- 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

Classical periodic trajectories and quantum scars in many-spin systems

Igor Ermakov1,2,3,*, Oleg Lychkovskiy3,2,1,†, and Boris V. Fine4,2,‡

  • *Contact author: ermakov1054@yandex.ru
  • †Contact author: lychkovskiy@gmail.com
  • ‡Contact author: boris.fine@uni-leipzig.de (B.V.F. had affiliation until 2022.)
PDF

Phys. Rev. E 112, 064109 – Published 4 December, 2025

DOI: https://doi.org/10.1103/zcgw-q34x

Abstract

To probe the limits of dynamic thermalization, we numerically investigate the stability of exceptional periodic classical trajectories in chaotic many-spin systems and explore a possible connection between these trajectories and exceptional nonthermal quantum eigenstates known as “quantum many-body scars.” The systems considered are chaotic spin chains with short-range interactions, both classical and quantum. On the classical side, the chosen periodic trajectories are such that all spins instantaneously point in the same direction, which evolves as a function of time. We find that the largest Lyapunov exponents characterizing the stability of these trajectories have surprisingly strong and nontrivial dependencies on the interaction constants and chain lengths. In particular, we identify rather long spin chains, where the above periodic trajectories are Lyapunov stable on many-body energy shells overwhelmingly dominated by chaotic motion. We show that the above phenomenology can be quantitatively described by connecting Lyapunov instabilities of translationally invariant periodic trajectories to irreducible representations of the translational symmetry group with well-defined wave vectors. We also find that instabilities around periodic trajectories in modestly large spin chains develop into a transient nearly quasiperiodic nonergodic regime. In some cases, the lifetime of this regime is extremely long, which we interpret as a manifestation of Arnold diffusion in the vicinity of integrable dynamics. On the quantum side, we numerically investigate the dynamics of quantum states starting with all spins initially pointing in the same direction: These are the quantum counterparts of the initial conditions for the above periodic classical trajectories. Our investigation reveals the existence of quantum many-body scars for numerically accessible finite chains of spins-3/2 and higher. No evidence of quantum scars was observed for spin-1/2 chains, while spin-1 chains were found to be transitional in this respect. The dynamic thermalization process dominated by quantum scars is shown to exhibit a slowdown in comparison with generic thermalization at the same energy. Finally, we identify quantum signatures of the proximity to a classical separatrix of the periodic motion.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (173)

  1. J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many-body systems out of equilibrium, Nat. Phys. 11, 124 (2015).
  2. 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).
  3. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  4. M. Berry, Quantum chaology, not quantum chaos, Phys. Scr. 40, 335 (1989).
  5. P. Reimann and M. Evstigneev, Quantum versus classical foundation of statistical mechanics under experimentally realistic conditions, Phys. Rev. E 88, 052114 (2013).
  6. P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge, UK, 1998).
  7. A. Vulpiani, F. Cecconi, and M. Cencini, Chaos: From Simple Models to Complex Systems (World Scientific, Singapore, 2009), Vol. 17.
  8. T. Kinoshita, T. Wenger, and D. S. Weiss, A quantum Newton's cradle, Nature (Lond.) 440, 900 (2006).
  9. M. C. Bañuls, J. I. Cirac, and M. B. Hastings, Strong and weak thermalization of infinite nonintegrable quantum systems, Phys. Rev. Lett. 106, 050405 (2011).
  10. M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer, Relaxation and prethermalization in an isolated quantum system, Science 337, 1318 (2012).
  11. B. Neyenhuis, J. Zhang, P. W. Hess, J. Smith, A. C. Lee, P. Richerme, Z.-X. Gong, A. V. Gorshkov, and C. Monroe, Observation of prethermalization in long-range interacting spin chains, Sci. Adv. 3, e1700672 (2017).
  12. D. J. Luitz, N. Laflorencie, and F. Alet, Extended slow dynamical regime close to the many-body localization transition, Phys. Rev. B 93, 060201(R) (2016).
  13. D. J. Luitz and Y. B. Lev, The ergodic side of the many-body localization transition, Ann. Phys. 529, 1600350 (2017).
  14. P. Peng, C. Yin, X. Huang, C. Ramanathan, and P. Cappellaro, Floquet prethermalization in dipolar spin chains, Nat. Phys. 17, 444 (2021).
  15. G. M. Lando and S. Flach, Thermalization slowing down in multidimensional Josephson junction networks, Phys. Rev. E 108, L062301 (2023).
  16. J. Pawłowski, M. Panfil, J. Herbrych, and M. Mierzejewski, Long-living prethermalization in nearly integrable spin ladders, Phys. Rev. B 109, L161109 (2024).
  17. S. Palzer, C. Zipkes, C. Sias, and M. Köhl, Quantum transport through a Tonks-Girardeau gas, Phys. Rev. Lett. 103, 150601 (2009).
  18. G. P. Berman, F. Borgonovi, F. M. Izrailev, and A. Smerzi, Irregular dynamics in a one-dimensional Bose system, Phys. Rev. Lett. 92, 030404 (2004).
  19. M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons, Phys. Rev. Lett. 98, 050405 (2007).
  20. G. P. Brandino, J.-S. Caux, and R. M. Konik, Glimmers of a quantum KAM theorem: Insights from quantum quenches in one-dimensional bose gases, Phys. Rev. X 5, 041043 (2015).
  21. J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong, and C. Monroe, Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator, Nature (Lond.) 551, 601 (2017).
  22. L. Rademaker and D. A. Abanin, Slow nonthermalizing dynamics in a quantum spin glass, Phys. Rev. Lett. 125, 260405 (2020).
  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. V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007).
  25. A. Pal and D. A. Huse, Many-body localization phase transition, Phys. Rev. B 82, 174411 (2010).
  26. M. Serbyn and D. A. Abanin, Loschmidt echo in many-body localized phases, Phys. Rev. B 96, 014202 (2017).
  27. W. Morong, F. Liu, P. Becker, K. Collins, L. Feng, A. Kyprianidis, G. Pagano, T. You, A. Gorshkov, and C. Monroe, Observation of stark many-body localization without disorder, Nature (Lond.) 599, 393 (2021).
  28. R. Ketzmerick and W. Wustmann, Statistical mechanics of floquet systems with regular and chaotic states, Phys. Rev. E 82, 021114 (2010).
  29. K. Ji and B. V. Fine, Nonthermal statistics in isolated quantum spin clusters after a series of perturbations, Phys. Rev. Lett. 107, 050401 (2011).
  30. L. D'Alessio and M. Rigol, Long-time behavior of isolated periodically driven interacting lattice systems, Phys. Rev. X 4, 041048 (2014).
  31. A. Lazarides, A. Das, and R. Moessner, Fate of many-body localization under periodic driving, Phys. Rev. Lett. 115, 030402 (2015).
  32. M. Bukov, L. D'Alessio, and A. Polkovnikov, Universal high-frequency behavior of periodically driven systems: From dynamical stabilization to Floquet engineering, Adv. Phys. 64, 139 (2015).
  33. K. Ji and B. V. Fine, Suppression of heating in quantum spin clusters under periodic driving as a dynamic localization effect, Phys. Rev. Lett. 121, 050602 (2018).
  34. A. Pizzi, D. Malz, G. De Tomasi, J. Knolle, and A. Nunnenkamp, Time crystallinity and finite-size effects in clean Floquet systems, Phys. Rev. B 102, 214207 (2020).
  35. C. Rylands, E. B. Rozenbaum, V. Galitski, and R. Konik, Many-body dynamical localization in a kicked Lieb-Liniger gas, Phys. Rev. Lett. 124, 155302 (2020).
  36. A. Rakcheev and A. M. Läuchli, Estimating heating times in periodically driven quantum many-body systems via avoided crossing spectroscopy, Phys. Rev. Res. 4, 043174 (2022).
  37. N. Shiraishi and T. Mori, Systematic construction of counterexamples to the eigenstate thermalization hypothesis, Phys. Rev. Lett. 119, 030601 (2017).
  38. H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner et al., Probing many-body dynamics on a 51-atom quantum simulator, Nature (Lond.) 551, 579 (2017).
  39. C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Weak ergodicity breaking from quantum many-body scars, Nat. Phys. 14, 745 (2018).
  40. A. Dymarsky, Mechanism of macroscopic equilibration of isolated quantum systems, Phys. Rev. B 99, 224302 (2019).
  41. I. Ermakov and B. V. Fine, Almost complete revivals in quantum many-body systems, Phys. Rev. A 104, L050202 (2021).
  42. V. I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Dokl. 5, 581 (1964).
  43. A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer Science & Business Media, New York, 2013), Vol. 38.
  44. A. S. de Wijn, B. Hess, and B. V. Fine, Largest Lyapunov exponents for lattices of interacting classical spins, Phys. Rev. Lett. 109, 034101 (2012).
  45. A. E. Tarkhov and B. V. Fine, Estimating ergodization time of a chaotic many-particle system from a time reversal of equilibrium noise, New J. Phys. 20, 123021 (2018).
  46. A. N. Kolmogorov, On the conservation of quasiperiodic motions for a small change in the Hamiltonian function, Doklad. Acad. Nauk USSR 98, 527 (1954).
  47. J. Möser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen, II 1 (1962).
  48. V. I. Arnol'd, Small denominators and problems of stability of motion in classical and celestial mechanics, Russ. Math. Surv. 18, 85 (1963).
  49. J. R. Schmidt, A. Bäcker, and R. Ketzmerick, Classical drift in the Arnold web induces quantum delocalization transition, Phys. Rev. Lett. 131, 187201 (2023).
  50. A. V. Varma, A. Vardi, and D. Cohen, Characterization of hybrid quantum eigenstates in systems with mixed classical phase space, Phys. Rev. E 109, 064207 (2024).
  51. H. A. Weidenmüller and G. E. Mitchell, Random matrices and chaos in nuclear physics: Nuclear structure, Rev. Mod. Phys. 81, 539 (2009).
  52. O. Bohigas, M. J. Giannoni, and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52, 1 (1984).
  53. J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  54. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  55. V. V. Flambaum, F. M. Izrailev, and G. Casati, Towards a statistical theory of finite Fermi systems and compound states: Random two-body interaction approach, Phys. Rev. E 54, 2136 (1996).
  56. G. Benenti, G. Casati, and D. L. Shepelyansky, Emergence of Fermi-Dirac thermalization in the quantum computer core, Eur. Phys. J. D 17, 265 (2001).
  57. J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog. Phys. 81, 082001 (2018).
  58. B. V. Fine, T. A. Elsayed, C. M. Kropf, and A. S. de Wijn, Absence of exponential sensitivity to small perturbations in nonintegrable systems of spins 1/2, Phys. Rev. E 89, 012923 (2014).
  59. T. A. Elsayed and B. V. Fine, Sensitivity to small perturbations in systems of large quantum spins, Phys. Scr. 2015, 014011 (2015).
  60. I. Kukuljan, S. Grozdanov, and T. Prosen, Weak quantum chaos, Phys. Rev. B 96, 060301(R) (2017).
  61. B. Craps, M. De Clerck, D. Janssens, V. Luyten, and C. Rabideau, Lyapunov growth in quantum spin chains, Phys. Rev. B 101, 174313 (2020).
  62. OTOCs were first used in Ref. [63] to characterize one-particle scattering in a superconductor without reference to chaos or Lyapunov exponents. The connection between OTOCs and Lyapunov exponents in many-particle systems was first proposed in Refs. [58, 59] before the term “OTOC” was introduced and broadly popularized in the context of quantum modeling of black holes [64, 65]. The connection between the specific definition of OTOC used in Refs. [64, 65] and the one used in Refs. [58, 59] is explained in Ref. [45].
