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  • Letter
  • Open Access

Space-time duality between quantum chaos and non-Hermitian boundary effect

Tian-Gang Zhou1,*, Yi-Neng Zhou1,*, Pengfei Zhang2,†, and Hui Zhai1,‡

  • 1Institute for Advanced Study, Tsinghua University, Beijing 100084, China
  • 2Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA

  • *These authors contributed equally to this work.
  • †PengfeiZhang.physics@gmail.com
  • ‡hzhai@mail.tsinghua.edu.cn
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Phys. Rev. Research 4, L022039 – Published 20 May, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L022039

Abstract

Quantum chaos in Hermitian systems concerns the sensitivity of long-time dynamical evolution to initial conditions. The skin effect discovered recently in non-Hermitian systems reveals the sensitivity to the spatial boundary condition even deep in the bulk. In this Letter, we show that these two seemingly different phenomena can be unified through the space-time duality. The intuition is that the space-time duality maps unitary dynamics to nonunitary dynamics and exchanges the temporal direction and spatial direction. Therefore, the space-time duality can establish the connection between the sensitivity to the initial condition in the temporal direction and the sensitivity to the boundary condition in the spatial direction. Here, we demonstrate this connection by studying the space-time duality of the out-of-time-ordered correlator in a concrete chaotic Hermitian model. We show that the out-of-time-ordered correlator is mapped to a special two-point correlator of a non-Hermitian system in the dual picture. For comparison, we show that the sensitivity disappears when the non-Hermiticity is removed in the dual picture.

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References (63)

