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Towards a derivation of holographic entanglement entropy

  • Horacio Casini
  • Marina Huerta
  • Robert C. Myers
Article

Abstract

We provide a derivation of holographic entanglement entropy for spherical entangling surfaces. Our construction relies on conformally mapping the boundary CFT to a hyperbolic geometry and observing that the vacuum state is mapped to a thermal state in the latter geometry. Hence the conformal transformation maps the entanglement entropy to the thermodynamic entropy of this thermal state. The AdS/CFT dictionary allows us to calculate this thermodynamic entropy as the horizon entropy of a certain topological black hole. In even dimensions, we also demonstrate that the universal contribution to the entanglement entropy is given by A-type trace anomaly for any CFT, without reference to holography.

Keywords

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

References

  1. [1]
    M. Levin and X.G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613] [SPIRES].ADSCrossRefGoogle Scholar
  2. [2]
    A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002 [hep-th/0405152] [SPIRES].
  4. [4]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory: A non-technical introduction, Int. J. Quant. Inf. 4 (2006) 429 [quant-ph/0505193] [SPIRES].MATHCrossRefGoogle Scholar
  5. [5]
    B. Hsu, M. Mulligan, E. Fradkin and E.-A. Kim, Universal entanglement entropy in 2D conformal quantum critical points, arXiv:0812.0203 [SPIRES].
  6. [6]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    I.R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of confinement, Nucl. Phys. B 796 (2008) 274 [arXiv:0709. 2140] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [SPIRES].
  10. [10]
    M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [SPIRES].ADSMATHCrossRefGoogle Scholar
  11. [11]
    M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [SPIRES].ADSGoogle Scholar
  12. [12]
    D.V. Fursaev, Proof of the holographic formula for entanglement entropy, JHEP 09 (2006) 018 [hep-th/0606184] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905. 0932] [SPIRES].MathSciNetGoogle Scholar
  14. [14]
    L.-Y. Hung, R.C. Myers and M. Smolkin, On holographic entanglement entropy and higher curvature gravity, JHEP 04 (2011) 025 [arXiv:1101.5813] [SPIRES].ADSCrossRefGoogle Scholar
  15. [15]
    C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [SPIRES].MathSciNetADSGoogle Scholar
  16. [16]
    R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [SPIRES].ADSGoogle Scholar
  17. [17]
    R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    H. Casini and M. Huerta, Entanglement entropy for the n-sphere, Phys. Lett. B 694 (2010) 167 [arXiv:1007.1813] [SPIRES].MathSciNetADSGoogle Scholar
  19. [19]
    P. Candelas and J.S. Dowker, Field theories on conformally related space-times: some global considerations, Phys. Rev. D 19 (1979) 2902 [SPIRES].MathSciNetADSGoogle Scholar
  20. [20]
    S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [SPIRES].MathSciNetADSGoogle Scholar
  21. [21]
    M.J. Duff, Observations on conformal anomalies, Nucl. Phys. B 125 (1977) 334 [SPIRES].ADSCrossRefGoogle Scholar
  22. [22]
    M.J. Duff, Twenty years of the Weyl anomaly, Class. Quant. Grav. 11 (1994) 1387 [hep-th/9308075] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  23. [23]
    S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [SPIRES].MathSciNetADSGoogle Scholar
  24. [24]
    J.S. Dowker, Entanglement entropy for even spheres, arXiv:1009.3854 [SPIRES].
  25. [25]
    J.S. Dowker, Hyperspherical entanglement entropy, J. Phys. A 43 (2010) 445402 [arXiv:1007.3865] [SPIRES].MathSciNetADSGoogle Scholar
  26. [26]
    S.N. Solodukhin, Entanglement entropy of round spheres, Phys. Lett. B 693 (2010) 605 [arXiv:1008.4314] [SPIRES].MathSciNetADSGoogle Scholar
  27. [27]
    J.S. Dowker, Entanglement entropy for odd spheres, arXiv:1012.1548 [SPIRES].
  28. [28]
