Advances in High Energy Physics

Volume 2019, Article ID 8270265, 8 pages

https://doi.org/10.1155/2019/8270265

## New Phase Transition Related to the Black Hole’s Topological Charge

Department of Physics, Lingnan Normal University, Zhanjiang, Guangdong 524048, China

Correspondence should be addressed to Shan-Quan Lan; moc.621@nalnauqnahs

Received 23 May 2018; Accepted 13 December 2018; Published 2 January 2019

Guest Editor: Farook Rahaman

Copyright © 2019 Shan-Quan Lan 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

The topological charge of AdS black hole is introduced by Tian et al. in their papers, where a complete thermodynamic first law is obtained. In this paper, we investigate a new phase transition related to the topological charge in Einstein-Maxwell theory. Firstly, we derive the explicit solutions corresponding to the divergence of specific heat and determine the phase transition critical point. Secondly, the curve and curve are investigated and they exhibit an interesting van der Waals system’s behavior. Critical physical quantities are also obtained which are consistent with those derived from the specific heat analysis. Thirdly, a van der Waals system’s swallow tail behavior is observed when in the graph. What is more, the analytic phase transition coexistence lines are obtained by using the Maxwell equal area law and free energy analysis, the results of which are consistent with each other.

#### 1. Introduction

Black hole is a complicated object; there are Hawking radiation, entropy, phase transition, etc. Although black hole’s microscopic mechanism is still not clear, its thermodynamic properties can be systematically studied as it is a thermodynamic system which is described by only few physical quantities, such as mass, charge, angular momentum, temperature, and entropy. Since the establishing of the four laws of black hole by Bardeen, Carter, and Hawking [1], black hole thermodynamics has become an exciting and extensively studied topic. Especially, in the anti-de Sitter space, there exists Hawking-Page phase transition [2] which is explained as the confinement/deconfinement phase transition of a gauge field [3] due to the AdS/CFT duality [4–6] and phase transition between small and large charged black holes which is reminiscent to the liquid-gas phase transition of van der Waals system [7]. This close relation between AdS black holes and van der Waals liquid-gas system has been further enhanced by the seminal work of Kubiznak and Mann in [8] where the cosmological constant is identified as thermodynamic pressure and the mass of the black hole is identified as the enthalpy [9]. The curves and the Gibbs free energy graphs in this extended phase space are shown to exhibit an interesting van der Waals systems behavior. For a review on this topic, one can refer to [10] and the references therein. However, in this paper, we will treat the cosmological constant as a constant, leaving the other case for future investigation [11].

Generally, the thermodynamic quantities are described on the horizon and they are related by the first law. However, they can be generalised on surface out of the horizon [12–14]. This has gotten new attention with the development of AdS/CFT, since the black hole thermodynamics on holographic screen has acquired a new and interesting interpretation as a duality of the correspondence field theory [15]. In [16, 17], a maximally symmetric black hole thermodynamics on holographic screen are studied in Einstein-Maxwell’s gravity and Lovelock-Maxwell theory. The author found a topological charge naturally arisen in holography. Together with all other known charges (electric charge, mass, and entropy [18]), they satisfy an extended first law and the Gibbs-Duhem-like relation as a completeness. Based on the extended first law in Einstein-Maxwell’s gravity, we will investigate the black hole’s possible phase transition phenomenon related to the topological charge. Actually, we found that the curves and the free energy graphs also exhibit an interesting van der Waals systems behavior as the extended phase space case does. This result is unexpected, noting that the cosmological constant is not treated as thermodynamic pressure here.

This paper is organized as follows. In Section 2 we will briefly review how the extended first law is obtained in [17]. In Section 3, by analysing the specific heat, the phase transition of AdS black hole in 4-dimensional space-time is studied and the critical point is determined. Then the van der Waals like behavior of temperature is observed in both graph and graph in Section 4. In Section 5 we use the Maxwell equal area law and free energy to have obtained a consistent phase transition coexistence line. Conclusions are drawn in Section 6.

#### 2. Review of the Topologically Charged AdS Black Holes

A d-dimensional space-time AdS black hole solution with the extra topological charge in the Einstein-Maxwell theory was investigated in [16, 17]. The metric readswhere are related to the ADM mass , electric charge , and cosmological constant byand is the volume of the “unit" sphere, plane, or hyperbola and stands for the spatial curvature of the black hole. Under suitable compactifications for , we assume that the volume of the unit space is a constant hereafter [16, 17].

Following [17], the first law can be obtained. Considering an equipotential surface with fixed , which can be rewritten asdefining , we have Noting we obtainMultiplying both sides with a constant factor , the above equation becomeswhere is the Unruh-Verlinde temperature [12, 19], is the Wald-Padmanabhan entropy [18, 20], and is the electric potential. If we introduce a new “charge” as in [16, 17] and denote its conjugate potential as , then the generalized first law is This new charge is called the last (lost) charge of a black hole, and it together with all other known charges satisfies the Gibbs-Duhem-like relation as a completeness relation [17].

#### 3. A New Phase Transition of AdS Black Hole

From the generalized first law, we see there is a topological charge . In this section, we will investigate the phase transition of AdS black hole in dimensional space-time in canonical ensemble related to the topological charge rather than the electric charge. To do so, one can observe the behavior of the specific heat at constant topological charge [21].

The Unruh-Verlinde temperature is Setting hereafter, the corresponding specific heat with topological charge fixed can be calculated as From the denominator, we can conclude the following.

When , has two diverge points at which corresponds to When , has only one diverge point at which corresponds to The temperature is .

When , .

Figure 1 shows the behavior of specific heat for the cases , , . For , there are two divergent points on the specific heat curve; they divide the region into three parts: the large radius region, the medium radius region, and the small radius region. With positive specific heat, both the large radius region and the small radius region are thermodynamically stable. While with negative specific heat, the medium radius region is unstable. So there is a phase transition taking place between small black hole and large black hole. For , the curve of specific heat has only one divergent point and is always positive which denote that is the phase transition critical point. While, for , the curve of specific heat has no divergent point and is always positive, which denote that the black holes are stable and no phase transition will take place.