Advertisement

Time evolution after double trace deformation

  • Masamichi Miyaji
Open Access
Regular Article - Theoretical Physics

Abstract

In this paper, we consider double trace deformation to single CFT2, and study time evolution after the deformation. The double trace deformation we consider is nonlocal: composed of two local operators placed at separate points. We study two types of local operators: one is usual local operator in CFT, and the other is HKLL bulk local operator, which is still operator in CFT but has properties as bulk local operator. We compute null energy and averaged null energy in the bulk in both types of deformations. We confirmed that, with the suitable choice of couplings, averaged null energies are negative. This implies causal structure is modified in the bulk, from classical background. We then calculate time evolution of entanglement entropy and entanglement Rényi entropy after double trace deformation. We find both quantities are found to show peculiar shockwave-like time evolution.

Keywords

AdS-CFT Correspondence Conformal Field Theory 

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]
    P. Gao, D.L. Jafferis and A. Wall, Traversable wormholes via a double trace deformation, JHEP 12 (2017) 151 [arXiv:1608.05687] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J. Maldacena, D. Stanford and Z. Yang, Diving into traversable wormholes, Fortsch. Phys. 65 (2017) 1700034 [arXiv:1704.05333] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    A. Almheiri, A. Mousatov and M. Shyani, Escaping the interiors of pure boundary-state black holes, arXiv:1803.04434 [INSPIRE].
  4. [4]
    J. De Boer, S.F. Lokhande, E. Verlinde, R. Van Breukelen and K. Papadodimas, On the interior geometry of a typical black hole microstate, arXiv:1804.10580 [INSPIRE].
  5. [5]
    J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv:1804.00491 [INSPIRE].
  6. [6]
    E. Caceres, A.S. Misobuchi and M.-L. Xiao, Rotating traversable wormholes in AdS, arXiv:1807.07239 [INSPIRE].
  7. [7]
    W.R. Kelly and A.C. Wall, Holographic proof of the averaged null energy condition, Phys. Rev. D 90 (2014) 106003 [Erratum ibid. D 91 (2015) 069902] [arXiv:1408.3566] [INSPIRE].
  8. [8]
    T. Faulkner, R.G. Leigh, O. Parrikar and H. Wang, Modular Hamiltonians for deformed half-spaces and the averaged null energy condition, JHEP 09 (2016) 038 [arXiv:1605.08072] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    T. Hartman, S. Kundu and A. Tajdini, Averaged null energy condition from causality, JHEP 07 (2017) 066 [arXiv:1610.05308] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    R. Bousso, Z. Fisher, J. Koeller, S. Leichenauer and A.C. Wall, Proof of the quantum null energy condition, Phys. Rev. D 93 (2016) 024017 [arXiv:1509.02542] [INSPIRE].
  11. [11]
    J. Koeller and S. Leichenauer, Holographic proof of the quantum null energy condition, Phys. Rev. D 94 (2016) 024026 [arXiv:1512.06109] [INSPIRE].
  12. [12]
    A.C. Wall, Lower bound on the energy density in classical and quantum field theories, Phys. Rev. Lett. 118 (2017) 151601 [arXiv:1701.03196] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    S. Balakrishnan, T. Faulkner, Z.U. Khandker and H. Wang, A general proof of the quantum null energy condition, arXiv:1706.09432 [INSPIRE].
  14. [14]
    A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].
  15. [15]
    J. Maldacena, A. Milekhin and F. Popov, Traversable wormholes in four dimensions, arXiv:1807.04726 [INSPIRE].
  16. [16]
    I. Ichinose and Y. Satoh, Entropies of scalar fields on three-dimensional black holes, Nucl. Phys. B 447 (1995) 340 [hep-th/9412144] [INSPIRE].
  17. [17]
    I.I. Shapiro, Fourth test of general relativity, Phys. Rev. Lett. 13 (1964) 789 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    M. Visser, B. Bassett and S. Liberati, Superluminal censorship, Nucl. Phys. Proc. Suppl. 88 (2000) 267 [gr-qc/9810026] [INSPIRE].
  19. [19]
    N. Engelhardt and S. Fischetti, The gravity dual of boundary causality, Class. Quant. Grav. 33 (2016) 175004 [arXiv:1604.03944] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Continuous multiscale entanglement renormalization ansatz as holographic surface-state correspondence, Phys. Rev. Lett. 115 (2015) 171602 [arXiv:1506.01353] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    K. Goto, M. Miyaji and T. Takayanagi, Causal evolutions of bulk local excitations from CFT, JHEP 09 (2016) 130 [arXiv:1605.02835] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
  23. [23]
    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
  24. [24]
    P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].
  25. [25]
    P. Calabrese and J. Cardy, Entanglement and correlation functions following a local quench: a conformal field theory approach, J. Stat. Mech. 0710 (2007) P10004 [arXiv:0708.3750] [INSPIRE].
  26. [26]
    P. Caputa, M. Nozaki and T. Takayanagi, Entanglement of local operators in large-N conformal field theories, PTEP 2014 (2014) 093B06 [arXiv:1405.5946] [INSPIRE].
  27. [27]
    M. Nozaki, Notes on quantum entanglement of local operators, JHEP 10 (2014) 147 [arXiv:1405.5875] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    M. Nozaki, T. Numasawa and T. Takayanagi, Quantum entanglement of local operators in conformal field theories, Phys. Rev. Lett. 112 (2014) 111602 [arXiv:1401.0539] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    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].
  30. [30]
    S. Leichenauer, M. Moosa and M. Smolkin, Dynamics of the area law of entanglement entropy, JHEP 09 (2016) 035 [arXiv:1604.00388] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    N. Shiba and T. Takayanagi, Volume law for the entanglement entropy in non-local QFTs, JHEP 02 (2014) 033 [arXiv:1311.1643] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    A. Mollabashi, N. Shiba and T. Takayanagi, Entanglement between two interacting CFTs and generalized holographic entanglement entropy, JHEP 04 (2014) 185 [arXiv:1403.1393] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    J.J. Bisognano and E.H. Wichmann, On the duality condition for quantum fields, J. Math. Phys. 17 (1976) 303 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    V. Rosenhaus and M. Smolkin, Entanglement entropy: a perturbative calculation, JHEP 12 (2014) 179 [arXiv:1403.3733] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  35. [35]
    V. Rosenhaus and M. Smolkin, Entanglement entropy for relevant and geometric perturbations, JHEP 02 (2015) 015 [arXiv:1410.6530] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Yukawa Institute for Theoretical Physics (YITP)Kyoto UniversityKyotoJapan

Personalised recommendations