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Scaling Theory of Fading Ergodicity

Rafał Świętek1,2, Miroslav Hopjan1,3, Carlo Vanoni4,5, Antonello Scardicchio6,7, and Lev Vidmar1,2

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Phys. Rev. Lett. 135, 170401 – Published 21 October, 2025

DOI: https://doi.org/10.1103/4l42-l7pk

Abstract

In most noninteracting quantum systems, the scaling theory of localization predicts one-parameter scaling flow in both ergodic and localized regimes. A corresponding scaling theory of many-body ergodicity breaking is still missing. Here, we introduce a scaling theory of ergodicity breaking in interacting systems, in which the divergent relaxation time follows from the Fermi golden rule, and the observable fluctuations in proximity of the ergodicity breaking critical point are described by the recently introduced fading ergodicity scenario. We argue that, in general, the one-parameter scaling is insufficient, and we show that the scaling theory predicts the critical exponent ν=1 at the ergodicity breaking critical point. Our theoretical framework may serve as a building block for two-parameter scaling theories of many-body systems.

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References (80)

  1. J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  2. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  3. M. Srednicki, The approach to thermal equilibrium in quantized chaotic systems, J. Phys. A 32, 1163 (1999).
  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. 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).
  7. J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog. Phys. 81, 082001 (2018).
  8. S. Trotzky, Y.-A. Chen, A. Flesch, I. P. McCulloch, U. Schollwöck, J. Eisert, and I. Bloch, Probing the relaxation towards equilibrium in an isolated strongly correlated 1D Bose gas, Nat. Phys. 8, 325 (2012).
  9. T. Langen, R. Geiger, M. Kuhnert, B. Rauer, and J. Schmiedmayer, Local emergence of thermal correlations in an isolated quantum many-body system, Nat. Phys. 9, 640 (2013).
  10. J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many-body systems out of equilibrium, Nat. Phys. 11, 124 (2015).
  11. P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, Many-body localization in the age of classical computing, Rep. Prog. Phys. 88, 026502 (2025).
  12. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, New York, 2011).
  13. K. Binder and A. P. Young, Spin glasses: Experimental facts, theoretical concepts, and open questions, Rev. Mod. Phys. 58, 801 (1986).
  14. M. Mézard, G. Parisi, and M. A. Virasoro, Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications (World Scientific, Singapore, 1987), Vol. 9.
  15. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  16. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979).
  17. A. MacKinnon and B. Kramer, One-parameter scaling of localization length and conductance in disordered systems, Phys. Rev. Lett. 47, 1546 (1981).
  18. A. MacKinnon and B. Kramer, The scaling theory of electrons in disordered solids: Additional numerical results, Z. Phys. B 53, 1 (1983).
  19. K. Slevin and T. Ohtsuki, Corrections to scaling at the Anderson transition, Phys. Rev. Lett. 82, 382 (1999).
  20. A. Rodriguez, L. J. Vasquez, K. Slevin, and R. A. Römer, Critical parameters from a generalized multifractal analysis at the Anderson transition, Phys. Rev. Lett. 105, 046403 (2010).
  21. M. Lopez, J.-F. m. c. Clément, P. Szriftgiser, J. C. Garreau, and D. Delande, Experimental test of universality of the Anderson transition, Phys. Rev. Lett. 108, 095701 (2012).
  22. K. Slevin and T. Ohtsuki, Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class, New J. Phys. 16, 015012 (2014).
  23. E. Tarquini, G. Biroli, and M. Tarzia, Critical properties of the Anderson localization transition and the high-dimensional limit, Phys. Rev. B 95, 094204 (2017).
  24. T. Devakul and D. A. Huse, Anderson localization transitions with and without random potentials, Phys. Rev. B 96, 214201 (2017).
  25. K. Slevin and T. Ohtsuki, Critical exponent of the Anderson transition using massively parallel supercomputing, J. Phys. Soc. Jpn. 87, 094703 (2018).
