- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

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

Critically Slow Hilbert-Space Ergodicity in Quantum Morphic Drives

Saúl Pilatowsky-Cameo1, Soonwon Choi1, and Wen Wei Ho2,3

PDF

Phys. Rev. Lett. 135, 140402 – Published 29 September, 2025

DOI: https://doi.org/10.1103/dmfd-lgcq

Abstract

The maximum entropy principle is foundational for statistical analyses of complex dynamics. This principle has been challenged by the findings of a previous work [Phys. Rev. X 7, 031034 (2017)], where it was argued that a quantum system driven in time by a certain aperiodic sequence without any explicit symmetries, dubbed the Thue-Morse drive, gives rise to emergent nonergodic steady states which are underpinned by effective conserved quantities. Here, we resolve this apparent tension. We rigorously prove that the Thue-Morse drive achieves a very strong notion of quantum ergodicity in the long-time limit: The time evolution of any initial state uniformly visits every corner of its Hilbert space. On the other hand, we find the dynamics also approximates a Floquet drive for arbitrarily long, albeit finite periods of time with no characteristic timescale, resulting in a scale-free ergodic dynamics we call critically slow complete Hilbert-space ergodicity. Furthermore, numerical studies reveal that critically slow complete Hilbert-space ergodicity is not specific to the Thue-Morse drive and is, in fact, exhibited by many other aperiodic drives derived from morphic sequences, i.e., words derived from repeatedly applying substitution rules on basic characters. Our Letter presents a new class of dynamics in time-dependent quantum systems where full ergodicity is eventually attained but only after astronomically long times.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (47)

  1. E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106, 620 (1957).
  2. D. A. Roberts and B. Yoshida, Chaos and complexity by design, J. High Energy Phys. 04 (2017) 121.
  3. S. Pilatowsky-Cameo, C. B. Dag, W. W. Ho, and S. Choi, Complete Hilbert-space ergodicity in quantum dynamics of generalized Fibonacci drives, Phys. Rev. Lett. 131, 250401 (2023).
  4. S. Pilatowsky-Cameo, I. Marvian, S. Choi, and W. W. Ho, Hilbert-space ergodicity in driven quantum systems: Obstructions and designs, Phys. Rev. X 14, 041059 (2024).
  5. A. L. Shaw, D. K. Mark, J. Choi, R. Finkelstein, P. Scholl, S. Choi, and M. Endres, Experimental signatures of Hilbert-space ergodicity: Universal bitstring distributions and applications in noise learning, Phys. Rev. X 15, 031001 (2025).
  6. D. K. Mark, F. Surace, A. Elben, A. L. Shaw, J. Choi, G. Refael, M. Endres, and S. Choi, Maximum entropy principle in deep thermalization and in Hilbert-space ergodicity, Phys. Rev. X 14, 041051 (2024).
  7. S. Ghosh, C. M. Langlett, N. Hunter-Jones, and J. F. Rodriguez-Nieva, Late-time ensembles of quantum states in quantum chaotic systems, Phys. Rev. B 112, 094302 (2025).
  8. L. Logaric, J. Goold, and S. Dooley, Hilbert subspace ergodicity, Phys. Rev. B 111, 144310 (2025).
  9. W. Liu, Z. Pan, Y. Fu, W. Cheng, W. W. Ho, X. Rong, and J. Du, Observation of hierarchy of Hilbert space ergodicities in the quantum dynamics of a single spin, arXiv:2507.05706.
  10. 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).
  11. S. Nandy, A. Sen, and D. Sen, Steady states of a quasiperiodically driven integrable system, Phys. Rev. B 98, 245144 (2018).
  12. S. Ray, S. Sinha, and D. Sen, Dynamics of quasiperiodically driven spin systems, Phys. Rev. E 100, 052129 (2019).
  13. P. T. Dumitrescu, R. Vasseur, and A. C. Potter, Logarithmically slow relaxation in quasiperiodically driven random spin chains, Phys. Rev. Lett. 120, 070602 (2018).
  14. P. T. Dumitrescu, J. G. Bohnet, J. P. Gaebler, A. Hankin, D. Hayes, A. Kumar, B. Neyenhuis, R. Vasseur, and A. C. Potter, Dynamical topological phase realized in a trapped-ion quantum simulator, Nature (London) 607, 463 (2022).
  15. P. Das, D. S. Bhakuni, L. F. Santos, and A. Sharma, Periodically and quasiperiodically driven anisotropic Dicke model, Phys. Rev. A 108, 063716 (2023).
  16. S. Ghosh, S. Bhattacharjee, and S. Bandyopadhyay, Slow relaxation of quasi-periodically driven integrable quantum many-body systems, arXiv:2404.06667.
  17. V. Tiwari, D. S. Bhakuni, and A. Sharma, Dynamical localization and slow dynamics in quasiperiodically driven quantum systems, Phys. Rev. B 109, L161104 (2024).
  18. S. Nandy, A. Sen, and D. Sen, Aperiodically driven integrable systems and their emergent steady states, Phys. Rev. X 7, 031034 (2017).
  19. M. Combescure, Recurrent versus diffusive dynamics for a kicked quantum system, J. Stat. Phys. 62, 779 (1991).
  20. T. Mori, H. Zhao, F. Mintert, J. Knolle, and R. Moessner, Rigorous bounds on the heating rate in Thue-Morse quasiperiodically and randomly driven quantum many-body systems, Phys. Rev. Lett. 127, 050602 (2021).
  21. H. Zhao, M. S. Rudner, R. Moessner, and J. Knolle, Anomalous random multipolar driven insulators, Phys. Rev. B 105, 245119 (2022).
  22. J. Yan, R. Moessner, and H. Zhao, Prethermalization in aperiodically kicked many-body dynamics, Phys. Rev. B 109, 064305 (2024).
  23. W. Chen, P. D. Sacramento, and R. Mondaini, Multifractality and prethermalization in the quasiperiodically kicked Aubry-André-Harper model, Phys. Rev. B 109, 054202 (2024).
  24. L. J. I. Moon, P. M. Schindler, Y. Sun, E. Druga, J. Knolle, R. Moessner, H. Zhao, M. Bukov, and A. Ajoy, Experimental observation of a time rondeau crystal: Temporal disorder in spatiotemporal order, arXiv:2404.05620.
  25. 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).
  26. M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, England, 2002).
  27. C. Jonay and T. Zhou, Physical theory of two-stage thermalization, Phys. Rev. B 110, L020306 (2024).
  28. N. Y. Yao, C. R. Laumann, J. I. Cirac, M. D. Lukin, and J. E. Moore, Quasi-many-body localization in translation-invariant systems, Phys. Rev. Lett. 117, 240601 (2016).
  29. Z. Lan, M. van Horssen, S. Powell, and J. P. Garrahan, Quantum slow relaxation and metastability due to dynamical constraints, Phys. Rev. Lett. 121, 040603 (2018).
  30. Y. Han, X. Chen, and E. Lake, Exponentially slow thermalization and the robustness of Hilbert space fragmentation, arXiv:2401.11294.
  31. S. Balasubramanian, S. Gopalakrishnan, A. Khudorozhkov, and E. Lake, Glassy word problems: Ultraslow relaxation, Hilbert space jamming, and computational complexity, Phys. Rev. X 14, 021034 (2024).
  32. C. Wang, S. Balasubramanian, Y. Han, E. Lake, X. Chen, and Z.-C. Yang, Exponentially slow thermalization in 1d fragmented dynamics, arXiv:2501.13930.
  33. N. J. A. Sloane, Thue-Morse sequence, Entry A010060 in The On-Line Encyclopedia of Integer Sequences (2012), https://oeis.org/A010060.
  34. M. Kolář, M. K. Ali, and F. Nori, Generalized Thue-Morse chains and their physical properties, Phys. Rev. B 43, 1034 (1991).

