- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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

Investigating Stark many-body localization with continuous unitary transformation flows

Jan-Niklas Herre1,*, Qiyu Liu2, Roman Rausch2, Christoph Karrasch2, and Dante M. Kennes1,3,†

  • *Contact author: jan.herre@rwth-aachen.de
  • †Contact author: dante.kennes@rwth-aachen.de
PDF

Phys. Rev. Research 7, 033047 – Published 10 July, 2025

DOI: https://doi.org/10.1103/wrtc-vzhl

Abstract

We investigate the ergodicity-to-localization transition in interacting fermion systems subjected to a spatially uniform electric field. For that, we employ the recently proposed tensorflow equations (TFE), a type of continuous unitary flow equations. This enables us to iteratively determine an approximate diagonal basis of the quantum many-body system. We present improvements to the method, which achieves good accuracy at small to intermediate interaction strengths, even in the absence of an electric field or disorder. Then, we examine two quantities that reveal the fate of Stark MBL in 1D and 2D. First, we investigate the structure of the resulting basis to determine the crossover between ergodic and localized regimes with respect to electric field strength. Second, we simulate long-time dynamics at infinite temperature. Our results in 1D show a localization transition at nonzero field for finite interaction that vanishes with increasing system size leading to localization at infinitesimally small field even in the presence of interactions. In 2D, we find less clear signatures of localization and strong finite-size effects. We establish that the TFE work accurately up to intermediate times but cannot capture higher order effects in interaction strength that lead to delocalization at longer times in finite-size Stark MBL systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (100)

