Advances in High Energy Physics

Volume 2018, Article ID 2175818, 14 pages

https://doi.org/10.1155/2018/2175818

## Phase Transition in Horava Gravity

School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China

Correspondence should be addressed to Wei Xu; nc.ude.guc@iewux

Received 27 March 2018; Revised 5 July 2018; Accepted 22 July 2018; Published 9 August 2018

Academic Editor: Saibal Ray

Copyright © 2018 Wei Xu. 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

We present another example of superfluid black hole containing phase transition in Horava gravity. After studying the extended thermodynamics of general dimensional Horava-Lifshitz AdS black holes, it is found that only the one with spherical horizon in four and five dimensions has phase transition, which is a line of (continuous) second-order phase transitions and was famous in the discussion of superfluidity of liquid . The “superfluid” black hole phase and “normal” black hole phase are also distinguished. Particularly, six-dimensional Horava-Lifshitz AdS black holes exhibit infinitely many critical points in plane and the divergent points for specific heat, for which they only contain the “normal” black hole phase and the “superfluid” black hole phase disappears due to the physical temperature constraint; therefore there is no similar phase transition. In more than six dimensions, there is no critical behavior. After choosing the appropriate ordering field, we study the critical phenomena in different planes of thermodynamical phase space. We also calculate the critical exponents, which are the same as the van der Waals fluid.

#### 1. Introduction

Black hole thermodynamics always provides valuable insight into quantum properties of gravity, and it has been studied extensively for quite a long time, especially for the quantum and microscopic interpretation of black hole temperature and entropy (see [1, 2], for example). Besides, thermodynamics and phase transitions of AdS black holes have been of great interest since the Hawking-Page phase transition [3] between stable large black hole and thermal gas is explained as the confinement/deconfinement phase transition of gauge field [4] inspired by the AdS/CFT correspondence [5–7].

After treating the cosmological constant as a pressure with its conjugate quantity being the thermodynamic volume in thermodynamic phase space of charged AdS black holes [8–14], the small/large black hole phase transition is established in [15], which is exactly analogous to the liquid/gas phase transition of the van der Waals fluid. This kind of black hole phase transitions has attracted much attention (see the recent review papers [16, 17]). Besides, this semiclassical method of analogue is generalized to study the microscopic structure of black holes and sheds some light on the black hole molecules [18] and microscopic origin of the black hole reentrant phase transition [19]. The study is also extended to the quantum statistic viewpoint, as the superfluid black holes are reported recently [20]. In Lovelock gravity with conformally coupled scalar field, the authors present the first example of phase transition, which is a line of (continuous) second-order phase transitions and was famous for the successful quantum and microscopic interpretation of superfluidity of liquid He.

In this paper, we present another example of black holes containing phase transition in Horava gravity, which is a candidate of quantum gravity in ultrahigh energy [21]. The Horava-Lifshitz (HL) black hole solutions, thermodynamics, and phase transitions have attracted a lot of attention [22–29] (see [30] for a review on the recent development of various areas). The general dimensional HL black hole solutions are also introduced [31]. We will consider the extended thermodynamics of general dimensional HL AdS black holes. It is shown that only the one with spherical horizon in four and five dimensions has phase transition. Note that the first example of “superfluid” black holes always has a hyperbolic horizon [20]. Particularly, six-dimensional HL AdS black holes exhibit infinitely many critical points in plane and the divergent points for specific heat, for which they only contain the “normal” black hole phase and the “superfluid” black hole phase disappears due to the physical temperature constraint; therefore there is no similar phase transition. In more than six dimensions, there is no critical behavior. After identifying parameter as the ordering field instead of pressure and temperature, we study the critical phenomena in different planes of thermodynamical phase space. We also obtain the critical exponents, which are the same as the van der Waals fluid.

The paper is structured as follows: in next section, we present the extended thermodynamics of generalized topological HL black holes. Then we study criticality in Section 3. We show the phase transition for four and five dimensions and the discussion for six dimensions in Sections 4 and 5, respectively. In Sections 6 and 7, we discuss the critical phenomena and calculate the critical exponents in different planes. In final section, some concluding remarks are given.

#### 2. Extended Thermodynamics of Generalized HL Black Holes

In this section, we present the extended thermodynamics of generalized topological HL black holes in dimensions (). We begin with the action of HL gravity at the UV fixed point, which can be reexpressed as [31]where the first two terms in the are the kinetic actions, while the residue corresponds to the potential actions. is the Ricci tensor, is the Ricci scalar, and is defined by , which are based on the ADM decomposition of the higher dimensional metric, i.e., . Here the lapse, shift, and -metric , , and are all functions of and , and a dot denotes a derivative with respect to . There are five constant parameters in the action: , , , , and . is the cosmological constant. and have their origin as the Newton constant and the speed of light. represents a dynamical coupling constant which is susceptible to quantum corrections. We will fix in the following paper, only for which general relativity can be recovered in the large distance approximation. In addition, we will only consider the general values of in the region , as corresponds to the so-called detailed-balance condition, and HL gravity with returns back to general relativity.

Action Eq. (1) admits arbitrary dimensional topological AdS black holes with the metric and the horizon function [23, 31] where denotes the line element of a -dimensional manifold with constant scalar curvature , and indicates different topology of the spatial space. is integration constant, which is related to the black hole mass where we have chosen the natural units and .

In AdS space-time, the cosmological constant is introduced as the thermodynamical pressure [15]: Then we can rewrite the horizon function as and the mass is where denotes the event horizon which is the largest positive root of . The conjugate thermodynamic volume of pressure is The entropy and temperature are presented in [31] with the following forms:It is easy to check the first law of thermodynamics: while the Smarr relation always fails to exist. This can be easily found because of the existence of (and logarithmic term for ) in black hole entropy, which is not fixed and may be calculated by invoking the quantum theory of gravity as argued in [23]. Note that scales as , and there is no other dimensional quantity in extended thermodynamic phase space. In this meaning, the validity of the Smarr relation will bring another physical consideration on the parameter .

In order to analyze the global thermodynamic stability and phase transition of the HL black hole, it is always to study the Gibbs free energy: as the black hole mass should be considered as the enthalpy in the extended thermodynamic phase space. For the local thermodynamic stability, one can turn to the specific heat of black holeWe present the above two quantities in the Appendix, as their forms are very complicated.

#### 3. Criticality

According to (10), we can obtain the equation of state (EOS):which reflects the double-valuedness of the pressure in the extended thermodynamic phase space. On the other hand, we can derive the pressure aswhile another one is the negative pressure branch. After taking a series expansion, it leads to Comparing the above equation with the Van der Waals equation one can easily find the specific volume . Therefore we will just use the horizon radius in EOS instead of the specific volume and study the behavior in the following paper.

To consider the criticality, we can focus on to find the critical points. As the direct differentiation of (see (15)) is too complicated, we prefer to employ the implicit differentiation on EOS (see (14)) and the above equations, and we can derive two simple conditions: They lead to the critical points: by which, the EOS (see (14)) is simplified as an equation of and results in the following condition: where

It is interesting that there is no critical volume or horizon radius, but the critical parameter . To start the physical discussion, we should firstly calculate the physical critical points with real and positive pressure and temperature, which lead to the constraint Namely, there is criticality only in four, five, and six dimensions. Besides, the physical critical points are simplified aswhere . It is easy to find that positive pressure and temperature require the conditions . In four dimensions (), the critical behavior is studied in [29].

Totally, we conclude that only four-, five-, and six-dimensional HL AdS black holes with spherical horizon have physical critical points, as shown in Table 1.