Eigenstate thermalisation in the conformal Sachdev-Ye-Kitaev model: an analytic approach

  • Pranjal Nayak
  • Julian SonnerEmail author
  • Manuel Vielma
Open Access
Regular Article - Theoretical Physics


The Sachdev-Ye-Kitaev (SYK) model provides an uncommon example of a chaotic theory that can be analysed analytically. In the deep infrared limit, the original model has an emergent conformal (reparametrisation) symmetry that is broken both spontaneously and explicitly. The explicit breaking of this symmetry comes about due to pseudo-Nambu-Goldstone modes that are not exact zero-modes of the model. In this paper, we study a version of the model which preserves the reparametrisation symmetry at all length scales. We study the heavy-light correlation functions of the operators in the conformal spectrum of the theory. The three point functions of such operators allow us to demonstrate that matrix elements of primaries \( \mathcal{O} \)n of the CFT1 take the form postulated by the Eigenstate Thermalisation Hypothesis. We also discuss the implications of these results for the states in AdS2 gravity dual.


1/N Expansion AdS-CFT Correspondence Conformal Field Theory Field Theories in Lower Dimensions 


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]
    L. D’Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65 (2016) 239 [arXiv:1509.06411] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    G.T. Horowitz and N. Itzhaki, Black holes, shock waves and causality in the AdS/CFT correspondence, JHEP 02 (1999) 010 [hep-th/9901012] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    U.H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Black hole formation in AdS and thermalization on the boundary, JHEP 02 (2000) 039 [hep-th/9912209] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    P.M. Chesler and L.G. Yaffe, Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 102 (2009) 211601 [arXiv:0812.2053] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    S. Bhattacharyya and S. Minwalla, Weak field black hole formation in asymptotically AdS spacetimes, JHEP 09 (2009) 034 [arXiv:0904.0464] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    T. Anous, T. Hartman, A. Rovai and J. Sonner, Black hole collapse in the 1/c expansion, JHEP 07 (2016) 123 [arXiv:1603.04856] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    T. Anous, T. Hartman, A. Rovai and J. Sonner, From conformal blocks to path integrals in the Vaidya geometry, JHEP 09 (2017) 009 [arXiv:1706.02668] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    S. Sachdev and J.-W. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
  9. [9]
    A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, University of Santa Barbara, Santa Barbara, CA, U.S.A., 7 April 2015.
  10. [10]
    A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, University of Santa Barbara, Santa Barbara, CA, U.S.A., 27 May 2015.
  11. [11]
    J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    A. Jevicki, K. Suzuki and J. Yoon, Bi-local holography in the SYK model, JHEP 07 (2016) 007 [arXiv:1603.06246] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016)106002 [arXiv:1604.07818] [INSPIRE].
  14. [14]
    E. Witten, An SYK-like model without disorder, arXiv:1610.09758 [INSPIRE].
  15. [15]
    R. Gurau, The complete 1/N expansion of a SYK-like tensor model, Nucl. Phys. B 916 (2017) 386 [arXiv:1611.04032] [INSPIRE].
  16. [16]
    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].
  17. [17]
    K. Jensen, Chaos in AdS 2 holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS 2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    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].
  20. [20]
    M. Cvetič and I. Papadimitriou, AdS 2 holographic dictionary, JHEP 12 (2016) 008 [Erratum ibid. 01 (2017) 120] [arXiv:1608.07018] [INSPIRE].
  21. [21]
    G. Mandal, P. Nayak and S.R. Wadia, Coadjoint orbit action of Virasoro group and two-dimensional quantum gravity dual to SYK/tensor models, JHEP 11 (2017) 046 [arXiv:1702.04266] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    G. Sárosi, AdS 2 holography and the SYK model, PoS(Modave2017)001 (2018) [arXiv:1711.08482] [INSPIRE].
  23. [23]
    V. Rosenhaus, An introduction to the SYK model, arXiv:1807.03334 [INSPIRE].
  24. [24]
    A. Almheiri and J. Polchinski, Models of AdS 2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    P. Nayak, A. Shukla, R.M. Soni, S.P. Trivedi and V. Vishal, On the dynamics of near-extremal black holes, JHEP 09 (2018) 048 [arXiv:1802.09547] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    F. Larsen, A nAttractor mechanism for nAdS 2 /nCFT 1 holography, JHEP 04 (2019) 055 [arXiv:1806.06330] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    A. Castro, F. Larsen and I. Papadimitriou, 5D rotating black holes and the nAdS 2/nCFT 1 correspondence, JHEP 10 (2018) 042 [arXiv:1807.06988] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    U. Moitra, S.P. Trivedi and V. Vishal, Extremal and near-extremal black holes and near-CFT 1, JHEP 07 (2019) 055 [arXiv:1808.08239] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    Y.-Z. Li, S.-L. Li and H. Lü, Exact embeddings of JT gravity in strings and M-theory, Eur. Phys. J. C 78 (2018) 791 [arXiv:1804.09742] [INSPIRE].
  30. [30]
    F. Larsen and Y. Zeng, Black hole spectroscopy and AdS 2 holography, JHEP 04 (2019) 164 [arXiv:1811.01288] [INSPIRE].
  31. [31]
    R.R. Poojary, BTZ dynamics and chaos, arXiv:1812.10073 [INSPIRE].
  32. [32]
    I. Danshita, M. Hanada and M. Tezuka, Creating and probing the Sachdev-Ye-Kitaev model with ultracold gases: towards experimental studies of quantum gravity, PTEP 2017 (2017) 083I01 [arXiv:1606.02454] [INSPIRE].
  33. [33]
    L. García- Álvarez, I.L. Egusquiza, L. Lamata, A. del Campo, J. Sonner and E. Solano, Digital quantum simulation of minimal AdS/CFT, Phys. Rev. Lett. 119 (2017) 040501 [arXiv:1607.08560] [INSPIRE].
  34. [34]
    M. Franz and M. Rozali, Mimicking black hole event horizons in atomic and solid-state systems, Nature Rev. Mater. 3 (2018) 491 [arXiv:1808.00541] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.Google Scholar
  36. [36]
    A. Dymarsky, Bound on eigenstate thermalization from transport, arXiv:1804.08626 [INSPIRE].
  37. [37]
    J.R. Garrison and T. Grover, Does a single eigenstate encode the full Hamiltonian?, Phys. Rev. X 8 (2018) 021026 [arXiv:1503.00729] [INSPIRE].
  38. [38]
    N. Lashkari, A. Dymarsky and H. Liu, Eigenstate thermalization hypothesis in conformal field theory, J. Stat. Mech. 1803 (2018) 033101 [arXiv:1610.00302] [INSPIRE].
  39. [39]
    A. Dymarsky, N. Lashkari and H. Liu, Subsystem ETH, Phys. Rev. E 97 (2018) 012140 [arXiv:1611.08764] [INSPIRE].
  40. [40]
    N. Lashkari, A. Dymarsky and H. Liu, Universality of quantum information in chaotic CFTs, JHEP 03 (2018) 070 [arXiv:1710.10458] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    P. Basu, D. Das, S. Datta and S. Pal, Thermality of eigenstates in conformal field theories, Phys. Rev. E 96 (2017) 022149 [arXiv:1705.03001] [INSPIRE].
  42. [42]
    T. Faulkner and H. Wang, Probing beyond ETH at large c, JHEP 06 (2018) 123 [arXiv:1712.03464] [INSPIRE].
  43. [43]
    E.M. Brehm, D. Das and S. Datta, Probing thermality beyond the diagonal, Phys. Rev. D 98 (2018)126015 [arXiv:1804.07924] [INSPIRE].
  44. [44]
    A. Romero-Bermúdez, P. Sabella-Garnier and K. Schalm, A Cardy formula for off-diagonal three-point coefficients; or, how the geometry behind the horizon gets disentangled, JHEP 09 (2018)005 [arXiv:1804.08899] [INSPIRE].
  45. [45]
    J. Sonner and M. Vielma, Eigenstate thermalization in the Sachdev-Ye-Kitaev model, JHEP 11 (2017) 149 [arXiv:1707.08013] [INSPIRE].
  46. [46]
    M. Haque and P. McClarty, Eigenstate thermalization scaling in Majorana clusters: from integrable to chaotic SYK models, arXiv:1711.02360 [INSPIRE].
  47. [47]
    H.T. Lam, T.G. Mertens, G.J. Turiaci and H. Verlinde, Shockwave S-matrix from Schwarzian quantum mechanics, JHEP 11 (2018) 182 [arXiv:1804.09834] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    I. Kourkoulou and J. Maldacena, Pure states in the SYK model and nearly-AdS 2 gravity, arXiv:1707.02325 [INSPIRE].
  49. [49]
    A. Eberlein, V. Kasper, S. Sachdev and J. Steinberg, Quantum quench of the Sachdev-Ye-Kitaev model, Phys. Rev. B 96 (2017) 205123 [arXiv:1706.07803] [INSPIRE].
  50. [50]
    A. Dhar, A. Gaikwad, L.K. Joshi, G. Mandal and S.R. Wadia, Gravitational collapse in SYK models and Choptuik-like phenomenon, arXiv:1812.03979 [INSPIRE].
  51. [51]
    D.J. Gross and V. Rosenhaus, A line of CFTs: from generalized free fields to SYK, JHEP 07 (2017) 086 [arXiv:1706.07015] [INSPIRE].
  52. [52]
    P. Nayak, J. Sonner and M. Vielma, Extended eigenstate thermalization and the role of FZZT branes in the Schwarzian theory, arXiv:1907.10061 [INSPIRE].
  53. [53]
    A. Kitaev and S.J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05 (2018) 183 [arXiv:1711.08467] [INSPIRE].
  54. [54]
    S. Sachdev, Bekenstein-Hawking entropy and strange metals, Phys. Rev. X 5 (2015) 041025 [arXiv:1506.05111] [INSPIRE].
  55. [55]
    O. Parcollet and A. Georges, Non-Fermi-liquid regime of a doped Mott insulator, Phys. Rev. B 59 (1999) 5341 [cond-mat/9806119].
  56. [56]
    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].
  57. [57]
    A. Jevicki and K. Suzuki, Bi-local holography in the SYK model: perturbations, JHEP 11 (2016) 046 [arXiv:1608.07567] [INSPIRE].
  58. [58]
    R. de Mello Koch, A. Jevicki, K. Suzuki and J. Yoon, AdS maps and diagrams of bi-local holography, JHEP 03 (2019) 133 [arXiv:1810.02332] [INSPIRE].
  59. [59]
    D.J. Gross and V. Rosenhaus, The bulk dual of SYK: cubic couplings, JHEP 05 (2017) 092 [arXiv:1702.08016] [INSPIRE].
  60. [60]
    D.J. Gross and V. Rosenhaus, All point correlation functions in SYK, JHEP 12 (2017) 148 [arXiv:1710.08113] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  61. [61]
    V. Balasubramanian and S.F. Ross, Holographic particle detection, Phys. Rev. D 61 (2000) 044007 [hep-th/9906226] [INSPIRE].
  62. [62]
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
  63. [63]
    A. Sen, State operator correspondence and entanglement in AdS 2/CFT 1, Entropy 13 (2011) 1305 [arXiv:1101.4254] [INSPIRE].
  64. [64]
    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].
  65. [65]
    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].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of Physics & AstronomyUniversity of KentuckyLexingtonU.S.A.
  2. 2.Department of Theoretical PhysicsUniversity of GenevaGeneva 4Switzerland

Personalised recommendations