# Critical behavior and phase transition of dilaton black holes with nonlinear electrodynamics

## Abstract

In this paper, we take into account the dilaton black hole solutions of Einstein gravity in the presence of logarithmic and exponential forms of nonlinear electrodynamics. First of all, we consider the cosmological constant and nonlinear parameter as thermodynamic quantities which can vary. We obtain thermodynamic quantities of the system such as pressure, temperature and Gibbs free energy in an extended phase space. We complete the analogy of the nonlinear dilaton black holes with the Van der Waals liquid–gas system. We work in the canonical ensemble and hence we treat the charge of the black hole as an external fixed parameter. Moreover, we calculate the critical values of temperature, volume and pressure and show that they depend on the dilaton coupling constant as well as on the nonlinear parameter. We also investigate the critical exponents and find that they are universal and independent of the dilaton and nonlinear parameters, which is an expected result. Finally, we explore the phase transition of nonlinear dilaton black holes by studying the Gibbs free energy of the system. We find that in the case of \(T>T_c\), we have no phase transition. When \(T=T_c\), the system admits a second-order phase transition, while for \(T=T_\mathrm{f}<T_c\) the system experiences a first-order transition. Interestingly, for \(T_\mathrm{f}<T<T_c\) we observe a *zeroth-order* phase transition in the presence of a dilaton field. This novel *zeroth-order* phase transition occurs due to a finite jump in the Gibbs free energy which is generated by the dilaton–electromagnetic coupling constant, \(\alpha \), for a certain range of pressure.

## 1 Introduction

Nowadays, it is a general belief that there should be some deep connection between gravity and thermodynamics. Bekenstein [1, 2] was the first who disclosed that a black hole can be regarded as a thermodynamic system with entropy and temperature proportional, respectively, to the horizon area and surface gravity [1, 2, 3, 4]. The temperature *T* and entropy *S* together with the energy (mass) of the black holes satisfy the first law of thermodynamics \(\mathrm{d}M=T\mathrm{d}S\) [1, 2, 3, 4]. Historically, Hawking and Page were the first who reported the existence of a certain phase transition in the phase space of the Schwarzschild anti-de Sitter (AdS) black hole [5]. In recent years, the studies on the phase transition of gravitational systems have got renewed interest. It has been shown that one can extend the thermodynamic phase space of a Reissner–Nordstrom (RN) black holes in an AdS space, by considering the cosmological constant as a thermodynamic pressure, \(P=-\Lambda /8\pi \) and its conjugate quantity as a thermodynamic volume [6, 7, 8, 9, 10, 11]. In particular, it was argued that indeed there is a complete analogy for RN–AdS black holes with the van der Walls liquid–gas system with the same critical exponents [12]. The studies were also extended to nonlinear Born–Infeld electrodynamics [13]. In this case, one needs to introduce a new thermodynamic quantity conjugate to the Born–Infeld parameter, which is required for consistency of both the first law of thermodynamics and the corresponding Smarr relation [13]. Extended phase space thermodynamics and *P*–*V* criticality of the black holes with power-Maxwell electrodynamics were investigated in [14]. When the gauge field is in the form of logarithmic and exponential nonlinear electrodynamics, critical behavior of black hole solutions in Einstein gravity has also been explored [15]. Treating the cosmological constant as a thermodynamic pressure, the effects of higher curvature corrections from Lovelock gravity on the phase structure of asymptotically AdS black holes have also been explored. In this regard, critical behavior and phase transition of higher curvature corrections such as Gauss–Bonnet [16, 17] and Lovelock gravity have also been investigated [18, 19]. The studies were also extended to the rotating black holes, where phase transition and critical behavior of Myers–Perry black holes have been investigated [20]. Other studies on the critical behavior of the black hole spacetimes in an extended phase space have been carried out in [21, 22, 23, 24, 25, 26].

Although Maxwell theory is able to explain a variety of phenomena in electrodynamics, it suffers some important problems such as then divergence of the electric field of a point-like charged particle or infinity of its self energy. In order to solve these problems, one may get help from the nonlinear electrodynamics [27, 28, 29, 30]. Inspired by developments in string/M-theory, the investigation on the nonlinear electrodynamics has got a lot of attention in recent years.

On the other side, a scalar field called dilaton emerges in the low energy limit of string theory [31]. Breaking of spacetime supersymmetry in ten dimensions leads to one or more Liouville-type potentials, which exist in the action of dilaton gravity. In addition, the presence of the dilaton field is necessary if one couples the gravity to other gauge fields. Therefore, the dilaton field plays an essential role in string theory and it has attracted extensive attention in the literature [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. Critical behavior of the Einstein–Maxwell-dilaton black holes has been studied in [45]. In the context of Born–Infeld and power-Maxwell nonlinear electrodynamics coupled to the dilaton field, critical behavior of \((n+1)\)-dimensional topological black holes in an extended phase space has been explored in [46, 47], respectively. Although the asymptotic behavior of these solutions [46, 47] are neither flat nor anti-de Sitter (AdS), it was found that the critical exponents have the universal mean field values and do not depend on the details of the system, while the thermodynamic quantities depend on the dilaton coupling constant, nonlinear parameter and the dimension of the spacetime. In the present work, we would like to extend the study of the critical behavior of the black holes, in an extended phase space, to other nonlinear electrodynamics in the context of dilaton gravity such as exponential and logarithmic nonlinear electrodynamics. Following [13, 47], and in order to satisfy the Smarr relation we shall extend the phase space to include a nonlinear parameter as a thermodynamic variable and consider its conjugate quantity as polarization. We will complete the analogy of the nonlinear dilaton black holes with a Van der Waals liquid–gas system and work in the canonical ensemble. In addition, we calculate the critical exponents and show that they are universal and are independent of the dilaton and nonlinearity parameters. Finally, we shall explore the phase transition of dilaton black holes coupled to nonlinear electrodynamics by considering the discontinuity in the Gibss free energy of the system. We will see that in addition to the first- and second-order phase transition in charged black holes, the presence of the dilaton field admits a *zeroth-order* phase transition in the system. This phase transition is occurred due to a finite jump in the Gibbs free energy which is generated by dilaton–electromagnetic coupling constant, \(\alpha \), for a certain range of pressure. This novel behavior indicates a small/large black hole *zeroth-order* phase transition in which the response functions of the black holes thermodynamics diverge, e.g. the isothermal compressibility.

This paper is outlined as follows. In the next section, we present the action, basic field equations and our metric ansatz for dilaton black holes. In Sect. 3, we explore the critical behavior of dilaton black holes coupled to exponential nonlinear (EN) electrodynamics. In Sect. 4, we investigate *P*–*V* criticality of dilaton black holes when the gauge field is in the form of logarithmic nonlinear (LN) electrodynamics. In Sect. 5, we investigate the effects of nonlinear gauge field parameter in the strong nonlinear regime on the critical behavior of the system. In Sect. 6, we explore the phase transition of nonlinear dilaton black holes. We finish with closing remarks in Sect. 7.

## 2 Basic field equations

*string tension*, \(\beta =\frac{1}{2\pi \alpha '}\)[50]. In fact \(\beta \) determines the strength of the nonlinearity of the electrodynamics. In the limit of large \(\beta \) (\(\beta \rightarrow \infty \)), the system goes to the linear regime and the nonlinearity of the theory disappears and the nonlinear electrodynamic theory reduces to the linear Maxwell electrodynamics. On the other hand, as \(\beta \) decreases (\(\beta \rightarrow 0\)), we go to the strong nonlinear regime of the electromagnetic and thus the behavior of the system will be completely different (see Sect. 5). In Eq. (3) \(F^2=F_{\mu \nu }F^{\mu \nu }\), where \(F_{\mu \nu }\) is the electromagnetic field tensor. By varying the action (1) with respect to the gravitational field \(g_{\mu \nu }\), the dilaton field \(\Phi \) and the electromagnetic field \(A_{\mu }\), we arrive at the following field equations [48, 49]:

*f*(

*r*) and

*R*(

*r*) are functions of

*r*which should be determined, and \(\mathrm{d}\Omega _{k}^2\) is the line element of a two-dimensional hypersurface \(\Sigma \) with constant curvature,

In the remaining part of this paper, we consider the critical behavior of END and LND black holes.

## 3 Critical behavior of END black holes

In this section, first of all, we review the solution of dilatonic black holes coupled to EN electrodynamics [48]. Then we construct the Smarr relation and the equation of state of the system to study the critical behavior of the system.

### 3.1 Review of END black holes

*f*(

*r*),

*R*(

*r*) and \(\Phi (r)\), we make the ansatz [38]

*m*and

*q*are integration constants which are related to the mass and the charge of the black holes. Also, \(L_W(x)={LambertW(x)}\) is the Lambert function and \( \digamma (a,b,z)\) is the hypergeometric function [51, 52]. Here \(\eta \) and \(\gamma \) are defined by

### 3.2 First law of thermodynamics and phase structure

*B*for large \(\beta \) for \(\alpha =0,1\). We find

### 3.3 Equation of state

*c*are the Boltzmann constant, the reduced Planck constant and the speed of light, respectively. Inserting Eq. (34) in Eq. (30), we can define the specific volume as

To summarize, our solution can face with a phase transition when temperature is below its critical value. One may predict this behavior by considering an isothermal *P*–*v* diagram. It is expected that the *P*–*v* diagrams for our solution and the Van der Waals gas have a similar behavior. In Fig. 1 we have plotted the behavior of *P* in terms of *v*. From these figures we see that, in the absence/presence of a dilaton field, the nonlinear black hole resemble the Van der Waals fluid behavior.

### 3.4 Gibbs free energy

### 3.5 Critical exponents

## 4 Critical behavior of LND black holes

*U*:

### 4.1 Equation of state

*P*–

*v*diagrams. According to Fig. 2 we can observe that, for specific values of the parameters, a phase transition exists below a critical temperature. It occurs for both large (Fig. 2a) and small (Fig. 2b) values of the nonlinear parameter in the presence of a dilaton field.

### 4.2 Gibbs free energy

### 4.3 Critical exponents

## 5 Effects of nonlinear gauge field

We have calculated the critical quantities in the limit of large \(\beta \) where the nonlinearity of the theory is small. However, it is clear from the *P*–*v* and Gibbs diagrams that there is a similar phase transition in the limit of small \(\beta \) where the nonlinearity of the theory is large. In the limit of small \(\beta \) it is nearly impossible to calculate analytically the critical quantities. Also, in the presence of the dilaton field, it will be very difficult to calculate them even numerically. For some numeric calculations (in the absence of a dilaton field) one may see [15].

A close look at the critical temperature in both END and LND given in Eqs. (37) and (65) shows that the presence of the nonlinear field makes the critical temperature larger and it will increase with decreasing \(\beta \). One may observe that the increasing in \(T_{c}\) and \(p_{c}\) in LND is stronger than END. In Fig. 4 we have plotted the critical quantities \(T_{c}\) and \(p_{c}\) of LND, END and Maxwell-dilaton (MD) theory in terms of the nonlinear parameter \(\beta \) and we showed that they will go to a same value in the large limit of \(\beta \) where the effects of nonlinearity disappear. Clearly, the linear MD theory is independent of the nonlinear parameter \(\beta \), as can be seen from Fig. 4. It is notable that the critical quantities in LND are the same as those in END for large \(\beta \). However, for small \(\beta \) (nonlinear regime), their behavior is quite different. The behavior of the critical temperature in terms of \(\alpha \) is shown in Fig. 5, for \(0\le \alpha <1\). From these figures, one can see that the behavior of the diagrams differs as the nonlinear parameter \(\beta \) decreases. This implies that in a very strong nonlinear regime, the nonlinearity nature of the theory plays a crucial role. When \(\alpha \rightarrow 1\), the critical temperatures in different types of electrodynamic fields grow toward each other, but it is completely unlike the critical pressure. As one see in Fig. 6, for \(\alpha \rightarrow 1\), the critical pressures become more different.

*P*–

*v*behavior in Figs. 7 and 8, to see the difference between the nonlinear theories we have considered. It is clear from these diagrams that the behavior of END, LND and BID black holes is very similar when

*T*or \(\beta \) are large enough. As one expects, for the same

*T*, the differences between the diagrams increase as \(\beta \) decreases (see Fig. 9).

It was extensively argued in [13] that in the absence of a dilaton field, a black hole with BI nonlinear electrodynamics may have two, one or zero critical points, which depends on the strength of the nonlinear and charge parameters. For BID black holes, only for small values of the dilaton–electromagnetic coupling \(\alpha \) one may see a second critical point. Interestingly enough, as the dilaton parameter \(\alpha \) increases, the second critical point disappears. As an example, we compare *P*–*v* diagrams of BID black holes for three values of the dilaton coupling \(\alpha \) in Fig. 10. It is clear from these diagrams that in the absence of a dilaton field (Fig. 10a) or for a weak dilaton field (Fig. 10b), there are two critical points but when the dilaton field increases (Fig. 10c) the second critical point vanishes and we have only one critical point. In the other types of nonlinear electrodynamics such as logarithmic, exponential or power-law Maxwell fields, the second critical point is never seen; neither in the absence nor in the presence of a dilaton field. Also it is worthwhile to mention that for very small value of the nonlinear parameter \(\beta \) there is not any critical point in all types of the above electrodynamics.

## 6 Zeroth-order phase transition

Let us emphasize that the observed phase transitions in the previous sections, which were similar to the Van der Waals phase transition, are called the first-order phase transition in the literature. It occurs where the Gibbs free energy is continuous, but its first derivative with respect to the temperature and pressure is discontinuous. Now we want to mention that another interesting type of phase transition occurs in a certain range of the metric parameters. This discontinuity in the Gibbs free energy, known as a *zeroth-order* phase transition, is observed in superfluidity and superconductivity [55]. It is important to note that, due to this transition, the response functions of the black holes thermodynamics diverge, e.g. the isothermal compressibility. Recently, a *zeroth-order* phase transition was observed in the context of Einstein–Maxwell-dilaton black holes [56]. It was confirmed that the presence of a dilaton field plays a crucial role for such a phase transition [56]. Indeed, a direct relation exists between the zeroth-order portion of the transition curve and the dilaton parameter \(\alpha \) [56]. In other words, we have no zeroth-order phase transition for Einstein–Maxwell (Reissner–Nordstrom) black holes. Moreover, for nonlinear BI electrodynamics, it was shown that a zeroth-order phase transition may occur even in the absence of the dilaton field [57], which means that the nonlinearity of the gauge field can also cause a zeroth-order phase transition in black hole thermodynamics.

Here we would like to explore the possibility of a *zeroth-order* phase transition in END and LND black holes, where both nonlinearity and the dilaton field are taken into account. In order to see the finite jump in the Gibbs free energy, we plot the diagrams of the Gibbs free energy with respect to the pressure in Figs. 11, 12 and 13 for different values of the metric parameters.

For completeness, we also investigate the phase transition of BID solutions presented in [46]. An interesting case in the BID theory is plotted in Fig. 11. From this figure, we see that, for certain values of the pressure and a special range of dilaton field parameters, both zeroth- and first-order phase transitions may be observed in one diagram. Based on this figure, by increasing the pressure until \(P_1\) a first-order transition occurs. For \(P>P_1\), the Gibbs free energy has two values, and as one can see the acceptable values of the energy are shown with the blue curve, since it includes smaller values of the energy. At point \(P_2\), one can see a discontinuity in the Gibbs free energy, which demonstrates a zeroth-order phase transition.

Also, Fig. 12 shows different critical behaviors of dilatonic black holes in the presence of three nonlinear versions of electrodynamics with respect to the changes in the temperature values when other metric parameters are fixed. In the case of \(T>T_c\), we have no phase transition. When \(T=T_c\), the system experiences a second-order phase transition as we have discussed before. As temperature decreases to \(T_\mathrm{f}<T<T_c\), a zeroth-order phase transition is observed. Finally, at \(T=T_\mathrm{f}\) the first-order phase transition occurs. It is worth mentioning that this behavior is repeated in the Gibbs free energy of all three types of black holes in the presence of nonlinear electrodynamics and non-zero values of the dilaton field.

It is important to note that, looking at Fig. 13, one may wonder that, for fixed values of the parameters, and in the absence of the dilaton field (\(\alpha =0\)), we do not observe a zeroth-order phase transition in END and LND theories. This is in contrast to the BID theory where a zeroth-order phase transition occurred in the small range of nonlinear parameters \(\beta \) even in the absence of a dilaton field (see Fig. 13a). In this figure, the red portion curve shows this behavior as we explained in the close-up of Fig. 11. It is one of the main differences between these three versions of nonlinear electrodynamics, which implies that their behaviors in the case of small values of \(\beta \) completely differ. This indicates that, while the nonlinearity can lead to a zeroth-order phase transition in BI theory, this is not the case for EN and LN theories. In other words, the presence of the dilaton field plays a crucial role for the occurrence of a zeroth-order phase transition in the context of END and LND electrodynamics.

## 7 Closing remarks

*P*–

*v*diagrams of this theory are similar to those of the Van der Waals gas. By applying the approach of the Van der Waals gas to find the critical point, we concluded that this point is exactly the same as in BID black holes. Besides, the Gibbs free energy diagram confirmed the existence of a phase transition, and finally the critical exponents were obtained, which are exactly the same as the mean field theory.

We also investigated the critical behavior of LND black holes. Again, for \(\beta \rightarrow \infty \), the series expansion of the LND Lagrangian is similar to the END and BID cases, so one expects the critical behavior of this theory to be similar to the BID and END theories in this limit. Our calculations confirmed that the critical behavior of LND theory is exactly the same as those of a Van der Waals gas system.

It is important to note that although the critical behavior of END and LND electrodynamics, in the limit of large nonlinear parameter \(\beta \), is similar to BID black holes explored in Ref. [46]; however, for small values of \(\beta \), the situation quite differs and the behavior of these three types of nonlinear electrodynamics are completely different. For example, it was argued in [13] that BI black holes may have two, one or zero critical points, however, this behavior is not seen for the logarithmic and exponential cases, namely the second critical point is never seen in the absence/presence of the dilaton field.

We also investigated the phase transition of END and LND black holes. In addition to the usual critical (second-order) and first-order phase transitions in END and LND black holes, we observed that a finite jump in the Gibbs free energy is generated by the dilaton–electromagnetic coupling constant, \(\alpha \), for a certain range of pressure. This novel behavior indicates a small/large black hole *zeroth-order* phase transition in which the response functions of the black holes thermodynamics diverge. It is worthy to note that, for temperature in the range \(T_\mathrm{f}<T<T_c\), a discontinuity occurs in the Gibbs free energy diagram which leads to a zeroth-order phase transition. We find that in the absence of a dilaton field, we do not observe a zeroth-order phase transition in END and LND theories. This is in contrast to the BI theory where a zeroth-order phase transition occurred in a small range of the nonlinear parameters \(\beta \) even in the absence of the dilaton field. We conclude that, while in BI black holes the nonlinearity can lead to a zeroth-order phase transition, this is not the case for EN and LN black holes. In other words, the presence of the dilaton field plays a crucial role for the occurrence of a zeroth-order phase transition in the context of EN and LN electrodynamics.

Finally, we would like to mention that the jump in the Gibbs free energy is observed for three types of dilatonic nonlinear electrodynamics, namely BID, END and LND. However, in the absence of a dilaton field, a zeroth-order phase transition occurs only for BI black holes, which means that the nonlinearity is responsible for this phase transition. However, for LND and END black holes, it seems that the dilaton field is responsible for this type of zeroth-order phase transition. However, for BID theory, both dilaton field and nonlinear electrodynamics can lead to a zeroth-order phase transition. This behavior and the physical reasons behind it need further investigations in future studies.

## Notes

### Acknowledgements

We are grateful to the referee for constructive comments which helped us improve our paper significantly. We also thank Shiraz University Research Council. The work of AS been supported financially by Research Institute for Astronomy and Astrophysics of Maragha, Iran.

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