- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

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

Power-law banded random matrix ensemble as a model for quantum many-body Hamiltonians

Wouter Buijsman1,*, Masudul Haque1,2,†, and Ivan M. Khaymovich3,‡

  • *Contact author: buijsman@pks.mpg.de
  • †Contact author: masudul.haque@tu-dresden.de
  • ‡Contact author: ivan.khaymovich@gmail.com
PDF

Phys. Rev. E 113, 034116 – Published 13 March, 2026

DOI: https://doi.org/10.1103/tvms-89w5

Abstract

We explore interpretations of the power-law banded random matrix (PLBRM) ensemble as Hamiltonians of one-dimensional quantum many-body systems. We introduce and compare a number of labeling schemes for assigning random matrix basis indices to many-body basis vectors. We compare the physical properties of the resulting Hamiltonians, focusing on the half-system eigenstate bipartite entanglement entropy. We show and quantify how the different PLBRM phases (ergodic, weakly ergodic, localized), known from the single-particle interpretation, can be interpreted as entanglement transitions in the quantum many-body interpretation. For the weakly ergodic phase, where spectral edge and bulk eigenstates show distinct behavior, we perform a detailed scaling analysis to provide a quantitative picture of the boundaries between different types of entanglement scaling behaviors. In particular, we identify and characterize an intermediate set of eigenstates whose entanglement entropy have volume law scaling but nonvanishing deviation from the Page value expected for maximally ergodic states.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (119)

  1. M. L. Mehta, Random Matrices, 2nd ed. (Academic Press, London, UK, 1991).
  2. T. Guhr, A. Müller-Groeling, and H. A. Weidemüller, Random-matrix theories in quantum physics: Common concepts, Phys. Rep. 299, 189 (1998).
  3. F. Haake, Quantum Signatures of Chaos (Springer-Verlag, Berlin, 2010).
  4. J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  5. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  6. M. Rigol and L. F. Santos, Quantum chaos and thermalization in gapped systems, Phys. Rev. A 82, 011604(R) (2010).
  7. P. Reimann, Generalization of von Neumann's approach to thermalization, Phys. Rev. Lett. 115, 010403 (2015).
  8. 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).
  9. F. Borgonovi, F. M. Izrailev, L. F. Santos, and V. G. Zelevinsky, Quantum chaos and thermalization in isolated systems of interacting particles, Phys. Rep. 626, 1 (2016).
  10. C. Gogolin and J. Eisert, Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems, Rep. Prog. Phys. 79, 056001 (2016).
  11. R. Mondaini and M. Rigol, Eigenstate thermalization in the two-dimensional transverse field Ising model. II. Off-diagonal matrix elements of observables, Phys. Rev. E 96, 012157 (2017).
  12. J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog. Phys. 81, 082001 (2018).
  13. I. M. Khaymovich, M. Haque, and P. A. McClarty, Eigenstate thermalization, random matrix theory, and behemoths, Phys. Rev. Lett. 122, 070601 (2019).
  14. S. Sugimoto, R. Hamazaki, and M. Ueda, Test of the eigenstate thermalization hypothesis based on local random matrix theory, Phys. Rev. Lett. 126, 120602 (2021).
  15. N. Rosenzweig and C. E. Porter, “Repulsion of energy levels” in complex atomic spectra, Phys. Rev. 120, 1698 (1960).
  16. 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).
  17. 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).
  18. K. Truong and A. Ossipov, Eigenvectors under a generic perturbation: Non-perturbative results from the random matrix approach, Europhys. Lett. 116, 37002 (2016).
  19. M. Amini, Spread of wave packets in disordered hierarchical lattices, Europhys. Lett. 117, 30003 (2017).
  20. P. von Soosten and S. Warzel, Non-ergodic delocalization in the Rosenzweig–Porter model, Lett. Math. Phys. 109, 905 (2018).
  21. 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).
  22. V. E. Kravtsov, I. M. Khaymovich, B. L. Altshuler, and L. B. Ioffe, Localization transition on the random regular graph as an unstable tricritical point in a log-normal Rosenzweig-Porter random matrix ensemble, arXiv:2002.02979.
  23. 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).
  24. G. Biroli and M. Tarzia, Lévy-Rosenzweig-Porter random matrix ensemble, Phys. Rev. B 103, 104205 (2021).
  25. I. M. Khaymovich and V. E. Kravtsov, Dynamical phases in a “multifractal” Rosenzweig-Porter model, SciPost Phys. 11, 045 (2021).
  26. W. Buijsman and Y. Bar Lev, Circular Rosenzweig-Porter random matrix ensemble, SciPost Phys. 12, 082 (2022).
  27. 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).
  28. D. Venturelli, L. F. Cugliandolo, G. Schehr, and M. Tarzia, Replica approach to the generalized Rosenzweig-Porter model, SciPost Phys. 14, 110 (2023).
  29. 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).
  30. A. Kutlin and I. M. Khaymovich, Anatomy of the eigenstates distribution: A quest for a genuine multifractality, SciPost Phys. 16, 008 (2024).
  31. W. Buijsman, Long-range spectral statistics of the Rosenzweig-Porter model, Phys. Rev. B 109, 024205 (2024).
  32. R. Ghosh, M. Sarkar, and I. M. Khaymovich, Reentrant localization induced by short-range hopping in the fractal Rosenzweig-Porter model, Phys. Rev. B 111, L220102 (2025).
  33. I. Dumitriu and A. Edelman, Matrix models for beta ensembles, J. Math. Phys. 43, 5830 (2002).
  34. W. Buijsman, V. Cheianov, and V. Gritsev, Random matrix ensemble for the level statistics of many-body localization, Phys. Rev. Lett. 122, 180601 (2019).
  35. V. Balasubramanian, J. M. Magan, and Q. Wu, Tridiagonalizing random matrices, Phys. Rev. D 107, 126001 (2023).
  36. A. K. Das and A. Ghosh, Nonergodic extended states in the β ensemble, Phys. Rev. E 105, 054121 (2022).
  37. 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).
  38. 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).
  39. E. P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. Math. 62, 548 (1955).
  40. G. Casati, L. Molinari, and F. Izrailev, Scaling properties of band random matrices, Phys. Rev. Lett. 64, 1851 (1990).
  41. Y. V. Fyodorov and A. D. Mirlin, Scaling properties of localization in random band matrices: A σ-model approach, Phys. Rev. Lett. 67, 2405 (1991).
  42. Y. V. Fyodorov and A. D. Mirlin, Analytical derivation of the scaling law for the inverse participation ratio in quasi-one-dimensional disordered systems, Phys. Rev. Lett. 69, 1093 (1992).
  43. 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).
  44. T. Prosen and M. Robnik, Energy level statistics and localization in sparsed banded random matrix ensemble, J. Phys. A: Math. Gen. 26, 1105 (1993).
  45. K. Frahm and A. Müller-Groeling, Analytical results for random band matrices with preferential basis, Europhys. Lett. 32, 385 (1995).
