Advances in High Energy Physics

Volume 2017 (2017), Article ID 2356174, 8 pages

https://doi.org/10.1155/2017/2356174

## Holographic van der Waals Phase Transition for a Hairy Black Hole

^{1}School of Material Science and Engineering, Chongqing Jiaotong University, Chongqing 400074, China^{2}State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China^{3}School of Computer Science and Information Engineering, Chongqing Technology and Business University, Chongqing 400070, China

Correspondence should be addressed to Xiao-Xiong Zeng

Received 2 March 2017; Accepted 29 May 2017; Published 12 July 2017

Academic Editor: Bum-Hoon Lee

Copyright © 2017 Xiao-Xiong Zeng and Yi-Wen Han. 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

The van der Waals (VdW) phase transition in a hairy black hole is investigated by analogizing its charge, temperature, and entropy as the temperature, pressure, and volume in the fluid, respectively. The two-point correlation function (TCF), which is dual to the geodesic length, is employed to probe this phase transition. We find the phase structure in the temperature-thermal entropy plane besides the scale of the horizontal coordinate (geodesic length plane resembles that in the temperature). In addition, we find the equal area law (EAL) for the first-order phase transition and critical exponent of the heat capacity for the second-order phase transition in the temperature-thermal entropy plane (geodesic length plane is consistent with that in temperature), which implies that the TCF is a good probe to probe the phase structure of the back hole.

#### 1. Introduction

Phase transition of the black holes is always a hot topic in theoretical physics for it provides a platform to relate the gravity, thermodynamics, and quantum theory. Phase transition in AdS space is more fascinated owing to the AdS/CFT correspondence. The Hawking-Page phase transition [1], which portrays the transition of thermal gas to the Schwarzschild black hole, can be used to describe the confinement to deconfinement transition of the quark-gluon plasma in Yang-Mills theory [2]. The phase transition of a scalar field condensation around charged AdS black holes can be used to describe the superconductor transition [3–5]. In particular, one often can use the nonlocal observables such as holographic entanglement entropy, Wilson loop, and TCF to probe these phase transitions [6, 7].

VdW phase transition is another important property of the charged AdS black hole. It was observed that a charged black hole will undergo a first-order phase transition and a second-order phase transition successively as the charge of the black hole increases from small to large, which is analogous to the van der Waals liquid-gas phase transition [8]. This phase transition was perfected recently by regarding the cosmological constant as the pressure for in this case we need not any analogy [9, 10].

Whether the VdW phase transition of the charged black hole can be probed by the nonlocal observables thus is worth exploring. Recently, Chamblin et al. [11] investigated the VdW phase transition of the Reissner-Nordström AdS black hole from the viewpoint of holography and found that the phase structure in the temperature-entanglement entropy plane resembles that in temperature-thermal entropy plane. Thereafter Nguyen [12] investigated exclusively the EAL in the temperature-entanglement entropy plane and found that the EAL holds regardless of the size of the entangling region. Now there have been some extensive studies [13–19] and all the results show that, in the case of thermal entropy, the entanglement entropy exhibits the similar VdW phase transition.

In this paper, we are going to use the equal time TCF to probe the phase structure of the hairy black holes. It has been shown that the TCF has the same effect as that of the holographic entanglement entropy as it was used to probe the thermalization behavior [20–33]; thus, it will be interesting to explore whether this observable can probe the phase structures of the black holes. The hairy AdS black hole is a solution of Einstein-Maxwell- theory conformally coupled to a scalar field [34]. This model has at least two advantages. One is that it is an ordinary and tractable model for studying superconducting phase transition with consideration of the back-reaction of the scalar field. Another is that it exhibits more fruitful phase transition behavior, namely, not only the VdW behavior in both the charged and uncharged cases, but also reentrant phase transition in the charged case [35]. In this paper, we mainly concentrate on the VdW phase transition behavior. Beside the thermal entropy-temperature plane, we will also study the EAL and critical exponent of the heat capacity in the geodesic length-temperature plane. We find that the results obtained in both frameworks are consistent.

Our paper is outlined as follows. In Section 1, we present the hairy AdS black hole solution and study the VdW phase transition in the thermal entropy-temperature plane. In Section 2, we employ the TCF to probe the VdW phase transition. In particular, we study the EAL and critical exponent of the heat capacity in the framework of holography and find that the result is similar as that obtained in the thermal entropy-temperature plane. The conclusion and discussion are presented in Section 3. Hereafter in this paper we use natural units () for simplicity.

#### 2. Thermodynamic Phase Transition of the Hairy Black Hole

The five-dimensional hairy black hole solution can be written as [34]in whichwhere is the electric charge, is the mass parameter, is the AdS radius that relates to the cosmology constant , and is related to the coupling constants of the conformal field , , and . For the planar solution, , while for the spherical symmetric black hole, can be expressed asin which takes the values , , and . In addition, to satisfy the field equations, the scalar coupling constants should obey the constraint As stressed in [34, 35], the hair parameter is not a conserved charge corresponding to some symmetry; it is a variable provided that the scalar field coupling constants are dynamic. In this paper, we will fix to investigate the phase structure of this black hole for has little effect on the phase structure.

The black hole event horizon is the largest root of the equation . At the event horizon, the Hawking temperature can be expressed asin which we have used the relation

In terms of the AdS/CFT correspondence, the temperature in (5) can be treated as the temperature of the dual conformal field theory. The Maxwell potential in this background is given by The entropy of the black hole isSubstituting (8) into (5), we can get the following relation:

Next, we will employ this equation to study the phase structure of the hairy black hole. The Helmholtz free energy of this system is given by [34] in which , where is the gravitational constant.

Now we concentrate on studying the phase structure of the hairy AdS black holes. In fluid, the VdW phase transition is depicted in the - plane, where correspond to pressure and volume of fluid. In black holes, there are two schemes to produce the VdW phase transition. One is presented in [9, 10] where the cosmology constant and curvature are treated as pressure and volume. The other is presented in [8] in which one should adopt analogy shown in Table 1.