Journal of High Energy Physics

, 2017:131 | Cite as

Permutation orbifolds and chaos

  • Alexandre BelinEmail author
Open Access
Regular Article - Theoretical Physics


We study out-of-time-ordered correlation functions in permutation orbifolds at large central charge. We show that they do not decay at late times for arbitrary choices of low-dimension operators, indicating that permutation orbifolds are non-chaotic theories. This is in agreement with the fact they are free discrete gauge theories and should be integrable rather than chaotic. We comment on the early-time behaviour of the correlators as well as the deformation to strong coupling.


AdS-CFT Correspondence Black Holes Conformal Field Theory 


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.


  1. [1]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    S. Leichenauer, Disrupting Entanglement of Black Holes, Phys. Rev. D 90 (2014) 046009 [arXiv:1405.7365] [INSPIRE].ADSGoogle Scholar
  4. [4]
    A. Kitaev, A simple model of quantum holography, talks at KITP (2015).Google Scholar
  5. [5]
    D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    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
  7. [7]
    S. Jackson, L. McGough and H. Verlinde, Conformal Bootstrap, Universality and Gravitational Scattering, Nucl. Phys. B 901 (2015) 382 [arXiv:1412.5205] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    J. Polchinski, Chaos in the black hole S-matrix, arXiv:1505.08108 [INSPIRE].
  11. [11]
    P. Caputa, J. Simón, A. Štikonas, T. Takayanagi and K. Watanabe, Scrambling time from local perturbations of the eternal BTZ black hole, JHEP 08 (2015) 011 [arXiv:1503.08161] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02 (2016) 004 [arXiv:1511.04021] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    D. Stanford, Many-body chaos at weak coupling, JHEP 10 (2016) 009 [arXiv:1512.07687] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    G. Gur-Ari, M. Hanada and S.H. Shenker, Chaos in Classical D0-Brane Mechanics, JHEP 02 (2016) 091 [arXiv:1512.00019] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    E. Berkowitz, M. Hanada and J. Maltz, Chaos in Matrix Models and Black Hole Evaporation, Phys. Rev. D 94 (2016) 126009 [arXiv:1602.01473] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  16. [16]
    J. Polchinski and V. Rosenhaus, The Spectrum in the Sachdev-Ye-Kitaev Model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    A.L. Fitzpatrick and J. Kaplan, A Quantum Correction To Chaos, JHEP 05 (2016) 070 [arXiv:1601.06164] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    B. Michel, J. Polchinski, V. Rosenhaus and S.J. Suh, Four-point function in the IOP matrix model, JHEP 05 (2016) 048 [arXiv:1602.06422] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    E. Perlmutter, Bounding the Space of Holographic CFTs with Chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    G. Turiaci and H. Verlinde, On CFT and Quantum Chaos, JHEP 12 (2016) 110 [arXiv:1603.03020] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    P. Caputa, Y. Kusuki, T. Takayanagi and K. Watanabe, Out-of-Time-Ordered Correlators in (T 2)n/ℤn, Phys. Rev. D 96 (2017) 046020 [arXiv:1703.09939] [INSPIRE].ADSGoogle Scholar
  22. [22]
    Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the Fast Scrambling Conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Einstein gravity 3-point functions from conformal field theory, arXiv:1610.09378 [INSPIRE].
  25. [25]
    A. Belin, C.A. Keller and A. Maloney, String Universality for Permutation Orbifolds, Phys. Rev. D 91 (2015) 106005 [arXiv:1412.7159] [INSPIRE].ADSMathSciNetGoogle Scholar
  26. [26]
    F.M. Haehl and M. Rangamani, Permutation orbifolds and holography, JHEP 03 (2015) 163 [arXiv:1412.2759] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    A. Belin, C.A. Keller and A. Maloney, Permutation Orbifolds in the large-N Limit, Annales Henri Poincaré (2016) 1 [arXiv:1509.01256] [INSPIRE].
  28. [28]
    C.A. Keller, Phase transitions in symmetric orbifold CFTs and universality, JHEP 03 (2011) 114 [arXiv:1101.4937] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    P. Kraus, A. Sivaramakrishnan and R. Snively, Black holes from CFT: Universality of correlators at large c, JHEP 08 (2017) 084 [arXiv:1706.00771] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    V. Balasubramanian, B. Craps, B. Czech and G. Sárosi, Echoes of chaos from string theory black holes, JHEP 03 (2017) 154 [arXiv:1612.04334] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    A. Belin, J. de Boer, J. Kruthoff, B. Michel, E. Shaghoulian and M. Shyani, Universality of sparse d > 2 conformal field theory at large-N, JHEP 03 (2017) 067 [arXiv:1610.06186] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    P.J. Cameron, Transitivity of permutation groups on unordered sets, Math. Z. 148 (1976) 127.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    P.J. Cameron, Oligomorphic permutation groups, in Perspectives in mathematical sciences. II, Stat. Sci. Interdiscip. Res. 8 (2009) 37, World Scientific Publ., Hackensack, NJ (2009).Google Scholar
  35. [35]
    P.J. Cameron, Oligomorphic permutation groups, Lond. Math. Soc. Lect. Note Ser. 152 (1990) 1, Cambridge University Press, Cambridge (1990).Google Scholar
  36. [36]
    A. Belin, C.A. Keller and I.G. Zadeh, Genus two partition functions and Rényi entropies of large c conformal field theories, J. Phys. A 50 (2017) 435401 [arXiv:1704.08250] [INSPIRE].ADSzbMATHGoogle Scholar
  37. [37]
    O. Lunin and S.D. Mathur, Correlation functions for M N /S N orbifolds, Commun. Math. Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  38. [38]
    A. Pakman, L. Rastelli and S.S. Razamat, Diagrams for Symmetric Product Orbifolds, JHEP 10 (2009) 034 [arXiv:0905.3448] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    B.A. Burrington, A.W. Peet and I.G. Zadeh, Twist-nontwist correlators in M N /S N orbifold CFTs, Phys. Rev. D 87 (2013) 106008 [arXiv:1211.6689] [INSPIRE].ADSGoogle Scholar
  40. [40]
    E. Dyer and G. Gur-Ari, 2D CFT Partition Functions at Late Times, JHEP 08 (2017) 075 [arXiv:1611.04592] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [arXiv:1611.04650] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    P. Caputa, T. Numasawa and A. Veliz-Osorio, Out-of-time-ordered correlators and purity in rational conformal field theories, PTEP 2016 (2016) 113B06 [arXiv:1602.06542] [INSPIRE].
  43. [43]
    S.G. Avery, B.D. Chowdhury and S.D. Mathur, Deforming the D1D5 CFT away from the orbifold point, JHEP 06 (2010) 031 [arXiv:1002.3132] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    M.R. Gaberdiel, C. Peng and I.G. Zadeh, Higgsing the stringy higher spin symmetry, JHEP 10 (2015) 101 [arXiv:1506.02045] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations