Skip to main content
SpringerLink
Log in

General properties of the spectral form factor in open quantum systems

  • Research Article
  • Published:
Frontiers of Physics Aims and scope Submit manuscript

Abstract

The spectral form factor (SFF) can probe the eigenvalue statistic at different energy scales as its time variable varies. In closed quantum chaotic systems, the SFF exhibits a universal dip-ramp-plateau behavior, which reflects the spectrum rigidity of the Hamiltonian. In this work, we explore the general properties of SFF in open quantum systems. We find that in open systems the SFF first decays exponentially, followed by a linear increase at some intermediate time scale, and finally decreases to a saturated plateau value. We derive general relations between (i) the early-time decay exponent and Lindblad operators; (ii) the long-time plateau value and the number of steady states. We also explain the effective field theory perspective of general behaviors. We verify our theoretical predictions by numerically simulating the Sachdev–Ye–Kitaev (SYK) model, random matrix theory (RMT), and the Bose–Hubbard model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References and notes

  1. E. Brézin and A. Zee, Universality of the correlations between eigenvalues of large random matrices, Nucl. Phys. B 402(3), 613 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  2. E. Brézin and S. Hikami, Correlations of nearby levels induced by a random potential, Nucl. Phys. B 479(3), 697 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  3. S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, Periodic-orbit theory of universality in quantum chaos, Phys. Rev. E 72(4), 046207 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  4. J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka, Black holes and random matrices, J. High Energy Phys. 2017(5), 118 (2017)

    Article  MathSciNet  Google Scholar 

  5. J. Liu, Spectral form factors and late time quantum chaos, Phys. Rev. D 98(8), 086026 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  6. P. Kos, M. Ljubotina, and T. Prosen, Many-body quantum chaos: Analytic connection to random matrix theory, Phys. Rev. X 8(2), 021062 (2018)

    Google Scholar 

  7. B. Bertini, P. Kos, and T. Prosen, Exact spectral form factor in a minimal model of many-body quantum chaos, Phys. Rev. Lett. 121(26), 264101 (2018)

    Article  ADS  Google Scholar 

  8. A. Chan, A. De Luca, and J. T. Chalker, Spectral statistics in spatially extended chaotic quantum many-body systems, Phys. Rev. Lett. 121(6), 060601 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  9. J. Kudler-Flam, L. Nie, and S. Ryu, Conformal field theory and the web of quantum chaos diagnostics, J. High Energy Phys. 2020(1), 175 (2020)

    Article  MathSciNet  Google Scholar 

  10. D. Roy and T. Prosen, Random matrix spectral form factor in kicked interacting fermionic chains, Phys. Rev. E 102(6), 060202 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  11. M. Winer and B. Swingle, Hydrodynamic theory of the connected spectral form factor, Phys. Rev. X 12(2), 021009 (2022)

    Google Scholar 

  12. D. Roy, D. Mishra, and T. Prosen, Spectral form factor in a minimal bosonic model of many-body quantum chaos, Phys. Rev. E 106(2), 024208 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  13. R. Barney, M. Winer, C. L. Baldwin, B. Swingle, and V. Galitski, Spectral statistics of a minimal quantum glass model, SciPost Phys. 15, 084 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  14. P. Saad, S. H. Shenker, and D. Stanford, A semi-classical ramp in SYK and in gravity, arXiv: 1806.06840 (2018)

  15. H. Gharibyan, M. Hanada, S. H. Shenker, and M. Tezuka, Onset of random matrix behavior in scrambling systems, J. High Energy Phys. 2018(7), 124 (2018)

    Article  MathSciNet  Google Scholar 

  16. M. Winer, S. K. Jian, and B. Swingle, Exponential ramp in the quadratic Sachdev–Ye–Kitaev model, Phys. Rev. Lett. 125(25), 250602 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  17. R. Shir, P. Martinez-Azcona, and A. Chenu, Full range spectral correlations and their spectral form factors in chaotic and integrable models, arXiv: 2311.09292 (2023)

  18. Y. N. Zhou, L. Mao, and H. Zhai, Rényi entropy dynamics and Lindblad spectrum for open quantum systems, Phys. Rev. Res. 3(4), 043060 (2021)

    Article  Google Scholar 

  19. G. Mazzucchi, W. Kozlowski, S. F. Caballero-Benitez, T. J. Elliott, and I. B. Mekhov, Quantum measurement-induced dynamics of many-body ultracold bosonic and fermionic systems in optical lattices, Phys. Rev. A 93(2), 023632 (2016)

