Abstract
Entanglement entropy of holographic CFTs is expected to play a crucial role in the reconstruction of semiclassical bulk gravity. We consider the entanglement entropy of spherical regions of vacuum, which is known to contain universal contributions. After perturbing the CFT with a relevant scalar operator, also the first order change of this quantity gives a universal term which only depends on a discrete set of basic CFT parameters. We show that in gravity this statement corresponds to the uniqueness of the ghost-free graviton propagator on an AdS background geometry. While the gravitational dynamics in this context contains little information about the structure of the bulk theory, there is a discrete set of dimensionless parameters of the theory which determines the entanglement entropy. We argue that for every (not necessarily holographic) CFT, any reasonable gravity model can be used to compute this particular entanglement entropy. We elucidate how this statement is consistent with AdS/CFT and also give various generalizations. On the one hand this illustrates the remarkable usefulness of geometric concepts for understanding entanglement in general CFTs. On the other hand, it provides hints as to what entanglement data can be expected to provide enough information to distinguish, e.g., bulk theories with different higher curvature couplings.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement ‘thermodynamics’, JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].
D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative entropy and holography, JHEP 08 (2013) 060 [arXiv:1305.3182] [INSPIRE].
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].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
T. Faulkner, Bulk emergence and the RG flow of entanglement entropy, JHEP 05 (2015) 033 [arXiv:1412.5648] [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, Some calculable contributions to holographic entanglement entropy, JHEP 08 (2011) 039 [arXiv:1105.6055] [INSPIRE].
H. Liu and M. Mezei, A refinement of entanglement entropy and the number of degrees of freedom, JHEP 04 (2013) 162 [arXiv:1202.2070] [INSPIRE].
T. Nishioka, Relevant perturbation of entanglement entropy and stationarity, Phys. Rev. D 90 (2014) 045006 [arXiv:1405.3650] [INSPIRE].
V. Rosenhaus and M. Smolkin, Entanglement entropy: a perturbative calculation, JHEP 12 (2014) 179 [arXiv:1403.3733] [INSPIRE].
V. Rosenhaus and M. Smolkin, Entanglement entropy for relevant and geometric perturbations, JHEP 02 (2015) 015 [arXiv:1410.6530] [INSPIRE].
D.V. Fursaev, A. Patrushev and S.N. Solodukhin, Distributional geometry of squashed cones, Phys. Rev. D 88 (2013) 044054 [arXiv:1306.4000] [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, On holographic entanglement entropy and higher curvature gravity, JHEP 04 (2011) 025 [arXiv:1101.5813] [INSPIRE].
J. de Boer, M. Kulaxizi and A. Parnachev, Holographic entanglement entropy in Lovelock gravities, JHEP 07 (2011) 109 [arXiv:1101.5781] [INSPIRE].
X. Dong, Holographic entanglement entropy for general higher derivative gravity, JHEP 01 (2014) 044 [arXiv:1310.5713] [INSPIRE].
J. Camps, Generalized entropy and higher derivative gravity, JHEP 03 (2014) 070 [arXiv:1310.6659] [INSPIRE].
A. Bhattacharyya, M. Sharma and A. Sinha, On generalized gravitational entropy, squashed cones and holography, JHEP 01 (2014) 021 [arXiv:1308.5748] [INSPIRE].
A. Bhattacharyya and M. Sharma, On entanglement entropy functionals in higher derivative gravity theories, JHEP 10 (2014) 130 [arXiv:1405.3511] [INSPIRE].
B. Swingle and M. Van Raamsdonk, Universality of gravity from entanglement, arXiv:1405.2933 [INSPIRE].
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].
G. Wong, I. Klich, L.A. Pando Zayas and D. Vaman, Entanglement temperature and entanglement entropy of excited states, JHEP 12 (2013) 020 [arXiv:1305.3291] [INSPIRE].
D. Allahbakhshi, M. Alishahiha and A. Naseh, Entanglement thermodynamics, JHEP 08 (2013) 102 [arXiv:1305.2728] [INSPIRE].
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].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].
R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].
A. Schwimmer and S. Theisen, Entanglement entropy, trace anomalies and holography, Nucl. Phys. B 801 (2008) 1 [arXiv:0802.1017] [INSPIRE].
J. Erdmenger and H. Osborn, Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions, Nucl. Phys. B 483 (1997) 431 [hep-th/9605009] [INSPIRE].
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
V. Ogievetsky and I. Polubarinov, Interacting field of spin 2 and the Einstein equations, Annals Phys. 35 (1965) 167.
R.M. Wald, Spin-2 fields and general covariance, Phys. Rev. D 33 (1986) 3613 [INSPIRE].
K.S. Stelle, Classical gravity with higher derivatives, Gen. Rel. Grav. 9 (1978) 353 [INSPIRE].
K.S. Stelle, Renormalization of higher derivative quantum gravity, Phys. Rev. D 16 (1977) 953 [INSPIRE].
