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Black holes and random matrices

An Erratum to this article was published on 03 September 2018

This article has been updated

A preprint version of the article is available at arXiv.

Abstract

We argue that the late time behavior of horizon fluctuations in large anti-de Sitter (AdS) black holes is governed by the random matrix dynamics characteristic of quantum chaotic systems. Our main tool is the Sachdev-Ye-Kitaev (SYK) model, which we use as a simple model of a black hole. We use an analytically continued partition function |Z(β + it)|2 as well as correlation functions as diagnostics. Using numerical techniques we establish random matrix behavior at late times. We determine the early time behavior exactly in a double scaling limit, giving us a plausible estimate for the crossover time to random matrix behavior. We use these ideas to formulate a conjecture about general large AdS black holes, like those dual to 4D super-Yang-Mills theory, giving a provisional estimate of the crossover time. We make some preliminary comments about challenges to understanding the late time dynamics from a bulk point of view.

Change history

  • 03 September 2018

    We have found a minor normalization error in some of the plots in this paper. This error has no effect on the qualitative or quantitative conclusions of the paper.

References

  1. J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  2. L. Dyson, J. Lindesay and L. Susskind, Is there really a de Sitter/CFT duality?, JHEP 08 (2002) 045 [hep-th/0202163] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  3. G.T. Horowitz and V.E. Hubeny, Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D 62 (2000) 024027 [hep-th/9909056] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

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

  5. L. Dyson, M. Kleban and L. Susskind, Disturbing implications of a cosmological constant, JHEP 10 (2002) 011 [hep-th/0208013] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  6. 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].

  7. A. Kitaev, A simple model of quantum holography, talks at KITP, 7 April 2015 and 27 May 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.

  8. E. Dyer and G. Gur-Ari, 2D CFT Partition Functions at Late Times, arXiv:1611.04592 [INSPIRE].

  9. J.L.F. Barbon and E. Rabinovici, Very long time scales and black hole thermal equilibrium, JHEP 11 (2003) 047 [hep-th/0308063] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  10. A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, On information loss in AdS 3 /CF T 2, JHEP 05 (2016) 109 [arXiv:1603.08925] [INSPIRE].

    ADS  Article  Google Scholar 

  11. A.L. Fitzpatrick and J. Kaplan, On the Late-Time Behavior of Virasoro Blocks and a Classification of Semiclassical Saddles, JHEP 04 (2017) 072 [arXiv:1609.07153] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, KITP seminar, 12 February 2015, http://online.kitp.ucsb.edu/online/joint98/kitaev/.

  13. J. Polchinski and V. Rosenhaus, The Spectrum in the Sachdev-Ye-Kitaev Model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  14. 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 

  15. 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 

  16. 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 

  17. A. Kitaev, Hidden Correlations in the Hawking Radiation and Thermal Noise, talk given at the Fundamental Physics Prize Symposium, 10 November 2014, https://www.youtube.com/watch?v=OQ9qN8j7EZI.

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. A. Almheiri and J. Polchinski, Models of AdS 2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  20. J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].

  21. K. Jensen, Chaos in AdS 2 Holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].

    ADS  Article  Google Scholar 

  22. J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS 2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  23. Y. Gu, X.-L. Qi and D. Stanford, Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, arXiv:1609.07832 [INSPIRE].

  24. M. Berkooz, P. Narayan, M. Rozali and J. Simón, Higher Dimensional Generalizations of the SYK Model, JHEP 01 (2017) 138 [arXiv:1610.02422] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  25. W. Fu, D. Gaiotto, J. Maldacena and S. Sachdev, Supersymmetric Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 026009 [arXiv:1610.08917] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  26. D.J. Gross and V. Rosenhaus, A Generalization of Sachdev-Ye-Kitaev, JHEP 02 (2017) 093 [arXiv:1610.01569] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. A. Jevicki, K. Suzuki and J. Yoon, Bi-Local Holography in the SYK Model, JHEP 07 (2016) 007 [arXiv:1603.06246] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. A. Almheiri and B. Kang, Conformal Symmetry Breaking and Thermodynamics of Near-Extremal Black Holes, JHEP 10 (2016) 052 [arXiv:1606.04108] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  29. 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 

  30. M. Cvetič and I. Papadimitriou, AdS 2 holographic dictionary, JHEP 12 (2016) 008 [Erratum ibid. 01 (2017) 120] [arXiv:1608.07018] [INSPIRE].

  31. W. Fu and S. Sachdev, Numerical study of fermion and boson models with infinite-range random interactions, Phys. Rev. B 94 (2016) 035135 [arXiv:1603.05246] [INSPIRE].

    ADS  Article  Google Scholar 

  32. 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 

  33. 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].

    ADS  Google Scholar 

  34. V. Balasubramanian, M. Berkooz, S.F. Ross and J. Simon, Black Holes, Entanglement and Random Matrices, Class. Quant. Grav. 31 (2014) 185009 [arXiv:1404.6198] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  35. 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 

  36. M.L. Mehta, Random matrices, volume 142, Academic Press (2004).

  37. C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  38. R. Jackiw, Lower Dimensional Gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].

    ADS  Article  Google Scholar 

  39. L. Fidkowski and A. Kitaev, Topological phases of fermions in one dimension, Phys. Rev. B 83 (2011) 075103.

    ADS  Article  Google Scholar 

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

  41. 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].

  42. E.P. Wigner, Results and theory of resonance absorption, in Proceedings of the conference on neutron physics by time-of-flight, Gatlinburg, Tennessee (1956).

  43. O. Bohigas and M.-J. Giannoni, Chaotic motion and random matrix theories, in Mathematical and computational methods in nuclear physics, Springer (1984), pg. 1-99.

  44. M.L. Mehta, On the statistical properties of the level-spacings in nuclear spectra, Nucl. Phys. 18 (1960) 395.

    MathSciNet  Article  MATH  Google Scholar 

  45. M. Gaudin, Sur la loi limite de l’espacement des valeurs propres d’une matrice aléatoire, Nucl. Phys. 25 (1961) 447.

    Article  MATH  Google Scholar 

  46. S. Sachdev, Bekenstein-Hawking Entropy and Strange Metals, Phys. Rev. X 5 (2015) 041025 [arXiv:1506.05111] [INSPIRE].

    Article  Google Scholar 

  47. O. Parcollet and A. Georges, Non-fermi-liquid regime of a doped mott insulator, Phys. Rev. B 59 (1999) 5341 [cond-mat/9806119].

  48. A. Georges, O. Parcollet and S. Sachdev, Quantum fluctuations of a nearly critical heisenberg spin glass, Phys. Rev. B 63 (2001) 134406.

    ADS  Article  Google Scholar 

  49. J. Polchinski and A. Streicher, unpublished.

  50. F.J. Dyson, Statistical theory of the energy levels of complex systems. III, J. Math. Phys. 3 (1962) 166.

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

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

    ADS  Google Scholar 

  53. B. Altshuler and B. Shklovskii, Repulsion of energy levels and conductivity of small metal samples, Sov. Phys. JETP 64 (1986) 127.

    Google Scholar 

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  55. A. Andreev and B. Altshuler, Spectral statistics beyond random matrix theory, Phys. Rev. Lett. 75 (1995) 902 [cond-mat/9503141].

  56. A. Kamenev and M. Mézard, Wigner-Dyson statistics from the replica method, J. Phys. A 32 (1999) 4373 [cond-mat/9901110].

  57. D. Stanford and E. Witten, Fermionic Localization of the Schwarzian Theory, arXiv:1703.04612 [INSPIRE].

  58. L. Erdős and D. Schröder, Phase transition in the density of states of quantum spin glasses, Math. Phys. Anal. Geom. 17 (2014) 9164 [arXiv:1407.1552].

    MathSciNet  MATH  Google Scholar 

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

    ADS  Article  Google Scholar 

  60. 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 

  61. 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 

  62. A. Kitaev, private communication.

  63. J. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.

    ADS  Article  Google Scholar 

  64. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888 [cond-mat/9403051].

  65. R.E. Prange, The spectral form factor is not self-averaging, Phys. Rev. Lett. 78 (1997) 2280 [chao-dyn/9606010].

  66. E. Witten, An SYK-Like Model Without Disorder, arXiv:1610.09758 [INSPIRE].

  67. R. Emparan, C.V. Johnson and R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  68. Ó.J. Dias, J.E. Santos and B. Way, Localised AdS5 × S5 Black Holes, Phys. Rev. Lett. 117 (2016) 151101 [arXiv:1605.04911] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  69. L. Susskind, Some speculations about black hole entropy in string theory, hep-th/9309145 [INSPIRE].

  70. G.T. Horowitz, Comments on black holes in string theory, Class. Quant. Grav. 17 (2000) 1107 [hep-th/9910082] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  71. G.T. Horowitz and J. Polchinski, A correspondence principle for black holes and strings, Phys. Rev. D 55 (1997) 6189 [hep-th/9612146] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  72. G. Rodgers and A. Bray, Density of states of a sparse random matrix, Phys. Rev. B 37 (1988) 3557.

    ADS  MathSciNet  Article  Google Scholar 

  73. J.P. Keating, N. Linden and H.J. Wells, Random matrices and quantum spin chains, Markov Process. Relat. 21 (2014) 537 [arXiv:1403.1114].

    MathSciNet  Google Scholar 

  74. L. Erdős and H.-T. Yau, Universality of local spectral statistics of random matrices, Bull. Am. Math. Soc. 49 (2012) 377 [arXiv:1106.4986].

    MathSciNet  Article  MATH  Google Scholar 

  75. F.D.M. Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnetic identification with the O(3) nonlinear σ-model, Phys. Lett. A 93 (1983) 464 [INSPIRE].

    ADS  Article  Google Scholar 

  76. M.E. Ismail, D. Stanton and G. Viennot, The combinatorics of q-hermite polynomials and the askey wilson integral, Eur. J. Combinator. 8 (1987) 379.

    MathSciNet  Article  MATH  Google Scholar 

  77. J.M. Magan, Random free fermions: An analytical example of eigenstate thermalization, Phys. Rev. Lett. 116 (2016) 030401 [arXiv:1508.05339] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  78. D. Anninos, T. Anous and F. Denef, Disordered Quivers and Cold Horizons, JHEP 12 (2016) 071 [arXiv:1603.00453] [INSPIRE].

    ADS  Article  Google Scholar 

  79. C. Itzykson and J.B. Zuber, The Planar Approximation. 2, J. Math. Phys. 21 (1980) 411 [INSPIRE].

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Correspondence to Guy Gur-Ari.

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Cotler, J.S., Gur-Ari, G., Hanada, M. et al. Black holes and random matrices. J. High Energ. Phys. 2017, 118 (2017). https://doi.org/10.1007/JHEP05(2017)118

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Keywords

  • 1/N Expansion
  • AdS-CFT Correspondence
  • Field Theories in Lower Dimensions
  • Random Systems