Advances in High Energy Physics

Volume 2017 (2017), Article ID 6402101, 8 pages

https://doi.org/10.1155/2017/6402101

## Van der Waals-Like Phase Transition from Holographic Entanglement Entropy in Lorentz Breaking Massive Gravity

^{1}School of Science, Hubei University for Nationalities, Enshi 445000, China^{2}College of Science, Agricultural University of Hebei, Baoding 071000, China^{3}School of Material Science and Engineering, Chongqing Jiaotong University, Chongqing 400074, China

Correspondence should be addressed to Xiao-Xiong Zeng

Received 12 June 2017; Revised 15 August 2017; Accepted 24 August 2017; Published 2 October 2017

Academic Editor: Li Li

Copyright © 2017 Xian-Ming Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Phase transition of AdS black holes in Lorentz breaking massive gravity has been studied in the framework of holography. We find that there is a first-order phase transition (FPT) and second-order phase transition (SPT) both in Bekenstein-Hawking entropy- (BHE-) temperature plane and in holographic entanglement entropy- (HEE-) temperature plane. Furthermore, for the FPT, the equal area law is checked and for the SPT, the critical exponent of the heat capacity is also computed. Our results confirm that the phase structure of HEE is similar to that of BHE in Lorentz breaking massive gravity, which implies that HEE and BHE have some potential underlying relationship.

#### 1. Introduction

The study of HEE and quantum phase transitions of black holes has attracted a lot of interest in recent years. On one hand, HEE can be used as a perfect probe to study quantum information science [1–3], strongly correlated quantum systems [4–13], and Many-Body Systems [14, 15]. On the other hand, investigation on HEE of black holes may shed some light on understanding the nature of BHE [16, 17].

Nearly ten years ago, a holographic derivation of the HEE in conformal quantum field theories was proposed by Ryu and Takayanagi using the famous AdS/CFT correspondence [18, 19]. Recently the HEE has been used as a probe to investigate the phase structure of the Reissner-Nordstrom AdS black hole [20]. The results showed that there is a Van der Waals-like (VDW) phase transition at the same critical temperature in both the fixed charge ensemble and chemical potential ensemble in the HEE-temperature plane. They also found that the SPT occurs for the HEE at the same critical point as the BHE with nearly the same critical exponent. This work was soon generalized to the extended phase space where the cosmological constant is considered as a thermodynamical variable [21]. Very recently, the equal area law of HEE was proved to hold for the FPT in the HEE-temperature plane [22]. Based on [20], VDW phase transition of HEE in various AdS black holes has been studied in [23–32]. All of these works showed that the HEE undergoes the same VDW phase transition as that of the BHE.

Massive gravity theories have attracted considerable interest recently. One of these reasons is that these alternative theories of gravity could explain the accelerated expansion of the universe without dark energy. The graviton behaves like a lattice excitation and exhibits a Drude peak in this theory. Current experimental data from the observation of gravitational waves by advanced LIGO require the graviton mass to be smaller than the inverse period of orbital motion of the binary system; that is, eV/*c*^{2} [33]. Another important reason for the interest in massive gravity is that the possibility of the mass graviton could help to understand the quantum gravity effect. The first to introduce a mass to the graviton is in [34]. However this primitive linear massive gravity theory contains the so-called Boulware-Deser ghosts problem [35] that was solved by a nonlinear massive gravity theory [36, 37], where the mass terms are obtained by introducing a reference metric. Recently Vegh proposed a new reference metric to describe a class of strongly interacting quantum field theories with broken translational symmetry in the holographic framework [38]. The recent progress in massive gravity can be found in [39, 40].

Here, we consider AdS black holes in Lorentz breaking massive gravity. In the massive gravity, the graviton acquires a mass by Lorentz symmetry breaking, which is very similar to the Higgs mechanism. A review of Lorentz-violating massive gravity theory can be found in [41, 42]. In this paper, we focus on the study of the VDW phase transition of AdS black holes in Lorentz breaking massive gravity using the HEE. The main motivation of this paper is to explore whether the BHE phase transition can also be described by HEE in Lorentz breaking massive gravity. Firstly, we would like to extend proposals in [20] to study VDW phase transitions in AdS black hole with a spherical horizon in Lorentz-violating massive gravity with the HEE as a probe. What is more, we also would like to check Maxwell’s equal area law and critical exponent of the heat capacity, which are two characteristic quantities in VDW phase transition.

The organization of this paper is as follows. In the next section, we shall provide a brief review of the black hole solution in Lorentz breaking massive gravity firstly. Then we will study the VDW phase transitions and critical phenomena for the AdS black hole in the BHE-temperature plane. In Section 3, we mainly concentrate on the VDW phase transition and critical phenomena in the framework of HEE. The last section is devoted to our discussions and conclusions.

#### 2. Phase Transition and Critical Phenomena of AdS Black Holes in Lorentz Breaking Massive Gravity

##### 2.1. Review of AdS Black Holes in Lorentz Breaking Massive Gravity

The four-dimensional Lorentz breaking massive gravity can be obtained by adding nonderivative coupling scalar fields to the standard Einstein gravity theory. As a matter field is considered, the theory can be described by the following action [41, 42]:here the first two terms are the curvature and ordinary matter minimally coupled to gravity, respectively, and the third term contains two functions and which relate to the four scalar fields, and asWhen the four scalar fields get a space-time depending vacuum expectation value, the system will break the Lorentz symmetry. What is more, the action can also be taken as a low-energy effective theory with the ultraviolet cutoff scale . Here the scale parameter has the dimension of mass and is in the order of , where and are the graviton mass and the Plank mass, respectively.

The AdS black hole solutions can be obtained from the above theory [43, 44]. The metric corresponding to the AdS black holes is given bywithHere, the four scalar fields, and , for this particular solution are given by in whichin which the scalar charge is related to massive gravity and the constant which is determined by the cosmological constant and the graviton mass with the relation . In this paper, we will set such that leading to Anti-de Sitter black holes. The constant is a positive integration constant. When , the ADM mass of the black hole solution diverges. For , the metric approaches the Schwarzschild-AdS black holes with a finite mass as . Thus we set in this paper. The constant . When , the black hole only has a single horizon , which is the root of the equation . The function for this case is given in Figure 1, which is similar to the Schwarzschild-AdS black hole. For , the black hole is very similar to the Reissner-Nordstrom-AdS black hole. The function for this case is given in Figure 2. The black hole event horizon is the largest root of the equation .