Journal of High Energy Physics

, 2016:116 | Cite as

Thermality and excited state Rényi entropy in two-dimensional CFT

  • Feng-Li Lin
  • Huajia Wang
  • Jia-ju Zhang
Open Access
Regular Article - Theoretical Physics


We evaluate one-interval Rényi entropy and entanglement entropy for the excited states of two-dimensional conformal field theory (CFT) on a cylinder, and examine their differences from the ones for the thermal state. We assume the interval to be short so that we can use operator product expansion (OPE) of twist operators to calculate Rényi entropy in terms of sum of one-point functions of OPE blocks. We find that the entanglement entropy for highly excited state and thermal state behave the same way after appropriate identification of the conformal weight of the state with the temperature. However, there exists no such universal identification for the Rényi entropy in the short-interval expansion. Therefore, the highly excited state does not look thermal when comparing its Rényi entropy to the thermal state one. As the Rényi entropy captures the higher moments of the reduced density matrix but the entanglement entropy only the average, our results imply that the emergence of thermality depends on how refined we look into the entanglement structure of the underlying pure excited state.


Conformal Field Theory AdS-CFT Correspondence Holography and condensed matter physics (AdS/CMT) 


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]
    J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.ADSGoogle Scholar
  3. [3]
    S. Goldstein, J.L. Lebowitz, R. Tumulka and N. Zanghi, Canonical typicality, Phys. Rev. Lett. 96 (2006) 050403 [cond-mat/0511091] [INSPIRE].
  4. [4]
    S. Popescu, A.J. Short and A. Winter, Entanglement and the foundations of statistical mechanics, Nature Phys. 2 (2006) 754 [quant-ph/0511225].
  5. [5]
    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
  6. [6]
    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].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    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].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    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
  9. [9]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].MathSciNetMATHGoogle Scholar
  10. [10]
    B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, A stereoscopic look into the bulk, JHEP 07 (2016) 129 [arXiv:1604.03110] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].ADSGoogle Scholar
  12. [12]
    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].MathSciNetGoogle Scholar
  13. [13]
    B. Chen and J.-J. Zhang, On short interval expansion of Rényi entropy, JHEP 11 (2013) 164 [arXiv:1309.5453] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    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
  15. [15]
    E. Perlmutter, Comments on Rényi entropy in AdS 3 /CFT 2, JHEP 05 (2014) 052 [arXiv:1312.5740] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    B. Chen, F.-y. Song and J.-j. Zhang, Holographic Rényi entropy in AdS 3 /LCFT 2 correspondence, JHEP 03 (2014) 137 [arXiv:1401.0261] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    M. Beccaria and G. Macorini, On the next-to-leading holographic entanglement entropy in AdS 3 /CF T 2, JHEP 04 (2014) 045 [arXiv:1402.0659] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    J.-j. Zhang, Holographic Rényi entropy for two-dimensional \( \mathcal{N}=\left(1,1\right) \) superconformal field theory, JHEP 12 (2015) 027 [arXiv:1510.01423] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    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].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    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].ADSCrossRefGoogle Scholar
  21. [21]
    S. He, T. Numasawa, T. Takayanagi and K. Watanabe, Quantum dimension as entanglement entropy in two dimensional conformal field theories, Phys. Rev. D 90 (2014) 041701 [arXiv:1403.0702] [INSPIRE].ADSGoogle Scholar
  22. [22]
    W.-Z. Guo and S. He, Rényi entropy of locally excited states with thermal and boundary effect in 2D CFTs, JHEP 04 (2015) 099 [arXiv:1501.00757] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    B. Chen, W.-Z. Guo, S. He and J.-q. Wu, Entanglement entropy for descendent local operators in 2D CFTs, JHEP 10 (2015) 173 [arXiv:1507.01157] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    M.M. Sheikh-Jabbari and H. Yavartanoo, Excitation entanglement entropy in 2D conformal field theories, arXiv:1605.00341 [INSPIRE].
  25. [25]
    R.C. Rashkov, Notes on entanglement entropy for excites holographic states in 2D, arXiv:1607.08373 [INSPIRE].
  26. [26]
    N. Lashkari, A. Dymarsky and H. Liu, Eigenstate thermalization hypothesis in conformal field theory, arXiv:1610.00302 [INSPIRE].
  27. [27]
    T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].MathSciNetMATHGoogle Scholar
  28. [28]
    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
  29. [29]
    B. Chen and J.-q. Wu, Single interval Rényi entropy at low temperature, JHEP 08 (2014) 032 [arXiv:1405.6254] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    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].ADSMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Department of PhysicsNational Taiwan Normal UniversityTaipeiTaiwan
  2. 2.Department of PhysicsUniversity of IllinoisUrbana-ChampaignU.S.A.
  3. 3.Dipartimento di FisicaUniversità degli Studi di Milano-BicoccaMilanoItaly
  4. 4.Theoretical Physics Division, Institute of High Energy PhysicsChinese Academy of SciencesBeijingP.R. China
  5. 5.Theoretical Physics Center for Science FacilitiesChinese Academy of SciencesBeijingP.R. China

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