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...geometry with arbitrary boundary conditions at τ = 0, 2pi. It can be computed using the bulk Hamiltonian for τ evolution. 2.2 Quantum argument This is a generalization of the black hole discussion in =-=[14, 15]-=- to situations without the U(1) symmetry. At the quantum level, the replica trick instructs us to compute the partition function of all the bulk quantum fields around the black hole geometry. This inv...

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...phase transitions, the entanglement entropy can act as a diagnostic for the nature of the phase transition. Entanglement entropy has been the subject of various recent investigations (including Refs. =-=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]-=-). A central feature of the entanglement entropy of fields in spatial dimensions d > 1 is that the entropy is strictly cutoff dependent (unlike the d = 1 case which involves only a logarithm [11, 12])...

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...ear in (53) coincide with those of [23]. Another issue is temperature-related. We used the zero-temperature field theory setup even though the black hole has Hawking temperature. (See, e.g., [41] and =-=[43]-=- for the finite temperature approach.) As wellknown, the black hole temperature was an induced effect in the original paper by Hawking. Therefore it is natural to expect that temperature should enter ...

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...anding the entanglement entropy in the quantum field theories (for a review see [3]), in studying its holographic interpretation [4] and in the promising applications to black holes (for a review see =-=[5]-=-). In the standard set-up the entanglement entropy is defined with respect to a given surface Σ that can be any co-dimension two surface lying in the hypersurface of constant time. The intrinsic and e...

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... Section 6.2, where we discuss this in more detail. • Some examples of surfaces that turn on fc(n) are black hole horizons, as black hole backgrounds generically have a non vanishing Weyl tensor; see =-=[20]-=- for a detailed overview. • One can show, using (3.17), that the only pair (fa(n), fc(n)) for which fa(n)/fc(n) is independent of n is that of the free scalar [19], fc(n) = 3fa(n) = 3a (1 + n)(1 + n2)...

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...It is a natural generalization of Gibbons-Hawking’s analysis of black hole thermodynamics[27]. 3. Conical singularity method[31]. The gravitational entropy can be calculated by the conical singularity=-=[32, 33]-=- method, the authors in [31] found an extension to the higher spin gravity. 4. Wilson line approach[28, 39]. This method can be generalized to calculate classical holographic entanglement entropy stra...

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...ment entropy is the entropy S = −∑i p ln pi of the eigenvalues of the density matrix M . Entanglement entropy has been studied in many works with seminal papers including [35, 36]. Useful reviews are =-=[37, 38]-=-. For our purposes the most important result is that the dominant contribution to entanglement entropy depends on the area of the surface bounding the two regions of space and the ultraviolet cutoff, ...

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...ting this out and for the reference [21]. 3 membrane, i.e. a causal boundary of the spacetime. This is at the basis of trying to interpret the black hole entropy as an entanglement entropy (see, e.g. =-=[30]-=-). With the conformal symmetries mentioned above, the horizon could be more than just a causal boundary. For a two dimensional statistical system, conformal symmetry usually arises at critical points ...