  35. We note that in previous physics literature it has been mischaracterized as such.

  36. An analogous definition can apply for dynamics over continuous time as well.

  37. The trace norm is defined as ||ρ||1=trρρ†=∑iλi, where λi are the singular values of ρ.

  38. The trace distance ΔT(k) is used because it gives the probability of distinguishing |ψ⟩ from a Haar-random state using an optimal, possibly entangling measurement over k copies, so the requirement limT→∞ΔT(k)=0 for all k means that the late-time state is indistinguishable from Haar-random, even with access to an arbitrarily large number of system replicas and entangling measurements.

  39. See Supplemental Material at http://link.aps.org/supplemental/10.1103/dmfd-lgcq for complete proofs and technical calculations, which additionally includes Refs. [40,41].
  40. A. W. Harrow, The church of the symmetric subspace, arXiv:1308.6595.
  41. A. Zygmund, Trigonometric Series, Cambridge Mathematical Library No. v. 1 (Cambridge University Press, Cambridge, England, 2002).
  42. S. Redner, A Guide to First-Passage Processes, A Guide to First-passage Processes (Cambridge University Press, Cambridge, England, 2001).

  43. With respect to the uniform measure on the semi-infinite cylinder.

  44. F. Nazarov, Mathoverflow answer to: Radial behavior of dynamical map xn+1=2xnyn, yn+1=1–2xn2 (2024), https://mathoverflow.net/a/461832/163674.
  45. N. J. A. Sloane, The “Octanacci” sequence, entry A324772 in The On-Line Encyclopedia of Integer Sequences (2019), https://oeis.org/A324772.
  46. N. J. A. Sloane, Period-doubling sequence (or period-doubling word), entry A096268 in The On-Line Encyclopedia of Integer Sequences (2004), https://oeis.org/A096268.

  47. This is unlike the TMW, which is cube-free [33]—no word appears repeated more that twice.

Outline

Information

Phys. Rev. Lett. 135, 140402– Published 29 September, 2025
  • Received 19 February 2025
  • Revised 28 August 2025
  • Accepted 8 September 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