Skip to main content
Download PDF

Onset of random matrix behavior in scrambling systems

Journal of High Energy Physics volume 2018, Article number: 124 (2018) Cite this article

An Erratum to this article was published on 28 February 2019

This article has been updated

A preprint version of the article is available at arXiv.

Abstract

The fine grained energy spectrum of quantum chaotic systems is widely believed to be described by random matrix statistics. A basic scale in such a system is the energy range over which this behavior persists. We define the corresponding time scale by the time at which the linearly growing ramp region in the spectral form factor begins. We call this time tramp. The purpose of this paper is to study this scale in many-body quantum systems that display strong chaos, sometimes called scrambling systems. We focus on randomly coupled qubit systems, both local and k-local (all-to-all interactions) and the Sachdev-Ye-Kitaev (SYK) model. Using numerical results, analytic estimates for random quantum circuits, and a heuristic analysis of Hamiltonian systems we find the following results. For geometrically local systems with a conservation law we find tramp is determined by the diffusion time across the system, order N2 for a 1D chain of N qubits. This is analogous to the behavior found for local one-body chaotic systems. For a k-local system like SYK the time is order log N but with a different prefactor and a different mechanism than the scrambling time. In the absence of any conservation laws, as in a generic random quantum circuit, we find tramp ∼ log N, independent of connectivity.

Change history

  • 28 February 2019

    We have found an error in section 6 of this paper. In that section we gave a heuristic argument estimating the ramp time of Hamiltonian systems by assuming that the slowest decay in eq. (105) was that of simple operators.

  • 28 February 2019

    We have found an error in section 6 of this paper. In that section we gave a heuristic argument estimating the ramp time of Hamiltonian systems by assuming that the slowest decay in eq. (105) was that of simple operators.

  • 28 February 2019

    We have found an error in section 6 of this paper. In that section we gave a heuristic argument estimating the ramp time of Hamiltonian systems by assuming that the slowest decay in eq. (105) was that of simple operators.

  • 28 February 2019

    We have found an error in section 6 of this paper. In that section we gave a heuristic argument estimating the ramp time of Hamiltonian systems by assuming that the slowest decay in eq. (105) was that of simple operators.

References

  1. F. Haake, Quantum Signatures of Chaos, Springer-Verlag New York, Inc. (2006).

  2. M. Serbyn and J.E. Moore, Spectral statistics across the many-body localization transition, Phys. Rev. B 93 (2016) 041424 [arXiv:1508.07293].

    ADS  Article  Google Scholar 

  3. A. Altland and D. Bagrets, Quantum ergodicity in the SYK model, Nucl. Phys. B 930 (2018) 45 [arXiv:1712.05073] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  4. A. Chan, A. De Luca and J.T. Chalker, Spectral statistics in spatially extended chaotic quantum many-body systems, arXiv:1803.03841 [INSPIRE].

  5. P. Kos, M. Ljubotina and T. Prosen, Many-body quantum chaos: Analytic connection to random matrix theory, Phys. Rev. X 8 (2018) 021062 [arXiv:1712.02665] [INSPIRE].

    Article  Google Scholar 

  6. 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 (2017) 115150 [arXiv:1602.06964] [INSPIRE].

    ADS  Article  Google Scholar 

  7. A.M. García-García and J.J.M. Verbaarschot, Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 126010 [arXiv:1610.03816] [INSPIRE].

  8. J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [arXiv:1611.04650] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].

  10. A. Kitaev, A Simple Model of Quantum Holography, talks at KITP (2015) [http://online.kitp.ucsb.edu/online/entangled15/kitaev/] [http://online.kitp.ucsb.edu/online/entangled15/kitaev2/].

  11. A. Kitaev and S.J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05 (2018) 183 [arXiv:1711.08467] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  12. J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  13. A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. Phys. JETP 28 (1969) 1200 [http://www.jetp.ac.ru/cgi-bin/dn/e_028_06_1200.pdf].

  14. P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  15. Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].

    ADS  Article  Google Scholar 

  16. A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An Apologia for Firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].

    ADS  Article  Google Scholar 

  17. A. Kitaev, Hidden Correlations in the Hawking Radiation and Thermal Noise, talk given at Fundamental Physics Prize Symposium, Nov. 10, 2014, and Stanford SITP seminars Nov. 11, 2014 and Dec. 18, 2014 [https://www.youtube.com/watch?v=OQ9qN8j7EZI].

  18. N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the Fast Scrambling Conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  21. D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  22. S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  23. J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  24. B. Altshuler and B. Shklovskii, Repulsion of energy levels and conductivity of small metal samples, Sov. Phys. JETP 64 (1986) 127 [http://www-thphys.physics.ox.ac.uk/talks/CMTjournalclub/sources/AltshulerShklovskii.pdf].

  25. L. Erdős and A. Knowles, The Altshuler-Shklovskii Formulas for Random Band Matrices I: the Unimodular Case, Commun. Math. Phys. 333 (2015) 1365.

    ADS  MathSciNet  Article  MATH  Google Scholar 

  26. A.M. García-García, Y. Jia and J.J.M. Verbaarschot, Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev Model, Phys. Rev. D 97 (2018) 106003 [arXiv:1801.01071] [INSPIRE].

  27. T. Banks, L. Susskind and M.E. Peskin, Difficulties for the Evolution of Pure States Into Mixed States, Nucl. Phys. B 244 (1984) 125.

    ADS  MathSciNet  Article  Google Scholar 

  28. J. Emerson, E. Livine and S. Lloyd, Convergence conditions for random quantum circuits, Phys. Rev. A 72 (2005) 060302 [quant-ph/0503210].

  29. R. Oliveira, O.C.O. Dahlsten and M.B. Plenio, Efficient Generation of Generic Entanglement, Phys. Rev. Lett. 98 (2007) 130502 [quant-ph/0605126].

  30. J. Emerson, Y.S. Weinstein, M. Saraceno, S. Lloyd and D.G. Cory, Pseudo-random unitary operators for quantum information processing, Science 302 (2003) 2098.

    ADS  MathSciNet  Article  MATH  Google Scholar 

  31. D. Gross, K. Audenaert and J. Eisert, Evenly distributed unitaries: On the structure of unitary designs, J. Math. Phys. 48 (2007) 052104 [quant-ph/0611002].

  32. L. Arnaud and D. Braun, Efficiency of producing random unitary matrices with quantum circuits, Phys. Rev. A 78 (2008) 062329 [arXiv:0807.0775].

    ADS  Article  Google Scholar 

  33. A. Harrow and R. Low. Efficient quantum tensor product expanders and k-designs, Lect. Notes Comput. Sci. 5687 (2009) 548 [arXiv:0811.2597].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. A.W. Harrow and R.A. Low, Random quantum circuits are approximate 2-designs, Commun. Math. Phys. 291 (2009) 257 [arXiv:0802.1919].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  35. W.G. Brown and L. Viola, Convergence rates for arbitrary statistical moments of random quantum circuits, Phys. Rev. Lett. 104 (2010) 250501 [arXiv:0910.0913].

    ADS  Article  Google Scholar 

  36. W. Brown and O. Fawzi, Scrambling speed of random quantum circuits, arXiv:1210.6644 [INSPIRE].

  37. F. Brandao, A. Harrow and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Commun. Math. Phys. 346 (2016) 397 [arXiv:1208.0692].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  38. E. Onorati, O. Buerschaper, M. Kliesch, W. Brown, A.H. Werner and J. Eisert, Mixing properties of stochastic quantum Hamiltonians, Commun. Math. Phys. 355 (2017) 905 [arXiv:1606.01914] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  39. Y. Nakata, C. Hirche, M. Koashi and A. Winter, Efficient Quantum Pseudorandomness with Nearly Time-Independent Hamiltonian Dynamics, Phys. Rev. X 7 (2017) 021006 [arXiv:1609.07021] [INSPIRE].

    Article  Google Scholar 

  40. L. Banchi, D. Burgarth and M.J. Kastoryano, Driven Quantum Dynamics: Will It Blend?, Phys. Rev. X 7 (2017) 041015 [arXiv:1704.03041] [INSPIRE].

    Article  Google Scholar 

  41. A. Nahum, S. Vijay and J. Haah, Operator Spreading in Random Unitary Circuits, Phys. Rev. X 8 (2018) 021014 [arXiv:1705.08975] [INSPIRE].

    Article  Google Scholar 

  42. C. von Keyserlingk, T. Rakovszky, F. Pollmann and S. Sondhi, Operator hydrodynamics, OTOCs and entanglement growth in systems without conservation laws, Phys. Rev. X 8 (2018) 021013 [arXiv:1705.08910] [INSPIRE].

    Article  Google Scholar 

  43. V. Khemani, A. Vishwanath and D.A. Huse, Operator spreading and the emergence of dissipation in unitary dynamics with conservation laws, arXiv:1710.09835 [INSPIRE].

  44. J. Preskill, Quantum computing and the entanglement frontier, arXiv:1203.5813 [INSPIRE].

  45. C. Neill et al., A blueprint for demonstrating quantum supremacy with superconducting qubits, Science 360 (2018) 195 [arXiv:1709.06678].

    ADS  MathSciNet  Article  Google Scholar 

  46. A.W. Harrow and A. Montanaro, Quantum computational supremacy, Nature 549 (2017) 203.

    ADS  Article  Google Scholar 

  47. A. Bouland, B. Fefferman, C. Nirkhe, U. Vazirani Quantum Supremacy and the Complexity of Random Circuit Sampling, arXiv:1803.04402.

  48. T. Rakovszky, F. Pollmann and C.W. von Keyserlingk, Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation, arXiv:1710.09827 [INSPIRE].

  49. D.A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP 04 (2017) 121 [arXiv:1610.04903] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  50. J. Cotler, N. Hunter-Jones, J. Liu and B. Yoshida, Chaos, Complexity and Random Matrices, JHEP 11 (2017) 048 [arXiv:1706.05400] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  51. F.J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3 (1962) 140 [INSPIRE].

  52. M.L. Mehta, Random matrices, vol. 142, Academic Press (2004).

  53. T. Guhr, A. Müller-Groeling and H.A. Weidenmuller, Random matrix theories in quantum physics: Common concepts, Phys. Rept. 299 (1998) 189 [cond-mat/9707301] [INSPIRE].

  54. J. Flores, M. Horoi, M. Müller and T.H. Seligman, Spectral statistics of the two-body random ensemble revisited, Phys. Rev. E 63 (2001) 026204 [cond-mat/0006144] [INSPIRE].

  55. A. Altland and D. Bagrets, Quantum ergodicity in the SYK model, Nucl. Phys. B 930 (2018) 45 [arXiv:1712.05073] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  56. L. Erdős and D. Schröder, Phase transition in the density of states of quantum spin glasses, Math. Phys. Anal. Geom. 17 (2014) 441 [arXiv:1407.1552].

    MathSciNet  Article  MATH  Google Scholar 

  57. E. Brézin and S. Hikami, Spectral form factor in a random matrix theory, articlePhys. Rev. E 55 (1997) 4067 [cond-mat/9608116].

  58. E. Brézin and S. Hikami, Extension of level-spacing universality, Phys. Rev. E 56 (1997) 264 [INSPIRE].

    ADS  Google Scholar 

  59. K. Papadodimas and S. Raju, Local Operators in the Eternal Black Hole, Phys. Rev. Lett. 115 (2015) 211601 [arXiv:1502.06692] [INSPIRE].

    ADS  Article  Google Scholar 

  60. A. Gaikwad and R. Sinha, Spectral Form Factor in Non-Gaussian Random Matrix Theories, arXiv:1706.07439 [INSPIRE].

  61. D. Bagrets, A. Altland and A. Kamenev, Sachdev-Ye-Kitaev model as Liouville quantum mechanics, Nucl. Phys. B 911 (2016) 191 [arXiv:1607.00694] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  62. D. Stanford and E. Witten, Fermionic Localization of the Schwarzian Theory, JHEP 10 (2017) 008 [arXiv:1703.04612] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  63. F. Haake, H.-J. Sommers and J. Weber, Fluctuations and ergodicity of the form factor of quantum propagators and random unitary matrices, J. Phys. A 32 (1999) 6903.

    ADS  MathSciNet  MATH  Google Scholar 

  64. P. Diaconis and S.N. Evans, Linear Functionals of Eigenvalues of Random Matrices, Trans. Am. Math. Soc. 353 (2001) 2615.

    MathSciNet  Article  MATH  Google Scholar 

  65. J.P. Keating, N. Linden and H.J. Wells, Random matrices and quantum spin chains, Markov Process. Related Fields 21 (2015) 537 [arXiv:1403.1114].

    MathSciNet  Google Scholar 

  66. J.P. Keating, N. Linden and H.J. Wells, Spectra and Eigenstates of Spin Chain Hamiltonians, Commun. Math. Phys. 338 (2015) 81 [arXiv:1403.1121].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  67. A. Pal and D.A. Huse, The many-body localization transition, arXiv:1003.2613.

  68. D.J. Luitz, N. Laflorencie and F. Alet, Many-body localization edge in the random-field Heisenberg chain, Phys. Rev. B 91 (2015) 081103 [arXiv:1411.0660].

    ADS  Article  Google Scholar 

  69. K. Agarwal, E. Altman, E. Demler, S. Gopalakrishnan, D.A. Huse and M. Knap, Rare-region effects and dynamics near the many-body localization transition, Annalen Phys. 529 (2017) 1600326 [arXiv:1611.00770].

    ADS  MathSciNet  Article  Google Scholar 

  70. P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity, in progress.

  71. L.L. Ng, Heisenberg Model, Bethe Ansatz and Random Walks, Senior Honors Thesis, Harvard University (1996) [https://services.math.duke.edu/ng/math/papers/senior-thesis.pdf].

  72. M. Karabach, G. Müller, H. Gould and J. Tobochnik, Introduction to the Bethe Ansatz I, Comput. Phys. 11 (1997) 36 [cond-mat/9809162].

  73. J.R.G. Mendonca, Exact eigenspectrum of the symmetric simple exclusion process on the complete, complete bipartite and related graphs, J. Phys. A 46 (2013) 295001 [arXiv:1207.4106].

    MathSciNet  MATH  Google Scholar 

  74. E.H. Lieb and D.W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28 (1972) 251 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  75. D.A. Roberts, D. Stanford and A. Streicher, Operator growth in the SYK model, JHEP 06 (2018) 122 [arXiv:1802.02633] [INSPIRE].

    ADS  Article  Google Scholar 

  76. J.T. Chalker, I.V. Lerner and R.A. Smith, Random Walks through the Ensemble: Linking Spectral Statistics with Wave-Function Correlations in Disordered Metals, Phys. Rev. Lett. 77 (1996) 554 [INSPIRE].

    ADS  Article  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Author information

Authors and Affiliations

  1. Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA, 94305, U.S.A.

    Hrant Gharibyan, Masanori Hanada & Stephen H. Shenker

  2. Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan

    Masanori Hanada

  3. The Hakubi Center for Advanced Research, Kyoto University, Kyoto, 606-8501, Japan

    Masanori Hanada

  4. Department of Physics, University of Colorado, Boulder, Colorado, 80309, U.S.A.

    Masanori Hanada

  5. Department of Physics, Kyoto University, Kyoto, 606-8502, Japan

    Masaki Tezuka

Authors
  1. Hrant Gharibyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masanori Hanada
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stephen H. Shenker
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Masaki Tezuka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hrant Gharibyan.

Additional information

ArXiv ePrint: 1803.08050

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gharibyan, H., Hanada, M., Shenker, S.H. et al. Onset of random matrix behavior in scrambling systems. J. High Energ. Phys. 2018, 124 (2018). https://doi.org/10.1007/JHEP07(2018)124

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP07(2018)124

Keywords

  • AdS-CFT Correspondence
  • Field Theories in Lower Dimensions
  • Quantum Dissipative Systems
  • Random Systems