Advances in High Energy Physics

Volume 2019, Article ID 6870793, 13 pages

https://doi.org/10.1155/2019/6870793

## Anti-de-Sitter-Maxwell-Yang-Mills Black Holes Thermodynamics from Nonlocal Observables Point of View

^{1}EPTHE, Physics Department, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco^{2}High Energy Physics and Astrophysics Laboratory, Faculty of Science Semlalia, Cadi Ayyad University, 40000 Marrakesh, Morocco

Correspondence should be addressed to H. El Moumni; am.acu.ude@inmuomle.nasah

Received 29 June 2018; Revised 28 October 2018; Accepted 23 December 2018; Published 20 January 2019

Guest Editor: Saibal Ray

Copyright © 2019 H. El Moumni. 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

In this paper we analyze the thermodynamic properties of the Anti-de-Sitter black hole in the Einstein-Maxwell-Yang-Mills-AdS gravity (EMYM) via many approaches and in different thermodynamical ensembles (canonical/grand canonical). First, we give a concise overview of this phase structure in the entropy-thermal diagram for fixed charges and then we investigate this thermodynamical structure in fixed potentials ensemble. The next relevant step is recalling the nonlocal observables such as holographic entanglement entropy and two-point correlation function to show that both observables exhibit a Van der Waals-like behavior in our numerical accuracy and just near the critical line as the case of the thermal entropy for fixed charges by checking Maxwell’s equal area law and the critical exponent. In the light of the grand canonical ensemble, we also find a newly phase structure for such a black hole where the critical behavior disappears in the thermal picture as well as in the holographic one.

#### 1. Introduction

Over the last years, a great emphasis has been put on the application of the Anti-de-Sitter/conformal field theory correspondence [1, 2] which plays a pivotal role in recent developments of many physical themes [3–5]; in this particular context the thermodynamics of Anti-de-Sitter black holes become more attractive for investigation [6].

In general, black hole thermodynamics has emerged as a fascinating laboratory for testing the predictions of candidate theories of quantum gravity. It has been figured that black holes are associated thermodynamically with a entropy and a temperature [7] and a pressure [8]. This association has led to a rich structure of phase picture and a remarkable critical behavior similar to van der Waals liquid/gas phase transition [9–19]. Another confirmation of this similarity appears when we employ nonlocal observables such as entanglement entropy, Wilson loop, two-point correlation function, and the complexity growth rate [20–33]. Meanwhile, these tools are used extensively in quantum information and to characterize phases and thermodynamical behavior [21, 23, 34–41].

The black hole charge finds a deep interpretation in the context of the AdS/CFT correspondence linked to condensed matter physics; the charged black hole introduces a charge density/chemical potential and temperature in the quantum field theory defined on the boundary [42]. In this background, the charged black hole can be viewed as an uncondensed unstable phase which develops a scalar hair at low temperature and breaks symmetry near the black hole horizon reminiscing the second-order phase transition between conductor and superconductor phases [43]; this situation is called the "s-wave" holographic superconductor. It has also been shown that "p-wave" holographic superconductor corresponds to vector hair models [44, 45]. The simplest example of p-wave holographic superconductors may be provided by an Einstein-Yang-Mills theory with gauge group and no scalar fields, where the electromagnetic gauge symmetry is identified with an subgroup of . The other components of the gauge field play the role of charged fields dual to some vector operators that break the symmetry, leading to a phase transition in the dual field theory.

Motived by all the ideas described above, although the Yang-Mills fields are confined to acting inside nuclei while the Maxwell field dominates outside, the consideration of such theory where the two kinds of field live is encouraged by the existence of exotic and highly dense matter in our universe. In this work, we try to contribute to this rich area by revisiting the phase transition of Anti-de-Sitter black holes in Einstein-Maxwell-Yang-Mills (EMYM) gravity. More especially, we investigate the first- and second-order phase transition by different approaches including the holographic one and in different canonical ensembles.

This work is organized as follow: First, we present some thermodynamic properties and phase structure of the EMYM-AdS black holes in (temperature, entropy)-plane in canonical and grand canonical ensemble. Next, we show in Section 3 that the holographic approach exhibits the same behavior; in other words we recall the entanglement entropy and two-point correlation function to check the Maxwell’s equal area law and calculate the critical exponent of the specific heat capacity which is consistent with that of the mean field theory of the Van der Waals in the canonical ensemble near the critical point. In the grand canonical one, a new phase structure arises where the critical behavior disappears in the thermal as well as the holographic framework. The last section is devoted to a conclusion.

#### 2. Critical Behavior of Einstein-Maxwell-Yang-Mills-AdS Black Holes in Thermal Picture

##### 2.1. Canonical Ensemble

We start this section by writing the -dimensional for Einstein-Maxwell-Yang-Mills gravity with a cosmological constant described by the following action [46, 47]where is the Ricci scalar while is the cosmological constant. Also and are the Maxwell invariant and the Yang-Mills invariant, respectively; the trace element stands for . Varying the action (1) with respect to the metric tensor , the Faraday tensor , and the YM tensor , one can obtain the following field equationswhere is the Einstein tensor, the quantity ’s stands for the structure constants of the -parameters Lie group , is coupling constant, and denotes the gauge groupe YM potential. We also note that the internal indices do not differ whether in covariant or contravariant form. In addition, and are the energy momentum tensor of Maxwell and YM fields with the following formulawhere is the usual Maxwell potential. The metric for such dimensional spherical black hole may be chosen to be [47]in which represents the volume of the unit -sphere which can be expressed in the standard spherical form

In order to find the electromagnetic field, we recall the following radial gauge potential ansatz which obeys the Maxwell field equations (3) with the following solutionwhere is an integration constant related to electric charge of the solutions. To solve the YM field, (4), we use the magnetic Wu–Yang ansatz of the gauge potential [48, 49] given bywhere we imply (to have a systematic process) that the super indices are chosen according to the values of and in order. For instance, we present some of themin which The YM field 2-forms are defined by the expression

In general for we must have gauge potentials. The integrability conditionsare easily satisfied by using (28). The YM equationsalso are all satisfied. The energy-momentum tensor (4) becomes afterwith the nonzero components

Using (2) and after some simplifications, one can find that the metric function has the following form given bywhere ; one can note in the particular case for that the last term of (19) diverges, involving an unusual logarithmic term in Yang-Mills charge [46]. For this gravity background the parameter is related to the mass of such black hole, while and are the charges of Maxwell field and Yang-Mills field, respectively. Following previous literature [8, 9], one can find a close connection between the cosmological constant and pressure as , leading to the following expressions of Hawking temperature, mass, and entropy of such black hole in terms of the horizon radius

The Yang-Mills potential and the electromagnetic one can be written aswhere is the volume of the unit -sphere. In fact, according to the interpretation of the black hole mass as an enthalpy [8] in the extended phase space context, the free energy of black hole can be written asand the heat capacity is given by

It is straightforward to show that obtained quantities (20), (21), and (22) obey the first law of black hole thermodynamics in the extended phase spacewhere is the Legendre transform of the pressure, which denotes the thermodynamic volume with In addition to this, using scaling argument, the corresponding Smarr formula is Without loss of generality, inserting (22) into (20), we can get the entropy Hawking temperature relation of such black hole, namely,

This relation is depicted in Figure 1; indeed it has been shown that there is a Van der Waals-like phase transition; furthermore a direct confirmation comes from the solution of the following systemwhich reveal the existence of a critical point. The critical charge, entropy, and temperature are given in Table 1; for all the rest of the paper we keep .