  26. X. Luo and T. Ohtsuki, Universality classes of the Anderson transitions driven by quasiperiodic potential in the three-dimensional Wigner-Dyson symmetry classes, Phys. Rev. B 106, 104205 (2022).
  27. P. Sierant, D. Delande, and J. Zakrzewski, Thouless time analysis of Anderson and many-body localization transitions, Phys. Rev. Lett. 124, 186601 (2020).
  28. J. Šuntajs, T. Prosen, and L. Vidmar, Spectral properties of three-dimensional Anderson model, Ann. Phys. (Amsterdam) 435, 168469 (2021).
  29. J. Šuntajs, T. Prosen, and L. Vidmar, Localization challenges quantum chaos in the finite two-dimensional Anderson model, Phys. Rev. B 107, 064205 (2023).
  30. C. Vanoni, B. L. Altshuler, V. E. Kravtsov, and A. Scardicchio, Renormalization group analysis of the Anderson model on random regular graphs, Proc. Natl. Acad. Sci. U.S.A. 121, e2401955121 (2024).
  31. B. L. Altshuler, V. E. Kravtsov, A. Scardicchio, P. Sierant, and C. Vanoni, Renormalization group for Anderson localization on high-dimensional lattices, Proc. Natl. Acad. Sci. U.S.A. 122, e2423763122 (2025).
  32. I. García-Mata, O. Giraud, B. Georgeot, J. Martin, R. Dubertrand, and G. Lemarié, Scaling theory of the Anderson transition in random graphs: Ergodicity and universality, Phys. Rev. Lett. 118, 166801 (2017).
  33. I. García-Mata, J. Martin, R. Dubertrand, O. Giraud, B. Georgeot, and G. Lemarié, Two critical localization lengths in the Anderson transition on random graphs, Phys. Rev. Res. 2, 012020 (2020).
  34. M. Pino, Scaling up the Anderson transition in random-regular graphs, Phys. Rev. Res. 2, 042031 (2020).
  35. I. García-Mata, J. Martin, O. Giraud, B. Georgeot, R. Dubertrand, and G. Lemarié, Critical properties of the Anderson transition on random graphs: Two-parameter scaling theory, Kosterlitz-Thouless type flow, and many-body localization, Phys. Rev. B 106, 214202 (2022).
  36. P. Sierant, M. Lewenstein, and A. Scardicchio, Universality in Anderson localization on random graphs with varying connectivity, SciPost Phys. 15, 045 (2023).
  37. J. M. Kosterlitz, Kosterlitz–Thouless physics: A review of key issues, Rep. Prog. Phys. 79, 026001 (2016).
  38. B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Levitov, Quasiparticle lifetime in a finite system: A nonperturbative approach, Phys. Rev. Lett. 78, 2803 (1997).
  39. D. Basko, I. Aleiner, and B. Altshuler, Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states, Ann. Phys. (Amsterdam) 321, 1126 (2006).
  40. S. Roy and D. E. Logan, The Fock-space landscape of many-body localisation, J. Phys. Condens. Matter 37, 073003 (2024).
  41. J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, Quantum chaos challenges many-body localization, Phys. Rev. E 102, 062144 (2020).
  42. J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, Ergodicity breaking transition in finite disordered spin chains, Phys. Rev. B 102, 064207 (2020).
  43. T. LeBlond, D. Sels, A. Polkovnikov, and M. Rigol, Universality in the onset of quantum chaos in many-body systems, Phys. Rev. B 104, L201117 (2021).
  44. H. Kim and A. Polkovnikov, Integrability as an attractor of adiabatic flows, Phys. Rev. B 109, 195162 (2024).
  45. B. Bhattacharjee, A. Andreanov, and S. Flach, Thermalization slowing down of weakly nonintegrable quantum spin dynamics, Phys. Rev. Res. 7, 023149 (2025).
  46. R. K. Panda, A. Scardicchio, M. Schulz, S. R. Taylor, and M. Žnidarič, Can we study the many-body localisation transition?, Europhys. Lett. 128, 67003 (2019).
  47. P. Sierant, M. Lewenstein, and J. Zakrzewski, Polynomially filtered exact diagonalization approach to many-body localization, Phys. Rev. Lett. 125, 156601 (2020).
  48. N. Laflorencie, G. Lemarié, and N. Macé, Chain breaking and Kosterlitz-Thouless scaling at the many-body localization transition in the random-field Heisenberg spin chain, Phys. Rev. Res. 2, 042033 (2020).
  49. D. Sels and A. Polkovnikov, Dynamical obstruction to localization in a disordered spin chain, Phys. Rev. E 104, 054105 (2021).
  50. D. Sels and A. Polkovnikov, Thermalization of dilute impurities in one-dimensional spin chains, Phys. Rev. X 13, 011041 (2023).
