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Journal of High Energy Physics

, 2019:257 | Cite as

Scrambling in hyperbolic black holes: shock waves and pole-skipping

  • Yongjun Ahn
  • Viktor Jahnke
  • Hyun-Sik Jeong
  • Keun-Young KimEmail author
Open Access
Regular Article - Theoretical Physics
  • 4 Downloads

Abstract

We study the scrambling properties of (d + 1)-dimensional hyperbolic black holes. Using the eikonal approximation, we calculate out-of-time-order correlators (OTOCs) for a Rindler-AdS geometry with AdS radius , which is dual to a d-dimensional conformal field theory (CFT) in hyperbolic space with temperature T = 1/(2π ℓ). We find agreement between our results for OTOCs and previously reported CFT calculations. For more generic hyperbolic black holes, we compute the butterfly velocity in two different ways, namely: from shock waves and from a pole-skipping analysis, finding perfect agreement between the two methods. The butterfly velocity vB (T) nicely interpolates between the Rindler-AdS result \( {v}_B\left(T=\frac{1}{2\pi \ell}\right)=\frac{1}{d-1} \) and the planar result \( {v}_B\left(T\gg \frac{1}{\ell}\right)=\sqrt{\frac{d}{2\left(d-1\right)}} \).

Keywords

Gauge-gravity correspondence Holography and condensed matter physics (AdS/CMT) 

Notes

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

References

  1. [1]
    J. Cotler, N. Hunter-Jones, J. Liu and B. Yoshida, Chaos, Complexity, and Random Matrices, JHEP 11 (2017) 048 [arXiv:1706.05400] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    R. de Mello Koch, J.-H. Huang, C.-T. Ma and H.J.R. Van Zyl, Spectral Form Factor as an OTOC Averaged over the Heisenberg Group, Phys. Lett. B 795 (2019) 183 [arXiv:1905.10981] [INSPIRE].
  3. [3]
    C. Murthy and M. Srednicki, Bounds on chaos from the eigenstate thermalization hypothesis, arXiv:1906.10808 [INSPIRE].
  4. [4]
    C.-T. Ma, Early-Time and Late-Time Quantum Chaos, arXiv:1907.04289 [INSPIRE].
  5. [5]
    T. Nosaka, D. Rosa and J. Yoon, The Thouless time for mass-deformed SYK, JHEP 09 (2018) 041 [arXiv:1804.09934] [INSPIRE].
  6. [6]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
  7. [7]
    S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    G. Sárosi, AdS 2 holography and the SYK model, PoS(Modave2017)001 [arXiv:1711.08482] [INSPIRE].
  11. [11]
    V. Jahnke, Recent developments in the holographic description of quantum chaos, Adv. High Energy Phys. 2019 (2019) 9632708 [arXiv:1811.06949] [INSPIRE].CrossRefGoogle Scholar
  12. [12]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  13. [13]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
  14. [14]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    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].ADSCrossRefGoogle Scholar
  16. [16]
    R.R. Poojary, BTZ dynamics and chaos, arXiv:1812.10073 [INSPIRE].
  17. [17]
    V. Jahnke, K.-Y. Kim and J. Yoon, On the Chaos Bound in Rotating Black Holes, JHEP 05 (2019) 037 [arXiv:1903.09086] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    J. Cotler and K. Jensen, A theory of reparameterizations for AdS3 gravity, JHEP 02 (2019) 079 [arXiv:1808.03263] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    F.M. Haehl and M. Rozali, Effective Field Theory for Chaotic CFTs, JHEP 10 (2018) 118 [arXiv:1808.02898] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
  21. [21]
    A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
  22. [22]
    J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
  23. [23]
    K. Jensen, Chaos in AdS 2 Holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    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].
  25. [25]
    J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS 2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    E. Perlmutter, Bounding the Space of Holographic CFTs with Chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, Rindler Quantum Gravity, Class. Quant. Grav. 29 (2012) 235025 [arXiv:1206.1323] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    D.N. Kabat and M. Ortiz, Eikonal quantum gravity and Planckian scattering, Nucl. Phys. B 388 (1992) 570 [hep-th/9203082] [INSPIRE].
  29. [29]
    H.S. Cohl and E.G. Kalnins, Fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry, arXiv:1201.4406.
  30. [30]
    M. Ammon and J. Erdmenger, Gauge/gravity duality, Cambridge University Press, (2015).Google Scholar
  31. [31]
    L. Susskind, Why do Things Fall?, arXiv:1802.01198 [INSPIRE].
  32. [32]
    R.B. Mann, Topological black holes: Outside looking in, Annals Israel Phys. Soc. 13 (1997) 311 [gr-qc/9709039] [INSPIRE].
  33. [33]
    D.R. Brill, J. Louko and P. Peldan, Thermodynamics of (3+1)-dimensional black holes with toroidal or higher genus horizons, Phys. Rev. D 56 (1997) 3600 [gr-qc/9705012] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2+1) black hole, Phys. Rev. D 48 (1993) 1506 [Erratum ibid. D 88 (2013) 069902] [gr-qc/9302012] [INSPIRE].
  35. [35]
    S. Grozdanov, K. Schalm and V. Scopelliti, Black hole scrambling from hydrodynamics, Phys. Rev. Lett. 120 (2018) 231601 [arXiv:1710.00921] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    M. Blake, H. Lee and H. Liu, A quantum hydrodynamical description for scrambling and many-body chaos, JHEP 10 (2018) 127 [arXiv:1801.00010] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    M. Blake, R.A. Davison, S. Grozdanov and H. Liu, Many-body chaos and energy dynamics in holography, JHEP 10 (2018) 035 [arXiv:1809.01169] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    S. Grozdanov, P.K. Kovtun, A.O. Starinets and P. Tadíc, The complex life of hydrodynamic modes, arXiv:1904.12862 [INSPIRE].
  39. [39]
    M. Blake, R.A. Davison and D. Vegh, Horizon constraints on holographic Green’s functions, arXiv:1904.12883 [INSPIRE].
  40. [40]
    M. Natsuume and T. Okamura, Nonuniqueness of Green’s functions at special points, arXiv:1905.12015 [INSPIRE].
  41. [41]
    S. Grozdanov, On the connection between hydrodynamics and quantum chaos in holographic theories with stringy corrections, JHEP 01 (2019) 048 [arXiv:1811.09641] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    P.K. Kovtun and A.O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72 (2005) 086009 [hep-th/0506184] [INSPIRE].
  43. [43]
    M. Mezei, On entanglement spreading from holography, JHEP 05 (2017) 064 [arXiv:1612.00082] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    V. Jahnke, Delocalizing entanglement of anisotropic black branes, JHEP 01 (2018) 102 [arXiv:1708.07243] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    D. Avila, V. Jahnke and L. Patiño, Chaos, Diffusivity and Spreading of Entanglement in Magnetic Branes and the Strengthening of the Internal Interaction, JHEP 09 (2018) 131 [arXiv:1805.05351] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    W. Fischler, V. Jahnke and J.F. Pedraza, Chaos and entanglement spreading in a non-commutative gauge theory, JHEP 11 (2018) 072 [arXiv:1808.10050] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    M. Baggioli, B. Padhi, P.W. Phillips and C. Setty, Conjecture on the Butterfly Velocity across a Quantum Phase Transition, JHEP 07 (2018) 049 [arXiv:1805.01470] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    M. Alishahiha, A. Davody, A. Naseh and S.F. Taghavi, On butterfly effect in Higher Derivative Gravities, JHEP 11 (2016) 032 [arXiv:1610.02890] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Yongjun Ahn
    • 1
  • Viktor Jahnke
    • 1
  • Hyun-Sik Jeong
    • 1
  • Keun-Young Kim
    • 1
    Email author
  1. 1.School of Physics and ChemistryGwangju Institute of Science and TechnologyGwangjuKorea

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