Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

The Lyapunov exponent as a signature of dissipative many-body quantum chaos (2403.12359v2)

Published 19 Mar 2024 in hep-th, cond-mat.str-el, and quant-ph

Abstract: A distinct feature of Hermitian quantum chaotic dynamics is the exponential increase of certain out-of-time-order-correlation (OTOC) functions around the Ehrenfest time with a rate given by a Lyapunov exponent. Physically, the OTOCs describe the growth of quantum uncertainty that crucially depends on the nature of the quantum motion. Here, we employ the OTOC in order to provide a precise definition of dissipative quantum chaos. For this purpose, we compute analytically the Lyapunov exponent for the vectorized formulation of the large $q$-limit of a $q$-body Sachdev-Ye-Kitaev model coupled to a Markovian bath. These analytic results are confirmed by an explicit numerical calculation of the Lyapunov exponent for several values of $q \geq 4$ based on the solutions of the Schwinger-Dyson and Bethe-Salpeter equations. We show that the Lyapunov exponent decreases monotonically as the coupling to the bath increases and eventually becomes negative at a critical value of the coupling signaling a transition to a dynamics which is no longer quantum chaotic. Therefore, a positive Lyapunov exponent is a defining feature of dissipative many-body quantum chaos. The observation of the breaking of the exponential growth for sufficiently strong coupling suggests that dissipative quantum chaos may require in certain cases a sufficiently weak coupling to the environment.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (73)
  1. Y. Sekino and L. Susskind, Fast scramblers, Journal of High Energy Physics 10, 065 (2008).
  2. D. Bagrets, A. Altland, and A. Kamenev, Power-law out of time order correlation functions in the SYK model, Nuclear Physics B 921, 727 (2017).
  3. A. Larkin and Y. N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov Phys JETP 28, 1200 (1969).
  4. G. Berman and G. Zaslavsky, Condition of stochasticity in quantum nonlinear systems, Physica A: Statistical Mechanics and its Applications 91, 450 (1978).
  5. 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).
  6. E. Wigner, On the statistical distribution of the widths and spacings of nuclear resonance levels, Math. Proc. Cam. Phil. Soc. 49, 790 (1951).
  7. F. J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3, 140 (1962a).
  8. F. Dyson, Statistical Theory of the Energy Levels of Complex Systems. II, J. Math. Phys. 3, 157 (1962b).
  9. F. Dyson, Statistical Theory of the Energy Levels of Complex Systems. III, J. Math. Phys. 3, 166 (1962c).
  10. F. Dyson, The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics, J. Math. Phys. 3, 1199 (1962d).
  11. F. Dyson, A Class of Matrix Ensembles, J. Math. Phys. 13, 90 (1972).
  12. M. L. Mehta, Random matrices (Academic press, 2004).
  13. J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, Journal of High Energy Physics 08, 106 (2016), arXiv:1503.01409 [hep-th] .
  14. J. French and S. Wong, Validity of random matrix theories for many-particle systems, Physics Letters B 33, 449 (1970).
  15. O. Bohigas and J. Flores, Two-body random hamiltonian and level density, Physics Letters B 34, 261 (1971a).
  16. J. French and S. Wong, Some random-matrix level and spacing distributions for fixed-particle-rank interactions, Physics Letters B 35, 5 (1971).
  17. O. Bohigas and J. Flores, Spacing and individual eigenvalue distributions of two-body random hamiltonians, Physics Letters B 35, 383 (1971b).
  18. L. Benet, T. Rupp, and H. A. Weidenmüller, Nonuniversal behavior of the k𝑘\mathit{k}italic_k-body embedded gaussian unitary ensemble of random matrices, Phys. Rev. Lett. 87, 010601 (2001), arXiv:cond-mat/0010425 [cond-mat] .
  19. S. Sachdev and J. Ye, Gapless spin-fluid ground state in a random quantum Heisenberg magnet, Phys. Rev. Lett. 70, 3339 (1993), arXiv:cond-mat/9212030 [cond-mat] .
  20. S. Sachdev, Holographic Metals and the Fractionalized Fermi Liquid, Phys. Rev. Lett. 105, 151602 (2010).
  21. J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016).
  22. A. M. Garc\́mathrm{i}a-Garc\́mathrm{i}a, C. Liu, and J. J. M. Verbaarschot, Sparsity independent Lyapunov exponent in the Sachdev-Ye-Kitaev model (2023), arXiv:2311.00639 [hep-th] .
  23. A. M. Garc\́mathrm{i}a-Garc\́mathrm{i}a and J. J. M. Verbaarschot, Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 126010 (2016), arXiv:1610.03816 [hep-th] .
  24. C. M. Bender and S. Boettcher, Real spectra in non-hermitian hamiltonians having P⁢T𝑃𝑇PTitalic_P italic_T symmetry, Phys. Rev. Lett. 80, 5243 (1998).
  25. J. Wiersig, Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle detection, Phys. Rev. Lett. 112, 203901 (2014).
  26. K. Kawabata, Y. Ashida, and M. Ueda, Information retrieval and criticality in parity-time-symmetric systems, Phys. Rev. Lett. 119, 190401 (2017).
  27. A. M. Garc\́mathrm{i}a-Garc\́mathrm{i}a and V. Godet, Euclidean wormhole in the Sachdev-Ye-Kitaev model, Phys. Rev. D 103, 046014 (2021), arXiv:2010.11633 [hep-th] .
  28. Y. Ashida, Z. Gong, and M. Ueda, Non-hermitian physics, Advances in Physics 69, 249 (2020).
  29. R. Grobe, F. Haake, and H.-J. Sommers, Quantum Distinction of Regular and Chaotic Dissipative Motion, Phys. Rev. Lett. 61, 1899 (1988).
  30. Y. V. Fyodorov, B. A. Khoruzhenko, and H.-J. Sommers, Almost Hermitian Random Matrices: Crossover from Wigner-Dyson to Ginibre Eigenvalue Statistics, Physical Review Letters 79, 557 (1997), cond-mat/9703152 .
  31. L. Sá, P. Ribeiro, and T. Prosen, Spectral and steady-state properties of random Liouvillians, J. Phys. A: Math. Theor. 53, 305303 (2020).
  32. A. Rubio-Garc\́mathrm{i}a, R. Molina, and J. Dukelsky, From integrability to chaos in quantum liouvillians, SciPost Physics Core 5, 10.21468/scipostphyscore.5.2.026 (2022).
  33. A. M. Garc\́mathrm{i}a-Garc\́mathrm{i}a, L. Sá, and J. J. M. Verbaarschot, Symmetry Classification and Universality in Non-Hermitian Many-Body Quantum Chaos by the Sachdev-Ye-Kitaev Model, Phys. Rev. X 12, 021040 (2022a).
  34. J. Li, T. Prosen, and A. Chan, Spectral statistics of non-hermitian matrices and dissipative quantum chaos, Phys. Rev. Lett. 127, 170602 (2021).
  35. A. Altland, M. Fleischhauer, and S. Diehl, Symmetry classes of open fermionic quantum matter, Physical Review X 11, 021037 (2021).
  36. A. M. Garc\́mathrm{i}a-Garc\́mathrm{i}a, S. M. Nishigaki, and J. J. M. Verbaarschot, Critical statistics for non-hermitian matrices, Phys. Rev. E 66, 016132 (2002).
  37. A. M. Garc\́mathrm{i}a-Garc\́mathrm{i}a, L. Sá, and J. J. M. Verbaarschot, Universality and its limits in non-hermitian many-body quantum chaos using the Sachdev-Ye-Kitaev model, Phys. Rev. D 107, 066007 (2023a).
  38. L. Sá, P. Ribeiro, and T. Prosen, Lindbladian dissipation of strongly-correlated quantum matter, Phys. Rev. Research 4, L022068 (2022).
  39. A. Kulkarni, T. Numasawa, and S. Ryu, Lindbladian dynamics of the Sachdev-Ye-Kitaev model, Phys. Rev. B 106, 075138 (2022).
  40. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  41. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N𝑁Nitalic_N-level systems, J. Math. Phys. 17, 821 (1976).
  42. D. Manzano, A short introduction to the Lindblad master equation, AIP Advances 10, 025106 (2020).
  43. P. D. Bergamasco, G. G. Carlo, and A. M. Rivas, Quantum lyapunov exponent in dissipative systems, Physical Review E 108, 024208 (2023).
  44. B. Yoshida and N. Y. Yao, Disentangling scrambling and decoherence via quantum teleportation, Physical Review X 9, 011006 (2019).
  45. J. Tuziemski, Out-of-time-ordered correlation functions in open systems: A Feynman-Vernon influence functional approach, Physical Review A 100, 062106 (2019).
  46. P. Zanardi and N. Anand, Information scrambling and chaos in open quantum systems, Physical Review A 103, 062214 (2021).
  47. S. Syzranov, A. Gorshkov, and V. Galitski, Out-of-time-order correlators in finite open systems, Physical Review B 97, 161114 (2018).
  48. L.-J. Zhai and S. Yin, Out-of-time-ordered correlator in non-hermitian quantum systems, Phys. Rev. B 102, 054303 (2020).
  49. G. J. Turiaci, An inelastic bound on chaos, Journal of High Energy Physics 2019, 10.1007/jhep07(2019)099 (2019).
  50. T. Schuster and N. Y. Yao, Operator Growth in Open Quantum Systems, Phys. Rev. Lett. 131, 160402 (2023), arXiv:2208.12272 [quant-ph] .
  51. B. Bhattacharjee, P. Nandy, and T. Pathak, Operator dynamics in Lindbladian SYK: a Krylov complexity perspective (2023b), arXiv:2311.00753 [quant-ph] .
  52. C. Liu, H. Tang, and H. Zhai, Krylov complexity in open quantum systems, Phys. Rev. Res. 5, 033085 (2023).
  53. N. S. Srivatsa and C. von Keyserlingk, The operator growth hypothesis in open quantum systems (2023), arXiv:2310.15376 [quant-ph] .
  54. P. Zhang and Z. Yu, Dynamical transition of operator size growth in quantum systems embedded in an environment, Phys. Rev. Lett. 130, 250401 (2023).
  55. J. Liu, R. Meyer, and Z.-Y. Xian, Operator size growth in Lindbladian SYK (2024), arXiv:2403.07115 [hep-th] .
  56. Y. Chen, H. Zhai, and P. Zhang, Tunable quantum chaos in the Sachdev-Ye-Kitaev model coupled to a thermal bath, Journal of High Energy Physics 2017, 10.1007/jhep07(2017)150 (2017).
  57. J. Maldacena and X.-L. Qi, Eternal traversable wormhole, eprint  (2018), arXiv:1804.00491 [hep-th] .
  58. E. Cáceres, A. Misobuchi, and R. Pimentel, Sparse SYK and traversable wormholes, Journal of High Energy Physics 2021, 11 (2021), arXiv:2108.08808 [hep-th] .
  59. T. Nosaka and T. Numasawa, Chaos exponents of SYK traversable wormholes, Journal of High Energy Physics 2021, 10.1007/jhep02(2021)150 (2021).
  60. T. Nosaka and T. Numasawa, On SYK traversable wormhole with imperfectly correlated disorders, Journal of High Energy Physics 2023, 10.1007/jhep04(2023)145 (2023).
  61. A. O. Caldeira and A. J. Leggett, Influence of Dissipation on Quantum Tunneling in Macroscopic Systems, Phys. Rev. Lett. 46, 211 (1981).
  62. A. Kamenev, Field theory of non-equilibrium systems (Cambridge University Press, 2011).
  63. L. M. Sieberer, M. Buchhold, and S. Diehl, Keldysh field theory for driven open quantum systems, Reports on Progress in Physics 79, 096001 (2016).
  64. M. Khramtsov and E. Lanina, Spectral form factor in the double-scaled SYK model, Journal of High Energy Physics 2021, 10.1007/jhep03(2021)031 (2021).
  65. G. Tarnopolsky, Large q𝑞qitalic_q expansion in the Sachdev-Ye-Kitaev model, Phys. Rev. D 99, 026010 (2019), arXiv:1801.06871 [hep-th] .
  66. C.-J. Lin and O. I. Motrunich, Out-of-time-ordered correlators in a quantum Ising chain, Phys. Rev. B 97, 144304 (2018).
  67. R. A. Jalabert, I. Garc\́mathrm{i}a-Mata, and D. A. Wisniacki, Semiclassical theory of out-of-time-order correlators for low-dimensional classically chaotic systems, Phys. Rev. E 98, 062218 (2018).
  68. I. Garcia-Mata, R. Jalabert, and D. Wisniacki, Out-of-time-order correlations and quantum chaos, Scholarpedia 18, 55237 (2023).
  69. D. J. Luitz, N. Laflorencie, and F. Alet, Many-body localization edge in the random-field Heisenberg chain, Phys. Rev. B 91, 081103 (2015), arXiv:1411.0660 [cond-mat.dis-nn] .
  70. J. Maldacena, G. J. Turiaci, and Z. Yang, Two dimensional nearly de Sitter gravity (2020), arXiv:1904.01911 [hep-th] .
  71. J. Cotler, K. Jensen, and A. Maloney, Low-dimensional de Sitter quantum gravity, Journal of High Energy Physics 2020 (2020).
  72. L. Aalsma and G. Shiu, Chaos and complementarity in de Sitter space, Journal of High Energy Physics 2020, 10.1007/jhep05(2020)152 (2020).
  73. Y. Gu and A. Kitaev, On the relation between the magnitude and exponent of OTOCs, Journal of High Energy Physics 2019, 10.1007/jhep02(2019)075 (2019).
Citations (5)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now