- 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
  • Letter
  • Open Access

Complexity measure diagnostics of ergodic to many-body localization transition

Khen Cohen1, Yaron Oz1, and De-liang Zhong (钟德亮)2

PDF

Phys. Rev. B 110, L180101 – Published 13 November, 2024

DOI: https://doi.org/10.1103/PhysRevB.110.L180101

Abstract

We introduce diagnostics of the transition between the ergodic and many-body localization phases, which are based on complexity measures defined via the probability distribution function of the Lanczos coefficients of the tridiagonalized Hamiltonian. We use these complexity measures to analyze the power-law random banded matrix model as a function of the correlation strength and show that the moments and the entropy of the distribution diagnose the ergodic to many-body transition, as well as the distinctive feature of the phases concerning the memory of the initial conditions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (70)

  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. R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  4. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  5. P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, Many-body localization in the age of classical computing, arXiv:2403.07111.
  6. Y. Avishai, J. Richert, and R. Berkovits, Level statistics in a Heisenberg chain with random magnetic field, Phys. Rev. B 66, 052416 (2002).
  7. V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007).
  8. V. Oganesyan, A. Pal, and D. A. Huse, Energy transport in disordered classical spin chains, Phys. Rev. B 80, 115104 (2009).
  9. A. Pal and D. A. Huse, Many-body localization phase transition, Phys. Rev. B 82, 174411 (2010).
  10. S. Iyer, V. Oganesyan, G. Refael, and D. A. Huse, Many-body localization in a quasiperiodic system, Phys. Rev. B 87, 134202 (2013).
  11. 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).
  12. N. Regnault and R. Nandkishore, Floquet thermalization: Symmetries and random matrix ensembles, Phys. Rev. B 93, 104203 (2016).
  13. S. D. Geraedts, R. Nandkishore, and N. Regnault, Many-body localization and thermalization: Insights from the entanglement spectrum, Phys. Rev. B 93, 174202 (2016).
  14. C. L. Bertrand and A. M. Garcia-Garcia, Anomalous Thouless energy and critical statistics on the metallic side of the many-body localization transition, Phys. Rev. B 94, 144201 (2016).
  15. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
  16. J. L. F. Barbón, E. Rabinovici, R. Shir, and R. Sinha, On the evolution of operator complexity beyond scrambling, J. High Energy Phys. 10 (2019) 264.
  17. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Operator complexity: A journey to the edge of Krylov space, J. High Energy Phys. 06 (2021) 062.
  18. V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106, 046007 (2022).
  19. V. Viswanath and G. Müller, The Recursion Method (Springer-Verlag, Berlin, 1994).
  20. S.-K. Jian, B. Swingle, and Z.-Y. Xian, Complexity growth of operators in the SYK model and in JT gravity, J. High Energy Phys. 03 (2021) 014.
  21. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Krylov localization and suppression of complexity, J. High Energy Phys. 03 (2022) 211.
  22. B. Bhattacharjee, X. Cao, P. Nandy, and T. Pathak, Krylov complexity in saddle-dominated scrambling, J. High Energy Phys. 05 (2022) 174.
  23. N. Hörnedal, N. Carabba, A. S. Matsoukas-Roubeas, and A. del Campo, Ultimate speed limits to the growth of operator complexity, Commun. Phys. 5, 207 (2022).
  24. F. B. Trigueros and C.-J. Lin, Krylov complexity of many- body localization: Operator localization in Krylov basis, SciPost Phys. 13, 037 (2022).
  25. J. D. Noh, Operator growth in the transverse-field Ising spin chain with integrability-breaking longitudinal field, Phys. Rev. E 104, 034112 (2021).
  26. J. Kim, J. Murugan, J. Olle, and D. Rosa, Operator delocalization in quantum networks, Phys. Rev. A 105, L010201 (2022).
  27. P. Caputa and S. Liu, Quantum complexity and topological phases of matter, Phys. Rev. B 106, 195125 (2022).
  28. A. Kar, L. Lamprou, M. Rozali, and J. Sully, Random matrix theory for complexity growth and black hole interiors, J. High Energy Phys. 01 (2022) 016.
  29. Z.-Y. Fan, Universal relation for operator complexity, Phys. Rev. A 105, 062210 (2022).
  30. P. Caputa, J. M. Magan, and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res. 4, 013041 (2022).
  31. A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, Operator growth and Krylov construction in dissipative open quantum systems, J. High Energy Phys. 12 (2022) 081.
  32. A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D 104, L081702 (2021).
  33. P. Caputa and S. Datta, Operator growth in 2D CFT, J. High Energy Phys. 12 (2021) 188.
  34. W. Mück and Y. Yang, Krylov complexity and orthogonal polynomials, Nucl. Phys. B 984, 115948 (2022).
  35. K. Adhikari and S. Choudhury, Cosmological Krylov Complexity, Fortschr. Phys. 70, 2200126 (2022).
  36. K. Adhikari, S. Choudhury, and A. Roy, Krylov complexity in quantum field theory, Nucl. Phys. B 993, 116263 (2023).
  37. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Krylov complexity from integrability to chaos, J. High Energy Phys. 07 (2022) 151..
  38. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, A bulk manifestation of Krylov complexity, J. High Energy Phys. 08 (2023) 213.
  39. P. Caputa, J. M. Magan, D. Patramanis, and E. Tonni, Krylov complexity of modular Hamiltonian evolution, arXiv:2306.14732.
  40. P. Caputa, H.-S. Jeong, S. Liu, J. F. Pedraza, and L.-C. Qu, Krylov complexity of density matrix operators, J. High Energy Phys. 05 (2024) 337.
  41. V. Balasubramanian, J. M. Magan, and Q. Wu, Quantum chaos, integrability, and late times in the Krylov basis, arXiv:2312.03848.
  42. V. Balasubramanian, J. M. Magan, and Q. Wu, Tridiagonalizing random matrices, Phys. Rev. D 107, 126001 (2023).
  43. A. Dymarsky and A. Gorsky, Quantum chaos as delocalization in Krylov space, Phys. Rev. B 102, 085137 (2020).
  44. A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Phys. Rev. Res. 2, 043234 (2020).
  45. H. G. Menzler and R. Jha, Krylov localization as a probe for ergodicity breaking, Phys. Rev. B 110, 125137 (2024).
  46. H. A. Camargo, V. Jahnke, H.-S. Jeong, K.-Y. Kim, and M. Nishida, Spectral and Krylov complexity in billiard systems, Phys. Rev. D 109, 046017 (2024).
  47. K.-B. Huh, H.-S. Jeong, and J. F. Pedraza, Spread complexity in saddle-dominated scrambling, J. High Energy Phys. 05 (2024) 137.
  48. 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).
  49. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.110.L180101, where Supplemental Material A discusses analytical moments for GOE ensembles, Supplemental Material B provides LCD plots for all states, Supplemental Material C examines LCD moments summand plots, Supplemental Material D covers the large μ ensemble, Supplemental Material E includes error analysis, Supplemental Material F explains the Rényi entropy of the LCD distribution, Supplemental Material G presents an alternative fit of the moments, Supplemental Material H explores the unnormalized Lanczos coefficients' dependence on μ, and Supplemental Material I analyzes size dependence of the LCD moments and Rényi entropy.
  50. This proposal is imprecise and there exist counterexamples, see, e.g., Ref. [22].
  51. I. Varga and D. Braun, Critical statistics in a power-law random-banded matrix ensemble, Phys. Rev. B 61, R11859 (2000).
  52. F. Evers and A. D. Mirlin, Fluctuations of the inverse participation ratio at the Anderson transition, Phys. Rev. Lett. 84, 3690 (2000).
  53. A. Mirlin and F. Evers, Multifractality and critical fluctuations at the Anderson transition, Phys. Rev. B 62, 7920 (2000).
  54. F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
  55. V. E. Kravtsov and K. A. Muttalib, New class of random matrix ensembles with multifractal eigenvectors, Phys. Rev. Lett. 79, 1913 (1997).
  56. J. Mendez-Bermudez, A. Alcazar-Lopez, and I. Varga, Multifractal dimensions for critical random matrix ensembles, Europhys. Lett. 98, 37006 (2012).
  57. I. Varga, Fluctuation of correlation dimension and inverse participation number at the Anderson transition, Phys. Rev. B 66, 094201 (2002).
  58. J. A. Mendez-Bermudez and I. Varga, Scattering at the Anderson transition: Power-law banded random matrix model, Phys. Rev. B 74, 125114 (2006).
  59. J. A. Mendez-Bermudez, A. Alcazar-Lopez, and I. Varga, On the generalized dimensions of multifractal eigenstates, J. Stat. Mech. (2014) P11012.
  60. V. Khemani, D. N. Sheng, and D. A. Huse, Two universality classes for the many-body localization transition, Phys. Rev. Lett. 119, 075702 (2017).
  61. 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).
  62. M. Carrera-Núñez, A. Martinez-Argüello, and J. Méndez-Bermúdez, Multifractal dimensions and statistical properties of critical ensembles characterized by the three classical Wigner-Dyson symmetry classes, Physica A 573, 125965 (2021).
  63. Note, that the reason for not considering ensemble averages of the Lanczos coefficients aℓ is the reflection symmetry of the random matrix model H→−H makes them vanish.
  64. I. Dumitriu and A. Edelman, Matrix models for beta ensembles, J. Math. Phys. 43, 5830 (2002).
  65. V. Balasubramanian, J. M. Magan, and Q. Wu, A tale of two Hungarians: Tridiagonalizing random matrices, arXiv:2208.08452.
  66. The μ points that we use for the numerics are given by μ∈{0.0,0.2,0.4,0.6,0.8,0.9,0.95,1.0,1.05,1.1,1.15,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9,2.0,2.3,2.5,2.7,3.0,3.3,3.5,4.0,4.5,5.0,5.5,6,7,8,9,10}.Unless specified otherwise, the data points colored in blue, orange, purple, green, pink, brown, and red correspond to the initial states |ψ1〉,...|ψ7〉, respectively, as defined in (10). The matrix size is N=500 and the ensemble consists of M=9600 samples.
  67. M. Serbyn and J. E. Moore, Spectral statistics across the many-body localization transition, Phys. Rev. B 93, 041424(R) (2016).
  68. P. Sierant and J. Zakrzewski, Level statistics across the many- body localization transition, Phys. Rev. B 99, 104205 (2019).
  69. R. Khasseh, J. Zhang, M. Heyl, and M. A. Rajabpour, Identifying quantum many-body integrability and chaos using eigenstate trace distances, Phys. Rev. Lett. 131, 216701 (2023).
  70. DOI: 10.14469/hpc/2232.

Outline

Information

Phys. Rev. B 110, L180101– Published 13 November, 2024
  • Received 2 June 2024
  • Revised 25 October 2024
  • Accepted 28 October 2024
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