  1. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  2. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  3. G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett. 93, 040502 (2004).
  4. S. R. White and A. E. Feiguin, Real-time evolution using the density matrix renormalization group, Phys. Rev. Lett. 93, 076401 (2004).
  5. A. J. Daley, C. Kollath, U. Schollwöck, and G. Vidal, Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces, J. Stat. Mech. (2004) P04005.
  6. A. E. Feiguin and S. R. White, Time-step targeting methods for real-time dynamics using the density matrix renormalization group, Phys. Rev. B 72, 020404 (2005).
  7. P. Schmitteckert, Nonequilibrium electron transport using the density matrix renormalization group method, Phys. Rev. B 70, 121302 (2004).
  8. G. Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension, Phys. Rev. Lett. 98, 070201 (2007).
  9. D. Kennes and C. Karrasch, Extending the range of real time density matrix renormalization group simulations, Comput. Phys. Commun. 200, 37 (2016).
  10. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  11. D.-L. Deng, X. Li, and S. Das Sarma, Quantum entanglement in neural network states, Phys. Rev. X 7, 021021 (2017).
  12. O. Sharir, A. Shashua, and G. Carleo, Neural tensor contractions and the expressive power of deep neural quantum states, Phys. Rev. B 106, 205136 (2022).
  13. X.-Q. Sun, T. Nebabu, X. Han, M. O. Flynn, and X.-L. Qi, Entanglement features of random neural network quantum states, Phys. Rev. B 106, 115138 (2022).
  14. D. Luo and J. Halverson, Infinite neural network quantum states: Entanglement and training dynamics, Mach. Learn.: Sci. Technol. 4, 025038 (2023).
  15. I. L. Gutiérrez and C. B. Mendl, Real time evolution with neural-network quantum states, Quantum 6, 627 (2022).
  16. G. Passetti, D. Hofmann, P. Neitemeier, L. Grunwald, M. A. Sentef, and D. M. Kennes, Can neural quantum states learn volume-law ground states? Phys. Rev. Lett. 131, 036502 (2023).
  17. A. Goremykina, R. Vasseur, and M. Serbyn, Analytically solvable renormalization group for the many-body localization transition, Phys. Rev. Lett. 122, 040601 (2019).
  18. S. Balasubramanian, Y. Liao, and V. Galitski, Many-body localization landscape, Phys. Rev. B 101, 014201 (2020).
  19. F. Monteiro, T. Micklitz, M. Tezuka, and A. Altland, Minimal model of many-body localization, Phys. Rev. Res. 3, 013023 (2021).
  20. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  21. F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
  22. L. Fleishman and P. W. Anderson, Interactions and the Anderson transition, Phys. Rev. B 21, 2366 (1980).
  23. 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).
  24. 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).
  25. 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).
  26. V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007).
  27. A. Pal and D. A. Huse, Many-body localization phase transition, Phys. Rev. B 82, 174411 (2010).
  28. V. Ros, M. Müller, and A. Scardicchio, Integrals of motion in the many-body localized phase, Nucl. Phys. B 891, 420 (2015).
  29. J. A. Kjäll, J. H. Bardarson, and F. Pollmann, Many-body localization in a disordered quantum Ising chain, Phys. Rev. Lett. 113, 107204 (2014).
  30. D. J. Luitz, N. Laflorencie, and F. Alet, Many-body localization edge in the random-field Heisenberg chain, Phys. Rev. B 91, 081103 (2015).
  31. D. J. Luitz and Y. B. Lev, The ergodic side of the many-body localization transition, Ann. der Phys. 529, 1600350 (2017).
  32. R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  33. E. Altman and R. Vosk, Universal dynamics and renormalization in many-body-localized systems, Annu. Rev. Condens. Matter Phys. 6, 383 (2015).
  34. M. Serbyn, Z. Papić, and D. A. Abanin, Criterion for many-body localization-delocalization phase transition, Phys. Rev. X 5, 041047 (2015).
  35. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  36. P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, Many-body localization in the age of classical computing, arXiv:2403.07111
  37. 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).
  38. S. Iyer, V. Oganesyan, G. Refael, and D. A. Huse, Many-body localization in a quasiperiodic system, Phys. Rev. B 87, 134202 (2013).
  39. M. Lee, T. R. Look, S. P. Lim, and D. N. Sheng, Many-body localization in spin chain systems with quasiperiodic fields, Phys. Rev. B 96, 075146 (2017).
  40. A. Chandran and C. Laumann, Localization and symmetry breaking in the quantum quasiperiodic ising glass, Phys. Rev. X 7, 031061 (2017).
  41. S. Nag and A. Garg, Many-body mobility edges in a one-dimensional system of interacting fermions, Phys. Rev. B 96, 060203 (2017).
