller, Correlations in anticrossing spectra and scattering theory. analytical aspects, Chem. Phys. 146, 21 (1990).
  76. J. Wang, L. Wang, J. Ma, A. Yang, L. Yuan, and L. Ying, Many-body quantum chaos, localization, and multiphoton entanglement in optical synthetic frequency dimension, Phys. Rev. Appl. 23, 034076 (2025).
  77. D. V. Vasilyev, A. Grankin, M. A. Baranov, L. M. Sieberer, and P. Zoller, Monitoring quantum simulators via quantum nondemolition couplings to atomic clock qubits, PRX Quantum 1, 020302 (2020).
  78. L. K. Joshi, A. Elben, A. Vikram, B. Vermersch, V. Galitski, and P. Zoller, Probing many-body quantum chaos with quantum simulators, Phys. Rev. X 12, 011018 (2022).
  79. F. J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3, 140 (1962).
  80. F. J. Dyson, A brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys. 3, 1191 (1962).
  81. P. J. Forrester, Beta random matrix ensembles, in Random Matrix Theory And Its Applications: Multivariate Statistics and Wireless Communications (World Scientific, Singapore, 2009), pp. 27–68.
  82. P. Forrester, Beta ensembles, in The Oxford Handbook of Random Matrix Theory (Oxford University Press, Oxford, 2011).
  83. G. Borot, B. Eynard, S. N. Majumdar, and C. Nadal, Large deviations of the maximal eigenvalue of random matrices, J. Stat. Mech. (2011) P11024.
  84. P. J. Forrester, Large deviation eigenvalue density for the soft edge Laguerre and Jacobi β-ensembles, J. Phys. A: Math. Theor. 45, 145201 (2012).
  85. R. Allez and L. Dumaz, Tracy-Widom at high temperature, J. Stat. Phys. 156, 1146 (2014).
  86. A. Edelman, P.-O. Persson, and B. D. Sutton, Low-temperature random matrix theory at the soft edge, J. Math. Phys. 55, 063302 (2014).
  87. A. Edelman and B. D. Sutton, From random matrices to stochastic operators, J. Stat. Phys. 127, 1121 (2007).
  88. P. Bourgade, L. Erdös, and H.-T. Yau, Edge universality of beta ensembles, Commun. Math. Phys. 332, 261 (2014).
  89. F. Bekerman, A. Figalli, and A. Guionnet, Transport maps for β-matrix models and universality, Commun. Math. Phys. 338, 589 (2015).
  90. M. Krishnapur, B. Rider, and B. Virág, Universality of the stochastic airy operator, Comm. Pure Appl. Math. 69, 145 (2016).
  91. P. Desrosiers and P. J. Forrester, Hermite and Laguerre β-ensembles: Asymptotic corrections to the eigenvalue density, Nucl. Phys. B 743, 307 (2006).
  92. P. J. Forrester, Spectral density asymptotics for Gaussian and Laguerre β-ensembles in the exponentially small region, J. Phys. A: Math. Theor. 45, 075206 (2012).
  93. R. Allez, J.-P. Bouchaud, and A. Guionnet, Invariant beta ensembles and the Gauss-Wigner crossover, Phys. Rev. Lett. 109, 094102 (2012).
  94. N. S. Witte and P. J. Forrester, Moments of the Gaussian β ensembles and the large-n expansion of the densities, J. Math. Phys. 55, 083302 (2014).
  95. P. J. Forrester, A. A. Rahman, and N. S. Witte, Large n expansions for the Laguerre and Jacobi β-ensembles from the loop equations, J. Math. Phys. 58, 113303 (2017).
  96. G. Livan, M. Novaes, and P. Vivo, Introduction to Random Matrices (Springer International Publishing, Cham, 2018).
  97. A. K. Das and A. Ghosh, Eigenvalue statistics for generalized symmetric and Hermitian matrices, J. Phys. A: Math. Theor. 52, 395001 (2019).
  98. D. Chafaï and G. Ferré, Simulating coulomb and log-gases with hybrid Monte Carlo algorithms, J. Stat. Phys. 174, 692 (2019).
  99. A. Pandey, A. Kumar, and S. Puri, Finite-range Coulomb gas models of banded random matrices and quantum kicked rotors, Phys. Rev. E 96, 052211 (2017).
  100. V. Balasubramanian, J. M. Magan, and Q. Wu, Tridiagonalizing random matrices, Phys. Rev. D 107, 126001 (2023).
  101. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
  102. J. Beugnon, C. Tuchendler, H. Marion, A. Gaëtan, Y. Miroshnychenko, Y. R. Sortais, A. M. Lance, M. P. Jones, G. Messin, A. Browaeys et al., Two-dimensional transport and transfer of a single atomic qubit in optical tweezers, Nat. Phys. 3, 696 (2007).
  103. A. M. Kaufman and K.-K. Ni, Quantum science with optical tweezer arrays of ultracold atoms and molecules, Nat. Phys. 17, 1324 (2021).
  104. F. J. Dyson, The dynamics of a disordered linear chain, Phys. Rev. 92, 1331 (1953).
  105. J. Gao, I. M. Khaymovich, A. Iovan, X.-W. Wang, G. Krishna, Z.-S. Xu, E. Tortumlu, A. V. Balatsky, V. Zwiller, and A. W. Elshaari, Coexistence of extended and localized states in finite-sized mosaic Wannier-Stark lattices, Phys. Rev. B 108, L140202 (2023).
  106. J. Gao, I. M. Khaymovich, X.-W. Wang, Z.-S. Xu, A. Iovan, G. Krishna, J. Jieensi, A. Cataldo, A. V. Balatsky, V. Zwiller et al., Probing multi-mobility edges in quasiperiodic mosaic lattices, Sci. Bull. 70, 58 (2025).
  107. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  108. J. A. Cuesta and A. Sánchez, General non-existence theorem for phase transitions in one-dimensional systems with short range interactions, and physical examples of such transitions, J. Stat. Phys. 115, 869 (2004).
  109. F. Izrailev, S. Ruffo, and L. Tessieri, Classical representation of the one-dimensional anderson model, J. Phys. A: Math. Gen. 31, 5263 (1998).
  110. F. Nakano and K. D. Trinh, Gaussian beta ensembles at high temperature: Eigenvalue fluctuations and bulk statistics, J. Stat. Phys. 173, 295 (2018).
  111. J. T. Albrecht, C. P. Chan, and A. Edelman, Sturm sequences and random eigenvalue distributions, Found. Comput. Math. 9, 461 (2009).
  112. 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, 084101 (2013).
  113. D. Facoetti, P. Vivo, and G. Biroli, From non-ergodic eigenvectors to local resolvent statistics and back: A random matrix perspective, Europhys. Lett. 115, 47003 (2016).
  114. K. Truong and A. Ossipov, Eigenvectors under a generic perturbation: Non-perturbative results from the random matrix approach, Europhys. Lett. 116, 37002 (2016).
  115. E. Bogomolny and M. Sieber, Eigenfunction distribution for the Rosenzweig-Porter model, Phys. Rev. E 98, 032139 (2018).
  116. P. von Soosten and S. Warzel, Non-ergodic delocalization in the Rosenzweig-Porter model, Lett. Math. Phys. 109, 905 (2019).
  117. I. M. Khaymovich, V. E. Kravtsov, B. L. Altshuler, and L. B. Ioffe, Fragile extended phases in the log-normal Rosenzweig-Porter model, Phys. Rev. Res. 2, 043346 (2020).
  118. G. Biroli and M. Tarzia, Lévy-Rosenzweig-Porter random matrix ensemble, Phys. Rev. B 103, 104205 (2021).
  119. W. Buijsman and Y. B. Lev, Circular Rosenzweig-Porter random matrix ensemble, SciPost Phys. 12, 082 (2022).
  120. G. De Tomasi and I. M. Khaymovich, Non-Hermitian Rosenzweig-Porter random-matrix ensemble: Obstruction to the fractal phase, Phys. Rev. B 106, 094204 (2022).
  121. M. Sarkar, R. Ghosh, and I. M. Khaymovich, Tuning the phase diagram of a rosenzweig-porter model with fractal disorder, Phys. Rev. B 108, L060203 (2023).
  122. B. Altshuler and V. Kravtsov, Random cantor sets and mini-bands in local spectrum of quantum systems, Ann. Phys. 456, 169300 (2023).
  123. M. Serbyn, Z. Papić, and D. A. Abanin, Thouless energy and multifractality across the many-body localization transition, Phys. Rev. B 96, 104201 (2017).
  124. P. Sierant and J. Zakrzewski, Model of level statistics for disordered interacting quantum many-body systems, Phys. Rev. B 101, 104201 (2020).
  125. W. Buijsman, V. Cheianov, and V. Gritsev, Sensitivity of the spectral form factor to short-range level statistics, Phys. Rev. E 102, 042216 (2020).
  126. P. Kos, M. Ljubotina, and T. Prosen, Many-body quantum chaos: Analytic connection to random matrix theory, Phys. Rev. X 8, 021062 (2018).
  127. A. Bäcker, F. Steiner, and P. Stifter, Spectral statistics in the quantized cardioid billiard, Phys. Rev. E 52, 2463 (1995).
  128. R. Berkovits, Super-poissonian behavior of the Rosenzweig-Porter model in the nonergodic extended regime, Phys. Rev. B 102, 165140 (2020).
  129. A. Ossipov, I. Rushkin, and E. Cuevas, Level-number variance and spectral compressibility in a critical two-dimensional random-matrix model, Phys. Rev. E 85, 021127 (2012).
  130. K. Efetov, Supersymmetry and theory of disordered metals, Adv. Phys. 32, 53 (1983).
  131. T. Mondal, S. Sadhukhan, and P. Shukla, Extended states with poisson spectral statistics, Phys. Rev. E 95, 062102 (2017).
  132. V. Kumar, T. Prosen, and D. Roy, Leading and beyond leading-order spectral form factor in chaotic quantum many-body systems across all dyson symmetry classes, arXiv:2502.04152.
  133. E. J. Torres-Herrera and L. F. Santos, Local quenches with global effects in interacting quantum systems, Phys. Rev. E 89, 062110 (2014).
  134. E. J. Torres-Herrera and L. F. Santos, Nonexponential fidelity decay in isolated interacting quantum systems, Phys. Rev. A 90, 033623 (2014).
  135. 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).
  136. K. Urbanowski, General properties of the evolution of unstable states at long times, Eur. Phys. J. D 54, 25 (2009).
  137. L. A. Khalfin, Contribution to the decay theory of a quasi-stationary state, Sov. Phys. JETP 6, 1053 (1958).
  138. M. Hopjan and L. Vidmar, Scale-invariant critical dynamics at eigenstate transitions, Phys. Rev. Res. 5, 043301 (2023).
  139. This is also visible in the total SFF at the minima of the oscillations of the disconnected SFF part, see, e.g., Fig. 6.
  140. J. Liu, Spectral form factors and late time quantum chaos, Phys. Rev. D 98, 086026 (2018).
  141. J. Cotler, N. Hunter-Jones, J. Liu, and B. Yoshida, Chaos, complexity, and random matrices, J. High Energy Phys. 11 (2017) 048.
  142. F. Fritzsch and M. F. I. Kieler, Universal spectral correlations in interacting chaotic few-body quantum systems, Phys. Rev. E 109, 014202 (2024).
  143. A. Kumar, A. Pandey, and S. Puri, Finite-range Coulomb gas models. II. Applications to quantum kicked rotors and banded random matrices, Phys. Rev. E 101, 022218 (2020).
  144. B. V. Chirikov, An example of chaotic eigenstates in a complex atom, Phys. Lett. A 108, 68 (1985).
  145. S. Iida, H. Weidenmüller, and J. Zuk, Statistical scattering theory, the supersymmetry method and universal conductance fluctuations, Ann. Phys. 200, 219 (1990).
  146. Y. V. Fyodorov and A. D. Mirlin, Level-to-level fluctuations of the inverse participation ratio in finite quasi 1D disordered systems, Phys. Rev. Lett. 71, 412 (1993).

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Phys. Rev. B 112, 094206– Published 23 September, 2025
  • Received 30 June 2025
  • Revised 3 September 2025
  • Accepted 5 September 2025
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