    Article  ADS  Google Scholar 

  20. Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98(20), 205136 (2018)

    Article  ADS  Google Scholar 

  21. B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9(3), 031009 (2019)

    Google Scholar 

  22. Y. Li, X. Chen, and M. P. A. Fisher, Measurement-driven entanglement transition in hybrid quantum circuits, Phys. Rev. B 100(13), 134306 (2019)

    Article  ADS  Google Scholar 

  23. M. Szyniszewski, A. Romito, and H. Schomerus, Entanglement transition from variable-strength weak measurements, Phys. Rev. B 100(6), 064204 (2019)

    Article  ADS  Google Scholar 

  24. A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Unitary-projective entanglement dynamics, Phys. Rev. B 99(22), 224307 (2019)

    Article  ADS  Google Scholar 

  25. R. Vasseur, A. C. Potter, Y. Z. You, and A. W. W. Ludwig, Entanglement transitions from holographic random tensor networks, Phys. Rev. B 100(13), 134203 (2019)

    Article  ADS  Google Scholar 

  26. T. Zhou and A. Nahum, Emergent statistical mechanics of entanglement in random unitary circuits, Phys. Rev. B 99(17), 174205 (2019)

    Article  ADS  Google Scholar 

  27. M. J. Gullans and D. A. Huse, Scalable probes of measurement-induced criticality, Phys. Rev. Lett. 125(7), 070606 (2020)

    Article  ADS  Google Scholar 

  28. C. M. Jian, Y. Z. You, R. Vasseur, and A. W. W. Ludwig, Measurement-induced criticality in random quantum circuits, Phys. Rev. B 101(10), 104302 (2020)

    Article  ADS  Google Scholar 

  29. Y. Fuji and Y. Ashida, Measurement-induced quantum criticality under continuous monitoring, Phys. Rev. B 102(5), 054302 (2020)

    Article  ADS  Google Scholar 

  30. A. Zabalo, M. J. Gullans, J. H. Wilson, S. Gopalakrishnan, D. A. Huse, and J. H. Pixley, Critical properties of the measurement-induced transition in random quantum circuits, Phys. Rev. B 101(6), 060301 (2020)

    Article  ADS  Google Scholar 

  31. M. J. Gullans and D. A. Huse, Dynamical purification phase transition induced by quantum measurements, Phys. Rev. X 10(4), 041020 (2020)

    Google Scholar 

  32. S. Choi, Y. Bao, X. L. Qi, and E. Altman, Quantum error correction in scrambling dynamics and measurement-induced phase transition, Phys. Rev. Lett. 125(3), 030505 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  33. Y. Bao, S. Choi, and E. Altman, Theory of the phase transition in random unitary circuits with measurements, Phys. Rev. B 101(10), 104301 (2020)

    Article  ADS  Google Scholar 

  34. A. Nahum, S. Roy, B. Skinner, and J. Ruhman, Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau–Ginsburg theory, PRX Quantum 2(1), 010352 (2021)

    Article  Google Scholar 

  35. R. Fan, S. Vijay, A. Vishwanath, and Y. Z. You, Self-organized error correction in random unitary circuits with measurement, Phys. Rev. B 103(17), 174309 (2021)

    Article  ADS  Google Scholar 

  36. S. Sang and T. H. Hsieh, Measurement-protected quantum phases, Phys. Rev. Res. 3(2), 023200 (2021)

    Article  Google Scholar 

  37. O. Alberton, M. Buchhold, and S. Diehl, Entanglement transition in a monitored free-fermion chain: From extended criticality to area law, Phys. Rev. Lett. 126(17), 170602 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  38. A. Lavasani, Y. Alavirad, and M. Barkeshli, Measurement-induced topological entanglement transitions in symmetric random quantum circuits, Nat. Phys. 17(3), 342 (2021)

    Article  Google Scholar 

  39. X. Turkeshi, A. Biella, R. Fazio, M. Dalmonte, and M. Schiró, Measurement-induced entanglement transitions in the quantum Ising chain: From infinite to zero clicks, Phys. Rev. B 103(22), 224210 (2021)