  42. V. Khemani, D. Sheng, and D. A. Huse, Two universality classes for the many-body localization transition, Phys. Rev. Lett. 119, 075702 (2017).
  43. Y. B. Lev, D. M. Kennes, C. Klöckner, D. R. Reichman, and C. Karrasch, Transport in quasiperiodic interacting systems: From superdiffusion to subdiffusion, Europhys. Lett. 119, 37003 (2017).
  44. 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).
  45. H. P. Lüschen, P. Bordia, S. Scherg, F. Alet, E. Altman, U. Schneider, and I. Bloch, Observation of slow dynamics near the many-body localization transition in one-dimensional quasiperiodic systems, Phys. Rev. Lett. 119, 260401 (2017).
  46. P. Bordia, H. Lüschen, S. Scherg, S. Gopalakrishnan, M. Knap, U. Schneider, and I. Bloch, Probing slow relaxation and many-body localization in two-dimensional quasiperiodic systems, Phys. Rev. X 7, 041047 (2017).
  47. W. De Roeck and F. Huveneers, Stability and instability towards delocalization in many-body localization systems, Phys. Rev. B 95, 155129 (2017).
  48. T. Thiery, F. Huveneers, M. Müller, and W. De Roeck, Many-body delocalization as a quantum avalanche, Phys. Rev. Lett. 121, 140601 (2018).
  49. G. H. Wannier, Dynamics of band electrons in electric and magnetic fields, Rev. Mod. Phys. 34, 645 (1962).
  50. M. Schulz, C. Hooley, R. Moessner, and F. Pollmann, Stark many-body localization, Phys. Rev. Lett. 122, 040606 (2019).
  51. E. van Nieuwenburg, Y. Baum, and G. Refael, From bloch oscillations to many-body localization in clean interacting systems, Proc. Natl. Acad. Sci. USA 116, 9269 (2019).
  52. D. H. Dunlap and V. M. Kenkre, Dynamic localization of a charged particle moving under the influence of an electric field, Phys. Rev. B 34, 3625 (1986).
  53. P. Ponte, Z. Papić, F. Huveneers, and D. A. Abanin, Many-body localization in periodically driven systems, Phys. Rev. Lett. 114, 140401 (2015).
  54. D. A. Abanin, W. De Roeck, and F. Huveneers, Theory of many-body localization in periodically driven systems, Ann. Phys. 372, 1 (2016).
  55. E. Bairey, G. Refael, and N. H. Lindner, Driving induced many-body localization, Phys. Rev. B 96, 020201 (2017).
  56. W. Morong, F. Liu, P. Becker, K. S. Collins, L. Feng, A. Kyprianidis, G. Pagano, T. You, A. V. Gorshkov, and C. Monroe, Observation of stark many-body localization without disorder, Nature (London) 599, 393 (2021).
  57. S. Scherg, T. Kohlert, P. Sala, F. Pollmann, B. Hebbe Madhusudhana, I. Bloch, and M. Aidelsburger, Observing non-ergodicity due to kinetic constraints in tilted Fermi-Hubbard chains, Nat. Commun. 12, 4490 (2021).
  58. Y.-Y. Wang, Z.-H. Sun, and H. Fan, Stark many-body localization transitions in superconducting circuits, Phys. Rev. B 104, 205122 (2021).
  59. Q. Guo, C. Cheng, H. Li, S. Xu, P. Zhang, Z. Wang, C. Song, W. Liu, W. Ren, H. Dong, R. Mondaini, and H. Wang, Stark many-body localization on a superconducting quantum processor, Phys. Rev. Lett. 127, 240502 (2021).
  60. F. J. Wegner, Flow equations for Hamiltonians, Nucl. Phys. B 90, 141 (2000).
  61. S. Kehrein, The Flow Equation Approach to Many-Particle Systems (Springer, Berlin, Heidelberg, 2006).
  62. D. Pekker, B. K. Clark, V. Oganesyan, and G. Refael, Fixed points of Wegner-Wilson flows and many-body localization, Phys. Rev. Lett. 119, 075701 (2017).
  63. S. J. Thomson and J. Eisert, Unraveling long-time quantum dynamics using flow equations, Nat. Phys. 20, 1401 (2024).
  64. S. J. Thomson and M. Schirò, Local integrals of motion in quasiperiodic many-body localized systems, SciPost Phys. 14, 125 (2023).
  65. S. R. Taylor, M. Schulz, F. Pollmann, and R. Moessner, Experimental probes of Stark many-body localization, Phys. Rev. B 102, 054206 (2020).
  66. G. Zisling, D. M. Kennes, and Y. Bar Lev, Transport in Stark many-body localized systems, Phys. Rev. B 105, L140201 (2022).
  67. E. V. H. Doggen, I. V. Gornyi, and D. G. Polyakov, Many-body localization in a tilted potential in two dimensions, Phys. Rev. B 105, 134204 (2022).
  68. G. Cataldi, A. Abedi, G. Magnifico, S. Notarnicola, N. Dalla Pozza, V. Giovannetti, and S. Montangero, Hilbert curve vs hilbert space: Exploiting fractal 2D covering to increase tensor network efficiency, Quantum 5, 556 (2021).
  69. F. Verstraete and J. I. Cirac, Mapping local Hamiltonians of Fermions to local Hamiltonians of spins, J. Stat. Mech. (2005) P09012.
  70. A. Mielke, Flow equations for band-matrices, Eur. Phys. J. B 5, 605 (1998).
  71. J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete, Time-dependent variational principle for quantum lattices, Phys. Rev. Lett. 107, 070601 (2011).
  72. P. Weinberg and M. Bukov, Quspin: A python package for dynamics and exact diagonalisation of quantum many body systems part i: Spin chains, SciPost Phys. 2, 003 (2017).
