- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

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

Quantum chaos without false positives

Dmitrii A. Trunin*

  • Moscow Institute of Physics and Technology, Institutskiy pereulok 9, Dolgoprudny 141701, Russia and Lebedev Physical Institute, Leninskiy prospect 53, Moscow 119991, Russia

  • *dmitriy.trunin@phystech.edu
PDF

Phys. Rev. D 108, L101703 – Published 29 November, 2023

DOI: https://doi.org/10.1103/PhysRevD.108.L101703

Abstract

Out-of-time-order correlators are widely used as an indicator of quantum chaos but give false-positive quantum Lyapunov exponents for integrable systems with isolated saddle points. We propose an alternative indicator that fixes this drawback and retains all advantages of out-of-time-order correlators. In particular, the new indicator correctly predicts the average Lyapunov exponent and the Ehrenfest time in the semiclassical limit, can be calculated analytically using the replica trick, and satisfies the bound on chaos.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (90)

  1. A. I. Larkin and Y. N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Zh. Eksp. Teor. Fiz. 55, 2262 (1968) [Sov. Phys. JETP 28, 1200 (1969)].
  2. J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, J. High Energy Phys. 08 (2016) 106.
  3. A. Kitaev, A simple model of quantum holography, KITP seminar, https://online.kitp.ucsb.edu/online/entangled15/kitaev/ (2015), and https://online.kitp.ucsb.edu/online/entangled15/kitaev2/ (2015).
  4. B. Swingle, Unscrambling the physics of out-of-time-order correlators, Nat. Phys. 14, 988 (2018).
  5. I. García-Mata, R. A. Jalabert, and D. A. Wisniacki, Out-of-time-order correlators and quantum chaos, Scholarpedia 18, 55237 (2023).

  6. We consider a Hamiltonian system and introduce the canonical coordinates on the phase space z=(q,p).

  7. S. H. Shenker and D. Stanford, Black holes and the butterfly effect, J. High Energy Phys. 03 (2014) 067.
  8. S. H. Shenker and D. Stanford, Multiple shocks, J. High Energy Phys. 12 (2014) 046.
  9. D. A. Roberts, D. Stanford, and L. Susskind, Localized shocks, J. High Energy Phys. 03 (2015) 051.
  10. S. H. Shenker and D. Stanford, Stringy effects in scrambling, J. High Energy Phys. 05 (2015) 132.
  11. B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hayden, Measuring the scrambling of quantum information, Phys. Rev. A 94, 040302 (2016).
  12. D. A. Roberts and B. Swingle, Lieb-Robinson bound and the butterfly effect in quantum field theories, Phys. Rev. Lett. 117, 091602 (2016).
  13. A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X 8, 021014 (2018).
  14. X. Mi et al., Information scrambling in quantum circuits, Science 374, 1479 (2021).
  15. S. Xu and B. Swingle, Accessing scrambling using matrix product operators, Nat. Phys. 16, 199 (2019).
  16. S. Xu and B. Swingle, Scrambling dynamics and out-of-time ordered correlators in quantum many-body systems: A tutorial, arXiv:2202.07060.
  17. P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, J. High Energy Phys. 09 (2007) 120.
  18. Y. Sekino and L. Susskind, Fast scramblers, J. High Energy Phys. 10 (2008) 065.
  19. N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden, Towards the fast scrambling conjecture, J. High Energy Phys. 04 (2013) 022.
  20. L. García-Álvarez, I. L. Egusquiza, L. Lamata, A. del Campo, J. Sonner, and E. Solano, Digital quantum simulation of minimal AdS/CFT, Phys. Rev. Lett. 119, 040501 (2017).
  21. D. Jafferis, A. Zlokapa, J. D. Lykken, D. K. Kolchmeyer, S. I. Davis, N. Lauk, H. Neven, and M. Spiropulu, Traversable wormhole dynamics on a quantum processor, Nature (London) 612, 51 (2022).
  22. M. Gärttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger, and A. M. Rey, Measuring out-of-time-order correlations and multiple quantum spectra in a trapped ion quantum magnet, Nat. Phys. 13, 781 (2017).
  23. J. Li, R. Fan, H. Wang, B. Ye, B. Zeng, H. Zhai, X. Peng, and J. Du, Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator, Phys. Rev. X 7, 031011 (2017).
  24. A. M. Green, A. Elben, C. H. Alderete, L. K. Joshi, N. H. Nguyen, T. V. Zache, Y. Zhu, B. Sundar, and N. M. Linke, Experimental measurement of out-of-time-ordered correlators at finite temperature, Phys. Rev. Lett. 128, 140601 (2022).
  25. J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, J. High Energy Phys. 04 (2016) 001.
  26. J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016).
  27. A. Kitaev and S. J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, J. High Energy Phys. 05 (2018) 183.
  28. G. Sárosi, AdS2 holography and the SYK model, Proc. Sci. Modave2017 (2018) 001.
  29. V. Rosenhaus, An introduction to the SYK model, J. Phys. A 52, 323001 (2019).
  30. D. A. Trunin, Pedagogical introduction to the Sachdev-Ye-Kitaev model and two-dimensional dilaton gravity, Usp. Fiz. Nauk 191, 225 (2021) [Phys. Usp. 64, 219 (2021)].
  31. J. Maldacena, D. Stanford, and Z. Yang, Conformal symmetry and its breaking in two dimensional nearly anti-de Sitter space, Prog. Theor. Exp. Phys. 2016, 12C104 (2016).
  32. K. Jensen, Chaos in AdS2 holography, Phys. Rev. Lett. 117, 111601 (2016).
  33. J. Engelsöy, T. G. Mertens, and H. Verlinde, An investigation of AdS2 backreaction and holography, J. High Energy Phys. 07 (2016) 139.
  34. D. J. Gross and V. Rosenhaus, The bulk dual of SYK: Cubic couplings, J. High Energy Phys. 05 (2017) 092.
  35. D. J. Gross and V. Rosenhaus, All point correlation functions in SYK, J. High Energy Phys. 12 (2017) 148.
  36. D. A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett. 115, 131603 (2015).
  37. A. L. Fitzpatrick and J. Kaplan, A quantum correction to chaos, J. High Energy Phys. 05 (2016) 070.
  38. G. Turiaci and H. Verlinde, On CFT and quantum chaos, J. High Energy Phys. 12 (2016) 110.
  39. D. Stanford, Many-body chaos at weak coupling, J. High Energy Phys. 10 (2016) 009.
  40. S. Grozdanov, K. Schalm, and V. Scopelliti, Kinetic theory for classical and quantum many-body chaos, Phys. Rev. E 99, 012206 (2019).
  41. K. Hashimoto, K. Murata, and R. Yoshii, Out-of-time-order correlators in quantum mechanics, J. High Energy Phys. 10 (2017) 138.
  42. T. Akutagawa, K. Hashimoto, T. Sasaki, and R. Watanabe, Out-of-time-order correlator in coupled harmonic oscillators, J. High Energy Phys. 08 (2020) 013.
  43. N. Kolganov and D. A. Trunin, Classical and quantum butterfly effect in nonlinear vector mechanics, Phys. Rev. D 106, 025003 (2022).
  44. P. V. Buividovich, M. Hanada, and A. Schäfer, Quantum chaos, thermalization, and entanglement generation in real-time simulations of the Banks-Fischler-Shenker-Susskind matrix model, Phys. Rev. D 99, 046011 (2019).
  45. P. V. Buividovich, Quantum chaos in supersymmetric quantum mechanics: An exact diagonalization study, Phys. Rev. D 106, 046001 (2022).
  46. A. Bohrdt, C. B. Mendl, M. Endres, and M. Knap, Scrambling and thermalization in a diffusive quantum many-body system, New J. Phys. 19, 063001 (2017).
  47. M. J. Klug, M. S. Scheurer, and J. Schmalian, Hierarchy of information scrambling, thermalization, and hydrodynamic flow in graphene, Phys. Rev. B 98, 045102 (2018).
  48. D. Chowdhury and B. Swingle, Onset of many-body chaos in the O(N) model, Phys. Rev. D 96, 065005 (2017).
  49. A. A. Patel, D. Chowdhury, S. Sachdev, and B. Swingle, Quantum butterfly effect in weakly interacting diffusive metals, Phys. Rev. X 7, 031047 (2017).
  50. M. Tikhanovskaya, S. Sachdev, and A. A. Patel, Maximal quantum chaos of the critical fermi surface, Phys. Rev. Lett. 129, 060601 (2022).
  51. B. Kent, S. Racz, and S. Shashi, Scrambling in quantum cellular automata, Phys. Rev. B 107, 144306 (2023).
  52. E. B. Rozenbaum, S. Ganeshan, and V. Galitski, Lyapunov exponent and out-of-time-ordered correlator’s growth rate in a chaotic system, Phys. Rev. Lett. 118, 086801 (2017).
  53. E. B. Rozenbaum, L. A. Bunimovich, and V. Galitski, Early-time exponential instabilities in nonchaotic quantum systems, Phys. Rev. Lett. 125, 014101 (2020).
  54. T. Xu, T. Scaffidi, and X. Cao, Does scrambling equal chaos?, Phys. Rev. Lett. 124, 140602 (2020).
  55. S. Pilatowsky-Cameo, J. Chávez-Carlos, M. A. Bastarrachea-Magnani, P. Stránský, S. Lerma-Hernández, L. F. Santos, and J. G. Hirsch, Positive quantum Lyapunov exponents in experimental systems with a regular classical limit, Phys. Rev. E 101, 010202 (2020).
  56. Q. Wang and F. Pérez-Bernal, Probing an excited-state quantum phase transition in a quantum many-body system via an out-of-time-order correlator, Phys. Rev. A 100, 062113 (2019).
  57. S. Pappalardi, A. Russomanno, B. Žunkovič, F. Iemini, A. Silva, and R. Fazio, Scrambling and entanglement spreading in long-range spin chains, Phys. Rev. B 98, 134303 (2018).
  58. Q. Hummel, B. Geiger, J. D. Urbina, and K. Richter, Reversible quantum information spreading in many-body systems near criticality, Phys. Rev. Lett. 123, 160401 (2019).
  59. K. Hashimoto, K.-B. Huh, K.-Y. Kim, and R. Watanabe, Exponential growth of out-of-time-order correlator without chaos: inverted harmonic oscillator, J. High Energy Phys. 11 (2020) 068.
  60. M. Steinhuber, P. Schlagheck, J.-D. Urbina, and K. Richter, A dynamical transition from localized to uniform scrambling in locally hyperbolic systems, Phys. Rev. E 108, 024216 (2023).
  61. N. Dowling, P. Kos, and K. Modi, Scrambling is necessary but not sufficient for chaos, Phys. Rev. Lett. 131, 180403 (2023).

  62. Note that the operator under logarithm is Hermitian and positive definite, so the logarithm is well defined.

  63. D. Shepelyansky, Ehrenfest time and chaos, Scholarpedia 15, 55031 (2020).
  64. G. M. Zaslavsky, Stochasticity in quantum systems, Phys. Rep. 80, 157 (1981).
  65. B. V. Chirikov, F. M. Izrailev, and D. L. Shepelyansky, Quantum chaos: Localization vs ergodicity, Physica (Amsterdam) 33D, 77 (1988).
  66. I. L. Aleiner and A. I. Larkin, Divergence of classical trajectories and weak localization, Phys. Rev. B 54, 14423 (1996).
  67. M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction (Wiley-Interscience, New York, 1989).
  68. Y. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Usp. Mat. Nauk 32, 55 (1977) [Russ. Math. Surv. 32, 55 (1977)].
  69. I. L. Aleiner, L. Faoro, and L. B. Ioffe, Microscopic model of quantum butterfly effect: Out-of-time-order correlators and traveling combustion waves, Ann. Phys. (Amsterdam) 375, 378 (2016).
  70. F. M. Haehl, R. Loganayagam, P. Narayan, and M. Rangamani, Classification of out-of-time-order correlators, SciPost Phys. 6, 001 (2019).
  71. A. Romero-Bermúdez, K. Schalm, and V. Scopelliti, Regularization dependence of the OTOC. Which Lyapunov spectrum is the physical one?, J. High Energy Phys. 07 (2019) 107.
  72. D. A. Trunin, companion paper, Refined quantum Lyapunov exponents from replica out-of-time-order correlators, Phys. Rev. D 108, 105023 (2023).
  73. H. J. Lipkin, N. Meshkov, and A. J. Glick, Validity of many-body approximation methods for a solvable model: (I). Exact solutions and perturbation theory, Nucl. Phys. 62, 188 (1965); N. Meshkov, A. J. Glick, and H. J. Lipkin, Validity of many-body approximation methods for a solvable model: (II). Linearization procedures, 62, 199 (1965); A. J. Glick, H. J. Lipkin, and N. Meshkov, Validity of many-body approximation methods for a solvable model: (III). Diagram summations, 62, 211 (1965).

  74. On a Kähler manifold, classical LEs do not depend on the space parametrization, and we believe this property to be preserved after the semiclassical quantization.

  75. M. Feingold and A. Peres, Regular and chaotic motion of coupled rotators, Physica (Amsterdam) 9D, 433 (1983).
  76. M. Feingold, N. Moiseyev, and A. Peres, Ergodicity and mixing in quantum theory. II, Phys. Rev. A 30, 509 (1984).
  77. Y. Fan, S. Gnutzmann, and Y. Liang, Quantum chaos for nonstandard symmetry classes in the Feingold-Peres model of coupled tops, Phys. Rev. E 96, 062207 (2017).
  78. M. D. Esposti and S. Graffi, Mathematical aspects of quantum maps, in Mathematical Aspects of Quantum Maps, Lecture Notes in Physics Vol. 618. (Springer-Verlag, Berlin, 2003), p. 49.
  79. J. H. Hannay and M. V. Berry, Quantization of linear maps on a torus-Fresnel diffraction by a periodic grating, Physica (Amsterdam) 1D, 267 (1980).
  80. I. García-Mata, M. Saraceno, R. A. Jalabert, A. J. Roncaglia, and D. A. Wisniacki, Chaos signatures in the short and long time behavior of the out-of-time ordered correlator, Phys. Rev. Lett. 121, 210601 (2018).
  81. S. Moudgalya, T. Devakul, C. W. von Keyserlingk, and S. L. Sondhi, Operator spreading in quantum maps, Phys. Rev. B 99, 094312 (2019).
  82. S. G. Matinyan, G. K. Savvidy, and N. G. Ter-Arutunian-Savvidy, Classical Yang-Mills mechanics. Nonlinear color oscillations, Zh. Eksp. Teor. Fiz. 80, 830 (1981) [Sov. Phys. JETP 53, 421 (1981)].
  83. B. V. Chirikov and D. L. Shepelyansky, Stochastic oscillations of classical Yang-Mills fields, Pis’ma Zh. Eksp. Teor. Fiz. 34, 171 (1981) [JETP Lett. 34, 163 (1981)], http://jetpletters.ru/ps/1516/article_23158.shtml.
  84. G. K. Savvidy, Classical and quantum mechanics of non-Abelian gauge fields, Nucl. Phys. B246, 302 (1984).
  85. G. Savvidy, Maximally chaotic dynamical systems of Anosov–Kolmogorov and fundamental interactions, Int. J. Mod. Phys. A 37, 2230001 (2022).
  86. S. Pappalardi and J. Kurchan, Quantum bounds on the generalized Lyapunov exponents, Entropy 25, 246 (2023).
  87. N. Tsuji, T. Shitara, and M. Ueda, Bound on the exponential growth rate of out-of-time-ordered correlators, Phys. Rev. E 98, 012216 (2018).
  88. B. Yoshida and A. Kitaev, Efficient decoding for the Hayden-Preskill protocol, arXiv:1710.03363.
  89. P. Gao and D. L. Jafferis, A traversable wormhole teleportation protocol in the SYK model, J. High Energy Phys. 07 (2021) 097.
  90. M. Sandri, Numerical calculation of Lyapunov exponents, Math. J. 6, 78 (1996), https://www.msandri.it/docs/Numerical_Calculation_of_Lyapunov_Exponents.pdf.

Outline

Information

Phys. Rev. D 108, L101703– Published 29 November, 2023
  • Received 10 August 2023
  • Accepted 1 November 2023
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. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation