Nonlocal multi-trace sources and bulk entanglement in holographic conformal field theories

  • Felix M. HaehlEmail author
  • Eric Mintun
  • Jason Pollack
  • Antony J. Speranza
  • Mark Van Raamsdonk
Open Access
Regular Article - Theoretical Physics


We consider CFT states defined by adding nonlocal multi-trace sources to the Euclidean path integral defining the vacuum state. For holographic theories, we argue that these states correspond to states in the gravitational theory with a good semiclassical description but with a more general structure of bulk entanglement than states defined from single-trace sources. We show that at leading order in large N , the entanglement entropies for any such state are precisely the same as those of another state defined by appropriate single-trace effective sources; thus, if the leading order entanglement entropies are geometrical for the single-trace states of a CFT, they are geometrical for all the multi-trace states as well. Next, we consider the perturbative calculation of 1/N corrections to the CFT entanglement entropies, demonstrating that these show qualitatively different features, including non-analyticity in the sources and/or divergences in the naive perturbative expansion. These features are consistent with the expectation that the 1/N corrections include contributions from bulk entanglement on the gravity side. Finally, we investigate the dynamical constraints on the bulk geometry and the quantum state of the bulk fields which must be satisfied so that the entropies can be reproduced via the quantum-corrected Ryu-Takayanagi formula.


AdS-CFT Correspondence 1/N Expansion Conformal Field Theory Models of Quantum Gravity 


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. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  5. [5]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
  7. [7]
    M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
  8. [8]
    M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  10. [10]
    J. Bhattacharya, M. Nozaki, T. Takayanagi and T. Ugajin, Thermodynamical Property of Entanglement Entropy for Excited States, Phys. Rev. Lett. 110 (2013) 091602 [arXiv:1212.1164] [INSPIRE].
  11. [11]
    N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglementthermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].
  12. [12]
    T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from Entanglement in Holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    A. Lewkowycz and O. Parrikar, The holographic shape of entanglement and Einsteins equations, JHEP 05 (2018) 147 [arXiv:1802.10103] [INSPIRE].
  14. [14]
    T. Faulkner, F.M. Haehl, E. Hijano, O. Parrikar, C. Rabideau and M. Van Raamsdonk, Nonlinear Gravity from Entanglement in Conformal Field Theories, JHEP 08 (2017) 057 [arXiv:1705.03026] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    F.M. Haehl, E. Hijano, O. Parrikar and C. Rabideau, Higher Curvature Gravity from Entanglement in Conformal Field Theories, Phys. Rev. Lett. 120 (2018) 201602 [arXiv:1712.06620] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  16. [16]
    F.M. Haehl, Comments on universal properties of entanglement entropy and bulk reconstruction, JHEP 10 (2015) 159 [arXiv:1508.00766] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    M. Botta-Cantcheff, P. Martínez and G.A. Silva, On excited states in real-time AdS/CFT, JHEP 02 (2016) 171 [arXiv:1512.07850] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    D. Marolf, O. Parrikar, C. Rabideau, A. Izadi Rad and M. Van Raamsdonk, From Euclidean Sources to Lorentzian Spacetimes in Holographic Conformal Field Theories, JHEP 06 (2018) 077 [arXiv:1709.10101] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].CrossRefzbMATHGoogle Scholar
  20. [20]
    N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].CrossRefGoogle Scholar
  21. [21]
    E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
  22. [22]
    M. Berkooz, A. Sever and A. Shomer, ‘Double tracedeformations, boundary conditions and space-time singularities, JHEP 05 (2002) 034 [hep-th/0112264] [INSPIRE].
  23. [23]
    G. Sárosi and T. Ugajin, Modular Hamiltonians of excited states, OPE blocks and emergent bulk fields, JHEP 01 (2018) 012 [arXiv:1705.01486] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    N. Lashkari, H. Liu and S. Rajagopal, Perturbation Theory for the Logarithm of a Positive Operator, arXiv:1811.05619 [INSPIRE].
  25. [25]
    T. Ugajin, Perturbative expansions of Rényi relative divergences and holography, arXiv:1812.01135 [INSPIRE].
  26. [26]
    B. Swingle and M. Van Raamsdonk, Universality of Gravity from Entanglement, arXiv:1405.2933 [INSPIRE].
  27. [27]
    A. Belin, N. Iqbal and S.F. Lokhande, Bulk entanglement entropy in perturbative excited states, SciPost Phys. 5 (2018) 024 [arXiv:1805.08782] [INSPIRE].CrossRefGoogle Scholar
  28. [28]
    A. Goel, H.T. Lam, G.J. Turiaci and H. Verlinde, Expanding the Black Hole Interior: Partially Entangled Thermal States in SYK, JHEP 02 (2019) 156 [arXiv:1807.03916] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    W. Cottrell, B. Freivogel, D.M. Hofman and S.F. Lokhande, How to Build the Thermofield Double State, JHEP 02 (2019) 058 [arXiv:1811.11528] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    M. Miyaji, Time Evolution after Double Trace Deformation, JHEP 10 (2018) 074 [arXiv:1806.10807] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    C. Ecker, D. Grumiller, W. van der Schee, M.M. Sheikh-Jabbari and P. Stanzer, Quantum Null Energy Condition and its (non)saturation in 2d CFTs, SciPost Phys. 6 (2019) 036 [arXiv:1901.04499] [INSPIRE].
  32. [32]
    D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    K. Skenderis and B.C. van Rees, Real-time gauge/gravity duality: Prescription, Renormalization and Examples, JHEP 05 (2009) 085 [arXiv:0812.2909] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  34. [34]
    A. Christodoulou and K. Skenderis, Holographic Construction of Excited CFT States, JHEP 04 (2016) 096 [arXiv:1602.02039] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  35. [35]
    P. Gao, D.L. Jafferis and A. Wall, Traversable Wormholes via a Double Trace Deformation, JHEP 12 (2017) 151 [arXiv:1608.05687] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    J. Maldacena, D. Stanford and Z. Yang, Diving into traversable wormholes, Fortsch. Phys. 65 (2017) 1700034 [arXiv:1704.05333] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  37. [37]
    O. Aharony, A.B. Clark and A. Karch, The CFT/AdS correspondence, massive gravitons and a connectivity index conjecture, Phys. Rev. D 74 (2006) 086006 [hep-th/0608089] [INSPIRE].
  38. [38]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    X. Dong, The Gravity Dual of Renyi Entropy, Nature Commun. 7 (2016) 12472 [arXiv:1601.06788] [INSPIRE].CrossRefGoogle Scholar
  40. [40]
    X. Dong, Holographic Rényi Entropy at High Energy Density, Phys. Rev. Lett. 122 (2019) 041602 [arXiv:1811.04081] [INSPIRE].
  41. [41]
    V. Rosenhaus and M. Smolkin, Entanglement Entropy: A Perturbative Calculation, JHEP 12 (2014) 179 [arXiv:1403.3733] [INSPIRE].CrossRefzbMATHGoogle Scholar
  42. [42]
    V. Rosenhaus and M. Smolkin, Entanglement Entropy for Relevant and Geometric Perturbations, JHEP 02 (2015) 015 [arXiv:1410.6530] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    T. Faulkner, Bulk Emergence and the RG Flow of Entanglement Entropy, JHEP 05 (2015) 033 [arXiv:1412.5648] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    G. Sárosi and T. Ugajin, Relative entropy of excited states in two dimensional conformal field theories, JHEP 07 (2016) 114 [arXiv:1603.03057] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    N. Lashkari, H. Liu and S. Rajagopal, Modular Flow of Excited States, arXiv:1811.05052 [INSPIRE].
  46. [46]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    M.J. Schlosser, Multiple Hypergeometric Series: Appell Series and Beyond, in Computer Algebra in Quantum Field Theory. Integration, Summation and Special Functions. Proceedings, LHCPhenoNet School, Linz, Austria, 9-13 July 2012, pp. 305-324 (2013) [] [arXiv:1305.1966] [INSPIRE].
  48. [48]
    X. Dong and A. Lewkowycz, Entropy, Extremality, Euclidean Variations and the Equations of Motion, JHEP 01 (2018) 081 [arXiv:1705.08453] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    S. Hollands and R.M. Wald, Stability of Black Holes and Black Branes, Commun. Math. Phys. 321 (2013) 629 [arXiv:1201.0463] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    N. Lashkari, J. Lin, H. Ooguri, B. Stoica and M. Van Raamsdonk, Gravitational positive energy theorems from information inequalities, PTEP 2016 (2016) 12C109 [arXiv:1605.01075] [INSPIRE].
  51. [51]
    V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
  52. [52]
    H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].
  53. [53]
    W. Donnelly, Entanglement entropy and nonabelian gauge symmetry, Class. Quant. Grav. 31 (2014) 214003 [arXiv:1406.7304] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].CrossRefGoogle Scholar
  55. [55]
    W. Donnelly and A.C. Wall, Geometric entropy and edge modes of the electromagnetic field, Phys. Rev. D 94 (2016) 104053 [arXiv:1506.05792] [INSPIRE].
  56. [56]
    S.N. Solodukhin, Newton constant, contact terms and entropy, Phys. Rev. D 91 (2015) 084028 [arXiv:1502.03758] [INSPIRE].
  57. [57]
    S.D. Mathur, A Proposal to resolve the black hole information paradox, Int. J. Mod. Phys. D 11 (2002) 1537 [hep-th/0205192] [INSPIRE].
  58. [58]
    S.B. Giddings, Black hole information, unitarity and nonlocality, Phys. Rev. D 74 (2006) 106005 [hep-th/0605196] [INSPIRE].
  59. [59]
    S.B. Giddings, Nonviolent nonlocality, Phys. Rev. D 88 (2013) 064023 [arXiv:1211.7070] [INSPIRE].
  60. [60]
    N. Bao, S.M. Carroll, A. Chatwin-Davies, J. Pollack and G.N. Remmen, Branches of the Black Hole Wave Function Need Not Contain Firewalls, Phys. Rev. D 97 (2018) 126014 [arXiv:1712.04955] [INSPIRE].
  61. [61]
    A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: A Boundary view of horizons and locality, Phys. Rev. D 73 (2006) 086003 [hep-th/0506118] [INSPIRE].
  62. [62]
    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].
  63. [63]
    H. Chen, On some trigonometric power sums, Int. J. Math. Math. Sci. 30 (2001) 185.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Felix M. Haehl
    • 1
    Email author
  • Eric Mintun
    • 1
  • Jason Pollack
    • 1
  • Antony J. Speranza
    • 2
  • Mark Van Raamsdonk
    • 1
  1. 1.Department of Physics and AstronomyUniversity of British ColumbiaVancouverCanada
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada

Personalised recommendations