Holographic entanglement entropy in Lovelock gravities

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Article

Abstract

We study entanglement entropies of simply connected surfaces in field theories dual to Lovelock gravities. We consider Gauss-Bonnet and cubic Lovelock gravities in detail. In the conformal case the logarithmic terms in the entanglement entropy are governed by the conformal anomalies of the CFT; we verify that the holographic calculations are consistent with this property. We also compute the holographic entanglement entropy of a slab in the Gauss-Bonnet examples dual to relativistic and non-relativistic CFTs and discuss its properties. Finally, we discuss features of the entanglement entropy in the backgrounds dual to renormalization group flows between fixed points and comment on the implications for a possible c-theorem in four spacetime dimensions.

Keywords

Gauge-gravity correspondence Anomalies in Field and String Theories 

References

  1. [1]
    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
  2. [2]
    S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [SPIRES].ADSCrossRefGoogle Scholar
  4. [4]
    A.B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2D field theory, JETP Lett. 43 (1986) 730 [Pisma Zh. Eksp. Teor. Fiz. 43 (1986) 565] [SPIRES].MathSciNetADSGoogle Scholar
  5. [5]
    J.L. Cardy, Is there a c theorem in four-dimensions?, Phys. Lett. B 215 (1988) 749 [SPIRES].MathSciNetADSGoogle Scholar
  6. [6]
    D. Erkal and D. Kutasov, a-maximization, global symmetries and RG flows, arXiv:1007.2176 [SPIRES].
  7. [7]
    D. Poland and D. Simmons-Duffin, Bounds on 4D conformal and superconformal field theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [SPIRES].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    D. Gaiotto, N. Seiberg and Y. Tachikawa, Comments on scaling limits of 4D N = 2 theories, JHEP 01 (2011) 078 [arXiv:1011.4568] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [SPIRES].ADSGoogle Scholar
  10. [10]
    R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    J.T. Liu, W. Sabra and Z. Zhao, Holographic c-theorems and higher derivative gravity, arXiv:1012.3382 [SPIRES].
  12. [12]
    M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, Viscosity bound violation in higher derivative gravity, Phys. Rev. D 77 (2008) 126006 [arXiv:0712.0805] [SPIRES].ADSGoogle Scholar
  13. [13]
    M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, The viscosity bound and causality violation, Phys. Rev. Lett. 100 (2008) 191601 [arXiv:0802.3318] [SPIRES].ADSCrossRefGoogle Scholar
  14. [14]
    A. Buchel and R.C. Myers, Causality of holographic hydrodynamics, JHEP 08 (2009) 016 [arXiv:0906.2922] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    D.M. Hofman, Higher derivative gravity, causality and positivity of energy in a UV complete QFT, Nucl. Phys. B 823 (2009) 174 [arXiv:0907.1625] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    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
  17. [17]
    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
  18. [18]
    A. Buchel et al., Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [SPIRES].ADSCrossRefGoogle Scholar
  19. [19]
    J.deBoer, M. Kulaxizi and A. Parnachev, Holographic Lovelock gravities and black holes, JHEP 06 (2010) 008 [arXiv:0912.1877] [SPIRES].CrossRefGoogle Scholar
  20. [20]
    X.O. Camanho and J.D. Edelstein, Causality in AdS/CFT and Lovelock theory, JHEP 06 (2010) 099 [arXiv:0912.1944] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    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
  22. [22]
    M. Kulaxizi and A. Parnachev, Energy flux positivity and unitarity in CFTs, Phys. Rev. Lett. 106 (2011) 011601 [arXiv:1007.0553] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    A. Buchel and S. Cremonini, Viscosity bound and causality in superfluid plasma, JHEP 10 (2010) 026 [arXiv:1007.2963] [SPIRES].ADSCrossRefGoogle Scholar
  24. [24]
    R.C. Myers, S. Sachdev and A. Singh, Holographic quantum critical transport without self-duality, Phys. Rev. D 83 (2011) 066017 [arXiv:1010.0443] [SPIRES].ADSGoogle Scholar
  25. [25]
    X.O. Camanho, J.D. Edelstein and M.F. Paulos, Lovelock theories, holography and the fate of the viscosity bound, JHEP 05 (2011) 127 [arXiv:1010.1682] [SPIRES].ADSCrossRefMathSciNetGoogle Scholar
  26. [26]
    S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [SPIRES].MathSciNetADSGoogle Scholar
  27. [27]
    D.V. Fursaev, Proof of the holographic formula for entanglement entropy, JHEP 09 (2006) 018 [hep-th/0606184] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    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
  29. [29]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [SPIRES].ADSCrossRefMathSciNetGoogle Scholar
  30. [30]
    L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [SPIRES].MathSciNetADSGoogle Scholar
  31. [31]
    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  32. [32]
    M.B. Plenio, J. Eisert, J. Dreissig and M. Cramer, Entropy, entanglement and area: analytical results for harmonic lattice systems, Phys. Rev. Lett. 94 (2005) 060503 [quant-ph/0405142] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    M. Cramer, J. Eisert, M.B. Plenio and J. Dreissig, An entanglement-area law for general bosonic harmonic lattice systems, Phys. Rev. A 73 (2006) 012309 [quant-ph/0505092] [SPIRES].ADSGoogle Scholar
  34. [34]
    M. Ahmadi, S. Das and S. Shankaranarayanan, Is entanglement entropy proportional to area?, Can. J. Phys. 84 (2006) 493 [hep-th/0507228] [SPIRES].ADSCrossRefGoogle Scholar
  35. [35]
    S. Das and S. Shankaranarayanan, How robust is the entanglement entropy-area relation?, Phys. Rev. D 73 (2006) 121701 [gr-qc/0511066] [SPIRES].ADSGoogle Scholar
  36. [36]
    H. Casini, Geometric entropy, area and strong subadditivity, Class. Quant. Grav. 21 (2004) 2351 [hep-th/0312238] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  37. [37]
    H. Casini and M. Huerta, Analytic results on the geometric entropy for free fields, J. Stat. Mech. (2008) P01012 [arXiv:0707.1300] [SPIRES].
  38. [38]
    M.M. Wolf, Violation of the entropic area law for fermions, Phys. Rev. Lett. 96 (2004) 010404 [quant-ph/0503219].CrossRefGoogle Scholar
  39. [39]
    D. Gioev and I. Klich, Entanglement entropy of fermions in any dimension and the Widom conjecture, Phys. Rev. Lett. 96 (2006) 100503 [quant-ph/0504151].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    T. Barthel, M. Chung and U. Schollwoeck, Entanglement scaling in critical two-dimensional fermionic and bosonic systems, Phys. Rev. A 74 (2006) 022329 [cond-mat/0602077].ADSGoogle Scholar
  41. [41]
    W. Li, L. Leitian Ding, R. Yu, T. Roscilde and S. Haas, Scaling behavior of entanglement in two-and three-dimensional free fermions, Phys. Rev. B 74 (2007) 0703103 [quant-ph/0602094].Google Scholar
  42. [42]
    H. Casini and M. Huerta, Universal terms for the entanglement entropy in 2 + 1 dimensions, Nucl. Phys. B 764 (2007) 183 [hep-th/0606256] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    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
  44. [44]
    H. Casini and M. Huerta, Entanglement entropy for the n-sphere, Phys. Lett. B 694 (2010) 167 [arXiv:1007.1813] [SPIRES].MathSciNetADSGoogle Scholar
  45. [45]
    J.S. Dowker, Hyperspherical entanglement entropy, J. Phys. A 43 (2010) 445402 [arXiv:1007.3865] [SPIRES].MathSciNetADSGoogle Scholar
  46. [46]
    J.S. Dowker, Entanglement entropy for even spheres, arXiv:1009.3854 [SPIRES].
  47. [47]
    J.S. Dowker, Determinants and conformal anomalies of GJMS operators on spheres, J. Phys. A 44 (2011) 115402 [arXiv:1010.0566] [SPIRES].MathSciNetADSGoogle Scholar
  48. [48]
    J.S. Dowker, Entanglement entropy for odd spheres, arXiv:1012.1548 [SPIRES].
  49. [49]
