- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

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
  • Access by Xinjiang University

Phase transition from localization to chaos in a classical many-body system

Yusuf Kasim1, Pavel Orlov1,2, and Tomaž Prosen1,3

PDF

Phys. Rev. E 114, 014140 – Published 21 July, 2026

DOI: https://doi.org/10.1103/6y6k-35l9

Abstract

We report a dynamical phase transition in the information spreading within a classical 2D deterministic interacting many-body system. Specifically, the transition is observed in a recently introduced momentum-conserving parity check cellular automaton (MCPCA) on the square lattice [Kasim and Prosen, Phys. Rev. Res. 7, 023230 (2025)]. We characterize the transition using information-theoretic quantities such as the Hamming distance and the classical decorrelator. By introducing conserved local charges of the momentum-conserving parity-check cellular automata, we show that selecting initial ensembles with specific charge values allows the system to transition from a localized information phase to a chaotic regime with ballistic information spreading. Importantly, our findings indicate that this transition is of second order, highlighting a sharp change in information spreading behavior. Furthermore, we revisit the multifractal behavior of the dynamical structure factor and show that, although present across both phases, it originates from effective local periodicities enforced by symmetry constraints.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (50)

  1. V. I. Arnold, V. V. Kozlov, A. I. Neishtadt, and I. Iacob, Mathematical Aspects of Classical and Celestial Mechanics (Springer, Berlin, Heidelberg, 2006), Vol. 3.
  2. E. Fermi, J. Pasta, S. Ulam, and M. Tsingou, Studies of Nonlinear Problems. I, Technical Report, Los Alamos Scientific Laboratory, University of California, Technical Report No. LA-1940 (1955).
  3. G. Berman and F. Izrailev, The Fermi–Pasta–Ulam problem: Fifty years of progress, Chaos 15, 015104 (2005).
  4. T. Prosen, Time evolution of a quantum many-body system: Transition from integrability to ergodicity in the thermodynamic limit, Phys. Rev. Lett. 80, 1808 (1998).
  5. T. Prosen, Chaos and complexity of quantum motion, J. Phys. A: Math. Theor. 40, 7881 (2007).
  6. J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, J. High Energy Phys. 08 (2016) 106.
  7. K. Hashimoto, K. Murata, and R. Yoshii, Out-of-time-order correlators in quantum mechanics, J. High Energy Phys. 10 (2017) 138.
  8. B. Swingle, Unscrambling the physics of out-of-time-order correlators, Nat. Phys. 14, 988 (2018).
  9. T. Bilitewski, S. Bhattacharjee, and R. Moessner, Temperature dependence of the butterfly effect in a classical many-body system, Phys. Rev. Lett. 121, 250602 (2018).
  10. A. Das, S. Chakrabarty, A. Dhar, A. Kundu, D. A. Huse, R. Moessner, S. S. Ray, and S. Bhattacharjee, Light-cone spreading of perturbations and the butterfly effect in a classical spin chain, Phys. Rev. Lett. 121, 024101 (2018).
  11. T. Bilitewski, S. Bhattacharjee, and R. Moessner, Classical many-body chaos with and without quasiparticles, Phys. Rev. B 103, 174302 (2021).
  12. S. D. Murugan, D. Kumar, S. Bhattacharjee, and S. S. Ray, Many-body chaos in thermalized fluids, Phys. Rev. Lett. 127, 124501 (2021).
  13. S.-W. Liu, J. Willsher, T. Bilitewski, J.-J. Li, A. Smith, K. Christensen, R. Moessner, and J. Knolle, Butterfly effect and spatial structure of information spreading in a chaotic cellular automaton, Phys. Rev. B 103, 094109 (2021).
  14. B. Bertini, K. Klobas, P. Kos, and D. Malz, Quantum and classical dynamics with random permutation circuits, Phys. Rev. X 15, 011015 (2025).
  15. R. W. Hamming, Error detecting and error correcting codes, Bell Syst. Tech. J. 29, 147 (1950).
  16. B. Derrida and D. Stauffer, Phase transitions in two-dimensional Kauffman cellular automata, Europhys. Lett. 2, 739 (1986).
  17. G. Weisbuch and D. Stauffer, Phase transition in cellular random Boolean nets, J. Phys. France 48, 11 (1987).
  18. S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys. 55, 601 (1983).
  19. G. Alfaro and M. A. F. Sanjuán, Classification of cellular automata based on the Hamming distance, Chaos 34, 083129 (2024).
  20. B. Buča, K. Klobas, and T. Prosen, Rule 54: Exactly solvable model of nonequilibrium statistical mechanics, J. Stat. Mech. (2021) 074001.
  21. M. Medenjak, K. Klobas, and T. Prosen, Diffusion in deterministic interacting lattice systems, Phys. Rev. Lett. 119, 110603 (2017).
  22. T. Prosen and C. Mejía-Monasterio, Integrability of a deterministic cellular automaton driven by stochastic boundaries, J. Phys. A: Math. Theor. 49, 185003 (2016).
  23. K. Klobas and T. Prosen, On two reversible cellular automata with two particle species, J. Phys. A: Math. Theor. 55, 094003 (2022).
  24. K. Klobas, B. Bertini, and L. Piroli, Exact thermalization dynamics in the “Rule 54” quantum cellular automaton, Phys. Rev. Lett. 126, 160602 (2021).
  25. J. W. P. Wilkinson, K. Klobas, T. Prosen, and J. P. Garrahan, Exact solution of the Floquet-PXP cellular automaton, Phys. Rev. E 102, 062107 (2020).
  26. T. Gombor and B. Pozsgay, Superintegrable cellular automata and dual unitary gates from Yang-Baxter maps, SciPost Phys. 12, 102 (2022).
  27. T. Prosen, On two non-ergodic reversible cellular automata, one classical, the other quantum, Entropy 25, 739 (2023).
  28. T. Gombor and B. Pozsgay, Integrable deformations of superintegrable quantum circuits, SciPost Phys. 16, 114 (2024).
  29. R. Sharipov, M. Koterle, S. Grozdanov, and T. Prosen, Ergodic behaviors in reversible 3-state cellular automata, SciPost Phys. 20, 145 (2026).
  30. Ž. Krajnik, J. Schmidt, V. Pasquier, E. Ilievski, and T. Prosen, Exact anomalous current fluctuations in a deterministic interacting model, Phys. Rev. Lett. 128, 160601 (2022).
  31. Ž. Krajnik, J. Schmidt, V. Pasquier, T. Prosen, and E. Ilievski, Universal anomalous fluctuations in charged single-file systems, Phys. Rev. Res. 6, 013260 (2024).
  32. Ž. Krajnik, K. Klobas, B. Bertini, and T. Prosen, Fluctuations of stochastic charged cellular automata, J. Stat. Mech. (2025) 053209.
  33. S. Gopalakrishnan, Operator growth and eigenstate entanglement in an interacting integrable Floquet system, Phys. Rev. B 98, 060302(R) (2018).
  34. S. Gopalakrishnan, D. A. Huse, V. Khemani, and R. Vasseur, Hydrodynamics of operator spreading and quasiparticle diffusion in interacting integrable systems, Phys. Rev. B 98, 220303(R) (2018).
  35. V. Alba, J. Dubail, and M. Medenjak, Operator entanglement in interacting integrable quantum systems: The case of the Rule 54 chain, Phys. Rev. Lett. 122, 250603 (2019).
  36. K. Klobas, Non-equilibrium dynamics of symmetry-resolved entanglement and entanglement asymmetry: Exact asymptotics in Rule 54, J. Phys. A: Math. Theor. 57, 505001 (2024).
  37. C. De Fazio, J. P. Garrahan, and K. Klobas, Exact results on the dynamics of the stochastic Floquet-East model, J. Phys. A: Math. Theor. 57, 505002 (2024).
  38. K. Klobas, C. De Fazio, and J. P. Garrahan, Exact pretransition effects in kinetically constrained circuits: Dynamical fluctuations in the Floquet-East model, Phys. Rev. E 110, L022101 (2024).
  39. Y. Kasim and T. Prosen, Deterministic many-body dynamics with multifractal response, Phys. Rev. Res. 7, 023230 (2025).
  40. P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Ergodicity breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians, Phys. Rev. X 10, 011047 (2020).
  41. S. Moudgalya, B. A. Bernevig, and N. Regnault, Quantum many-body scars and Hilbert space fragmentation: A review of exact results, Rep. Prog. Phys. 85, 086501 (2022).
  42. M. Serbyn, Z. Papić, and D. A. Abanin, Local conservation laws and the structure of the many-body localized states, Phys. Rev. Lett. 111, 127201 (2013).
  43. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  44. 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).
  45. The local map Φv can as well be identified with a 24×24 (permutation) matrix which is embedded in 2NE dimensional Hilbert space C2NE of NE qubits. Similarly, the complete dynamical map Φ is a 2NE×2NE matrix representing an element in S(2NE), i.e., deterministic dynamics. Let us define observables as functions over phase space a:Z2NE→C, or equivalently as vectors in C2NE or ΦTq=q in vector notation.
  46. P. Orlov, C. Jonay, and T. Prosen, Loop charges and fragmentation in pairwise difference conserving circuits, arXiv:2510.18992.
  47. In contrast with topological loop charges, which act on N degrees of freedom, and so, the density Mγ/N for these charges can be well defined in the thermodynamic limit.
  48. P. Grassberger, Are damage spreading transitions generically in the universality class of directed percolation? J. Stat. Phys. 79, 13–23 (1995).
  49. H. E. Stanley, D. Stauffer, J. Kertész, and H. J. Herrmann, Dynamics of spreading phenomena in two-dimensional Ising models, Phys. Rev. Lett. 59, 2326 (1987).
  50. P. Grassberger, Damage spreading and critical exponents for “model A” Ising dynamics, Physica A 214, 547 (1995).

Outline

Information

Phys. Rev. E 114, 014140– Published 21 July, 2026
  • Received 5 December 2025
  • Accepted 8 June 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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation