Journal of High Energy Physics

, 2019:152 | Cite as

Note on ETH of descendant states in 2D CFT

  • Wu-zhong Guo
  • Feng-Li Lin
  • Jiaju ZhangEmail author
Open Access
Regular Article - Theoretical Physics


We investigate the eigenstate thermalization hypothesis (ETH) of highly excited descendant states in two-dimensional large central charge c conformal field theory. We use operator product expansion of twist operators to calculate the short interval expansions of entanglement entropy and relative entropy for an interval of length ℓ up to order 12. Using these results to ensure ETH of a heavy state when compared with the canonical ensemble state up to various orders of c, we get the constraints on the expectation values of the first few quasiprimary operators in the vacuum conformal family at the corresponding order of c. Similarly, we also obtain the constraints from the expectation values of the first few Korteweg-de Vries charges. We check these constraints for some types of special descendant excited states. Among the descendant states we consider, we find that at most only the leading order ones of the ETH constraints can be satisfied for the descendant states that are slightly excited on top of a heavy primary state. Otherwise, the ETH constraints are violated for the descendant states that are heavily excited on top of a primary state.


AdS-CFT Correspondence Conformal Field Theory Gauge-gravity correspondence 


Open Access

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  1. [1]
    J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.Google Scholar
  2. [2]
    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.Google Scholar
  3. [3]
    M. Srednicki, Thermal fluctuations in quantized chaotic systems, J. Phys. A 29 (1996) L75 [chao-dyn/9511001] [INSPIRE].
  4. [4]
    N. Lashkari, A. Dymarsky and H. Liu, Eigenstate thermalization hypothesis in conformal field theory, J. Stat. Mech. 1803 (2018) 033101 [arXiv:1610.00302] [INSPIRE].
  5. [5]
    A. Dymarsky, N. Lashkari and H. Liu, Subsystem ETH, Phys. Rev. E 97 (2018) 012140 [arXiv:1611.08764] [INSPIRE].
  6. [6]
    J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    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].
  9. [9]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    P. Kraus and A. Maloney, A Cardy formula for three-point coefficients or how the black hole got its spots, JHEP 05 (2017) 160 [arXiv:1608.03284] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    E.M. Brehm, D. Das and S. Datta, Probing thermality beyond the diagonal, Phys. Rev. D 98 (2018) 126015 [arXiv:1804.07924] [INSPIRE].
  13. [13]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Y. Hikida, Y. Kusuki and T. Takayanagi, Eigenstate thermalization hypothesis and modular invariance of two-dimensional conformal field theories, Phys. Rev. D 98 (2018) 026003 [arXiv:1804.09658] [INSPIRE].
  15. [15]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
  16. [16]
    M. Headrick, Entanglement Rényi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
  17. [17]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. 1101 (2011) P01021 [arXiv:1011.5482] [INSPIRE].
  18. [18]
    B. Chen and J.-J. Zhang, On short interval expansion of Rényi entropy, JHEP 11 (2013) 164 [arXiv:1309.5453] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    R. Sasaki and I. Yamanaka, Virasoro algebra, vertex operators, quantum sine-Gordon and solvable quantum field theories, in Conformal field theory and solvable lattice models, Elsevier, The Netherlands (1988), pg. 271 [Adv. Stud. Pure Math. 16 (1988) 271].Google Scholar
  20. [20]
    T. Eguchi and S.-K. Yang, Deformations of conformal field theories and soliton equations, Phys. Lett. B 224 (1989) 373 [INSPIRE].
  21. [21]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys. 177 (1996) 381 [hep-th/9412229] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of long-distance AdS physics from the CFT bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro conformal blocks and thermality from classical background fields, JHEP 11 (2015) 200 [arXiv:1501.05315] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    C.T. Asplund, A. Bernamonti, F. Galli and T. Hartman, Holographic entanglement entropy from 2d CFT: heavy states and local quenches, JHEP 02 (2015) 171 [arXiv:1410.1392] [INSPIRE].
  25. [25]
    P. Caputa, J. Simón, A. Štikonas and T. Takayanagi, Quantum entanglement of localized excited states at finite temperature, JHEP 01 (2015) 102 [arXiv:1410.2287] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    F.-L. Lin, H. Wang and J.-J. Zhang, Thermality and excited state Rényi entropy in two-dimensional CFT, JHEP 11 (2016) 116 [arXiv:1610.01362] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  27. [27]
    S. He, F.-L. Lin and J.-J. Zhang, Subsystem eigenstate thermalization hypothesis for entanglement entropy in CFT, JHEP 08 (2017) 126 [arXiv:1703.08724] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    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].
  29. [29]
    S. He, F.-L. Lin and J.-J. Zhang, Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis, JHEP 12 (2017) 073 [arXiv:1708.05090] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    N. Lashkari, A. Dymarsky and H. Liu, Universality of quantum information in chaotic CFTs, JHEP 03 (2018) 070 [arXiv:1710.10458] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    M. Rigol, V. Dunjko, V. Yurovsky and M. Olshanii, Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of 1d lattice hard-core bosons, Phys. Rev. Lett. 98 (2007) 050405 [cond-mat/0604476].
  32. [32]
    T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [INSPIRE].
  33. [33]
    G. Wong, I. Klich, L.A. Pando Zayas and D. Vaman, Entanglement temperature and entanglement entropy of excited states, JHEP 12 (2013) 020 [arXiv:1305.3291] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    W.-Z. Guo, F.-L. Lin and J. Zhang, Non-geometric states in a holographic conformal field theory, arXiv:1806.07595 [INSPIRE].
  35. [35]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    X. Dong, The gravity dual of Rényi entropy, Nature Commun. 7 (2016) 12472 [arXiv:1601.06788] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    M. Bañados, Three-dimensional quantum geometry and black holes, AIP Conf. Proc. 484 (1999) 147 [hep-th/9901148] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    W.-Z. Guo, F.-L. Lin and J. Zhang, Distinguishing black hole microstates using Holevo information, Phys. Rev. Lett. 121 (2018) 251603 [arXiv:1808.02873] [INSPIRE].CrossRefGoogle Scholar
  40. [40]
    J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
  41. [41]
    P.H. Ginsparg, Applied conformal field theory, in Les Houches summer school in theoretical physics: fields, strings, critical phenomena, Les Houches, France, 28 June-5 August 1988, pg. 1 [hep-th/9108028] [INSPIRE].
  42. [42]
    P.D. Francesco, P. Mathieu and D. Sénéchal, Quantum field theory, Springer, New York, U.S.A. (1997) [INSPIRE].
  43. [43]
    R. Blumenhagen and E. Plauschinn, Introduction to conformal field theory, Lect. Notes Phys. 779 (2009) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    B. Chen, J. Long and J.-J. Zhang, Holographic Rényi entropy for CFT with W symmetry, JHEP 04 (2014) 041 [arXiv:1312.5510] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    Z. Li and J.-J. Zhang, On one-loop entanglement entropy of two short intervals from OPE of twist operators, JHEP 05 (2016) 130 [arXiv:1604.02779] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    B. Chen, J.-B. Wu and J.-J. Zhang, Short interval expansion of Rényi entropy on torus, JHEP 08 (2016) 130 [arXiv:1606.05444] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  47. [47]
    J.L. Cardy, O.A. Castro-Alvaredo and B. Doyon, Form factors of branch-point twist fields in quantum integrable models and entanglement entropy, J. Statist. Phys. 130 (2008) 129 [arXiv:0706.3384] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    N. Lashkari, Relative entropies in conformal field theory, Phys. Rev. Lett. 113 (2014) 051602 [arXiv:1404.3216] [INSPIRE].
  49. [49]
    N. Lashkari, Modular Hamiltonian for excited states in conformal field theory, Phys. Rev. Lett. 117 (2016) 041601 [arXiv:1508.03506] [INSPIRE].
  50. [50]
    B. Chen, Z. Li and J.-J. Zhang, Corrections to holographic entanglement plateau, JHEP 09 (2017) 151 [arXiv:1707.07354] [INSPIRE].
  51. [51]
    J. Cardy and C.P. Herzog, Universal thermal corrections to single interval entanglement entropy for two dimensional conformal field theories, Phys. Rev. Lett. 112 (2014) 171603 [arXiv:1403.0578] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    B. Chen and J.-Q. Wu, Single interval Rényi entropy at low temperature, JHEP 08 (2014) 032 [arXiv:1405.6254] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    B. Chen, J.-Q. Wu and Z.-C. Zheng, Holographic Rényi entropy of single interval on torus: with W symmetry, Phys. Rev. D 92 (2015) 066002 [arXiv:1507.00183] [INSPIRE].
  54. [54]
    G. Sárosi and T. Ugajin, Relative entropy of excited states in two dimensional conformal field theories, JHEP 07 (2016) 114 [arXiv:1603.03057] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    G. Sárosi and T. Ugajin, Relative entropy of excited states in conformal field theories of arbitrary dimensions, JHEP 02 (2017) 060 [arXiv:1611.02959] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    D. Aharonov, A necessary and sufficient condition for univalence of a meromorphic function, Duke Math. J. 36 (1969) 599.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Physics Division, National Center for Theoretical SciencesNational Tsing Hua UniversityHsinchuTaiwan
  2. 2.Department of PhysicsNational Taiwan Normal UniversityTaipeiTaiwan
  3. 3.Dipartimento di Fisica G. OcchialiniUniversità degli Studi di Milano-BicoccaMilanoItaly
  4. 4.SISSA and INFNTriesteItaly

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