N.H. Barth and S.M. Christensen, Quantizing fourth order gravity theories. 1. The functional integral, Phys. Rev. D 28 (1983) 1876 [INSPIRE].
T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar, Towards singularity and ghost free theories of gravity, Phys. Rev. Lett. 108 (2012) 031101 [arXiv:1110.5249] [INSPIRE].
A. Conroy, A. Mazumdar and A. Teimouri, Wald entropy for ghost-free, infinite derivative theories of gravity, Phys. Rev. Lett. 114 (2015) 201101 [arXiv:1503.05568] [INSPIRE].
T.P. Sotiriou and V. Faraoni, f (R) theories of gravity, Rev. Mod. Phys. 82 (2010) 451 [arXiv:0805.1726] [INSPIRE].
D.G. Boulware, S. Deser and K. Stelle, Properties of energy in higher derivative gravity theories, in Quantum field theory and quantum statistics, vol. 2, I.A. Batalin et al. eds., U.S.A. (1987), pg. 101 [INSPIRE].
E. Alvarez, D. Blas, J. Garriga and E. Verdaguer, Transverse Fierz-Pauli symmetry, Nucl. Phys. B 756 (2006) 148 [hep-th/0606019] [INSPIRE].
P. Van Nieuwenhuizen, On ghost-free tensor Lagrangians and linearized gravitation, Nucl. Phys. B 60 (1973) 478 [INSPIRE].
D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].
R.C. Myers and B. Robinson, Black holes in quasi-topological gravity, JHEP 08 (2010) 067 [arXiv:1003.5357] [INSPIRE].
A. Buchel, J. Escobedo, R.C. Myers, M.F. Paulos, A. Sinha and M. Smolkin, Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [INSPIRE].
D.G. Boulware and S. Deser, String generated gravity models, Phys. Rev. Lett. 55 (1985) 2656 [INSPIRE].
S. Deser and B. Tekin, Energy in generic higher curvature gravity theories, Phys. Rev. D 67 (2003) 084009 [hep-th/0212292] [INSPIRE].
W.R. Kelly, K. Kuns and D. Marolf, ’t Hooft suppression and holographic entropy, JHEP 10 (2015) 059 [arXiv:1507.03654] [INSPIRE].
M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
M. Blau, K.S. Narain and E. Gava, On subleading contributions to the AdS/CFT trace anomaly, JHEP 09 (1999) 018 [hep-th/9904179] [INSPIRE].
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] [INSPIRE].
S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [INSPIRE].
T. Hirata and T. Takayanagi, AdS/CFT and strong subadditivity of entanglement entropy, JHEP 02 (2007) 042 [hep-th/0608213] [INSPIRE].
E. Fradkin and J.E. Moore, Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum, Phys. Rev. Lett. 97 (2006) 050404 [cond-mat/0605683] [INSPIRE].
H. Casini and M. Huerta, Universal terms for the entanglement entropy in 2 + 1 dimensions, Nucl. Phys. B 764 (2007) 183 [hep-th/0606256] [INSPIRE].
R.C. Myers and A. Singh, Entanglement entropy for singular surfaces, JHEP 09 (2012) 013 [arXiv:1206.5225] [INSPIRE].
P. Bueno, R.C. Myers and W. Witczak-Krempa, Universality of corner entanglement in conformal field theories, Phys. Rev. Lett. 115 (2015) 021602 [arXiv:1505.04804] [INSPIRE].
P. Bueno and R.C. Myers, Corner contributions to holographic entanglement entropy, JHEP 08 (2015) 068 [arXiv:1505.07842] [INSPIRE].
M. Alishahiha, A.F. Astaneh, P. Fonda and F. Omidi, Entanglement entropy for singular surfaces in hyperscaling violating theories, JHEP 09 (2015) 172 [arXiv:1507.05897] [INSPIRE].
R.-X. Miao, A holographic proof of the universality of corner entanglement for CFTs, JHEP 10 (2015) 038 [arXiv:1507.06283] [INSPIRE].
L.-Y. Hung, R.C. Myers, M. Smolkin and A. Yale, Holographic calculations of Rényi entropy, JHEP 12 (2011) 047 [arXiv:1110.1084] [INSPIRE].
D.A. Galante and R.C. Myers, Holographic Rényi entropies at finite coupling, JHEP 08 (2013) 063 [arXiv:1305.7191] [INSPIRE].
E. Perlmutter, A universal feature of CFT Rényi entropy, JHEP 03 (2014) 117 [arXiv:1308.1083] [INSPIRE].
A.J. Amsel and D. Gorbonos, Wald-like formula for energy, Phys. Rev. D 87 (2013) 024032 [arXiv:1209.1603] [INSPIRE].
K. Sen and A. Sinha, Holographic stress tensor at finite coupling, JHEP 07 (2014) 098 [arXiv:1405.7862] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1508.00766
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Haehl, F.M. Comments on universal properties of entanglement entropy and bulk reconstruction. J. High Energ. Phys. 2015, 159 (2015). https://doi.org/10.1007/JHEP10(2015)159
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2015)159