  51. J. Niedda, G. B. Testasecca, G. Magnifico, F. Balducci, C. Vanoni, and A. Scardicchio, Renormalization-group analysis of the many-body localization transition in the random-field XXZ chain, arXiv:2410.12430.
  52. V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007).
  53. M. Žnidarič, T. Prosen, and P. Prelovšek, Many-body localization in the Heisenberg XXZ magnet in a random field, Phys. Rev. B 77, 064426 (2008).
  54. A. De Luca and A. Scardicchio, Ergodicity breaking in a model showing many-body localization, Europhys. Lett. 101, 37003 (2013).
  55. F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics, C. R. Phys. 19, 498 (2018).
  56. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  57. D. J. Luitz, F. Huveneers, and W. De Roeck, How a small quantum bath can thermalize long localized chains, Phys. Rev. Lett. 119, 150602 (2017).
  58. J. Šuntajs and L. Vidmar, Ergodicity breaking transition in zero dimensions, Phys. Rev. Lett. 129, 060602 (2022).
  59. M. Kliczkowski, R. Świętek, M. Hopjan, and L. Vidmar, Fading ergodicity, Phys. Rev. B 110, 134206 (2024).
  60. 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).
  61. J. Šuntajs, M. Hopjan, W. De Roeck, and L. Vidmar, Similarity between a many-body quantum avalanche model and the ultrametric random matrix model, Phys. Rev. Res. 6, 023030 (2024).
  62. W. De Roeck and F. Huveneers, Stability and instability towards delocalization in many-body localization systems, Phys. Rev. B 95, 155129 (2017).
  63. T. Thiery, F. Huveneers, M. Müller, and W. De Roeck, Many-body delocalization as a quantum avalanche, Phys. Rev. Lett. 121, 140601 (2018).
  64. I.-D. Potirniche, S. Banerjee, and E. Altman, Exploration of the stability of many-body localization in d>1, Phys. Rev. B 99, 205149 (2019).
  65. P. J. D. Crowley and A. Chandran, Avalanche induced coexisting localized and thermal regions in disordered chains, Phys. Rev. Res. 2, 033262 (2020).
  66. P. J. D. Crowley and A. Chandran, Mean-field theory of failed thermalizing avalanches, Phys. Rev. B 106, 184208 (2022).
  67. D. Sels, Bath-induced delocalization in interacting disordered spin chains, Phys. Rev. B 106, L020202 (2022).
  68. A. Morningstar, L. Colmenarez, V. Khemani, D. J. Luitz, and D. A. Huse, Avalanches and many-body resonances in many-body localized systems, Phys. Rev. B 105, 174205 (2022).
  69. J. Léonard, S. Kim, M. Rispoli, A. Lukin, R. Schittko, J. Kwan, E. Demler, D. Sels, and M. Greiner, Probing the onset of quantum avalanches in a many-body localized system, Nat. Phys. 19, 481 (2023).
  70. K. Pawlik, P. Sierant, L. Vidmar, and J. Zakrzewski, Many-body mobility edge in quantum sun models, Phys. Rev. B 109, L180201 (2024).
  71. T. Szołdra, P. Sierant, M. Lewenstein, and J. Zakrzewski, Catching thermal avalanches in the disordered XXZ model, Phys. Rev. B 109, 134202 (2024).
  72. A. Kutlin and C. Vanoni, Investigating finite-size effects in random matrices by counting resonances, SciPost Phys. 18, 090 (2025).
  73. E. Bianchi, L. Hackl, M. Kieburg, M. Rigol, and L. Vidmar, Volume-law entanglement entropy of typical pure quantum states, PRX Quantum 3, 030201 (2022).
  74. See Supplemental Material at http://link.aps.org/supplemental/10.1103/4l42-l7pk for further details on numerical implementation of the model, the relationship between the entanglement entropy and matrix elements fluctuations, finite-size scaling of the entanglement entropies, and the numerical implementation of the beta function. It also includes Ref. [75].
  75. A. T. James, Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Stat. 35, 475 (1964).
  76. www.hpc-rivr.si.
  77. eurohpc-ju.europa.eu.
  78. www.izum.si.
  79. M. Hopjan and L. Vidmar, Scale-invariant survival probability at eigenstate transitions, Phys. Rev. Lett. 131, 060404 (2023).
  80. M. Hopjan and L. Vidmar, Scale-invariant critical dynamics at eigenstate transitions, Phys. Rev. Res. 5, 043301 (2023).

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Phys. Rev. Lett. 135, 170401– Published 21 October, 2025
  • Received 19 December 2024
  • Revised 7 July 2025
  • Accepted 24 September 2025
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