#### 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 .

An important remark that can be observed here is that both quantities and are insensible with the charge . The behavior of the free energy with respect to the temperature may be investigated by plotting in Figure 2 the graph for a fixed value of charge under the critical one. From this plot, we can observe the characteristic swallow-tail which guarantees the existence of the Van der Waals-like phase transition.

Using Figure 2, we can derive numerically the coexistence temperature needed in the Maxwell’s equal area law constructionwhere , , and are the solutions of in descending order, in addition to the numerical values of these points, and we report in Table 2 the areas and in Maxwell’s law (32)

Obviously, the area equals area for different , so the equal area law does not break. For the second phase transition, we know that, near the critical point, there is always a linear relation with slope equal to [47, 50]in this context, the heat capacity behaves like where the critical exponent of the second-order phase is , which is consistent with the mean field theory.

##### 2.2. Grand Canonical Ensemble

Having described the thermodynamical behavior of the EMYM-AdS black hole with a fixed charge, by showing the occurrence of the first and second phase transition, we will focus this section on the phase structure when the potentials are kept fixed.

To facilitate the calculation of relevant quantities, it is convenient to reexpress the Hawking temperature as a function of entropy, Yang-Mills, and electromagnetic potentials, inserting (23) and (24) into (30); one can write

In our assumptions, where and , (35) reduces towhich is in agreement with the result of [51, 52] if we set . Now we are able to write easily

The solution of can be derived aswhere the condition should be verified to ensure that the entropy in (39) is positive. In the other case , one cannot find meaningful root of the equation . Substituting (39) into (38) we obtain the following constraint

implying that no critical point is observed in the diagram. This observation differs from the result in the previous section where the charges are kept fixed, consolidating the assertion that the thermodynamics in the grand canonical ensemble is quite different from that in the canonical one. In Figure 3, we depict the Hawking temperature for both the cases and and one can see that there exists minimum temperature for the left panel. Substituting (39) into (36), the minimum temperature can be obtained asHowever, the Hawking temperature increases monotonically in the right panel.

Having obtained the phase picture of the thermal entropy of the AdS-Maxwell-Yang-Mills black hole, the canonical/grand canonical ensemble, we will now revisit the phase structure of the entanglement entropy and two-point correlation function to see whether they have similar phase structure in each thermodynamical ensemble.

#### 3. Phase Transitions of Einstein-Maxwell-Yang-Mills-AdS Black Holes in Holographic Picture

##### 3.1. Holographic Entanglement Entropy

First, let us provide a concise review of some generalities about the holographic entanglement entropy. For a given quantum field theory described by a density matrix , with being some region of a Cauchy surface of spacetime and standing for its complement, the von Neumann entropy traduces the entanglement between these two regions with being the reduced density matrix of given by . Ryu and Takayanagi propose a simple geometric way to evaluate the entanglement entropy as [53, 54]in which denotes a codimension-2 minimal surface with boundary condition and stands for the gravitational Newton’s constant. In our black hole model we choose the region to be a spherical cap on the boundary delimited by , and the minimal surface can be parametrized by the function . According to definition of the area and (9) and (43), one can show that the holographic entanglement entropy is governed bywhere the notation prime denotes the derivative with respect to ; e.g., . Treat (44) as a Lagrangian and solve the equation of motion given byDue to the difficulty to find an analytical form of the solution , we will perform a numerical calculation with adopting the following boundary conditions

Knowing that the entanglement entropy is divergent at the boundary, we regularized it by subtracting the area of the minimal surface in pure AdS whose boundary is also withWe label the regularized entanglement entropy by and for our numerical calculation we choose and while Ultra Violet cutoff is chosen to be and , respectively. To compare with the phase structure in the thermal picture, we will study the relation between the entanglement entropy and the Hawking temperature representing the temperature of the dual field theory; this relation is depicted in Figure 4 for different values of Maxwell’s charge with a fixed Yang-Mills one near the critical point and the chosen .

For each panel, the red lines are associated with a charge less than the critical one while the ones equal to the critical charge are depicted in black dashed lines and green lines correspond to a charge upper than the critical one. Particularly, the first-order phase transition temperature and second-order phase transition temperature are plotted by magenta and orange dashed line, respectively. As can be seen in all these plots, the Van der Waals-like phase structure can also be observed in the diagram. Particularly, the coexistence temperature and second-order phase transition temperature are exactly the same as those in the thermal entropy structure.

Adopting the same steps as in the thermal picture, we will also check numerically whether Maxwell’s equal area law holds

with the quantities , , and being roots of the equation in ascending order. The Maxwell’s equal area law stipulates that We tabulate in Table 3 the values of the both areas and for the chosen and the charge as well as the relative error between and taken to be the difference between and divided by their average.

Based on Table 3, we can see that, as the pressure approaches the critical one, the relative error which translates the disagreement between Maxwell’s areas decreases. We can claim that the first-order phase transition of the holographic entanglement entropy obeys Maxwell’s equal area law just near the critical point and within our numerical accuracy.

The next obvious step in our investigation is the check of the critical exponent of the second-order phase transition by analyzing the slope of the relation between and , where is the critical entanglement entropy found numerically by an equation . We also introduce the definition of an analog to heat capacity by writing

Taking and for different charge , we plot the relationship between and in Figure 5, and the analytical relation can be fitted as

The slope of (52) is around 3 indicating that the critical exponent is in total concordance with that in (33); therefore the critical exponent for second-order phase transition of the holographic entanglement entropy agrees with that of the thermal entropy in the canonical ensemble [47, 50].

Now, we turn our attention to the grand canonical ensemble; we adopt the same analysis and the chosen values of the previous subsection, by writing (19) and (45) as a function of the potentials and ; taking the same boundary conditions (46), we perform the numerical calculations used in the plot of Hawking temperature as a function of the holographic entanglement entropy with fixed potentials in Figure 6.

Comparing Figure 3 with Figure 6, one may find that the thermal picture shares the same behavior of the holographic entanglement entropy; we can also observe the same minimum value of the temperature . Then the holographic framework reproduces the same attitude of the diagram in the grand canonical ensemble.

Now, after showing that the holographic entanglement entropy shears the same phase picture as that of the thermal entropy for grand canonical and just near the critical point for the canonical ensemble since the relative disagreement between Maxwell’s areas can become significantly large at low pressure, we attempt in the next section to explore whether the two-point correlation function has similar behavior to that of the entanglement entropy.

##### 3.2. Two-Point Correlation Function

According to the Anti-de-Sitter/conformal fields theory correspondence, the time two-point correlation function can be written under the saddle-point assumption and in the large limit of as [55]where is the conformal dimension of the scalar operator in the dual field theory and the quantity stands for the length of the bulk geodesic between the points and on the AdS boundary. Taking into account the symmetry of the considered black hole spacetime, we can simply use with the boundary and employ it to parameterize the trajectory. In this case, the proper length can be expressed as Treating as Lagrangian and as time, one can write the equation of motion for asRecalling the boundary conditions of (46) we attempt to solve this equation by choosing the same background of the previous section, in other words the same values of the parameter with the same UV cutoff values in the dual field theory. The regularized two-point correlation function is labeled as , where denotes the geodesic length in pure AdS under the same boundary region. In Figure 7 we depict the behavior of the temperature in function of ; all plots show the Van der Waals-like phase transition as in the case of the thermal and the holographic entanglement entropy portrait.

As in the case of the holographic entanglement entropy, the relevant calculated results are listed in Table 4 which are the values, the , and the areas and .

The results of Table 4 tell us that under our numerical accuracy and just near the critical point Maxwell’s equal area law still is verified implying that and are equal in the proximity to the critical pressure. These remarks consolidate the behavior of all panels of Figure 7. At this point, one can conclude that, like the entanglement entropy, the two-point correlation function also exhibits apparently a first-order phase transition as that of the thermal entropy. However, exploring broader ranges of the pressure has revealed the fact that the equal area law does not also hold.

For the second phase transition, we will be interested in the quantities and in which is obtained numerically by the equation . The relations between the logarithms of and are shown in Figure 8

The straight blue line in each panel of Figure 8 is fitted following the linear equations

Again, we found a slope around 3; then the critical exponent of the specific heat capacity is consistent with that of the mean field theory of the Van der Waals as in the thermal and entanglement entropy portraits [47, 50]. Therefore, we conclude that the two-point correlation function of the Anti-de-Sitter-Maxwell-Yang-Mills black hole exists in a second-order phase transition at the critical temperature .

For the grand canonical ensemble, we also plot the temperature in function of in Figure 9, from which we can see that thermodynamical behavior is held as the thermal and the holographic entanglement entropy frameworks.

At this level we remark that radical rupture appears when we change the thermodynamical ensemble (canonical/grand canonical). The complete comprehension of such different behaviors is not yet completely understood. We believe that it is typical of such system [41].

#### 4. Conclusion

In this work We have investigated the phase transition of Anti-de-Sitter black hole in the Einstein-Maxwell-Yang-Mills gravity considering the canonical and the grand canonical ensemble. We first studied the phase structure of the thermal entropy in the -plane for fixed charges and found that the phase structure agrees with the study made in [47] when the electrodynamics is linear. The authors consider the thermodynamics of such black hole in the -plane, notably the critical behavior and the analogy with the Van der Waals gas. We also have shown that this behavior disappears in the grand canonical ensemble where the potentials and are kept fixed.

Then, we found that this phase structure of the EMYM-AdS black hole can be probed by the two-point correlation function and holographic entanglement entropy in each thermodynamical ensemble, which reproduce the same thermodynamical behavior of the thermal portrait just for a range of the pressure near the critical one where the equal area law holds within our numerical accuracy; for broader ranges the disagreement between Maxwell’s areas becomes significant. These remarks remain open questions while this approach provides a new step in our understanding of the black hole phase structure from the point of view of holography. Considering the high dimensional solutions or additional hairs by adding Yang-Mills fields and taking into account their confinement can be the object of a future publication.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The author declares that they have no conflicts of interest.