Thermodynamics and holographic entanglement entropy for spherical black holes in 5D Gauss-Bonnet gravity
Received: June
Thermodynamics and holographic entanglement entropy for spherical black holes in 5D Gauss-Bonnet gravity
Yuan Sun 0 1
Hao Xu 0 1
Liu Zhao 0 1
0 Tianjin 300071 , China
1 School of Physics, Nankai University
The holographic entanglement entropy is studied numerically in (4+1)dimensional spherically symmetric Gauss-Bonnet AdS black hole spacetime with compact boundary. On the bulk side the black hole spacetime undergoes a van der Waals-like phase transition in the extended phase space, which is reviewed with emphasis on the behavior on the temperature-entropy plane. On the boundary, we calculated the regularized HEE of a disk region of di erent sizes. We nd strong numerical evidence for the failure of equal area law for isobaric curves on the temperature-HEE plane and for the correctness of rst law of entanglement entropy, and brie y give an explanation for why the latter may serve as a reason for the former, i.e. the failure of equal area law on the temperature-HEE plane.
Gauss-Bonnet; AdS-CFT Correspondence; Black Holes; Conformal Field Theory
1 Introduction
2
3
4
5
8 , the thermodynamics of black
hole can be established on the extended phase space [7, 8]. The physical meaning of the
thermodynamical volume that is conjugate to the e ective pressure P remains to be fully
understood, but it is conjectured to satisfy the reverse isoperimetric inequality [9]. In
this consideration, the black hole mass is taken as the enthalpy H rather than the internal
energy. The extended phase space thermodynamics has been investigated for many di erent
spacetimes [10{28], and in many cases the extended phase space thermodynamic behavior
is very similar to van der Waals liquid-gas system.
Up to now the dual eld theory interpretation of the van der Waals-like phase transition
remains unknown. However, progress has been made in this direction recently. In ref. [29],
it was found that the holographic entanglement entropy (HEE) as a function of temperature
behaves qualitatively the same as black hole entropy in the context of a charged black hole
in AdS background with
nite volume. In this case, the HEE undergoes van der Waals-like
phase transitions, and an in exion point appears on the temperature-HEE curve at the
same critical temperature. More recently, the similarity between the two kinds of entropies
has been investigated further by considering Maxwell's equal area law, which holds for black
hole entropy, and seems to be still valid on the HEE-temperature curve [30]. The numerical
{ 1 {
results show that for RN-AdS black holes this \equal area law" on the HEE-temperature
curve holds up to an accuracy of around 1%, however, it fails for dyonic RN-AdS black
holes. Therefore, to get a better understanding of the eld theory interpretation of the van
der Waals-like phase transitions, it is important to examine whether these ideas applies to
other gravity models. This connection has been extended to other cases [31{34], including
the extended phase space. It seems that the HEE can be a nice probe of the extended
phase space.
Motivated by the above considerations and progresses, we extended the study of van
der Waals-like behavior for HEE to Gauss-Bonnet AdS black holes with a spherical horizon
in (4+1)-dimensions. The thermodynamics of this particular black hole spacetime has been
studied in the extended phase space in [14]. It was shown that for GB-AdS black holes,
V criticality and phase transition only occurs when the black hole has a spherical
horizon. When the charge of the black hole is turned o , only in (4+1)-dimension the
P
V criticality and phase transition takes place.
The inclusion of Gauss-Bonnet term is a non-trivial generalization of Einstein gravity.
As a consequence, one must employ the HEE formula for general higher derivative
gravity [35{38]. We will show that that the equal area law on the temperature-HEE plane fails
but a van de Waals-like behavior on both the temperature-HEE and the temperature-black
hole mass curves indeed holds.
The rest of the paper is organized as follows: in section 2, we review the black hole
thermodynamics for spherically symmetric GB-AdS black holes, and discuss the critical
behavior and Maxwell equal area law on the entropy-temperature plane. In section 4, we
brie y review the holographic entanglement entropy in Gauss-Bonnet gravity and present
the HEE formula for our setup. In section 4, The numerical results are presented, which
include, in particular, the numerical evidence for the failure of the equal area law on
the temperature-HEE plane and the correctness of the rst law of entanglement entropy,
which has never been established before. By employing the linear relationship between
HEE and black hole mass, we give an explanation for why the equal area law fails on the
the temperature-HEE plane. In the nal section, we present some concluding remarks.
2
Thermodynamics for Gauss-Bonnet AdS black holes
In this section, we give a brief review of the therm (...truncated)