    Article  ADS  Google Scholar 

  40. Y. Le Gal, X. Turkeshi, and M. Schiró, Volume-to-area law entanglement transition in a non-Hermitian free fermionic Chain, SciPost Phys. 14, 138 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  41. S. K. Jian, C. Liu, X. Chen, B. Swingle, and P. Zhang, Measurement-induced phase transition in the monitored Sachdev–Ye–Kitaev model, Phys. Rev. Lett. 127(14), 140601 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  42. P. Zhang, C. Liu, S. K. Jian, and X. Chen, Universal entanglement transitions of free fermions with longrange non-unitary dynamics, Quantum 6, 723 (2022)

    Article  Google Scholar 

  43. C. Liu, P. Zhang, and X. Chen, Non-unitary dynamics of Sachdev–Ye–Kitaev chain, SciPost Phys. 10, 048 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  44. P. Zhang, S. K. Jian, C. Liu, and X. Chen, Emergent replica conformal symmetry in non-Hermitian SYK2 chains, Quantum 5, 579 (2021)

    Article  Google Scholar 

  45. P. Zhang, Quantum entanglement in the Sachdev–Ye–Kitaev model and its generalizations, Front. Phys. 17(4), 43201 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  46. S. Sahu, S. K. Jian, G. Bentsen, and B. Swingle, Entanglement phases in large-n hybrid Brownian circuits with long-range couplings, Phys. Rev. B 106(22), 224305 (2022)

    Article  ADS  Google Scholar 

  47. C. Liu, H. Tang, and H. Zhai, Krylov complexity in open quantum systems, Phys. Rev. Res. 5, 033085 (2023)

    Article  Google Scholar 

  48. A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, Operator growth and Krylov construction in dissipative open quantum systems, J. High Energy Phys. 2022(12), 81 (2022)

    Article  MathSciNet  Google Scholar 

  49. B. Bhattacharjee, X. Cao, P. Nandy, and T. Pathak, Operator growth in open quantum systems: Lessons from the dissipative SYK, J. High Energy Phys. 2023(3), 54 (2023)

    Article  MathSciNet  Google Scholar 

  50. A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, On Krylov complexity in open systems: An approach via bi-Lanczos algorithm, J. High Energy Phys. 2023, 66 (2023)

    Article  MathSciNet  Google Scholar 

  51. T. Can, Random Lindblad dynamics, J. Phys. A Math. Theor. 52(48), 485302 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  52. J. Li, T. Prosen, and A. Chan, Spectral statistics of non-Hermitian matrices and dissipative quantum chaos, Phys. Rev. Lett. 127(17), 170602 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  53. P. Kos, B. Bertini, and T. Prosen, Chaos and ergodicity in extended quantum systems with noisy driving, Phys. Rev. Lett. 126(19), 190601 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  54. K. Kawabata, A. Kulkarni, J. Li, T. Numasawa, and S. Ryu, Dynamical quantum phase transitions in SYK Lindbladians, Phys. Rev. B 108, 075110 (2023)

    Article  ADS  Google Scholar 

  55. Z. Xu, A. Chenu, T. Prosen, and A. del Campo, Thermofield dynamics: Quantum chaos versus decoherence, Phys. Rev. B 103(6), 064309 (2021)

    Article  ADS  Google Scholar 

  56. J. Cornelius, Z. Xu, A. Saxena, A. Chenu, and A. del Campo, Spectral filtering induced by non-Hermitian evolution with balanced gain and loss: Enhancing quantum chaos, Phys. Rev. Lett. 128(19), 190402 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  57. A. S. Matsoukas-Roubeas, F. Roccati, J. Cornelius, Z. Xu, A. Chenu, and A. del Campo, Non-Hermitian Hamiltonian deformations in quantum mechanics, J. High Energy Phys. 2023(1), 60 (2023)

    Article  MathSciNet  Google Scholar 

  58. F. Roccati, F. Balducci, R. Shir, and A. Chenu, Diagnosing non-Hermitian many-body localization and quantum chaos via singular value decomposition, arXiv: 2311.16229 (2023)

  59. If we simply generalize the definition of the SFF for non-Hermitian systems as follows: \(F_{\gamma}(t)={{1}\over{{[\cal{Z}(0)]^{2}}}}\sum\nolimits_{m,n}\rm{e}^{-\rm{i}(\epsilon_{m}-\epsilon_{n})t}\). where {ϵn} is the set of eigenvalues of the non-Hermitian system, and we denote the real and imaginary parts of the eigenvalues as αn and βn respectively. Since the energy eigenvalues of a general non-Hermitian system are complex, implying that the imaginary part βn is generally nonzero, from the definition we observe that \(F_{\gamma}(t)={{1}\over{{[\cal{Z}(0)]^{2}}}}\sum\nolimits_{m,n}\rm{e}^{-\rm{i}(\alpha_{m}-\alpha_{n})t}\rm{e}^{(\beta_{m}-\beta_{n})t}\). Hence, for the set of m, n that satisfies βm − βn > 0, there will be an exponential growth term \(\rm{e}^{(\beta_{m}-\beta_{n})t}\) in the above definition, resulting in the exponential growth of the SFF as time increases.