  46. G. Casati, B. Chirikov, I. Guarneri, and F. Izrailev, Quantum ergodicity and localization in conservative systems: The Wigner band random matrix model, Phys. Lett. A 223, 430 (1996).
  47. Y. V. Fyodorov, O. A. Chubykalo, F. M. Izrailev, and G. Casati, Wigner random banded matrices with sparse structure: Local spectral density of states, Phys. Rev. Lett. 76, 1603 (1996).
  48. 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).
  49. Y. V. Fyodorov, A. Ossipov, and A. Rodriguez, The Anderson localization transition and eigenfunction multifractality in an ensemble of ultrametric random matrices, J. Stat. Mech. (2009) L12001.
  50. E. Bogomolny and O. Giraud, Eigenfunction entropy and spectral compressibility for critical random matrix ensembles, Phys. Rev. Lett. 106, 044101 (2011).
  51. I. Rushkin, A. Ossipov, and Y. V. Fyodorov, Universal and non-universal features of the multifractality exponents of critical wavefunctions, J. Stat. Mech. (2011) L03001.
  52. B. Gutkin and V. A. Osipov, Spectral problem of block-rectangular hierarchical matrices, J. Stat. Phys. 143, 72 (2011).
  53. J. A. Méndez-Bermúdez, A. Alcázar-López, and I. Varga, Multifractal dimensions for critical random matrix ensembles, Europhys. Lett. 98, 37006 (2012).
  54. P. von Soosten and S. Warzel, The phase transition in the ultrametric ensemble and local stability of Dyson Brownian motion, Electron. J. Probab. 23, 1 (2018).
  55. E. Bogomolny and M. Sieber, Power-law random banded matrices and ultrametric matrices: Eigenvector distribution in the intermediate regime, Phys. Rev. E 98, 042116 (2018).
  56. P. von Soosten and S. Warzel, Delocalization and continuous spectrum for ultrametric random operators, Ann. Henri Poincaré 20, 2877 (2019).
  57. 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).
  58. A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada, and T. H. Seligman, Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices, Phys. Rev. E 54, 3221 (1996).
  59. V. E. Kravtsov and K. A. Muttalib, New class of random matrix ensembles with multifractal eigenvectors, Phys. Rev. Lett. 79, 1913 (1997).
  60. I. Varga and D. Braun, Critical statistics in a power-law random-banded matrix ensemble, Phys. Rev. B 61, R11859 (2000).
  61. A. D. Mirlin and F. Evers, Multifractality and critical fluctuations at the Anderson transition, Phys. Rev. B 62, 7920 (2000).
  62. E. Cuevas and V. E. Kravtsov, Two-eigenfunction correlation in a multifractal metal and insulator, Phys. Rev. B 76, 235119 (2007).
  63. F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
  64. G. De Tomasi, Algebraic many-body localization and its implications on information propagation, Phys. Rev. B 99, 054204 (2019).
  65. P. A. Nosov, I. M. Khaymovich, and V. E. Kravtsov, Correlation-induced localization, Phys. Rev. B 99, 104203 (2019).
  66. D. A. Vega-Oliveros, J. A. Méndez-Bermúdez, and F. A. Rodrigues, Multifractality in random networks with power-law decaying bond strengths, Phys. Rev. E 99, 042303 (2019).
  67. P. Łydżba, M. Rigol, and L. Vidmar, Eigenstate entanglement entropy in random quadratic Hamiltonians, Phys. Rev. Lett. 125, 180604 (2020).
  68. 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. E 105, 014109 (2022).
  69. W.-J. Rao, Power-law random banded matrix ensemble as the effective model for many-body localization transition, Eur. Phys. J. Plus 137, 398 (2022).
  70. M. Hopjan and L. Vidmar, Scale-invariant critical dynamics at eigenstate transitions, Phys. Rev. Res. 5, 043301 (2023).
  71. P. Łydżba, R. Świętek, M. Mierzejewski, M. Rigol, and L. Vidmar, Normal weak eigenstate thermalization, Phys. Rev. B 110, 104202 (2024).
  72. G. C. Santra, A. Windey, S. Bandyopadhyay, A. Legramandi, and P. Hauke, Complexity transitions in chaotic quantum systems, arXiv:2505.09707.
  73. A. D. Mirlin, Statistics of energy levels and eigenfunctions in disordered systems, Phys. Rep. 326, 259 (2000).
  74. F. Evers and A. D. Mirlin, Fluctuations of the inverse participation ratio at the Anderson transition, Phys. Rev. Lett. 84, 3690 (2000).
  75. V. L. Quito, P. Titum, D. Pekker, and G. Refael, Localization transition in one dimension using Wegner flow equations, Phys. Rev. B 94, 104202 (2016).
  76. G. De Tomasi and I. M. Khaymovich, Non-Hermiticity induces localization: Good and bad resonances in power-law random banded matrices, Phys. Rev. B 108, L180202 (2023).
  77. S. Ghosh, M. Kulkarni, and S. Roy, Eigenvector correlations across the localization transition in non-Hermitian power-law banded random matrices, Phys. Rev. B 108, L060201 (2023).
  78. 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).
  79. M. Prasad, S. H. Tekur, B. K. Agarwalla, and M. Kulkarni, Assessment of spectral phases of non-Hermitian quantum systems through complex and singular values, Phys. Rev. B 111, L161408 (2025).
  80. V. Tiwari, D. S. Bhakuni, and A. Sharma, Periodically and aperiodically Thue-Morse driven long-range systems: From dynamical localization to slow dynamics, Phys. Rev. B 111, 205109 (2025).
  81. J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010).
  82. L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80, 517 (2008).
  83. W. Beugeling, A. Andreanov, and M. Haque, Global characteristics of all eigenstates of local many-body Hamiltonians: Participation ratio and entanglement entropy, J. Stat. Mech. (2015) P02002.
