- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Scaling of many-body localization transitions: Quantum dynamics in Fock space and real space

Thibault Scoquart*, Igor V. Gornyi, and Alexander D. Mirlin

  • *Contact author: thibault.scoquart@kit.edu
PDF

Phys. Rev. B 112, 064203 – Published 7 August, 2025

DOI: https://doi.org/10.1103/qcvd-n2yk

Abstract

Many-body-localization (MBL) transitions are studied in a family of single-spin-flip spin-12 models, including the one-dimensional (1D) chain with nearest-neighbor interactions, the quantum dot (QD) model with all-to-all pair interactions, and the quantum random energy model (QREM). We investigate the generalized imbalance that characterizes propagation in Fock space out of an initial basis state and, at the same time, can be efficiently probed by real-space measurements. For all models considered, the average imbalance and its quantum and mesoscopic fluctuations provide excellent indicators for the position of the MBL transition Wc(n), where n is the number of spins. Combining these findings with earlier results on level statistics, we determine phase diagrams of the MBL transitions in the n−W plane. Our results provide evidence for a direct transition between the ergodic and MBL phases for each of the models, without any intermediate phase. For QREM and QD model, Wc(n) grows as a power law of n (with logarithmic corrections), in agreement with analytical predictions WcQREM(n)∼n1/2lnn and WcQD(n)≳n3/4ln1/2n. This growth is in stark contrast to the 1D model, where Wc(n) is essentially independent of n, consistent with the analytic expectation Wc1D(n→∞)=const. We also determine the scaling of the transition width ΔW(n)/Wc(n) and estimate the system size n needed to study the asymptotic scaling behavior. Although the corresponding values of n are larger than those accessible to exact simulations on a classical computer, they are within the reach of quantum simulators. Our results indicate feasibility of experimental studies of n−W phase diagrams and scaling properties of MBL transitions in models of 1D and QD type and in their extensions to other spatial geometry or distance-dependent interactions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (136)

  1. I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Interacting electrons in disordered wires: Anderson localization and low-T transport, Phys. Rev. Lett. 95, 206603 (2005).
  2. D. Basko, I. Aleiner, and B. Altshuler, Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states, Ann. Phys. 321, 1126 (2006).
  3. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  4. F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
  5. R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  6. F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics, C. R. Phys. 19, 498 (2018).
  7. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  8. S. Gopalakrishnan and S. Parameswaran, Dynamics and transport at the threshold of many-body localization, Phys. Rep. 862, 1 (2020).
  9. K. S. Tikhonov and A. D. Mirlin, From Anderson localization on random regular graphs to many-body localization, Ann. Phys. 435, 168525 (2021).
  10. E. V. Doggen, I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Many-body localization in large systems: Matrix-product-state approach, Ann. Phys. 435, 168437 (2021).
  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. A phase transition in a strict sense is a property of the system in the thermodynamic limit, n→∞. Clearly, the MBL transition becomes a crossover at a finite n. The disorder Wc(n) characterizes the location of this crossover, and ΔW(n) its width. The terms “critical disorder” for Wc(n) and “transition width” for ΔW should be understood in this sense. The crossover develops into a sharp transition, once ΔW(n)/Wc(n)→0 at n→∞. By the same token, we will speak about phase diagrams in the W−n plane. One can draw here an analogy to quantum phase transitions, where it is common to speak about phase diagrams with temperature T on one of the axes, while the phase transition strictly takes place at T=0 only. The situation of an MBL transition with Wc(n)→∞ at n→∞ was termed a “nonstandard thermodynamic limit” in Ref. [127].
  13. V. Ros, M. Müller, and A. Scardicchio, Integrals of motion in the many-body localized phase, Nucl. Phys. B 891, 420 (2015).
  14. J. Z. Imbrie, Diagonalization and many-body localization for a disordered quantum spin chain, Phys. Rev. Lett. 117, 027201 (2016).
