Advertisement

Journal of High Energy Physics

, 2019:35 | Cite as

Quantum correction to chaos in Schwarzian theory

  • Yong-Hui Qi
  • Sang-Jin Sin
  • Junggi YoonEmail author
Open Access
Regular Article - Theoretical Physics
  • 4 Downloads

Abstract

We discuss the quantum correction to chaos in the Schwarzian theory. We carry out the semi-classical analysis of the Schwarzian theory to study Feynman diagrams of the Schwarzian soft mode. We evaluate the contribution of the soft mode to the out-of- time-order correlator up to order 𝒪 (g4). We show that the quantum correction of order 𝒪 (g4) by the soft mode decreases the maximum Lyapunov exponent \( \frac{2\pi }{\beta } \).

Keywords

AdS-CFT Correspondence Effective Field Theories Field Theories in Lower Dimensions 2D Gravity 

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]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP03 (2014) 067 [arXiv:1306.0622] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    S.H. Shenker and D. Stanford, Multiple shocks, JHEP12 (2014) 046 [arXiv:1312.3296] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  3. [3]
    D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP03 (2015) 051 [arXiv:1409.8180] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP05 (2015) 132 [arXiv:1412.6087] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP08 (2016) 106 [arXiv:1503.01409] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    J. Yoon, A bound on chaos from stability, arXiv:1905.08815 [INSPIRE].
  7. [7]
    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].
  8. [8]
    A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, KITP seminar February 12 (2015).Google Scholar
  9. [9]
    A. Kitaev, A simple model of quantum holography, talks given at KITP, April 7 and May 27 (2015).Google Scholar
  10. [10]
    J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP04 (2016) 001 [arXiv:1601.06768] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    A. Jevicki, K. Suzuki and J. Yoon, Bi-local holography in the SYK model, JHEP07 (2016) 007 [arXiv:1603.06246] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev.D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
  13. [13]
    A. Jevicki and K. Suzuki, Bi-local holography in the SYK model: perturbations, JHEP11 (2016) 046 [arXiv:1608.07567] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    D.J. Gross and V. Rosenhaus, A generalization of Sachdev-Ye-Kitaev, JHEP02 (2017) 093 [arXiv:1610.01569] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    W. Fu, D. Gaiotto, J. Maldacena and S. Sachdev, Supersymmetric Sachdev-Ye-Kitaev models, Phys. Rev.D 95 (2017) 026009 [arXiv:1610.08917] [INSPIRE].
  16. [16]
    J. Yoon, SYK models and SYK-like tensor models with global symmetry, JHEP10 (2017) 183 [arXiv:1707.01740] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    J. Yoon, Supersymmetric SYK model: bi-local collective superfield/supermatrix formulation, JHEP10 (2017) 172 [arXiv:1706.05914] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    P. Narayan and J. Yoon, Supersymmetric SYK Model with global symmetry, JHEP08 (2018) 159 [arXiv:1712.02647] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    F. Ferrari and F.I. Schaposnik Massolo, Phases of melonic quantum mechanics, Phys. Rev.D 100 (2019) 026007 [arXiv:1903.06633] [INSPIRE].
  20. [20]
    R. Gurau, The 1/N expansion of colored tensor models, Annales Henri Poincaŕe12 (2011) 829 [arXiv:1011.2726] [INSPIRE].
  21. [21]
    S. Carrozza and A. Tanasa, O(N ) random tensor models, Lett. Math. Phys.106 (2016) 1531 [arXiv:1512.06718] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    E. Witten, An SYK-like model without disorder, arXiv:1610.09758 [INSPIRE].
  23. [23]
    R. Gurau, The complete 1/N expansion of a SYK–like tensor model, Nucl. Phys.B 916 (2017) 386 [arXiv:1611.04032] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    I.R. Klebanov and G. Tarnopolsky, Uncolored random tensors, melon diagrams and the Sachdev-Ye-Kitaev models, Phys. Rev.D 95 (2017) 046004 [arXiv:1611.08915] [INSPIRE].
  25. [25]
    J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
  26. [26]
    J. de Boer, E. Llabrés, J.F. Pedraza and D. Vegh, Chaotic strings in AdS/CFT, Phys. Rev. Lett.120 (2018) 201604 [arXiv:1709.01052] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    K. Murata, Fast scrambling in holographic Einstein-Podolsky-Rosen pair, JHEP11 (2017) 049 [arXiv:1708.09493] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    A. Banerjee, A. Kundu and R.R. Poojary, Strings, branes, Schwarzian action and maximal chaos, arXiv:1809.02090 [INSPIRE].
  29. [29]
    A.L. Fitzpatrick and J. Kaplan, A quantum correction to chaos, JHEP05 (2016) 070 [arXiv:1601.06164] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    H. Chen, A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, Degenerate operators and the 1/c expansion: Lorentzian resummations, high order computations and super-Virasoro blocks, JHEP03 (2017) 167 [arXiv:1606.02659] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    A.L. Fitzpatrick and J. Kaplan, On the late-time behavior of Virasoro blocks and a classification of semiclassical saddles, JHEP04 (2017) 072 [arXiv:1609.07153] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    E. Hijano, P. Kraus and R. Snively, Worldline approach to semi-classical conformal blocks, JHEP07 (2015) 131 [arXiv:1501.02260] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    E. Perlmutter, Virasoro conformal blocks in closed form, JHEP08 (2015) 088 [arXiv:1502.07742] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    Y. Hikida and T. Uetoko, Three point functions in higher spin AdS 3holography with 1/N corrections, Universe3 (2017) 70 [arXiv:1708.02017] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    D. Stanford and E. Witten, Fermionic localization of the Schwarzian theory, JHEP10 (2017) 008 [arXiv:1703.04612] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    T.G. Mertens, G.J. Turiaci and H.L. Verlinde, Solving the Schwarzian via the conformal bootstrap, JHEP08 (2017) 136 [arXiv:1705.08408] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    T.G. Mertens, The Schwarzian theory — Origins, JHEP05 (2018) 036 [arXiv:1801.09605] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    Y.-H. Qi, Y. Seo, S.-J. Sin and G. Song, Correlation functions in Schwarzian liquid, Phys. Rev.D 99 (2019) 066001 [arXiv:1804.06164] [INSPIRE].
  39. [39]
    H.T. Lam, T.G. Mertens, G.J. Turiaci and H. Verlinde, Shockwave S-matrix from Schwarzian quantum mechanics, JHEP11 (2018) 182 [arXiv:1804.09834] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    A. Blommaert, T.G. Mertens and H. Verschelde, The Schwarzian theory — A Wilson line perspective, JHEP12 (2018) 022 [arXiv:1806.07765] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    J. Cotler and K. Jensen, A theory of reparameterizations for AdS 3gravity, JHEP02 (2019) 079 [arXiv:1808.03263] [INSPIRE].ADSzbMATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    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].ADSzbMATHCrossRefGoogle Scholar
  43. [43]
    P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
  44. [44]
    C. Peng, M. Spradlin and A. Volovich, Correlators in the \( \mathcal{N} \) = 2 supersymmetric SYK model, JHEP10 (2017) 202 [arXiv:1706.06078] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  45. [45]
    A. Blommaert, T.G. Mertens and H. Verschelde, Fine structure of Jackiw-Teitelboim quantum gravity, JHEP09 (2019) 066 [arXiv:1812.00918] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  46. [46]
    T.G. Mertens and G.J. Turiaci, Defects in Jackiw-Teitelboim quantum gravity, JHEP08 (2019) 127 [arXiv:1904.05228] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  47. [47]
    L.V. Iliesiu, S.S. Pufu, H. Verlinde and Y. Wang, An exact quantization of Jackiw-Teitelboim gravity, arXiv:1905.02726 [INSPIRE].
  48. [48]
    V. Jahnke, K.-Y. Kim and J. Yoon, On the chaos bound in rotating black holes, JHEP05 (2019) 037 [arXiv:1903.09086] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    P. Narayan and J. Yoon, Chaos in three-dimensional higher spin gravity, JHEP07 (2019) 046 [arXiv:1903.08761] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  50. [50]
    M. Ammon, A. Castro and N. Iqbal, Wilson lines and entanglement entropy in higher spin gravity, JHEP10 (2013) 110 [arXiv:1306.4338] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  51. [51]
    A. Castro, N. Iqbal and E. Llabrés, Wilson lines and Ishibashi states in AdS 3/CFT 2, JHEP09 (2018) 066 [arXiv:1805.05398] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  52. [52]
    R.R. Poojary, BTZ dynamics and chaos, arXiv:1812.10073 [INSPIRE].
  53. [53]
    S. Grozdanov, K. Schalm and V. Scopelliti, Black hole scrambling from hydrodynamics, Phys. Rev. Lett.120 (2018) 231601 [arXiv:1710.00921] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    M. Blake, H. Lee and H. Liu, A quantum hydrodynamical description for scrambling and many-body chaos, JHEP10 (2018) 127 [arXiv:1801.00010] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  55. [55]
    F.M. Haehl and M. Rozali, Effective field theory for chaotic CFTs, JHEP10 (2018) 118 [arXiv:1808.02898] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  56. [56]
    M. Blake, R.A. Davison, S. Grozdanov and H. Liu, Many-body chaos and energy dynamics in holography, JHEP10 (2018) 035 [arXiv:1809.01169] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  57. [57]
    S. Grozdanov, On the connection between hydrodynamics and quantum chaos in holographic theories with stringy corrections, JHEP01 (2019) 048 [arXiv:1811.09641] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  58. [58]
    S. Grozdanov, P.K. Kovtun, A.O. Starinets and P. Tadić, The complex life of hydrodynamic modes, arXiv:1904.12862 [INSPIRE].
  59. [59]
    M. Blake, R.A. Davison and D. Vegh, Horizon constraints on holographic Green’s functions, arXiv:1904.12883 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of PhysicsHanyang UniversitySeoulKorea
  2. 2.School of PhysicsKorea Institute for Advanced StudySeoulRepublic of Korea

Personalised recommendations