    Q. Exirifard and M.M. Sheikh-Jabbari, Lovelock gravity at the crossroads of Palatini and metric formulations, Phys. Lett. B 661 (2008) 158 [arXiv:0705.1879] [SPIRES].MathSciNetADSGoogle Scholar
  50. [50]
    D.G. Boulware and S. Deser, String generated gravity models, Phys. Rev. Lett. 55 (1985) 2656 [SPIRES].ADSCrossRefGoogle Scholar
  51. [51]
    R.-G. Cai, Gauss-Bonnet black holes in AdS spaces, Phys. Rev. D 65 (2002) 084014 [hep-th/0109133] [SPIRES].ADSGoogle Scholar
  52. [52]
    M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    M. Henningson and K. Skenderis, Holography and the Weyl anomaly, Fortsch. Phys. 48 (2000) 125 [hep-th/9812032] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  54. [54]
    S. Nojiri and S.D. Odintsov, On the conformal anomaly from higher derivative gravity in AdS/CFT correspondence, Int. J. Mod. Phys. A 15 (2000) 413 [hep-th/9903033] [SPIRES].MathSciNetADSGoogle Scholar
  55. [55]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [SPIRES].MathSciNetGoogle Scholar
  57. [57]
    M. Headrick and T. Takayanagi, A holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D 76 (2007) 106013 [arXiv:0704.3719] [SPIRES].MathSciNetADSGoogle Scholar
  58. [58]
    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
  59. [59]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002 [hep-th/0405152] [SPIRES].
  60. [60]
    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
  61. [61]
    J.-R. Sun, Note on Chern-Simons term correction to holographic entanglement entropy, JHEP 05 (2009) 061 [arXiv:0810.0967] [SPIRES].ADSCrossRefGoogle Scholar
  62. [62]
    D.V. Fursaev and S.N. Solodukhin, On the description of the Riemannian geometry in the presence of conical defects, Phys. Rev. D 52 (1995) 2133 [hep-th/9501127] [SPIRES].MathSciNetADSGoogle Scholar
  63. [63]
    D.V. Fursaev and S.N. Solodukhin, On one loop renormalization of black hole entropy, Phys. Lett. B 365 (1996) 51 [hep-th/9412020] [SPIRES].MathSciNetADSGoogle Scholar
  64. [64]
    S.N. Solodukhin, On ’non-geometric’ contribution to the entropy of black hole due to quantum corrections, Phys. Rev. D 51 (1995) 618 [hep-th/9408068] [SPIRES].MathSciNetADSGoogle Scholar
  65. [65]
    S.N. Solodukhin, The conical singularity and quantum corrections to entropy of black hole, Phys. Rev. D 51 (1995) 609 [hep-th/9407001] [SPIRES].MathSciNetADSGoogle Scholar
  66. [66]
    D.V. Fursaev, Black hole thermodynamics and renormalization, Mod. Phys. Lett. A 10 (1995) 649 [hep-th/9408066] [SPIRES].MathSciNetADSGoogle Scholar
  67. [67]
    D.V. Fursaev, Spectral geometry and one loop divergences on manifolds with conical singularities, Phys. Lett. B 334 (1994) 53 [hep-th/9405143] [SPIRES].MathSciNetADSGoogle Scholar
  68. [68]
    D. Nesterov and S.N. Solodukhin, Short-distance regularity of Green’s function and UV divergences in entanglement entropy, JHEP 09 (2010) 041 [arXiv:1008.0777] [SPIRES].ADSCrossRefGoogle Scholar
  69. [69]
    T. Padmanabhan, Finite entanglement entropy from the zero-point-area of spacetime, Phys. Rev. D 82 (2010) 124025 [arXiv:1007.5066] [SPIRES].ADSGoogle Scholar
  70. [70]
    D. Nesterov and S.N. Solodukhin, Gravitational effective action and entanglement entropy in UV modified theories with and without Lorentz symmetry, Nucl. Phys. B 842 (2011) 141 [arXiv:1007.1246] [SPIRES].ADSCrossRefGoogle Scholar
  71. [71]
    M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [SPIRES].ADSGoogle Scholar
  72. [72]
    T. Jacobson and R.C. Myers, Black hole entropy and higher curvature interactions, Phys. Rev. Lett. 70 (1993) 3684 [hep-th/9305016] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  73. [73]
    M. Bañados, C. Teitelboim and J. Zanelli, Black hole entropy and the dimensional continuation of the Gauss-Bonnet theorem, Phys. Rev. Lett. 72 (1994) 957 [gr-qc/9309026] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  74. [74]
    W. Nelson, A comment on black hole entropy in string theory, Phys. Rev. D 50 (1994) 7400 [hep-th/9406011] [SPIRES].ADSGoogle Scholar
  75. [75]
    V. Iyer and R.M. Wald, A comparison of Noether charge and euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D 52 (1995) 4430 [gr-qc/9503052] [SPIRES].MathSciNetADSGoogle Scholar
  76. [76]
    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
  77. [77]
    M. Kulaxizi and A. Parnachev, Supersymmetry constraints in holographic gravities, Phys. Rev. D 82 (2010) 066001 [arXiv:0912.4244] [SPIRES].ADSGoogle Scholar
  78. [78]
    S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [SPIRES].MathSciNetADSGoogle Scholar
  79. [79]
    K. Balasubramanian and K. Narayan, Lifshitz spacetimes from AdS null and cosmological solutions, JHEP 08 (2010) 014 [arXiv:1005.3291] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  80. [80]
    A. Donos and J.P. Gauntlett, Lifshitz solutions of D = 10 and D = 11 supergravity, JHEP 12 (2010) 002 [arXiv:1008.2062] [SPIRES].ADSCrossRefMathSciNetGoogle Scholar
  81. [81]
    A. Donos, J.P. Gauntlett, N. Kim and O. Varela, Wrapped M5-branes, consistent truncations and AdS/CMT, JHEP 12 (2010) 003 [arXiv:1009.3805] [SPIRES].ADSCrossRefMathSciNetGoogle Scholar
  82. [82]
    R. Gregory, S.L. Parameswaran, G. Tasinato and I. Zavala, Lifshitz solutions in supergravity and string theory, JHEP 12 (2010) 047 [arXiv:1009.3445] [SPIRES].ADSCrossRefMathSciNetGoogle Scholar
  83. [83]
    A. Adams, A. Maloney, A. Sinha and S.E. Vazquez, 1/N effects in non-relativistic gauge-gravity duality, JHEP 03 (2009) 097 [arXiv:0812.0166] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  84. [84]
    M.H. Dehghani and R.B. Mann, Lovelock-Lifshitz black holes, JHEP 07 (2010) 019 [arXiv:1004.4397] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  85. [85]
    M.H. Dehghani and R.B. Mann, Thermodynamics of Lovelock-Lifshitz black branes, Phys. Rev. D 82 (2010) 064019 [arXiv:1006.3510] [SPIRES].ADSGoogle Scholar
  86. [86]
    A.H. Chamseddine, Topological gauge theory of gravity in five-dimensions and all odd dimensions, Phys. Lett. B 233 (1989) 291 [SPIRES].MathSciNetADSGoogle Scholar
  87. [87]
    S.N. Solodukhin, Entanglement entropy in non-relativistic field theories, JHEP 04 (2010) 101 [arXiv:0909.0277] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  88. [88]
    T. Azeyanagi, W. Li and T. Takayanagi, On string theory duals of Lifshitz-like fixed points, JHEP 06 (2009) 084 [arXiv:0905.0688] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  89. [89]
    C. Hoyos and P. Koroteev, On the null energy condition and causality in Lifshitz holography, Phys. Rev. D 82 (2010) 084002 [Erratum ibid. D 82 (2010) 109905] [arXiv:1007.1428] [SPIRES].ADSGoogle Scholar
  90. [90]
    M. Berg and H. Samtleben, An exact holographic RG flow between 2D conformal fixed points, JHEP 05 (2002) 006 [hep-th/0112154] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  91. [91]
    M. Taylor, More on counterterms in the gravitational action and anomalies, hep-th/0002125 [SPIRES].
  92. [92]
    M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [SPIRES].MathSciNetADSGoogle Scholar
  93. [93]
    I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, hep-th/0404176 [SPIRES].
  94. [94]
    H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [SPIRES].MathSciNetADSGoogle Scholar
  95. [95]
    H. Casini and M. Huerta, A c-theorem for the entanglement entropy, J. Phys. A 40 (2007) 7031 [cond-mat/0610375] [SPIRES].MathSciNetADSGoogle Scholar

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© The Author(s) 2011

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • Jan de Boer
    • 1
  • Manuela Kulaxizi
    • 2
  • Andrei Parnachev
    • 3
  1. 1.Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Theoretical Physics, Department of Physics and AstronomyUppsala UniversityUppsalaSweden
  3. 3.Institute of Cosmos Sciences and E.C.M., Facultat de FisicaUniversitat de BarcelonaBarcelonaSpain

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