  15. J. Z. Imbrie, On many-body localization for quantum spin chains, J. Stat. Phys. 163, 998 (2016).
  16. W. De Roeck and F. Huveneers, Stability and instability towards delocalization in many-body localization systems, Phys. Rev. B 95, 155129 (2017).
  17. T. Thiery, F. Huveneers, M. Müller, and W. De Roeck, Many-body delocalization as a quantum avalanche, Phys. Rev. Lett. 121, 140601 (2018).
  18. A. Goremykina, R. Vasseur, and M. Serbyn, Analytically solvable renormalization group for the many-body localization transition, Phys. Rev. Lett. 122, 040601 (2019).
  19. A. Morningstar and D. A. Huse, Renormalization-group study of the many-body localization transition in one dimension, Phys. Rev. B 99, 224205 (2019).
  20. A. Morningstar, D. A. Huse, and J. Z. Imbrie, Many-body localization near the critical point, Phys. Rev. B 102, 125134 (2020).
  21. A. Pal and D. A. Huse, Many-body localization phase transition, Phys. Rev. B 82, 174411 (2010).
  22. 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).
  23. E. V. H. Doggen, F. Schindler, K. S. Tikhonov, A. D. Mirlin, T. Neupert, D. G. Polyakov, and I. V. Gornyi, Many-body localization and delocalization in large quantum chains, Phys. Rev. B 98, 174202 (2018).
  24. P. Sierant, M. Lewenstein, and J. Zakrzewski, Polynomially filtered exact diagonalization approach to many-body localization, Phys. Rev. Lett. 125, 156601 (2020).
  25. J. Colbois, F. Alet, and N. Laflorencie, Interaction-driven instabilities in the random-field XXZ chain, Phys. Rev. Lett. 133, 116502 (2024).
  26. J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, Quantum chaos challenges many-body localization, Phys. Rev. E 102, 062144 (2020).
  27. J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, Ergodicity breaking transition in finite disordered spin chains, Phys. Rev. B 102, 064207 (2020).
  28. D. Sels and A. Polkovnikov, Dynamical obstruction to localization in a disordered spin chain, Phys. Rev. E 104, 054105 (2021).
  29. D. Sels, Bath-induced delocalization in interacting disordered spin chains, Phys. Rev. B 106, L020202 (2022).
  30. D. Sels and A. Polkovnikov, Thermalization of dilute impurities in one-dimensional spin chains, Phys. Rev. X 13, 011041 (2023).
  31. F. Weiner, F. Evers, and S. Bera, Slow dynamics and strong finite-size effects in many-body localization with random and quasiperiodic potentials, Phys. Rev. B 100, 104204 (2019).
  32. G. Biroli, A. K. Hartmann, and M. Tarzia, Large-deviation analysis of rare resonances for the many-body localization transition, Phys. Rev. B 110, 014205 (2024).
  33. J. Colbois, F. Alet, and N. Laflorencie, Statistics of systemwide correlations in the random-field XXZ chain: Importance of rare events in the many-body localized phase, Phys. Rev. B 110, 214210 (2024).
  34. 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).
  35. P. Jacquod and D. L. Shepelyansky, Emergence of quantum chaos in finite interacting Fermi systems, Phys. Rev. Lett. 79, 1837 (1997).
  36. A. D. Mirlin and Y. V. Fyodorov, Localization and fluctuations of local spectral density on treelike structures with large connectivity: Application to the quasiparticle line shape in quantum dots, Phys. Rev. B 56, 13393 (1997).
  37. P. G. Silvestrov, Decay of a quasiparticle in a quantum dot: The role of energy resolution, Phys. Rev. Lett. 79, 3994 (1997).
  38. P. G. Silvestrov, Chaos thresholds in finite Fermi systems, Phys. Rev. E 58, 5629 (1998).
  39. I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Many-body delocalization transition and relaxation in a quantum dot, Phys. Rev. B 93, 125419 (2016).