  1. A. Larkin and Y. N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Zh. Eksp. Teor. Fiz. 55, 2262 (1969) [Sov. Phys. JETP 28, 1200 (1969)].
  2. A. Almheiri, D. Marolf, J. Polchinski, D. Stanford, and J. Sully, Black holes: complementarity or firewalls? J. High Energy Phys. 02 (2013) 062.
  3. S. H. Shenker and D. Stanford, Black holes and the butterfly effect, J. High Energy Phys. 03 (2014) 067.
  4. D. A. Roberts and D. Stanford, Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory, Phys. Rev. Lett. 115, 131603 (2015).
  5. D. A. Roberts, D. Stanford, and L. Susskind, Localized shocks, J. High Energy Phys. 03 (2015) 051.
  6. S. H. Shenker and D. Stanford, Stringy effects in scrambling, J. High Energy Phys. 05 (2015) 132.
  7. P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Chaos in quantum channels, J. High Energy Phys. 02 (2016) 004.
  8. J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, J. High Energy Phys. 08 (2016) 106.
  9. D. Stanford, Many-body chaos at weak coupling, J. High Energy Phys. 10 (2016) 009.
  10. J. Maldacena, D. Stanford, and Z. Yang, Diving into traversable wormholes, Prog. Theor. Exp. Phys. 2016, 12C104 (2016).
  11. J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016).
  12. G. Zhu, M. Hafezi, and T. Grover, Measurement of many-body chaos using a quantum clock, Phys. Rev. A 94, 062329 (2016).
  13. 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).
  14. J. Li, R. Fan, H. Wang, B. Ye, B. Zeng, H. Zhai, X. Peng, J. Du, Measuring Out-of-Time-Order Correlators on a Nuclear Magnetic Resonance Quantum Simulator, Phys. Rev. X 7, 031011 (2017).
  15. Y. Gu, X.-L. Qi, and D. Stanford, Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, J. High Energy Phys. 05 (2017) 125.
  16. A. A. Patel, D. Chowdhury, S. Sachdev, and B. Swingle, Quantum Butterfly Effect in Weakly Interacting Diffusive Metals, Phys. Rev. X 7, 031047 (2017).
  17. Y. Chen, H. Zhai, and P. Zhang, Tunable quantum chaos in the Sachdev-Ye-Kitaev model coupled to a thermal bath, J. High Energy Phys. 07 (2017) 150.
  18. X. Chen, R. Fan, Y. Chen, H. Zhai, and P. Zhang, Competition between Chaotic and Nonchaotic Phases in a Quadratically Coupled Sachdev-Ye-Kitaev Model, Phys. Rev. Lett. 119, 207603 (2017).
  19. X.-Y. Song, C.-M. Jian, and L. Balents, Strongly Correlated Metal Built from Sachdev-Ye-Kitaev Models, Phys. Rev. Lett. 119, 216601 (2017).
  20. 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).
  21. S. Xu and B. Swingle, Locality, Quantum Fluctuations, and Scrambling, Phys. Rev. X 9, 031048 (2019).
  22. P. Zhang, Quantum chaos for the unitary Fermi gas from the generalized Boltzmann equations, J. Phys. B: At., Mol. Opt. Phys. 52, 135301 (2019).
  23. H. Guo, Y. Gu, and S. Sachdev, Transport and chaos in lattice Sachdev-Ye-Kitaev models, Phys. Rev. B 100, 045140 (2019).
  24. R. J. Lewis-Swan, A. Safavi-Naini, J. J. Bollinger, and A. M. Rey, Unifying scrambling, thermalization and entanglement through measurement of fidelity out-of-time-order correlators in the Dicke model, Nat. Commun. 10, 1581 (2019).
  25. Y. Gu and A. Kitaev, On the relation between the magnitude and exponent of OTOCs, J. High Energy Phys. 02 (2019) 075.
  26. P. Zhang, Y. Gu, and A. Kitaev, An obstacle to sub-AdS holography for SYK-like models, J. High Energy Phys. 03 (2021) 094.
  27. X. Mi, P. Roushan, C. Quintana, S. Mandra, J. Marshall, C. Neill, F. Arute, K. Arya, J. Atalaya, R. Babbush, J. C. Bardin, R. Barends, A. Bengtsson, S. Boixo, A. Bourassa, M. Broughton, B. B. Buckley, D. A. Buell, B. Burkett, N. Bushnell et al., Information scrambling in quantum circuits, Science 374, 1479 (2021).
  28. S. Yao and Z. Wang, Edge States and Topological Invariants of Non-Hermitian Systems, Phys. Rev. Lett. 121, 086803 (2018).
  29. F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems, Phys. Rev. Lett. 121, 026808 (2018).
  30. V. M. Martinez Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points, Phys. Rev. B 97, 121401(R) (2018).
  31. C. H. Lee and R. Thomale, Anatomy of skin modes and topology in non-Hermitian systems, Phys. Rev. B 99, 201103(R) (2019).
  32. F. Song, S. Yao, and Z. Wang, Non-Hermitian Topological Invariants in Real Space, Phys. Rev. Lett. 123, 246801 (2019).
  33. S. Longhi, Probing non-Hermitian skin effect and non-Bloch phase transitions, Phys. Rev. Research 1, 023013 (2019).
  34. K. Zhang, Z. Yang, and C. Fang, Correspondence between Winding Numbers and Skin Modes in Non-Hermitian Systems, Phys. Rev. Lett. 125, 126402 (2020).
  35. L. Li, C. H. Lee, and J. Gong, Topological Switch for Non-Hermitian Skin Effect in Cold-Atom Systems with Loss, Phys. Rev. Lett. 124, 250402 (2020).
  36. K. Kawabata, M. Sato, and K. Shiozaki, Higher-order non-Hermitian skin effect, Phys. Rev. B 102, 205118 (2020).
  37. N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological Origin of Non-Hermitian Skin Effects, Phys. Rev. Lett. 124, 086801 (2020).
  38. Y. Yi and Z. Yang, Non-Hermitian Skin Modes Induced by On-Site Dissipations and Chiral Tunneling Effect, Phys. Rev. Lett. 125, 186802 (2020).
  39. L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Non-Hermitian bulk-boundary correspondence in quantum dynamics, Nat. Phys. 16, 761 (2020).
  40. T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits, Nat. Phys. 16, 747 (2020).
  41. A. Ghatak, M. Brandenbourger, J. van Wezel, and C. Coulais, Observation of non-Hermitian topology and its bulk-edge correspondence in an active mechanical metamaterial, Proc. Natl. Acad. Sci. USA 117, 29561 (2020).
  42. M. Akila, D. Waltner, B. Gutkin, and T. Guhr, Particle-time duality in the kicked Ising chain I: The dual operator, J. Phys. A: Math. Theor. 49, 375101 (2016).
  43. B. Bertini, P. Kos, and T. Prosen, Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos, Phys. Rev. Lett. 121, 264101 (2018).
  44. B. Bertini, P. Kos, and T. Prosen, Entanglement Spreading in a Minimal Model of Maximal Many-Body Quantum Chaos, Phys. Rev. X 9, 021033 (2019).
  45. B. Bertini, P. Kos, and T. Prosen, Exact Correlation Functions for Dual-Unitary Lattice Models in 1+1 Dimensions, Phys. Rev. Lett. 123, 210601 (2019).
  46. S. Gopalakrishnan and A. Lamacraft, Unitary circuits of finite depth and infinite width from quantum channels, Phys. Rev. B 100, 064309 (2019).
  47. P. W. Claeys and A. Lamacraft, Maximum velocity quantum circuits, Phys. Rev. Research 2, 033032 (2020).
  48. L. Piroli, B. Bertini, J. I. Cirac, and T. Prosen, Exact dynamics in dual-unitary quantum circuits, Phys. Rev. B 101, 094304 (2020).
  49. R. Hamazaki, Exceptional dynamical quantum phase transitions in periodically driven systems, Nat. Commun. 12, 5108 (2021).
  50. P. Kos, B. Bertini, and T. Prosen, Correlations in Perturbed Dual-Unitary Circuits: Efficient Path-Integral Formula, Phys. Rev. X 11, 011022 (2021).
  51. M. Ippoliti and V. Khemani, Postselection-Free Entanglement Dynamics via Spacetime Duality, Phys. Rev. Lett. 126, 060501 (2021).
  52. F. Fritzsch and T. Prosen, Eigenstate Thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions, Phys. Rev. E 103, 062133 (2021).
  53. T.-C. Lu and T. Grover, Spacetime duality between localization transitions and measurement-induced transitions, PRX Quantum 2, 040319 (2021).
  54. M. Ippoliti, T. Rakovszky, and V. Khemani, Fractal, Logarithmic, and Volume-Law Entangled Nonthermal Steady States via Spacetime Duality, Phys. Rev. X 12, 011045 (2022).
  55. P. W. Claeys and A. Lamacraft, Ergodic and Nonergodic Dual-Unitary Quantum Circuits with Arbitrary Local Hilbert Space Dimension, Phys. Rev. Lett. 126, 100603 (2021)
  56. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.4.L022039 for deriving the relations between parameters before and after space-time duality.
  57. G. A. Alvarez, R. Kaiser, and D. Suter, Quantum simulations of localization effects with dipolar interactions, Ann. Phys. 525, 833 (2013)
  58. We note that, even in non-Hermitian systems under Hamiltonian dynamics, a physical observable such as a real-time correlator should not be sensitive to the boundary effect, as shown recently by L. Mao, T. Deng, and P. Zhang, Boundary condition independence of non-Hermitian Hamiltonian dynamics, Phys. Rev. B 104, 125435 (2021).
  59. The codes for the numerical results are available in https://github.com/tgzhou98/Dual-OTOC.
  60. R. Fan, P. Zhang, H. Shen, and H. Zhai, Out-of-time-order correlation for many-body localization, Sci. Bull. 62, 707 (2017).
  61. Y. Huang, Y.-L. Zhang, and X. Chen, Out-of-time-ordered correlators in many-body localized systems, Ann. Phys. 529, 1600318 (2017).
  62. B. Swingle and D. Chowdhury, Slow scrambling in disordered quantum systems, Phys. Rev. B 95, 060201(R) (2017).
  63. Y. Chen, Universal logarithmic scrambling in many body localization, arXiv:1608.02765.

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Phys. Rev. Research 4, L022039– Published 20 May, 2022
  • Received 6 August 2021
  • Accepted 27 April 2022
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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.

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