  60. P. Martinez-Azcona and A. Chenu, Analyticity constraints bound the decay of the spectral form factor, Quantum 6, 852 (2022)

    Article  Google Scholar 

  61. O. Agam, B. L. Altshuler, and A. V. Andreev, Spectral statistics: From disordered to chaotic systems, Phys. Rev. Lett. 75(24), 4389 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  62. E. B. Bogomolny and J. P. Keating, Gutzwiller’s trace formula and spectral statistics: Beyond the diagonal approximation, Phys. Rev. Lett. 77(8), 1472 (1996)

    Article  ADS  Google Scholar 

  63. J. E. Tyson, Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries, J. Phys. Math. Gen. 36(39), 10101 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  64. M. Zwolak and G. Vidal, Mixed-state dynamics in one-dimensional quantum lattice systems: A time-dependent superoperator renormalization algorithm, Phys. Rev. Lett. 93(20), 207205 (2004)

    Article  ADS  Google Scholar 

  65. In this supplementary, we show (A) alternative definitions of SFF; (B) the derivation of the pre-factor α in early decay region; (C, D, E) detailed calculation of SFF in three examples; (F) possible experimental realization of SFF.

  66. In general, we think the types of different Lindblad operators will not change the general properties of the normalized SFF regarding its short-time exponential decay and long-time plateau behavior. Since the argument we provide just below Eq. (9) does not resume some specific form of the Lindblad operators. Nevertheless, different Lindblad operators may lead to a different number of steady states, thereby altering the value of θ. For example, let us consider a Hamiltonian H with charge conservation, such as our Bose–Hubbard model. In the main text, we focus on Lindblad operators that preserve the particle number, ensuring that charge conservation is a strong U(1) symmetry of the open system. In this scenario, there is at least one steady state in each charge sector, resulting in at least N + 1 steady states in the full Fock space with arbitrary particle numbers. (Note that our discussions in the main text focus on a single charge sector.) In contrast, when some Lindblad operators couple different charge sectors, the system exhibits only a weak U(1) symmetry. Consequently, there may be only one steady state even in the full Fock space.

  67. P. Saad, S. H. Shenker, and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv: 1806.06840 (2018)

  68. L. Sá, P. Ribeiro, and T. Prosen, Lindbladian dissipation of strongly-correlated quantum matter, Phys. Rev. Res. 4(2), L022068 (2022)

    Article  Google Scholar 

  69. A. M. García-García, L. Sá, J. J. M. Verbaarschot, and J. P. Zheng, Keldysh wormholes and anomalous relaxation in the dissipative Sachdev–Ye–Kitaev model, Phys. Rev. D 107(10), 106006 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  70. K. Kawabata, A. Kulkarni, J. Li, T. Numasawa, and S. Ryu, Dynamical quantum phase transitions in Sachdev–Ye–Kitaev Lindbladians, Phys. Rev. B 108(7), 075110 (2023)

    Article  ADS  Google Scholar 

  71. H. Wang, C. Liu, P. Zhang, and A. M. García-García, Entanglement transition and replica wormholes in the dissipative Sachdev–Ye–Kitaev model, Phys. Rev. D 109(4), 046005 (2024)

    Article  ADS  MathSciNet  Google Scholar 

  72. Y. Z. You, A. W. W. Ludwig, and C. Xu, Sachdev–Ye–Kitaev model and thermalization on the boundary of many-body localized fermionic symmetry-protected topological states, Phys. Rev. B 95(11), 115150 (2017)

    Article  ADS  Google Scholar 

  73. I. Danshita and A. Polkovnikov, Superfluid-to-Mott-insulator transition in the one-dimensional Bose–Hubbard model for arbitrary integer filling factors, Phys. Rev. A 84(6), 063637 (2011)

    Article  ADS  Google Scholar 

  74. H. Shen, P. Zhang, R. Fan, and H. Zhai, Out-of-time-order correlation at a quantum phase transition, Phys. Rev. B 96(5), 054503 (2017)

    Article  ADS  Google Scholar 

  75. I. Boettcher, P. Bienias, R. Belyansky, A. J. Kollár, and A. V. Gorshkov, Quantum simulation of hyperbolic space with circuit quantum electrodynamics: From graphs to geometry, Phys. Rev. A 102(3), 032208 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  76. L. Pausch, A. Buchleitner, E. G. Carnio, and A. Rodríguez, Optimal route to quantum chaos in the Bose–Hubbard model, J. Phys. A Math. Theor. 55(32), 324002 (2022)

    Article  MathSciNet  Google Scholar 

  77. S. Denisov, T. Laptyeva, W. Tarnowski, D. Chruscinski, and K. Zyczkowski, Universal spectra of random Linblad operators, Phys. Rev. Lett. 123(14), 140403 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  78. D. Poulin, R. Laflamme, G. J. Milburn, and J. P. Paz, Testing integrability with a single bit of quantum information, Phys. Rev. A 68(2), 022302 (2003)

    Article  ADS  Google Scholar 

  79. D. V. Vasilyev, A. Grankin, M. A. Baranov, L. M. Sieberer, and P. Zoller, Monitoring quantum simulators via quantum nondemolition couplings to atomic clock qubits, PRX Quantum 1(2), 020302 (2020)

    Article  Google Scholar 

  80. L. K. Joshi, A. Elben, A. Vikram, B. Vermersch, V. Galitski, and P. Zoller, Probing many-body quantum chaos with quantum simulators, Phys. Rev. X 12(1), 011018 (2022)

    Google Scholar 

  81. L. Leviandier, M. Lombardi, R. Jost, and J. P. Pique, A tool to measure statistical level properties in very complex spectra, Phys. Rev. Lett. 56(23), 2449 (1986)

    Article  ADS  Google Scholar 

  82. J. P. Pique, Y. Chen, R. W. Field, and J. L. Kinsey, Chaos and dynamics on 0.5–300 ps time scales in vibrationally excited acetylene: Fourier transform of stimulated-emission pumping spectrum, Phys. Rev. Lett. 58(5), 475 (1987)

    Article  ADS  Google Scholar 

  83. T. Guhr and H. A. Weidenmuller, Correlations in anti-crossing spectra and scattering theory: Analytical aspects, Chem. Phys. 146(1–2), 21 (1990)

    Article  Google Scholar 

  84. M. Lombardi and T. H. Seligman, Universal and nonuniversal statistical properties of levels and intensities for chaotic Rydberg molecules, Phys. Rev. A 47(5), 3571 (1993)

    Article  ADS  Google Scholar 

  85. E. J. Torres-Herrera and L. F. Santos, Dynamical manifestations of quantum chaos: Correlation hole and bulge, Philos. Trans. Royal Soc. A 375(2108), 20160434 (2017)

    Article  ADS  Google Scholar 

  86. H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, Oxford, 2007

    Book  Google Scholar 

  87. Y. C. Cheng and R. J. Silbey, Markovian approximation in the relaxation of open quantum systems, J. Phys. Chem. B 109, 21399 (2005)

    Article  Google Scholar 

Download references

Acknowledgements

We thank Hui Zhai for the invaluable discussions and for carefully reading the manuscript. We thank Yingfei Gu, Haifeng Tang, Liang Mao, and Hanteng Wang for the helpful discussions. We especially thank Adolfo del Campo for assisting us in rectifying our estimation of the decay exponent in the early-time regime and for bringing to our attention several relevant papers that were overlooked in a previous version.

Author information

Authors and Affiliations

  1. Institute for Advanced Study, Tsinghua University, Beijing, 100084, China

    Yi-Neng Zhou & Tian-Gang Zhou

  2. Department of Physics, Fudan University, Shanghai, 200438, China

    Pengfei Zhang

Authors
  1. Yi-Neng Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tian-Gang Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pengfei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pengfei Zhang.

Ethics declarations

Declarations The authors declare that they have no competing interests and there are no conflicts.

Supplementary material

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, YN., Zhou, TG. & Zhang, P. General properties of the spectral form factor in open quantum systems. Front. Phys. 19, 31202 (2024). https://doi.org/10.1007/s11467-024-1406-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11467-024-1406-7

Keywords

Search

Navigation