  40. I. Gornyi, A. Mirlin, D. Polyakov, and A. Burin, Spectral diffusion and scaling of many-body delocalization transitions, Ann. Phys. 529, 1600360 (2017).
  41. B. Georgeot and D. L. Shepelyansky, Breit-Wigner width and inverse participation ratio in finite interacting Fermi systems, Phys. Rev. Lett. 79, 4365 (1997).
  42. X. Leyronas, P. G. Silvestrov, and C. W. J. Beenakker, Scaling at the chaos threshold for interacting electrons in a quantum dot, Phys. Rev. Lett. 84, 3414 (2000).
  43. D. Shepelyansky, Quantum chaos and quantum computers, Phys. Scr. T90, 112 (2001).
  44. P. H. Song, Scaling near the quantum chaos border in interacting Fermi systems, Phys. Rev. E 62, R7575 (2000).
  45. P. Jacquod and I. Varga, Duality between the weak and strong interaction limits of deformed fermionic two-body random ensembles, Phys. Rev. Lett. 89, 134101 (2002).
  46. A. M. F. Rivas, E. R. Mucciolo, and A. Kamenev, Numerical study of quasiparticle lifetime in quantum dots, Phys. Rev. B 65, 155309 (2002).
  47. V. B. Bulchandani, D. A. Huse, and S. Gopalakrishnan, Onset of many-body quantum chaos due to breaking integrability, Phys. Rev. B 105, 214308 (2022).
  48. A. M. García-García, B. Loureiro, A. Romero-Bermúdez, and M. Tezuka, Chaotic-integrable transition in the Sachdev-Ye-Kitaev model, Phys. Rev. Lett. 120, 241603 (2018).
  49. T. Micklitz, F. Monteiro, and A. Altland, Nonergodic extended states in the Sachdev-Ye-Kitaev model, Phys. Rev. Lett. 123, 125701 (2019).
  50. F. Monteiro, T. Micklitz, M. Tezuka, and A. Altland, Minimal model of many-body localization, Phys. Rev. Res. 3, 013023 (2021).
  51. F. Monteiro, M. Tezuka, A. Altland, D. A. Huse, and T. Micklitz, Quantum ergodicity in the many-body localization problem, Phys. Rev. Lett. 127, 030601 (2021).
  52. D. K. Nandy, T. Čadež, B. Dietz, A. Andreanov, and D. Rosa, Delayed thermalization in the mass-deformed Sachdev-Ye-Kitaev model, Phys. Rev. B 106, 245147 (2022).
  53. A. Larzul and M. Schiró, Quenches and (pre)thermalization in a mixed Sachdev-Ye-Kitaev model, Phys. Rev. B 105, 045105 (2022).
  54. J.-N. Herre, J. F. Karcher, K. S. Tikhonov, and A. D. Mirlin, Ergodicity-to-localization transition on random regular graphs with large connectivity and in many-body quantum dots, Phys. Rev. B 108, 014203 (2023).
  55. J. Dieplinger and S. Bera, Finite-size prethermalization at the chaos-to-integrable crossover, Phys. Rev. B 107, 224207 (2023).
  56. M. Serbyn, Z. Papić, and D. A. Abanin, Thouless energy and multifractality across the many-body localization transition, Phys. Rev. B 96, 104201 (2017).
  57. K. S. Tikhonov and A. D. Mirlin, Many-body localization transition with power-law interactions: Statistics of eigenstates, Phys. Rev. B 97, 214205 (2018).
  58. N. Macé, F. Alet, and N. Laflorencie, Multifractal scalings across the many-body localization transition, Phys. Rev. Lett. 123, 180601 (2019).
  59. K. S. Tikhonov and A. D. Mirlin, Eigenstate correlations around the many-body localization transition, Phys. Rev. B 103, 064204 (2021).
  60. S. Nag and A. Garg, Many-body localization in the presence of long-range interactions and long-range hopping, Phys. Rev. B 99, 224203 (2019).