  84. J. R. Garrison and T. Grover, Does a single eigenstate encode the full Hamiltonian? Phys. Rev. X 8, 021026 (2018).
  85. C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Quantum scarred eigenstates in a Rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations, Phys. Rev. B 98, 155134 (2018).
  86. S. Moudgalya, N. Regnault, and B. A. Bernevig, Entanglement of exact excited states of Affleck-Kennedy-Lieb-Tasaki models: Exact results, many-body scars, and violation of the strong eigenstate thermalization hypothesis, Phys. Rev. B 98, 235156 (2018).
  87. S. Ok, K. Choo, C. Mudry, C. Castelnovo, C. Chamon, and T. Neupert, Topological many-body scar states in dimensions one, two, and three, Phys. Rev. Res. 1, 033144 (2019).
  88. B. Mukherjee, S. Nandy, A. Sen, D. Sen, and K. Sengupta, Collapse and revival of quantum many-body scars via Floquet engineering, Phys. Rev. B 101, 245107 (2020).
  89. B. Mukherjee, A. Sen, D. Sen, and K. Sengupta, Dynamics of the vacuum state in a periodically driven Rydberg chain, Phys. Rev. B 102, 075123 (2020).
  90. T. Iadecola and M. Schecter, Quantum many-body scar states with emergent kinetic constraints and finite-entanglement revivals, Phys. Rev. B 101, 024306 (2020).
  91. D. K. Mark, C.-J. Lin, and O. I. Motrunich, Unified structure for exact towers of scar states in the Affleck-Kennedy-Lieb-Tasaki and other models, Phys. Rev. B 101, 195131 (2020).
  92. N. Shibata, N. Yoshioka, and H. Katsura, Onsager's scars in disordered spin chains, Phys. Rev. Lett. 124, 180604 (2020).
  93. S. Chattopadhyay, H. Pichler, M. D. Lukin, and W. W. Ho, Quantum many-body scars from virtual entangled pairs, Phys. Rev. B 101, 174308 (2020).
  94. D. K. Mark and O. I. Motrunich, η-pairing states as true scars in an extended Hubbard model, Phys. Rev. B 102, 075132 (2020).
  95. P. A. McClarty, M. Haque, A. Sen, and J. Richter, Disorder-free localization and many-body quantum scars from magnetic frustration, Phys. Rev. B 102, 224303 (2020).
  96. 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).
  97. K. Tamura and H. Katsura, Quantum many-body scars of spinless fermions with density-assisted hopping in higher dimensions, Phys. Rev. B 106, 144306 (2022).
  98. M. Kliczkowski, R. Świętek, L. Vidmar, and M. Rigol, Average entanglement entropy of midspectrum eigenstates of quantum-chaotic interacting Hamiltonians, Phys. Rev. E 107, 064119 (2023).
  99. R. Świętek, M. Kliczkowski, L. Vidmar, and M. Rigol, Eigenstate entanglement entropy in the integrable spin-12 XYZ model, Phys. Rev. E 109, 024117 (2024).
  100. S. Imai and N. Tsuji, Quantum many-body scars with unconventional superconducting pairing symmetries via multibody interactions, Phys. Rev. Res. 7, 013064 (2025).
  101. A. Bäcker, M. Haque, and I. M. Khaymovich, Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems, Phys. Rev. E 100, 032117 (2019).
  102. L. S. Levitov, Absence of localization of vibrational modes due to dipole-dipole interaction, Europhys. Lett. 9, 83 (1989).
  103. L. S. Levitov, Delocalization of vibrational modes caused by electric dipole interaction, Phys. Rev. Lett. 64, 547 (1990).
  104. E. Bogomolny and M. Sieber, Eigenfunction distribution for the Rosenzweig-Porter model, Phys. Rev. E 98, 032139 (2018).
  105. F. Gray, Pulse code communication, U.S. Patent No. 2,632,058A (1953).
  106. R. W. Doran, The Gray code, J. Universal Comput. Sci. 13, 1573 (2007).
  107. E. Lubkin, Entropy of an n‐system from its correlation with a k‐reservoir, J. Math. Phys. 19, 1028 (1978).
  108. D. N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71, 1291 (1993).
  109. T. Vojta, Disorder in quantum many-body systems, Annu. Rev. Condens. Matter Phys. 10, 233 (2019).
  110. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  111. 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).
  112. A. Gyenis, A. Di Paolo, J. Koch, A. Blais, A. A. Houck, and D. I. Schuster, Moving beyond the transmon: Noise-protected superconducting quantum circuits, PRX Quantum 2, 030101 (2021).
  113. C. Berke, E. Varvelis, S. Trebst, A. Altland, and D. P. DiVincenzo, Transmon platform for quantum computing challenged by chaotic fluctuations, Nat. Commun. 13, 2495 (2022).
  114. S.-D. Börner, C. Berke, D. P. DiVincenzo, S. Trebst, and A. Altland, Classical chaos in quantum computers, Phys. Rev. Res. 6, 033128 (2024).
  115. P. Sierant and X. Turkeshi, Universal behavior beyond multifractality of wave functions at measurement-induced phase transitions, Phys. Rev. Lett. 128, 130605 (2022).
  116. C. Liu, X. Chen, and L. Balents, Quantum entanglement of the Sachdev-Ye-Kitaev models, Phys. Rev. B 97, 245126 (2018).
  117. P. Łydżba, M. Rigol, and L. Vidmar, Entanglement in many-body eigenstates of quantum-chaotic quadratic Hamiltonians, Phys. Rev. B 103, 104206 (2021).
  118. E. Bianchi, L. Hackl, and M. Kieburg, Page curve for fermionic Gaussian states, Phys. Rev. B 103, L241118 (2021).
  119. Q. Miao and T. Barthel, Eigenstate entanglement: Crossover from the ground state to volume laws, Phys. Rev. Lett. 127, 040603 (2021).

Outline

Information

Phys. Rev. E 113, 034116– Published 13 March, 2026
  • Received 24 May 2025
  • Accepted 23 February 2026
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. Funded by Bibsam.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation