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Towards a derivation of holographic entanglement entropy

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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.

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References

  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].

    Article  ADS  Google Scholar 

  2. A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  3. P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002 [hep-th/0405152] [SPIRES].

  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].

    Article  MATH  Google Scholar 

  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. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  7. S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  9. M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [SPIRES].

  10. M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [SPIRES].

    Article  ADS  MATH  Google Scholar 

  11. M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [SPIRES].

    ADS  Google Scholar 

  12. D.V. Fursaev, Proof of the holographic formula for entanglement entropy, JHEP 09 (2006) 018 [hep-th/0606184] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  13. T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905. 0932] [SPIRES].

    MathSciNet  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  15. C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  16. R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [SPIRES].

    ADS  Google Scholar 

  17. R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  18. H. Casini and M. Huerta, Entanglement entropy for the n-sphere, Phys. Lett. B 694 (2010) 167 [arXiv:1007.1813] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  19. P. Candelas and J.S. Dowker, Field theories on conformally related space-times: some global considerations, Phys. Rev. D 19 (1979) 2902 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  20. S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  21. M.J. Duff, Observations on conformal anomalies, Nucl. Phys. B 125 (1977) 334 [SPIRES].

    Article  ADS  Google Scholar 

  22. M.J. Duff, Twenty years of the Weyl anomaly, Class. Quant. Grav. 11 (1994) 1387 [hep-th/9308075] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  24. J.S. Dowker, Entanglement entropy for even spheres, arXiv:1009.3854 [SPIRES].

  25. J.S. Dowker, Hyperspherical entanglement entropy, J. Phys. A 43 (2010) 445402 [arXiv:1007.3865] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  26. S.N. Solodukhin, Entanglement entropy of round spheres, Phys. Lett. B 693 (2010) 605 [arXiv:1008.4314] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  27. J.S. Dowker, Entanglement entropy for odd spheres, arXiv:1012.1548 [SPIRES].

  28. R. Haag, Local quantum physics: fields, particles, algebras, Springer, U.S.A. (1992) [SPIRES].

    MATH  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  30. P. Calabrese and A. Lefevre, Entanglement spectrum in one-dimensional systems, Phys. Rev. A 78 (2008) 032329.

    ADS  Google Scholar 

  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].

    ADS  Google Scholar 

  32. L. Fidkowski, Entanglement spectrum of topological insulators and superconductors, Phys. Rev. Lett. 104 (2010) 130502 [arXiv:0909.2654].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  34. J.J. Bisognano and E.H. Wichmann, On the duality condition for quantum fields, J. Math. Phys. 17 (1976) 303 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  35. J.J. Bisognano and E.H. Wichmann, On the duality condition for a hermitian scalar field, J. Math. Phys. 16 (1975) 985 [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [SPIRES].

    ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. H. Casini and M. Huerta, Remarks on the entanglement entropy for disconnected regions, JHEP 03 (2009) 048 [arXiv:0812.1773] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

  42. L. Vanzo, Black holes with unusual topology, Phys. Rev. D 56 (1997) 6475 [gr-qc/9705004] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  43. R.B. Mann, Pair production of topological Anti-de Sitter black holes, Class. Quant. Grav. 14 (1997) L109 [gr-qc/9607071] [SPIRES].

    Article  ADS  Google Scholar 

  44. D. Birmingham, Topological black holes in Anti-de Sitter space, Class. Quant. Grav. 16 (1999) 1197 [hep-th/9808032] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. R. Emparan, AdS membranes wrapped on surfaces of arbitrary genus, Phys. Lett. B 432 (1998) 74 [hep-th/9804031] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  46. D.J. Gross and J.H. Sloan, The quartic effective action for the heterotic string, Nucl. Phys. B 291 (1987) 41 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  48. R.C. Myers, M.F. Paulos and A. Sinha, Holographic studies of quasi-topological gravity, JHEP 08 (2010) 035 [arXiv:1004. 2055] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  49. A. Buchel et al., Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [SPIRES].

    Article  ADS  Google Scholar 

  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].

    Article  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  52. J. de Boer, M. Kulaxizi and A. Parnachev, Holographic Lovelock gravities and black holes, JHEP 06 (2010) 008 [arXiv:0912.1877] [SPIRES].

    Article  Google Scholar 

  53. X.O. Camanho and J.D. Edelstein, Causality in AdS/CFT and Lovelock theory, JHEP 06 (2010) 099 [arXiv:0912.1944] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  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. D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  56. D. Lovelock, Divergence-free tensorial concomitants, Aequat. Math. 4 (1970) 127.

    Article  MathSciNet  MATH  Google Scholar 

  57. R.C. Myers and B. Robinson, Black holes in quasi-topological gravity, JHEP 08 (2010) 067 [arXiv:1003.5357] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  58. R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  59. T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

  61. N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, Cambridge University Press, Cambridge U.K. (1982), p. 340.

    MATH  Google Scholar 

  62. R. Laflamme, Geometry and thermofields, Nucl. Phys. B 324 (1989) 233 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

  64. D.V. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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].

    ADS  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

  67. L.Y. Hung, R.C. Myers, M. Smolkin and A. Yale, Holographic calculations of Renyi entropy in higher dimensions, in preparation.

  68. M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  69. M. Henningson and K. Skenderis, Holography and the Weyl anomaly, Fortsch. Phys. 48 (2000) 125 [hep-th/9812032] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  70. J. de Boer, M. Kulaxizi and A. Parnachev, Holographic entanglement entropy in Lovelock gravities, arXiv:1101.5781 [SPIRES].

  71. J.T. Liu, W. Sabra and Z. Zhao, Holographic c-theorems and higher derivative gravity, arXiv:1012.3382 [SPIRES].

  72. A. Sinha, On higher derivative gravity, c-theorems and cosmology, Class. Quant. Grav. 28 (2011) 085002 [arXiv:1008.4315] [SPIRES].

    Article  ADS  Google Scholar 

  73. M.F. Paulos, Holographic phase space: c-functions and black holes as renormalization group flows, arXiv:1101.5993 [SPIRES].

  74. L. Susskind and E. Witten, The holographic bound in Anti-de Sitter space, hep-th/9805114 [SPIRES].

  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].

    Article  MathSciNet  ADS  Google Scholar 

  76. A. Schwimmer and S. Theisen, Entanglement entropy, trace anomalies and holography, Nucl. Phys. B 801 (2008) 1 [arXiv:0802.1017] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  77. H. Casini, Mutual information challenges entropy bounds, Class. Quant. Grav. 24 (2007) 1293 [gr-qc/0609126] [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  78. H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [SPIRES].

    MathSciNet  Google Scholar 

  79. B. Swingle, Mutual information and the structure of entanglement in quantum field theory, arXiv:1010.4038 [SPIRES].

  80. D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, arXiv:1012.3210 [SPIRES].

  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. K.A. Intriligator and B. Wecht, The exact superconformal R-symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  84. D.J. Gross and E. Witten, Superstring modifications of Einstein’s equations, Nucl. Phys. B 277 (1986) 1 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    ADS  Google Scholar 

  86. M.F. Paulos, Higher derivative terms including the Ramond-Ramond five-form, JHEP 10 (2008) 047 [arXiv:0804.0763] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    ADS  Google Scholar 

  88. M.B. Green and M. Gutperle, Effects of D-instantons, Nucl. Phys. B 498 (1997) 195 [hep-th/9701093] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    ADS  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    ADS  Google Scholar 

  93. A. Buchel, R.C. Myers and A. Sinha, Beyond η/s = 1/4π, JHEP 03 (2009) 084 [arXiv:0812.2521] [SPIRES].

    Article  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  96. J.M. Maldacena, Eternal black holes in Anti-de-Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

  99. R.B. Mann, Charged topological black hole pair creation, Nucl. Phys. B 516 (1998) 357 [hep-th/9705223] [SPIRES].

    Article  ADS  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

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  1. Centro Atómico Bariloche and Instituto Balseiro, 8400-S.C. de Bariloche, Río Negro, Argentina

    Horacio Casini & Marina Huerta

  2. Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada

    Robert C. Myers

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  1. Horacio Casini
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  2. Marina Huerta
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Casini, H., Huerta, M. & Myers, R.C. Towards a derivation of holographic entanglement entropy. J. High Energ. Phys. 2011, 36 (2011). https://doi.org/10.1007/JHEP05(2011)036

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