    R. Haag, Local quantum physics: fields, particles, algebras, Springer, U.S.A. (1992) [SPIRES].MATHGoogle Scholar
  29. [29]
    H. Li and F.D.M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: identification of topological order in non-abelian fractional quantum Hall effect states, Phys. Rev. Lett. 101 (2008) 010504 [arXiv:0805.0332].ADSCrossRefGoogle Scholar
  30. [30]
    P. Calabrese and A. Lefevre, Entanglement spectrum in one-dimensional systems, Phys. Rev. A 78 (2008) 032329.ADSGoogle Scholar
  31. [31]
    A.M. Turner, F. Pollmann and E. Berg, Topological phases of one-dimensional fermions: an entanglement point of view, Phys. Rev. B 83 (2011) 075102 [arXiv:1008.4346].ADSGoogle Scholar
  32. [32]
    L. Fidkowski, Entanglement spectrum of topological insulators and superconductors, Phys. Rev. Lett. 104 (2010) 130502 [arXiv:0909.2654].ADSCrossRefGoogle Scholar
  33. [33]
    H. Yao and X.-L. Qi, Entanglement entropy and entanglement spectrum of the Kitaev model, Phys. Rev. Lett. 105 (2010) 080501 [arXiv:1001.1165].ADSCrossRefGoogle Scholar
  34. [34]
    J.J. Bisognano and E.H. Wichmann, On the duality condition for quantum fields, J. Math. Phys. 17 (1976) 303 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    J.J. Bisognano and E.H. Wichmann, On the duality condition for a hermitian scalar field, J. Math. Phys. 16 (1975) 985 [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  36. [36]
    W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [SPIRES].ADSGoogle Scholar
  37. [37]
    P.D. Hislop and R. Longo, Modular structure of the local algebras associated with the free massless scalar field theory, Commun. Math. Phys. 84 (1982) 71 [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  38. [38]
    H. Casini and M. Huerta, Remarks on the entanglement entropy for disconnected regions, JHEP 03 (2009) 048 [arXiv:0812.1773] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    R. Emparan, AdS/CFT duals of topological black holes and the entropy of zero-energy states, JHEP 06 (1999) 036 [hep-th/9906040] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    S. Aminneborg, I. Bengtsson, S. Holst and P. Peldan, Making Anti-de Sitter black holes, Class. Quant. Grav. 13 (1996) 2707 [gr-qc/9604005] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  41. [41]
    D.R. Brill, J. Louko and P. Peldan, Thermodynamics of (3 + 1)-dimensional black holes with toroidal or higher genus horizons, Phys. Rev. D 56 (1997) 3600 [gr-qc/9705012] [SPIRES].MathSciNetADSGoogle Scholar
  42. [42]
    L. Vanzo, Black holes with unusual topology, Phys. Rev. D 56 (1997) 6475 [gr-qc/9705004] [SPIRES].MathSciNetADSGoogle Scholar
  43. [43]
    R.B. Mann, Pair production of topological Anti-de Sitter black holes, Class. Quant. Grav. 14 (1997) L109 [gr-qc/9607071] [SPIRES].ADSCrossRefGoogle Scholar
  44. [44]
    D. Birmingham, Topological black holes in Anti-de Sitter space, Class. Quant. Grav. 16 (1999) 1197 [hep-th/9808032] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  45. [45]
    R. Emparan, AdS membranes wrapped on surfaces of arbitrary genus, Phys. Lett. B 432 (1998) 74 [hep-th/9804031] [SPIRES].MathSciNetADSGoogle Scholar
  46. [46]
    D.J. Gross and J.H. Sloan, The quartic effective action for the heterotic string, Nucl. Phys. B 291 (1987) 41 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  47. [47]
    C.G. Callan Jr., E.J. Martinec, M.J. Perry and D. Friedan, Strings in background fields, Nucl. Phys. B 262 (1985) 593 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    R.C. Myers, M.F. Paulos and A. Sinha, Holographic studies of quasi-topological gravity, JHEP 08 (2010) 035 [arXiv:1004. 2055] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    A. Buchel et al., Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [SPIRES].ADSCrossRefGoogle Scholar
  50. [50]
    J. de Boer, M. Kulaxizi and A. Parnachev, AdS 7 /CFT 6 , Gauss-Bonnet gravity and viscosity bound, JHEP 03 (2010) 087 [arXiv:0910.5347] [SPIRES].CrossRefGoogle Scholar
  51. [51]
    X.O. Camanho and J.D. Edelstein, Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity, JHEP 04 (2010) 007 [arXiv:0911.3160] [SPIRES].ADSCrossRefGoogle Scholar
  52. [52]
    J. de Boer, M. Kulaxizi and A. Parnachev, Holographic Lovelock gravities and black holes, JHEP 06 (2010) 008 [arXiv:0912.1877] [SPIRES].CrossRefGoogle Scholar
  53. [53]
    X.O. Camanho and J.D. Edelstein, Causality in AdS/CFT and Lovelock theory, JHEP 06 (2010) 099 [arXiv:0912.1944] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    X.O. Camanho, J.D. Edelstein and M.F. Paulos, Lovelock theories, holography and the fate of the viscosity bound, arXiv:1010.1682 [SPIRES].
  55. [55]
    D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  56. [56]
    D. Lovelock, Divergence-free tensorial concomitants, Aequat. Math. 4 (1970) 127.MathSciNetMATHCrossRefGoogle Scholar
  57. [57]
    R.C. Myers and B. Robinson, Black holes in quasi-topological gravity, JHEP 08 (2010) 067 [arXiv:1003.5357] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  58. [58]
    R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [SPIRES].MathSciNetADSGoogle Scholar
  59. [59]
    T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [SPIRES].MathSciNetADSGoogle Scholar
  60. [60]
    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] [SPIRES].MathSciNetADSGoogle Scholar
  61. [61]
    N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, Cambridge University Press, Cambridge U.K. (1982), p. 340.MATHGoogle Scholar
  62. [62]
    R. Laflamme, Geometry and thermofields, Nucl. Phys. B 324 (1989) 233 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  63. [63]
    D.V. Fursaev and G. Miele, Finite temperature scalar field theory in static de Sitter space, Phys. Rev. D 49 (1994) 987 [hep-th/9302078] [SPIRES].MathSciNetADSGoogle Scholar
  64. [64]
    D.V. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  65. [65]
    L.S. Brown and J.P. Cassidy, Stress tensors and their trace anomalies in conformally flat space-times, Phys. Rev. D 16 (1977) 1712 [SPIRES].ADSGoogle Scholar
  66. [66]
    T.S. Bunch and P.C.W. Davies, Quantum field theory in de Sitter space: renormalization by point splitting, Proc. Roy. Soc. Lond. A 360 (1978) 117 [SPIRES].MathSciNetADSGoogle Scholar
  67. [67]
    L.Y. Hung, R.C. Myers, M. Smolkin and A. Yale, Holographic calculations of Renyi entropy in higher dimensions, in preparation.Google Scholar
  68. [68]
    M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  69. [69]
    M. Henningson and K. Skenderis, Holography and the Weyl anomaly, Fortsch. Phys. 48 (2000) 125 [hep-th/9812032] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  70. [70]
    J. de Boer, M. Kulaxizi and A. Parnachev, Holographic entanglement entropy in Lovelock gravities, arXiv:1101.5781 [SPIRES].
  71. [71]
    J.T. Liu, W. Sabra and Z. Zhao, Holographic c-theorems and higher derivative gravity, arXiv:1012.3382 [SPIRES].
  72. [72]
    A. Sinha, On higher derivative gravity, c-theorems and cosmology, Class. Quant. Grav. 28 (2011) 085002 [arXiv:1008.4315] [SPIRES].ADSCrossRefGoogle Scholar
  73. [73]
    M.F. Paulos, Holographic phase space: c-functions and black holes as renormalization group flows, arXiv:1101.5993 [SPIRES].
  74. [74]
    L. Susskind and E. Witten, The holographic bound in Anti-de Sitter space, hep-th/9805114 [SPIRES].
  75. [75]
    C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  76. [76]
    A. Schwimmer and S. Theisen, Entanglement entropy, trace anomalies and holography, Nucl. Phys. B 801 (2008) 1 [arXiv:0802.1017] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  77. [77]
    H. Casini, Mutual information challenges entropy bounds, Class. Quant. Grav. 24 (2007) 1293 [gr-qc/0609126] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  78. [78]
    H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [SPIRES].MathSciNetGoogle Scholar
  79. [79]
    B. Swingle, Mutual information and the structure of entanglement in quantum field theory, arXiv:1010.4038 [SPIRES].
  80. [80]
    D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, arXiv:1012.3210 [SPIRES].
  81. [81]
    D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-theorem: N = 2 field theories on the three-sphere, arXiv:1103.1181 [SPIRES].
  82. [82]
    K.A. Intriligator and B. Wecht, The exact superconformal R-symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  83. [83]
    E. Barnes, K.A. Intriligator, B. Wecht and J. Wright, Evidence for the strongest version of the 4D a-theorem, via a-maximization along RG flows, Nucl. Phys. B 702 (2004) 131 [hep-th/0408156] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  84. [84]
    D.J. Gross and E. Witten, Superstring modifications of Einstein’s equations, Nucl. Phys. B 277 (1986) 1 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  85. [85]
    M.T. Grisaru, A.E.M. van de Ven and D. Zanon, Four loop β-function for the N = 1 and N = 2 supersymmetric nonlinear σ-model in two-dimensions, Phys. Lett. B 173 (1986) 423 [SPIRES].ADSGoogle Scholar
  86. [86]
    M.F. Paulos, Higher derivative terms including the Ramond-Ramond five-form, JHEP 10 (2008) 047 [arXiv:0804.0763] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  87. [87]
    A. Buchel, R.C. Myers, M.F. Paulos and A. Sinha, Universal holographic hydrodynamics at finite coupling, Phys. Lett. B 669 (2008) 364 [arXiv:0808.1837] [SPIRES].ADSGoogle Scholar
  88. [88]
    M.B. Green and M. Gutperle, Effects of D-instantons, Nucl. Phys. B 498 (1997) 195 [hep-th/9701093] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  89. [89]
    R.C. Myers, M.F. Paulos and A. Sinha, Quantum corrections to η/s, Phys. Rev. D 79 (2009) 041901 [arXiv:0806.2156] [SPIRES].ADSGoogle Scholar
  90. [90]
    D. Anselmi, D.Z. Freedman, M.T. Grisaru and A.A. Johansen, Universality of the operator product expansions of SCFT(4), Phys. Lett. B 394 (1997) 329 [hep-th/9608125] [SPIRES].MathSciNetADSGoogle Scholar
  91. [91]
    D. Anselmi, D.Z. Freedman, M.T. Grisaru and A.A. Johansen, Nonperturbative formulas for central functions of supersymmetric gauge theories, Nucl. Phys. B 526 (1998) 543 [hep-th/9708042] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  92. [92]
    R. Lohmayer, H. Neuberger, A. Schwimmer and S. Theisen, Numerical determination of entanglement entropy for a sphere, Phys. Lett. B 685 (2010) 222 [arXiv:0911.4283] [SPIRES].ADSGoogle Scholar
  93. [93]
    A. Buchel, R.C. Myers and A. Sinha, Beyond η/s = 1/4π, JHEP 03 (2009) 084 [arXiv:0812.2521] [SPIRES].ADSCrossRefGoogle Scholar
  94. [94]
    Y. Kats and P. Petrov, Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory, JHEP 01 (2009) 044 [arXiv:0712.0743] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  95. [95]
    C.P. Bachas, P. Bain and M.B. Green, Curvature terms in D-brane actions and their M-theory origin, JHEP 05 (1999) 011 [hep-th/9903210] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  96. [96]
    J.M. Maldacena, Eternal black holes in Anti-de-Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  97. [97]
    D. Klemm, V. Moretti and L. Vanzo, Rotating topological black holes, Phys. Rev. D 57 (1998) 6127 [Erratum ibid. D 60 (1999) 109902] [gr-qc/9710123] [SPIRES].MathSciNetADSGoogle Scholar
  98. [98]
    M.H. Dehghani, Rotating topological black holes in various dimensions and AdS/CFT correspondence, Phys. Rev. D 65 (2002) 124002 [hep-th/0203049] [SPIRES].MathSciNetADSGoogle Scholar
  99. [99]
    R.B. Mann, Charged topological black hole pair creation, Nucl. Phys. B 516 (1998) 357 [hep-th/9705223] [SPIRES].ADSCrossRefGoogle Scholar
  100. [100]
    R.-G. Cai and A. Wang, Thermodynamics and stability of hyperbolic charged black holes, Phys. Rev. D 70 (2004) 064013 [hep-th/0406057] [SPIRES].MathSciNetADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  • Horacio Casini
    • 1
  • Marina Huerta
    • 1
  • Robert C. Myers
    • 2
  1. 1.Centro Atómico Bariloche and Instituto BalseiroRío NegroArgentina
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada

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