  63. A. Larkin and Y. N. Ovchinnikov, Sov. Phys. JETP 28, 1200 (1969).
  64. A. Kitaev, A simple model of quantum holography. Talks at KITP, University of California, Santa Barbara (2015), https://online.kitp.ucsb.edu/online/entangled15/kitaev/.
  65. J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, J. High Energy Phys. 08 (2016) 106.
  66. A. R. Kolovsky and D. L. Shepelyansky, Dynamical thermalization in isolated quantum dots and black holes, Europhys. Lett. 117, 10003 (2017).
  67. M. Schmitt, D. Sels, S. Kehrein, and A. Polkovnikov, Semiclassical echo dynamics in the Sachdev-Ye-Kitaev model, Phys. Rev. B 99, 134301 (2019).
  68. T. A. Elsayed and B. V. Fine, Effectiveness of classical spin simulations for describing NMR relaxation of quantum spins, Phys. Rev. B 91, 094424 (2015).
  69. G. A. Starkov and B. V. Fine, Hybrid quantum-classical method for simulating high-temperature dynamics of nuclear spins in solids, Phys. Rev. B 98, 214421 (2018).
  70. D. Schubert, J. Richter, F. Jin, K. Michielsen, H. D. Raedt, and R. Steinigeweg, Quantum versus classical dynamics in spin models: Chains, ladders, and square lattices, Phys. Rev. B 104, 054415 (2021).
  71. T. Bilitewski, S. Bhattacharjee, and R. Moessner, Classical many-body chaos with and without quasiparticles, Phys. Rev. B 103, 174302 (2021).
  72. Y. Lebel, L. F. Santos, and Y. Bar Lev, Chaos enhancement in large-spin chains, SciPost Phys. 15, 022 (2023).
  73. L. Benet, F. Borgonovi, F. M. Izrailev, and L. F. Santos, Quantum-classical correspondence of strongly chaotic many-body spin models, Phys. Rev. B 107, 155143 (2023).
  74. F. Borgonovi, F. M. Izrailev, and L. F. Santos, Timescales of quantum and classical chaotic spin models evolving toward equilibrium, Phys. Rev. E 111, 044210 (2025).
  75. S. Åberg, Onset of chaos in rapidly rotating nuclei, Phys. Rev. Lett. 64, 3119 (1990).
  76. L. F. Santos, F. Borgonovi, and F. M. Izrailev, Onset of chaos and relaxation in isolated systems of interacting spins: Energy shell approach, Phys. Rev. E 85, 036209 (2012).
  77. J. Gemmer, M. Michel, and G. Mahler, Quantum Thermodynamics (Springer-Verlag, Berlin, 2009).
  78. S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zanghì, Canonical typicality, Phys. Rev. Lett. 96, 050403 (2006).
  79. S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics, Nat. Phys. 2, 754 (2006).
  80. W. Hahn and B. V. Fine, Stability of quantum statistical ensembles with respect to local measurements, Phys. Rev. E 94, 062106 (2016).
  81. C. Bartsch and J. Gemmer, Dynamical typicality of quantum expectation values, Phys. Rev. Lett. 102, 110403 (2009).
  82. T. A. Elsayed and B. V. Fine, Regression relation for pure quantum states and its implications for efficient computing, Phys. Rev. Lett. 110, 070404 (2013).
  83. B. V. Fine, Long-time relaxation on spin lattice as a manifestation of chaotic dynamics, Int. J. Mod. Phys. B 18, 1119 (2004).
  84. B. V. Fine, Long-time behavior of spin echo, Phys. Rev. Lett. 94, 247601 (2005).
  85. O. Lychkovskiy, Necessary condition for the thermalization of a quantum system coupled to a quantum bath, Phys. Rev. E 82, 011123 (2010).
  86. A. Khodja, R. Steinigeweg, and J. Gemmer, Relevance of the eigenstate thermalization hypothesis for thermal relaxation, Phys. Rev. E 91, 012120 (2015).
  87. A. Teretenkov, F. Uskov, and O. Lychkovskiy, Pseudomode expansion of many-body correlation functions, Phys. Rev. B 111, 174308 (2025).
  88. T. A. Elsayed, B. Hess, and B. V. Fine, Signatures of chaos in time series generated by many-spin systems at high temperatures, Phys. Rev. E 90, 022910 (2014).
  89. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
  90. M. Pandey, P. W. Claeys, D. K. Campbell, A. Polkovnikov, and D. Sels, Adiabatic eigenstate deformations as a sensitive probe for quantum chaos, Phys. Rev. X 10, 041017 (2020).
  91. C. Lim, K. Matirko, A. Polkovnikov, and M. O. Flynn, Defining classical and quantum chaos through adiabatic transformations, arXiv:2401.01927.
  92. M. Campisi and J. Goold, Thermodynamics of quantum information scrambling, Phys. Rev. E 95, 062127 (2017).
  93. E. Iyoda and T. Sagawa, Scrambling of quantum information in quantum many-body systems, Phys. Rev. A 97, 042330 (2018).
  94. C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws, Phys. Rev. X 8, 021013 (2018).
  95. V. Khemani, A. Vishwanath, and D. A. Huse, Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws, Phys. Rev. X 8, 031057 (2018).
  96. J. Chávez-Carlos, B. López-del Carpio, M. A. Bastarrachea-Magnani, P. Stránský, S. Lerma-Hernández, L. F. Santos, and J. G. Hirsch, Quantum and classical lyapunov exponents in atom-field interaction systems, Phys. Rev. Lett. 122, 024101 (2019).
  97. X. Mi et al., Information scrambling in quantum circuits, Science 374, 1479 (2021).
  98. Q. Zhu, Z.-H. Sun, M. Gong, F. Chen, Y.-R. Zhang, Y. Wu, Y. Ye, C. Zha, S. Li, S. Guo, H. Qian, H.-L. Huang, J. Yu, H. Deng, H. Rong, J. Lin, Y. Xu, L. Sun, C. Guo, N. Li et al., Observation of thermalization and information scrambling in a superconducting quantum processor, Phys. Rev. Lett. 128, 160502 (2022).
  99. S. Pappalardi, L. Foini, and J. Kurchan, Microcanonical windows on quantum operators, Quantum 8, 1227 (2024).
  100. F. Uskov and O. Lychkovskiy, Quantum dynamics in one and two dimensions via the recursion method, Phys. Rev. B 109, L140301 (2024).
  101. P. Nandy, A. S. Matsoukas-Roubeas, P. Martínez-Azcona, A. Dymarsky, and A. del Campo, Quantum dynamics in Krylov space: Methods and applications, Phys. Rep. 1125-1128, 1 (2025).
  102. W.-B. Zhang, X. Yu, C.-L. Wang, H.-J. Sun, I.-F. Hsieh, Y. Li, X.-H. Dong, K. Yue, R. Van Horn, and S. Z. D. Cheng, Molecular nanoparticles are unique elements for macromolecular science: From “nanoatoms” to giant molecules, Macromolecules 47, 1221 (2014).
  103. B. V. Fine and T. Egami, Phase separation in the vicinity of quantum-critical doping concentration: Implications for high-temperature superconductors, Phys. Rev. B 77, 014519 (2008).
  104. A. Richaud and V. Penna, Phase separation can be stronger than chaos, New J. Phys. 20, 105008 (2018).
  105. O. Lychkovskiy and B. V. Fine, Spin excitation spectrum of high-temperature cuprate superconductors from finite cluster simulations, J. Phys.: Condens. Matter 30, 405801 (2018).
  106. C.-J. Lin and O. I. Motrunich, Exact quantum many-body scar states in the Rydberg-Blockaded atom chain, Phys. Rev. Lett. 122, 173401 (2019).
  107. M. Serbyn, D. A. Abanin, and Z. Papić, Quantum many-body scars and weak breaking of ergodicity, Nat. Phys. 17, 675 (2021).
  108. N. Shibata, N. Yoshioka, and H. Katsura, Onsager's scars in disordered spin chains, Phys. Rev. Lett. 124, 180604 (2020).
  109. A. A. Michailidis, C. J. Turner, Z. Papić, D. A. Abanin, and M. Serbyn, Stabilizing two-dimensional quantum scars by deformation and synchronization, Phys. Rev. Res. 2, 022065(R) (2020).
  110. K. Bull, A. Hallam, Z. Papić, and I. Martin, Tuning between continuous time crystals and many-body scars in long-range XYZ spin chains, Phys. Rev. Lett. 129, 140602 (2022).
  111. M. Ljubotina, B. Roos, D. A. Abanin, and M. Serbyn, Optimal steering of matrix product states and quantum many-body scars, PRX Quantum 3, 030343 (2022).
  112. S. Sinha and S. Sinha, Chaos and quantum scars in Bose-Josephson junction coupled to a bosonic mode, Phys. Rev. Lett. 125, 134101 (2020).
  113. D. Banerjee and A. Sen, Quantum scars from zero modes in an Abelian lattice gauge theory on ladders, Phys. Rev. Lett. 126, 220601 (2021).
  114. Q. Hummel, K. Richter, and P. Schlagheck, Genuine many-body quantum scars along unstable modes in Bose-Hubbard systems, Phys. Rev. Lett. 130, 250402 (2023).
  115. E. J. Heller, Bound-state eigenfunctions of classically chaotic Hamiltonian systems: Scars of periodic orbits, Phys. Rev. Lett. 53, 1515 (1984).
  116. M. V. Berry, Quantum scars of classical closed orbits in phase space, Proc. R. Soc. London A 423, 219 (1989).
  117. P. J. J. Luukko, B. Drury, A. Klales, L. Kaplan, E. J. Heller, and E. Räsänen, Strong quantum scarring by local impurities, Sci. Rep. 6, 37656 (2016).
  118. J. Keski-Rahkonen, P. J. J. Luukko, S. Åberg, and E. Räsänen, Effects of scarring on quantum chaos in disordered quantum wells, J. Phys.: Condens. Matter 31, 105301 (2019).
  119. W. W. Ho, S. Choi, H. Pichler, and M. D. Lukin, Periodic orbits, entanglement, and quantum many-body scars in constrained models: Matrix product state approach, Phys. Rev. Lett. 122, 040603 (2019).
  120. C. J. Turner, J.-Y. Desaules, K. Bull, and Z. Papić, Correspondence principle for many-body scars in ultracold Rydberg atoms, Phys. Rev. X 11, 021021 (2021).
  121. D. Mondal, S. Sinha, and S. Sinha, Dynamical route to ergodicity and quantum scarring in kicked coupled top, Phys. Rev. E 104, 024217 (2021).
  122. D. Mondal, S. Sinha, and S. Sinha, Quantum transitions, ergodicity, and quantum scars in the coupled top model, Phys. Rev. E 105, 014130 (2022).
  123. D. Mondal, S. Sinha, S. Ray, J. Kroha, and S. Sinha, Classical route to ergodicity and scarring phenomena in a two-component Bose-Josephson junction, Phys. Rev. A 106, 043321 (2022).
  124. I. Ermakov, Dynamics of exceptional states in many-body systems, Ph.D. thesis, Skoltech (2022) https://www.skoltech.ru/en/education/phd-defenses/igor-ermakov/.
  125. C.-J. Lin and O. I. Motrunich, Quasiparticle explanation of the weak-thermalization regime under quench in a nonintegrable quantum spin chain, Phys. Rev. A 95, 023621 (2017).
  126. C. Peng and X. Cui, Bridging quantum many-body scars and quantum integrability in Ising chains with transverse and longitudinal fields, Phys. Rev. B 106, 214311 (2022).
  127. A. S. de Wijn, B. Hess, and B. V. Fine, Lyapunov instabilities in lattices of interacting classical spins at infinite temperature, J. Phys. A: Math. Theor. 46, 254012 (2013).
  128. A. Katok, A. Katok, and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, Cambridge, UK, 1995), Vol. 54.
  129. G. Teschl, Ordinary Differential Equations and Dynamical Systems (American Mathematical Soceity, Providence, RI, 2012), Vol. 140.
  130. G. Benettin, L. Galgani, and J.-M. Strelcyn, Kolmogorov entropy and numerical experiments, Phys. Rev. A 14, 2338 (1976).
  131. G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; A method for computing all of them. Part 1: Theory, Meccanica 15, 9 (1980).
  132. S. Wimberger, Nonlinear Dynamics and Quantum Chaos, Graduate Texts in Physics (Springer, Cham, 2014).
  133. L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, and W. T. M. Irvine, Topological mechanics of gyroscopic metamaterials, Proc. Natl. Acad. Sci. USA 112, 14495 (2015).
  134. A. E. Tarkhov, S. Wimberger, and B. V. Fine, Extracting Lyapunov exponents from the echo dynamics of Bose-Einstein condensates on a lattice, Phys. Rev. A 96, 023624 (2017).
  135. C. Bingham, M. Godfrey, and J. Tukey, Modern techniques of power spectrum estimation, IEEE Trans. Audio Electroacoust. 15, 56 (1967).
  136. F. J. Harris, On the use of windows for harmonic analysis with the discrete Fourier transform, Proc. IEEE 66, 51 (1978).
  137. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978).
  138. M. Feingold and A. Peres, Regular and chaotic motion of coupled rotators, Physica D 9, 433 (1983).
  139. W. Schweizer, R. Niemeier, H. Friedrich, G. Wunner, and H. Ruder, Lyapunov exponents for classical orbits of the hydrogen atom in a magnetic field, Phys. Rev. A 38, 1724 (1988).
  140. P. Lochak, Arnold diffusion; A compendium of remarks and questions, in Hamiltonian systems with three or more degrees of freedom, in NATO ASI Series: C—Mathematical and Physical Sciences, Vol. 533 (Kluwer Academic, Dordrecht, 1954), pp. 527–530.
  141. D. Treschev, Arnold diffusion far from strong resonances in multidimensional a priori unstable Hamiltonian systems, Nonlinearity 25, 2717 (2012).
  142. V. Gelfreich and D. Turaev, Arnold diffusion in a priori chaotic symplectic maps, Commun. Math. Phys. 353, 507 (2017).
  143. A. Clarke and D. Turaev, Arnold diffusion in multidimensional convex billiards, Duke Math. J. 172, 1813 (2023).
  144. B. V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep. 52, 263 (1979).
  145. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).
  146. O. Piro and M. Feingold, Diffusion in three-dimensional Liouvillian maps, Phys. Rev. Lett. 61, 1799 (1988).
  147. B. Chirikov and V. Vecheslavov, Kam integrability, in Analysis, et cetera (Elsevier, Amsterdam, 1990), pp. 219–236.
  148. B. Chirikov and V. Vecheslavov, Theory of fast Arnold diffusion in many-frequency systems, J. Stat. Phys. 71, 243 (1993).
  149. B. Chirikov and V. Vecheslavov, Arnold diffusion in large systems, J. Exp. Theor. Phys. 85, 616 (1997).
  150. M. Mulansky, K. Ahnert, A. Pikovsky, and D. Shepelyansky, Strong and weak chaos in weakly nonintegrable many-body Hamiltonian systems, J. Stat. Phys. 145, 1256 (2011).
  151. A. Shapere and F. Wilczek, Classical time crystals, Phys. Rev. Lett. 109, 160402 (2012).
  152. F. Wilczek, Quantum time crystals, Phys. Rev. Lett. 109, 160401 (2012).
  153. K. Sacha and J. Zakrzewski, Time crystals: A review, Rep. Prog. Phys. 81, 016401 (2018).
  154. D. V. Else, C. Monroe, C. Nayak, and N. Y. Yao, Discrete time crystals, Annu. Rev. Condens. Matter Phys. 11, 467 (2020).
  155. J. Rovny, R. L. Blum, and S. E. Barrett, Observation of discrete-time-crystal signatures in an ordered dipolar many-body system, Phys. Rev. Lett. 120, 180603 (2018).
  156. A. Pizzi, J. Knolle, and A. Nunnenkamp, Period-n discrete time crystals and quasicrystals with ultracold bosons, Phys. Rev. Lett. 123, 150601 (2019).
  157. N. Y. Yao, C. Nayak, L. Balents, and M. P. Zaletel, Classical discrete time crystals, Nat. Phys. 16, 438 (2020).
  158. R. Daviet, C. P. Zelle, A. Rosch, and S. Diehl, Nonequilibrium criticality at the onset of time-crystalline order, Phys. Rev. Lett. 132, 167102 (2024).
  159. S. Autti, V. B. Eltsov, and G. E. Volovik, Observation of a time quasicrystal and its transition to a superfluid time crystal, Phys. Rev. Lett. 120, 215301 (2018).
  160. G. He, B. Ye, R. Gong, C. Yao, Z. Liu, K. W. Murch, N. Y. Yao, and C. Zu, Experimental realization of discrete time quasi-crystals, Phys. Rev. X 15, 011055 (2025).
  161. Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Distribution of the ratio of consecutive level spacings in random matrix ensembles, Phys. Rev. Lett. 110, 084101 (2013).
  162. A pure quantum state randomly sampling the infinite temperature equilibrium in a Hilbert space of dimension N≫1 is obtained as |inf〉=12L−1∑n=02L−1cn|bn〉, where |bn〉 are basis states, and cn=|cn|eiϕn are complex amplitudes with phases ϕn chosen randomly from the interval [0,2π) and absolute values |cn| chosen using the probability distribution P(|cn|2)=Ne−N|cn|2 (see Refs. [82, 163]).
  163. B. V. Fine, Typical state of an isolated quantum system with fixed energy and unrestricted participation of eigenstates, Phys. Rev. E 80, 051130 (2009).
  164. I. Ermakov, Operator growth in many-body systems of higher spins, arXiv:2504.07833.
  165. J. M. Radcliffe, Some properties of coherent spin states, J. Phys. A: Gen. Phys. 4, 313 (1971).
  166. B. V. Fine, Universal long-time relaxation on lattices of classical spins: Markovian behavior on non-Markovian timescales, J. Stat. Phys. 112, 319 (2003).
  167. S. W. Morgan, B. V. Fine, and B. Saam, Universal long-time behavior of nuclear spin decays in a solid, Phys. Rev. Lett. 101, 067601 (2008).
  168. E. G. Sorte, B. V. Fine, and B. Saam, Long-time behavior of nuclear spin decays in various lattices, Phys. Rev. B 83, 064302 (2011).
  169. B. Meier, J. Kohlrautz, and J. Haase, Eigenmodes in the long-time behavior of a coupled spin system measured with nuclear magnetic resonance, Phys. Rev. Lett. 108, 177602 (2012).
  170. L. Beringer, M. Steinhuber, J. D. Urbina, K. Richter, and S. Tomsovic, Controlling many-body quantum chaos: Bose–Hubbard systems, New J. Phys. 26, 073002 (2024).
  171. B. Evrard, A. Pizzi, S. I. Mistakidis, and C. B. Dag, Quantum scars and regular eigenstates in a chaotic spinor condensate, Phys. Rev. Lett. 132, 020401 (2024).
  172. B. Evrard, A. Pizzi, S. I. Mistakidis, and C. B. Dag, Quantum many-body scars from unstable periodic orbits, Phys. Rev. B 110, 144302 (2024).
  173. A. Pizzi, L.-H. Kwan, B. Evrard, C. B. Dag, and J. Knolle, Genuine quantum scars in many-body spin systems, Nat. Commun. 16, 6722 (2025), [the preprint arXiv:2408.10301v1 of the cited article without the results on higher quantum spins appeared eleven days prior to the preprint arXiv:2409.00258v1 of our article].

Outline

Information

Phys. Rev. E 112, 064109– Published 4 December, 2025
  • Received 30 August 2024
  • Revised 3 July 2025
  • Accepted 24 October 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