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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.

Over the last years, a great emphasis has been put on the application of the Anti-de-Sitter/conformal field theory correspondence [

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 [

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 [

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

We start this section by writing the

In order to find the electromagnetic field, we recall the following radial gauge potential ansatz

In general for

Using (

The Yang-Mills potential

It is straightforward to show that obtained quantities (

This relation is depicted in Figure

Coordinates of the critical points for different values of

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The temperature as a function of the entropy for different values of charge

An important remark that can be observed here is that both quantities

The Helmholtz free energy in function of the entropy for EMYM-AdS black holes for different values of the charge

Using Figure

Check of the equal area law in the

C | | | | | | |
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| 0.15376 | 0.638237 | 1.52994 | 3.13241 | 0.383499 | 0.38350 |

| ||||||

| 150965 | 1.05043 | 11.54616 | 2.25932 | 00.18248 | 0.1825 |

Obviously, the area

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 (

In our assumptions, where

The solution of

implying that no critical point is observed in the

The temperature as a function of the entropy for different potentials.

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.

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

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

Plot of isocharges on the

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

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

with the quantities

Comparison of

| | | | | | | | Relative error |
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| | 0.2 | 0.00339417 | 0.00490769 | 0.00755095 | | 2.218 × | 2.36 % |

0.3 | 0.011477 | 0.0164978 | 0.0254236 | | 6.407 × | 2.4% | ||

| 0.2 | 0.00161291 | 0.0042458 | 0.0130777 | | 0.00005560 | 28.81% | |

| ||||||||

| | 0.2 | 0.00425941 | 0.00502275 | 0.00617356 | | 1.023 × | |

0.3 | 0.0151665 | 0.0169359 | 0.0208834 | | 1.631 × | 1.85% | ||

| 0.2 | 0.00339968 | 0.00496717 | 0.00743883 | | 1.76781 × | 18.44% |

Based on Table

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

Taking

The relation between

The slope of (

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 (

The relation between

Comparing Figure

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.

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

Plot of isocharges on the

As in the case of the holographic entanglement entropy, the relevant calculated results are listed in Table

Comparison of

| | | | | | | | relative error |
---|---|---|---|---|---|---|---|---|

| | 0.2 | 1.34978 | 1.34987 | 1.35003 | 0.000287 | 0.000300 | 4.42% |

| 0.881705 | 0.881911 | 0.882274 | 0.000197 | 0.000208 | 5.43% | ||

| 0.2 | 1.34967 | 1.34983 | 1.35043 | 0.001431 | 0.001812 | 23.49% | |

| ||||||||

| | 0.2 | 1.34983 | 1.34988 | 1.34995 | 0.000142 | 0.000138 | 2.85% |

0.3 | 0.881822 | 0.881926 | 0.882085 | 0.000209 | 0.000216 | 3.29% | ||

| 0.2 | 1.34978 | 1.34987 | 1.35003 | 0.0003907 | 0.0004653 | 17.42% |

The results of Table

For the second phase transition, we will be interested in the quantities

The relation between

The straight blue line in each panel of Figure

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 [

For the grand canonical ensemble, we also plot the temperature

The relation between

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 [

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

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.

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

The author declares that they have no conflicts of interest.