  61. M. Tarzia, Many-body localization transition in Hilbert space, Phys. Rev. B 102, 014208 (2020).
  62. S. Roy and D. E. Logan, Fock-space anatomy of eigenstates across the many-body localization transition, Phys. Rev. B 104, 174201 (2021).
  63. P. J. D. Crowley and A. Chandran, A constructive theory of the numerically accessible many-body localized to thermal crossover, SciPost Phys. 12, 201 (2022).
  64. D. M. Long, P. J. D. Crowley, V. Khemani, and A. Chandran, Phenomenology of the prethermal many-body localized regime, Phys. Rev. Lett. 131, 106301 (2023).
  65. I. Creed, D. E. Logan, and S. Roy, Probability transport on the Fock space of a disordered quantum spin chain, Phys. Rev. B 107, 094206 (2023).
  66. N. Roy, J. Sutradhar, and S. Banerjee, Diagnostics of nonergodic extended states and many body localization proximity effect through real-space and Fock-space excitations, Phys. Rev. B 107, 115155 (2023).
  67. S. Ghosh, J. Sutradhar, S. Mukerjee, and S. Banerjee, Scaling of Fock space propagator in quasiperiodic many-body localizing systems, Ann. Phys. 478, 170001 (2025).
  68. 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).
  69. P. Roushan, C. Neill, J. Tangpanitanon, V. M. Bastidas, A. Megrant, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, B. Foxen, M. Giustina, E. Jeffrey, J. Kelly, E. Lucero, J. Mutus, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. White, H. Neven, D. G. Angelakis, and J. Martinis, Spectroscopic signatures of localization with interacting photons in superconducting qubits, Science 358, 1175 (2017).
  70. K. Xu, J.-J. Chen, Y. Zeng, Y.-R. Zhang, C. Song, W. Liu, Q. Guo, P. Zhang, D. Xu, H. Deng, K. Huang, H. Wang, X. Zhu, D. Zheng, and H. Fan, Emulating many-body localization with a superconducting quantum processor, Phys. Rev. Lett. 120, 050507 (2018).
  71. A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. Léonard, and M. Greiner, Probing entanglement in a many-body-localized system, Science 364, 256 (2019).
  72. Y. Yao, L. Xiang, Z. Guo, Z. Bao, Y.-F. Yang, Z. Song, H. Shi, X. Zhu, F. Jin, J. Chen, S. Xu, Z. Zhu, F. Shen, N. Wang, C. Zhang, Y. Wu, Y. Zou, P. Zhang, H. Li, Z. Wang, C. Song, C. Cheng, R. Mondaini, H. Wang, J. Q. You, S.-Y. Zhu, L. Ying, and Q. Guo, Observation of many-body Fock space dynamics in two dimensions, Nat. Phys. 19, 1459 (2023).
  73. Y.-Y. Wang, Y.-H. Shi, Z.-H. Sun, C.-T. Chen, Z.-A. Wang, K. Zhao, H.-T. Liu, W.-G. Ma, Z. Wang, H. Li, J.-C. Zhang, Y. Liu, C.-L. Deng, T.-M. Li, Y. He, Z.-H. Liu, Z.-Y. Peng, X. Song, G. Xue, H. Yu, K. Huang, Z. Xiang, D. Zheng, K. Xu, and H. Fan, Exploring Hilbert-space fragmentation on a superconducting processor, PRX Quantum 6, 010325 (2025).
  74. T. Scoquart, I. V. Gornyi, and A. D. Mirlin, Role of Fock-space correlations in many-body localization, Phys. Rev. B 109, 214203 (2024).
  75. G. Biroli, A. Ribeiro-Teixeira, and M. Tarzia, Difference between level statistics, ergodicity and localization transitions on the Bethe lattice, arXiv:1211.7334.
  76. A. De Luca, B. L. Altshuler, V. E. Kravtsov, and A. Scardicchio, Anderson localization on the Bethe lattice: Nonergodicity of extended states, Phys. Rev. Lett. 113, 046806 (2014).
  77. K. Tikhonov, A. Mirlin, and M. Skvortsov, Anderson localization and ergodicity on random regular graphs, Phys. Rev. B 94, 220203(R) (2016).
  78. 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).
  79. F. L. Metz and I. P. Castillo, Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase, Phys. Rev. B 96, 064202 (2017).
  80. G. Biroli and M. Tarzia, Delocalized glassy dynamics and many-body localization, Phys. Rev. B 96, 201114(R) (2017).
  81. V. Kravtsov, B. Altshuler, and L. Ioffe, Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph, Ann. Phys. 389, 148 (2018).
  82. G. Biroli and M. Tarzia, Delocalization and ergodicity of the Anderson model on Bethe lattices, arXiv:1810.07545.
  83. S. Bera, G. De Tomasi, I. M. Khaymovich, and A. Scardicchio, Return probability for the Anderson model on the random regular graph, Phys. Rev. B 98, 134205 (2018).
  84. K. S. Tikhonov and A. D. Mirlin, Statistics of eigenstates near the localization transition on random regular graphs, Phys. Rev. B 99, 024202 (2019).
  85. K. S. Tikhonov and A. D. Mirlin, Critical behavior at the localization transition on random regular graphs, Phys. Rev. B 99, 214202 (2019).
  86. 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(R) (2020).
  87. 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).
  88. 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).
  89. P. Sierant, M. Lewenstein, and A. Scardicchio, Universality in Anderson localization on random graphs with varying connectivity, SciPost Phys. 15, 045 (2023).
  90. O. Valba and A. Gorsky, Mobility edge in the Anderson model on partially disordered random regular graphs, JETP Lett. 116, 398 (2022).
  91. G. Biroli, A. K. Hartmann, and M. Tarzia, Critical behavior of the Anderson model on the Bethe lattice via a large-deviation approach, Phys. Rev. B 105, 094202 (2022).
  92. T. Rizzo and M. Tarzia, Localized phase of the Anderson model on the Bethe lattice, Phys. Rev. B 110, 184210 (2024).
  93. D. Kochergin, I. M. Khaymovich, O. Valba, and A. Gorsky, Robust extended states in Anderson model on partially disordered random regular graphs, SciPost Phys. 16, 106 (2024).
  94. A. Mirlin and Y. V. Fyodorov, Universality of level correlation function of sparse random matrices, J. Phys. A: Math. Gen. 24, 2273 (1991).
  95. Y. V. Fyodorov and A. D. Mirlin, Localization in ensemble of sparse random matrices, Phys. Rev. Lett. 67, 2049 (1991).
  96. The 1D model defined by Eqs. (1)– (5) is exactly the model that was studied numerically in Ref. [74]. At the same time, there was a slight inaccuracy in the definition of this model in Ref. [74]: in Eq. (27) of Ref. [74], the second term of Eq.  (3) of the present work was missing, which was compensated by an additional factor two in Eq. (30) of Ref. [74] compared to Eq. (5) of the present work. This did not influence the variances of Fock-space energies Eα and hoppings Tαβ, Eq.  (9), as well as the energy covariance matrix (CE1D)αβ, Eq. (10). However, this led to a small modification in the covariance matrix (CT1D)αβμν: the second term of Eq. (11) was missing in Eq. (37) of Ref. [74], compensated by an additional factor of two. This modified 1D model has the same properties as that defined by Eqs. (1)– (5), with a slight shift of numerical values of Wc(n). Since the model that was studied numerically in Ref.  [74] was exactly the model explored in the present work, we will directly compare the numerical values.
  97. C. R. Laumann, A. Pal, and A. Scardicchio, Many-body mobility edge in a mean-field quantum spin glass, Phys. Rev. Lett. 113, 200405 (2014).
  98. C. L. Baldwin, C. R. Laumann, A. Pal, and A. Scardicchio, The many-body localized phase of the quantum random energy model, Phys. Rev. B 93, 024202 (2016).
  99. Y. Y. Goldschmidt, Solvable model of the quantum spin glass in a transverse field, Phys. Rev. B 41, 4858 (1990).
  100. M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of interacting fermions in a quasirandom optical lattice, Science 349, 842 (2015).
  101. J.-Y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Exploring the many-body localization transition in two dimensions, Science 352, 1547 (2016).
  102. D. J. Luitz, N. Laflorencie, and F. Alet, Extended slow dynamical regime close to the many-body localization transition, Phys. Rev. B 93, 060201(R) (2016).
  103. P. Prelovšek, M. Mierzejewski, O. Barišić, and J. Herbrych, Density correlations and transport in models of many-body localization, Ann. Phys. 529, 1600362 (2017).
  104. E. V. H. Doggen and A. D. Mirlin, Many-body delocalization dynamics in long Aubry-André quasiperiodic chains, Phys. Rev. B 100, 104203 (2019).
  105. E. V. H. Doggen, I. V. Gornyi, and D. G. Polyakov, Stark many-body localization: Evidence for Hilbert-space shattering, Phys. Rev. B 103, L100202 (2021).
  106. P. Sierant and J. Zakrzewski, Challenges to observation of many-body localization, Phys. Rev. B 105, 224203 (2022).
  107. G. De Tomasi, I. M. Khaymovich, F. Pollmann, and S. Warzel, Rare thermal bubbles at the many-body localization transition from the Fock space point of view, Phys. Rev. B 104, 024202 (2021).
  108. Z.-H. Sun, Y.-Y. Wang, J. Cui, H. Fan, and M. Heyl, Characterizing dynamical criticality of many-body localization transitions from the Fock-space perspective, Phys. Rev. B 111, 094210 (2025).
  109. C. Monthus and C. Texier, Random walk on the Bethe lattice and hyperbolic Brownian motion, J. Phys. A: Math. Gen. 29, 2399 (1996).
  110. K. B. Efetov, Density–density correlator in a model of a disordered metal on a Bethe lattice, Zh. Eksp. Teor. Fiz. 92, 638 (1987) [JETP 65, 360 (1987)].
  111. A. D. Mirlin and Y. V. Fyodorov, Localization transition in the Anderson model on the Bethe lattice: Spontaneous symmetry breaking and correlation functions, Nucl. Phys. B 366, 507 (1991).
  112. S. Gopalakrishnan, K. Agarwal, E. A. Demler, D. A. Huse, and M. Knap, Griffiths effects and slow dynamics in nearly many-body localized systems, Phys. Rev. B 93, 134206 (2016).
  113. P. Pöpperl, I. V. Gornyi, and A. D. Mirlin, Memory effects in the density-wave imbalance in delocalized disordered systems, Phys. Rev. B 106, 094201 (2022).
  114. B. Villalonga and B. K. Clark, Eigenstates hybridize on all length scales at the many-body localization transition, arXiv:2005.13558.
  115. R. Ghosh and M. Žnidarič, Resonance-induced growth of number entropy in strongly disordered systems, Phys. Rev. B 105, 144203 (2022).
  116. 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).
  117. J.-N. Herre, Modelling many-body localization with Anderson localization on random regular graphs, Master thesis, Karlsruhe Institute of Technology, 2022.
  118. H. Ha, A. Morningstar, and D. A. Huse, Many-body resonances in the avalanche instability of many-body localization, Phys. Rev. Lett. 130, 250405 (2023).
  119. The n dependence in the extrapolation γI(W,n)=γI(W,∞)+c(W)/n is justified as follows. We perform a fit of I(t)¯ by a power law, i.e., by a linear function on the log-log scale, in the range from t=O(1) to t∼tH∼2n, which corresponds to a window of width ∼n for lnt. There is necessarily a finite-size correction to the obtained slope, which comes from region of times t∼O(1) near the left boundary of this range. Since the imbalance changes by an n-independent factor in this range, the resulting correction to the slope γI scales as 1/n.
  120. A. Chandran, C. R. Laumann, and V. Oganesyan, Finite size scaling bounds on many-body localized phase transitions, arXiv:1509.04285.
  121. J. T. Chayes, L. Chayes, D. S. Fisher, and T. Spencer, Correlation length bounds for disordered Ising ferromagnets, Commun. Math. Phys. 120, 501 (1989).
  122. As pointed out in Ref. [120], the numerical coefficient in front of n−1/2 in the bound formally diverges for a box distribution of εi. We used a Gaussian distribution of εi, with the same variance, to obtain the coefficient in the bound (88).
  123. J. A. Kjäll, J. H. Bardarson, and F. Pollmann, Many-body localization in a disordered quantum Ising chain, Phys. Rev. Lett. 113, 107204 (2014).
  124. While the observed saturation of numerical data for Wc1D(n)—in contrast to the fast power-law growth of WcQD(n) and WcQD(n)—provides a clear evidence of the value Wc1D(n→∞)≈6.5, it is worth mentioning that finite-size numerics cannot in principle provide a rigorous proof of the numerical value of Wc1D(n→∞).
  125. 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.
  126. F. Balducci, G. B. Testasecca, J. Niedda, A. Scardicchio, and C. Vanoni, Scaling analysis and renormalization group on the mobility edge in the quantum random energy model, Phys. Rev. B 111, 214206 (2025).
  127. S. Gopalakrishnan and D. A. Huse, Instability of many-body localized systems as a phase transition in a nonstandard thermodynamic limit, Phys. Rev. B 99, 134305 (2019).
  128. E. V. H. Doggen, I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Slow many-body delocalization beyond one dimension, Phys. Rev. Lett. 125, 155701 (2020).
  129. M. Kiefer-Emmanouilidis, R. Unanyan, M. Fleischhauer, and J. Sirker, Evidence for unbounded growth of the number entropy in many-body localized phases, Phys. Rev. Lett. 124, 243601 (2020).
  130. M. Kiefer-Emmanouilidis, R. Unanyan, M. Fleischhauer, and J. Sirker, Unlimited growth of particle fluctuations in many-body localized phases, Ann. Phys. 435, 168481 (2021).
  131. M. Kiefer-Emmanouilidis, R. Unanyan, M. Fleischhauer, and J. Sirker, Slow delocalization of particles in many-body localized phases, Phys. Rev. B 103, 024203 (2021).
  132. F. Evers, I. Modak, and S. Bera, Internal clock of many-body delocalization, Phys. Rev. B 108, 134204 (2023).
  133. D. Aceituno Chávez, C. Artiaco, T. Klein Kvorning, L. Herviou, and J. H. Bardarson, Ultraslow growth of number entropy in an ℓ-bit model of many-body localization, Phys. Rev. Lett. 133, 126502 (2024).
  134. C. Sanderson and R. Curtin, Armadillo: An efficient framework fornumerical linear algebra, in Proceedings of the 17th International Conference on Computer and Automation Engineering (ICCAE) (IEEE, New York, 2025), pp. 303–307.
  135. M. Serbyn, Z. Papić, and D. A. Abanin, Universal slow growth of entanglement in interacting strongly disordered systems, Phys. Rev. Lett. 110, 260601 (2013).
  136. D. B. Gutman, I. V. Protopopov, A. L. Burin, I. V. Gornyi, R. A. Santos, and A. D. Mirlin, Energy transport in the Anderson insulator, Phys. Rev. B 93, 245427 (2016).

Outline

Information

Phys. Rev. B 112, 064203– Published 7 August, 2025
  • Received 22 February 2025
  • Revised 20 May 2025
  • Accepted 14 July 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation