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We consider three-dimensional BTZ black holes with three models of nonlinear electrodynamics as source. Calculating heat capacity, we study the stability and phase transitions of these black holes. We show that Maxwell, logarithmic, and exponential theories yield only type one phase transition which is related to the root(s) of heat capacity, whereas, for correction form of nonlinear electrodynamics, heat capacity contains two roots and one divergence point. Next, we use geometrical approach for studying classical thermodynamical behavior of the system. We show that Weinhold and Ruppeiner metrics fail to provide fruitful results and the consequences of the Quevedo approach are not completely matched to the heat capacity results. Then, we employ a new metric for solving this problem. We show that this approach is successful and all divergencies of its Ricci scalar and phase transition points coincide. We also show that there is no phase transition for uncharged BTZ black holes.

One of the interesting subjects for recent gravitational studies is the investigation of three-dimensional black holes [

The Maxwell theory is in agreement with experimental results, but it fails regarding some important issues such as self-energy of point-like charges which motivates one to regard nonlinear electrodynamics (NED). There are some evidences that motivate one to consider NED theories: solving the problem of point-like charge self-energy, understanding the nature of different complex systems, obtaining more information and insight regarding quantum gravity, compatible with AdS/CFT correspondence and string theory frames, description of pair creation for Hawking radiation, and the behavior of the compact astrophysical objects such as neutron stars and pulsars [

On the other hand, thermodynamical structure of the black holes has been of great interest. It is due to the fact that, according to AdS/CFT correspondence, black hole thermodynamics provides a machinery to map a solution in AdS spacetime to a conformal field on the boundary of this spacetime [

Another approach for studying phase transition of black holes is through thermodynamical geometry. The concept is to construct a spacetime by employing the thermodynamical properties of the system. Then, by studying the divergence points of thermodynamical Ricci scalar of the metric, one can investigate phase transition points. In other words, it is expected that divergencies of thermodynamical Ricci scalar (TRS) coincide with phase transition points of the black holes. Firstly, Weinhold introduced differential geometric concepts into ordinary thermodynamics [

It is notable that these two approaches fail in order to describe phase transition of several black holes [

In this paper, we study thermal stability and phase transition of the BTZ black holes in the presence of several NED models in context of heat capacity. Then, we employ Weinhold, Ruppeiner, and Quevedo methods for studying geometrothermodynamics of these black holes. We will see that Weinhold and Ruppeiner metrics fail to provide fruitful results and the consequences of the Quevedo approach are not completely matched to the heat capacity results. Then, we employ the HPEM metric and study the phase transition of these black holes in context of geometrothermodynamics. We end the paper with some closing remarks.

The

The nonlinearly charged static black hole solutions can be introduced with the following line element:

The entropy and the electric charge of the obtained NED black hole solutions can be calculated with the following forms [

Regarding (

In order to investigate thermal stability and phase transition, one can usually adopt two different approaches to the matter at hand. In one method, the electric charge is considered as a fixed parameter and heat capacity of the black hole will be calculated. The positivity of the heat capacity is sufficient to ensure the local thermal stability of the solutions and its divergencies are corresponding to the phase transition points. This approach is known as canonical ensemble. Another approach for studying thermal stability of the black holes is grand canonical ensemble. In this approach, thermal stability is investigated by calculating the determinant of Hessian matrix of

It is notable that when we study heat capacity for investigating the phase transition, we encounter with two different phenomena. In one, the changes in the signature of the heat capacity are representing a phase transition of the system. In other words, if the heat capacity is negative, then the system is in thermally unstable phase, whereas, for the case of the positive

The other case of phase transition is the divergency of the heat capacity. In other words, the singular points of the heat capacity are representing places in which system goes under phase transition. This assumption leads to the fact that the roots of the denominator of the heat capacity are representing phase transitions. Therefore, we have the following relation for this type of phase transition:

In order to have an effective geometrical approach for studying phase transition of a system, one can build a suitable thermodynamical metric and investigate its Ricci scalar. Thermodynamical metrics were introduced based on the Hessian matrix of the mass (internal energy) with respect to the extensive variables. Therefore, although the electric charge is a fixed parameter for calculating the heat capacity in canonical ensemble, it may be an extensive variable for constructing thermodynamical metrics. In this method we expect that TRS diverges in both types of the mentioned phase transition points. In other words, the denominator of TRS must be constructed in a way that contains roots of the denominator and numerator of the heat capacity. In what follows, we will study the denominator of TRS of the several geometrical approaches and follow the recently proposed thermodynamical metric whose denominator only contains numerator and denominator of the heat capacity and, therefore, divergencies of TRS coincide with roots and divergences of the heat capacity.

In order to find the roots and divergence points of heat capacity, we should solve its numerator and denominator, separately. Solving the mentioned equations with respect to entropy leads to

It is evident from obtained equation for HNED and SNED that there is only one real positive entropy in which heat capacity vanishes. Interestingly, in case of CNED theory, we find two roots for heat capacity. It is evident that the roots are increasing functions of the electric charge in these theories. As for nonlinearity parameter, in case of the HNED and SNED theories, the root is an increasing function of

Now we are in a position to study the existence of the type two phase transition point which is related to divergency of the heat capacity. Considering HNED branch of (

The Weinhold metric was given in [

The Ruppeiner metric was defined as [

Taking into account thermodynamical metrics of Weinhold and Ruppeiner, one can obtain their Ricci scalars. Since we would like to investigate divergence points of TRS,

The Quevedo metrics have two kinds with the following forms [

In order to avoid any extra divergencies in TRS which may not coincide with phase transitions of types one and two and also ensure that all the divergencies of the TRS coincide with phase transition points of the both types, HPEM metric was introduced [

In this case, we have considered the total mass as thermodynamical potential and entropy and electric charge as extensive parameters. Calculations show that denominator of TRS leads to

Here, we investigate phase transitions of black holes using geometrothermodynamics. For this purpose, we used thermodynamical metrics introduced in previous section for the black holes solutions obtained in Section

For Weinhold metric, none of divergencies of the Ricci scalar coincide with roots of the heat capacity in every theories of NED models that we have considered in this paper. On the other hand, one of the divergencies of TRS and divergence point of the heat capacity in CNED theory coincide with each other. It is notable that, in cases of the HNED and SNED theories, there is one divergence point for TRS (up panels of Figure

Weinhold Ricci scalar (solid line), heat capacity (dotted line), and temperature (dashed line) versus

In case of Ruppeiner metric, for

Ruppeiner Ricci scalar (solid line), heat capacity (dotted line), and temperature (dashed line) versus

As for Quevedo metrics, for case I, similar behavior as Weinhold is observed for all three theories of NED (Figure

Quevedo Ricci scalar case I (solid line), heat capacity (dotted line), and temperature (dashed line) versus

Quevedo Ricci scalar case II (solid line), heat capacity (dotted line), and temperature (dashed line) versus

It is evident that, in case of HPEM, all types of phase transition points of heat capacity coincide with divergencies of TRS of HPEM method (Figure

HPEM Ricci scalar (solid line), heat capacity (dotted line), and temperature (dashed line) versus

In this paper, we have considered BTZ black holes, in the presence of three models of NED. We studied stability and phase transitions related to the heat capacity of the mentioned black holes. Next, we employed geometrical approach to study the thermodynamical behavior of the system. In other words, we have studied phase transitions of the system through Weinhold, Ruppeiner, and Quevedo methods. Also, we used the recently proposed approach to study geometrical thermodynamics.

We found that the Weinhold and Ruppeiner metrics for studying these BTZ solutions fail to provide a suitable result. In addition, the divergence points of the Quevedo TRS were not completely matched with the phase transition points of the heat capacity results. In other words, in these approaches, the existence of extra divergencies was observed which were not related to any phase transition point in the classical thermodynamics. In some of these approaches, no divergency of TRS coincided with phase transition points. In order to obtain a consistent results with the classical thermodynamic consequences (the heat capacity), we employed a new thermodynamical metric. In this approach, all the divergencies of TRS coincided with phase transition points. In other words, roots and divergence points of the heat capacity of the BTZ black holes in the presence of each nonlinear models matched with divergencies of TRS of this metric.

Also we found that, in case of HNED and SNED theories, there is no divergency for heat capacity. It means that, like Maxwell theory, these two theories have no second type phase transition. These two nonlinear theories of electrodynamics preserved the characteristic behavior of the Maxwell theory in case of heat capacity. On the other hand, for

Finally, it is worthwhile to mention a comment related to Legendre invariancy. It was shown that [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the anonymous referee for valuable suggestions. They also thank the Shiraz University Research Council. This work has been supported financially by the Research Institute for Astronomy and Astrophysics of Maragha, Iran.