  73. P. Weinberg and M. Bukov, Quspin: A python package for dynamics and exact diagonalisation of quantum many body systems. Part II: Bosons, fermions and higher spins, SciPost Phys. 7, 020 (2019).
  74. F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, Matrix product density operators: Simulation of finite-temperature and dissipative systems, Phys. Rev. Lett. 93, 207204 (2004).
  75. A. Nocera and G. Alvarez, Symmetry-conserving purification of quantum states within the density matrix renormalization group, Phys. Rev. B 93, 045137 (2016).
  76. P. W. Hess, P. Becker, H. B. Kaplan, A. Kyprianidis, A. C. Lee, B. Neyenhuis, G. Pagano, P. Richerme, C. Senko, J. Smith, W. L. Tan, J. Zhang, and C. Monroe, Non-thermalization in trapped atomic ion spin chains, Philos. Trans. R. Soc. A 375, 0107 (2017).
  77. T. Heitmann, J. Richter, D. Schubert, and R. Steinigeweg, Selected applications of typicality to real-time dynamics of quantum many-body systems, Zeitschrift für Naturforschung A 75, 421 (2020).
  78. K. S. Tikhonov and A. D. Mirlin, Many-body localization transition with power-law interactions: Statistics of eigenstates, Phys. Rev. B 97, 214205 (2018).
  79. J. Šuntajs, J. Bonča, T. C. V. Prosen, and L. Vidmar, Quantum chaos challenges many-body localization, Phys. Rev. E 102, 062144 (2020).
  80. A. Chandran, I. H. Kim, G. Vidal, and D. A. Abanin, Constructing local integrals of motion in the many-body localized phase, Phys. Rev. B 91, 085425 (2015).
  81. H. Singh, B. Ware, R. Vasseur, and S. Gopalakrishnan, Local integrals of motion and the quasiperiodic many-body localization transition, Phys. Rev. B 103, L220201 (2021).
  82. C. Bertoni, J. Eisert, A. Kshetrimayum, A. Nietner, and S. J. Thomson, Local integrals of motion and the stability of many-body localization in Wannier-stark potentials, Phys. Rev. B 109, 024206 (2024).
  83. 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).
  84. K. S. Tikhonov and A. D. Mirlin, Eigenstate correlations around the many-body localization transition, Phys. Rev. B 103, 064204 (2021).
  85. 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).
  86. S.-X. Zhang and H. Yao, Universal properties of many-body localization transitions in quasiperiodic systems, Phys. Rev. Lett. 121, 206601 (2018).
  87. A. S. Aramthottil, T. Chanda, P. Sierant, and J. Zakrzewski, Finite-size scaling analysis of the many-body localization transition in quasiperiodic spin chains, Phys. Rev. B 104, 214201 (2021).
  88. B. Kloss, J. C. Halimeh, A. Lazarides, and Y. Bar Lev, Absence of localization in interacting spin chains with a discrete symmetry, Nat. Commun. 14, 3778 (2023).
  89. P. Zhang, Subdiffusion in strongly tilted lattice systems, Phys. Rev. Res. 2, 033129 (2020).
  90. E. Guardado-Sanchez, A. Morningstar, B. M. Spar, P. T. Brown, D. A. Huse, and W. S. Bakr, Subdiffusion and heat transport in a tilted two-dimensional Fermi-Hubbard system, Phys. Rev. X 10, 011042 (2020).
  91. D. A. Abanin, W. De Roeck, and F. Huveneers, Exponentially slow heating in periodically driven many-body systems, Phys. Rev. Lett. 115, 256803 (2015).
  92. D. A. Abanin, W. De Roeck, W. W. Ho, and F. Huveneers, Effective hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems, Phys. Rev. B 95, 014112 (2017).
  93. D. Abanin, W. De Roeck, W. W. Ho, and F. Huveneers, A rigorous theory of many-body prethermalization for periodically driven and closed quantum systems, Commun. Math. Phys. 354, 809 (2017).
  94. A. Kaczmarek and A. S. Sajna, Slow semiclassical dynamics of a two-dimensional Hubbard model in disorder-free potentials, Phys. Rev. B 108, 134304 (2023).
  95. M. Qin, H. Shi, and S. Zhang, Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method, Phys. Rev. B 94, 085103 (2016).
  96. N. Laflorencie, G. Lemarié, and N. Macé, Topological order in random interacting Ising-Majorana chains stabilized by many-body localization, Phys. Rev. Res. 4, L032016 (2022).
  97. N. C. Costa, K. Seki, S. Yunoki, and S. Sorella, Phase diagram of the two-dimensional Hubbard-Holstein model, Commun. Phys. 3, 80 (2020).
  98. S. J. Thomson, D. Magano, and M. Schirò, Flow equations for disordered floquet systems, SciPost Phys. 11, 028 (2021).
  99. K. Decker, C. Karrasch, J. Eisert, and D. Kennes, Floquet engineering topological many-body localized systems, Phys. Rev. Lett. 124, 190601 (2020).
  100. S. A. Weidinger and M. Knap, Floquet prethermalization and regimes of heating in a periodically driven, interacting quantum system, Sci. Rep. 7, 45382 (2017).

Outline

Information

Phys. Rev. Research 7, 033047– Published 10 July, 2025
  • Received 2 December 2024
  • Accepted 24 June 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. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation