Skip to main content

Advertisement

SpringerLink
  1. Home
  2. Journal of High Energy Physics
  3. Article
A bound on chaos
Download PDF
Your article has downloaded

Similar articles being viewed by others

Slider with three articles shown per slide. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide.

Exponential growth of out-of-time-order correlator without chaos: inverted harmonic oscillator

13 November 2020

Koji Hashimoto, Kyoung-Bum Huh, … Ryota Watanabe

Extremal chaos

27 January 2022

Sandipan Kundu

On the Size of Chaos via Glauber Calculus in the Classical Mean-Field Dynamics

13 February 2021

Mitia Duerinckx

Subleading bounds on chaos

04 April 2022

Sandipan Kundu

Expansions in the Local and the Central Limit Theorems for Dynamical Systems

11 November 2021

Kasun Fernando & Françoise Pène

Quantum Bound to Chaos and the Semiclassical Limit

10 May 2018

Jorge Kurchan

Quantum complexity of time evolution with chaotic Hamiltonians

21 January 2020

Vijay Balasubramanian, Matthew DeCross, … Onkar Parrikar

An inelastic bound on chaos

17 July 2019

Gustavo J. Turiaci

Quantitative estimates of propagation of chaos for stochastic systems with $$W^{-1,\infty }$$ W - 1 , ∞ kernels

06 July 2018

Pierre-Emmanuel Jabin & Zhenfu Wang

Download PDF
  • Regular Article - Theoretical Physics
  • Open Access
  • Published: 17 August 2016

A bound on chaos

  • Juan Maldacena1,
  • Stephen H. Shenker2 &
  • Douglas Stanford1 

Journal of High Energy Physics volume 2016, Article number: 106 (2016) Cite this article

  • 7953 Accesses

  • 1188 Citations

  • 87 Altmetric

  • Metrics details

A preprint version of the article is available at arXiv.

Abstract

We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λ L ≤ 2πk B T/ℏ. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.

Download to read the full article text

Working on a manuscript?

Avoid the most common mistakes and prepare your manuscript for journal editors.

Learn more

References

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  4. A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, JETP 28 (1969) 1200.

    ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  7. M.B. Hastings and T. Koma, Spectral gap and exponential decay of correlations, Commun. Math. Phys. 265 (2006) 781 [math-ph/0507008] [INSPIRE].

  8. M.B. Hastings, Locality in quantum systems, arXiv:1008.5137.

  9. C. Dankert, R. Cleve, J. Emerson and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Phys. Rev. A 80 (2009) 012304 [quant-ph/0606161].

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  12. 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].

    Article  ADS  Google Scholar 

  13. I.T. Diniz and D. Jonathan, Comment on the paper ‘Random quantum circuits are approximate 2-designs’, Comm. Math. Phys. 304 (2011) 281 [arXiv:1006.4202].

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  18. A. Kitaev, Hidden Correlations in the Hawking Radiation and Thermal Noise, talk given at Fundamental Physics Prize Symposium, November 10, 2014.

  19. A. Kitaev, Stanford SITP seminars, November 11 and December 18, 2014.

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

    Article  ADS  MathSciNet  Google Scholar 

  21. D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, Phys. Rev. Lett. 115 (2015) 131603 [arXiv:1412.5123] [INSPIRE].

    Article  ADS  Google Scholar 

  22. S. Jackson, L. McGough and H. Verlinde, Conformal Bootstrap, Universality and Gravitational Scattering, Nucl. Phys. B 901 (2015) 382 [arXiv:1412.5205] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. G. ’t Hooft, The black hole interpretation of string theory, Nucl. Phys. B 335 (1990) 138 [INSPIRE].

  24. Y. Kiem, H.L. Verlinde and E.P. Verlinde, Black hole horizons and complementarity, Phys. Rev. D 52 (1995) 7053 [hep-th/9502074] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  25. P. Kovtun, D.T. Son and A.O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [INSPIRE].

    Article  ADS  Google Scholar 

  26. Y. Kats and P. Petrov, Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory, JHEP 01 (2009) 044 [arXiv:0712.0743] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, Viscosity Bound Violation in Higher Derivative Gravity, Phys. Rev. D 77 (2008) 126006 [arXiv:0712.0805] [INSPIRE].

    ADS  Google Scholar 

  28. M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, The Viscosity Bound and Causality Violation, Phys. Rev. Lett. 100 (2008) 191601 [arXiv:0802.3318] [INSPIRE].

    Article  ADS  Google Scholar 

  29. T. Banks and G. Festuccia, The Regge Limit for Green Functions in Conformal Field Theory, JHEP 06 (2010) 105 [arXiv:0910.2746] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. R.C. Brower, J. Polchinski, M.J. Strassler and C.-I. Tan, The pomeron and gauge/string duality, JHEP 12 (2007) 005 [hep-th/0603115] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. T. Dray and G. ’t Hooft, The Gravitational Shock Wave of a Massless Particle, Nucl. Phys. B 253 (1985) 173 [INSPIRE].

  32. K. Sfetsos, On gravitational shock waves in curved space-times, Nucl. Phys. B 436 (1995) 721 [hep-th/9408169] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  33. X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality Constraints on Corrections to the Graviton Three-Point Coupling, JHEP 02 (2016) 020 [arXiv:1407.5597] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  34. I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  35. L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions, JHEP 08 (2007) 019 [hep-th/0611122] [INSPIRE].

    Article  ADS  MathSciNet  MATH  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. School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ, USA

    Juan Maldacena & Douglas Stanford

  2. Stanford Institute for Theoretical Physics and Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA, USA

    Stephen H. Shenker

Authors
  1. Juan Maldacena
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephen H. Shenker
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Douglas Stanford
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Douglas Stanford.

Additional information

ArXiv ePrint: 1503.01409

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Maldacena, J., Shenker, S.H. & Stanford, D. A bound on chaos. J. High Energ. Phys. 2016, 106 (2016). https://doi.org/10.1007/JHEP08(2016)106

Download citation

  • Received: 11 August 2016

  • Accepted: 11 August 2016

  • Published: 17 August 2016

  • DOI: https://doi.org/10.1007/JHEP08(2016)106

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • 1/N Expansion
  • Black Holes
  • AdS-CFT Correspondence
Download PDF

Working on a manuscript?

Avoid the most common mistakes and prepare your manuscript for journal editors.

Learn more

Advertisement

Over 10 million scientific documents at your fingertips

Switch Edition
  • Academic Edition
  • Corporate Edition
  • Home
  • Impressum
  • Legal information
  • Privacy statement
  • California Privacy Statement
  • How we use cookies
  • Manage cookies/Do not sell my data
  • Accessibility
  • FAQ
  • Contact us
  • Affiliate program

Not affiliated

Springer Nature

© 2023 Springer Nature Switzerland AG. Part of Springer Nature.