Advances in High Energy Physics

Volume 2016 (2016), Article ID 6153435, 13 pages

http://dx.doi.org/10.1155/2016/6153435

## Holographic Phase Transition Probed by Nonlocal Observables

^{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}Center for Space Science and Applied Research, Chinese Academy of Sciences, Beijing 100190, China

Received 31 March 2016; Accepted 8 June 2016

Academic Editor: Davood Momeni

Copyright © 2016 Xiao-Xiong Zeng and Li-Fang Li. 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

From the viewpoint of holography, the phase structure of a 5-dimensional Reissner-Nordström-AdS black hole is probed by the two-point correlation function, Wilson loop, and entanglement entropy. As the case of thermal entropy, we find for all the probes that the black hole undergoes a Hawking-Page phase transition, a first-order phase transition, and a second-order phase transition successively before it reaches a stable phase. In addition, for these probes, we find that the equal area law for the first-order phase transition is valid always and the critical exponent of the heat capacity for the second-order phase transition coincides with that of the mean field theory regardless of the size of the boundary region.

#### 1. Introduction

Phase transition is a ubiquitous phenomenon for garden-variety thermodynamic systems. Due to the pioneering work by Hawking [1, 2], a black hole is also a thermodynamic system. Such a fact is further supported by AdS/CFT correspondence [3–5], where a black hole in the AdS bulk is dual to a thermal system without gravity. So one can naturally expect that a black hole can also undertake some interesting phase transitions as the general thermodynamic system. Actually it has been shown that a charged AdS black hole undergoes a Hawking-Page phase transition [6, 7], which is interpreted as the confinement/deconfinement phase transition in the dual gauge field theory [8] and a van der Waals-like phase transition before it reaches the stable state [9]. The Hawking-Page phase transition implies that the thermal AdS is unstable and it will transit to the stable Schwarzschild AdS black hole lastly. The van der Waals-like phase transition has been observed till now in many circumstances. The first observation was contributed by [9] in the plane. Specifically speaking, in a fixed charge ensemble, for a black hole endowed with small charge, there is an unstable black hole interpolating between the stable small hole and stable large hole, and the small stable hole will undertake a first-order phase transition to the large stable hole as the temperature of the black hole reaches a critical temperature. As the charge increases to the critical charge, the small hole and the large hole merge into one and squeeze out the unstable phase so that an inflection point emerges and the phase transition is second order. When the charge exceeds the critical charge, the black hole is always stable. Recently in the extended phase space, where the negative cosmological constant is treated as the pressure while its conjugate acts as the thermodynamical volume, the van der Waals-like phase transition has also been observed in the plane [10–16]. In addition, it was shown in [17] that the van der Waals-like phase transition also shows up in the plane. Particularly, in the Gauss-Bonnet gravity, it is found that the Gauss-Bonnet coupling parameter also affects the phase structure of the space time, and in the plane, a 5-dimensional neutral Gauss-Bonnet black hole also demonstrates the van der Waals-like phase transition [18].

In this paper, we intend to probe the Hawking-Page phase transition and van der Waals-like phase transition appeared in a 5-dimensional Reissner-Nordström-AdS black hole by the geodesic length, minimal area surface, and minimal surface area in the bulk, which are dual to the nonlocal observables on the boundary theory by holography, namely, the two-point correlation function, Wilson loop, and entanglement entropy, individually (recently these nonlocal observables have been used to probe the nonequilibrium thermalization process, and it has been found that all of them have the same effect [19–25]). In fact, there have been some similar works to probe the phase structure by holographic entanglement entropy. In [26], the phase structure of entanglement entropy is studied in the plane for both a fixed charge ensemble and a fixed chemical potential ensemble, and it is found that the phase structure of entanglement entropy is similar to that of the thermal entropy. In particular, the entanglement entropy is found to demonstrate the same second-order phase transition at the critical point as the thermal entropy. Soon after, it is found that the entanglement entropy can also probe the van der Waals-like phase transition in the plane [27]. In [28], Nguyen has investigated exclusively the equal area law of holographic entanglement entropy and found that the equal area law holds regardless of the size of the entangling region. Very recently [29] investigated entanglement entropy for a quantum system with infinite volume; their result showed that the entanglement entropy also exhibits the same van der Waals-like phase transition as the thermal entropy. They also checked the equal area law and obtained the critical exponent of the heat capacity near the critical point.

In this paper, we will further investigate whether one can probe the phase structure by two-point correlation function and Wilson loop besides the entanglement entropy. We intend to explore whether they exhibit the similar van der Waals-like phase transition as the entanglement entropy and thermal entropy. In addition, we also want to check whether these nonlocal observables can probe the Hawking-Page phase transition between the AdS black hole and thermal gas so that we can get a complete picture about the phase transition of the black holes in the framework of holography.

This paper is organized as follows. In Section 2, we will discuss the thermal entropy phase transition of a 5-dimensional Reissner-Nordström-AdS black hole in the plane in a fixed charge ensemble. Then in Section 3, we will probe these phase transitions by geodesic length, Wilson loop, and holographic entanglement entropy individually. In each subsection, the equal area law is checked and the critical exponent of the heat capacity is obtained for different sizes of the boundary region. Section 4 is devoted to discussions and conclusions.

#### 2. Thermodynamic Phase Transition of the 5-Dimensional Reissner-Nordström-AdS Black Hole

Starting from the actionwhere and is the AdS radius, we shall focus on the case of , in which the charged Reissner-Nordström-AdS black hole can be written as [9]where , , and are hyperspherical coordinates for the 3 spheres, andwith and being the mass and charge of the black hole. Whence we can get the Hawking temperature of this space time asIn addition, it follows from the Bekenstein-Hawking formula that the entropy of the black hole is given bywhere is the outer event horizon of the black hole, namely, the largest root of the equation . With this, the mass of the back hole can thus be expressed as the function of the event horizon:Substituting (5) and (6) into (4), we can get the relation between the temperature and entropy of the 5-dimensional Reissner-Nordström-AdS black hole:In addition, with the relation , the Helmholtz free energy can be expressed asNote that this formula for our free energy has implicitly chosen the pure AdS as the reference space time because the free energy vanishes for pure AdS by this formula. Now let us review the relevant phase transitions in the fixed charge ensemble by (7) and (8) in the plane.

To achieve this, we should first find the critical charge by the following equations:Inserting (7) into (9), we can get the values for the critical charge and critical entropy:Substituting these critical values into (7), we can get the critical temperature:

We plot the discharge curves for different charges in Figure 1. For the case , there is a minimum temperature [30], which is indicated by the red solid line in . When the temperature is lower than , we have only a thermal AdS. When the temperature is higher than , there are two additional black hole branches. The small branch is unstable while the large branch is stable. This can be justified by checking the corresponding heat capacities, which is related to their slopes. The Hawking-Page phase transition occurs at the temperature given by [30], which is higher than and indicated by the red dashed line. This can be observed by the relation in Figure 2(a), where is the horizontal coordinate of the cusp and is the horizontal coordinate for the intersection of the stable branch and the horizontal axis. Obviously, when the temperature is lower than , the thermal AdS is the most stable state. While when the temperature is higher than , the most stable state is taken over by the large black hole branch.