Extended phase space of Black Holes in Lovelock gravity with nonlinear electrodynamics

In this paper, we consider Lovelock gravity in presence of two Born-Infeld types of nonlinear electrodynamics and study their thermodynamical behavior. We extend the phase space by considering cosmological constant as a thermodynamical pressure. We obtain critical values of pressure, volume and temperature and investigate the effects of both the Lovelock gravity and the nonlinear electrodynamics on these values. We plot $P-v$, $T-v$ and $G-T$ diagrams to study the phase transition of these thermodynamical systems. We show that power of the nonlinearity and gravity have opposite effects. We also show how considering cosmological constant, nonlinearity and Lovelock parameters as thermodynamical variables will modify Smarr formula and first law of thermodynamics. In addition, we study the behavior of universal ratio of $\frac{P_{c}v_{c}}{T_{c}}$ for different values of nonlinearity power of electrodynamics as well as the Lovelock coefficients.


I. INTRODUCTION
In context of AdS/CFT correspondence, it was proposed that variation of cosmological constant corresponds to varying the number of colors in the boundary field theory of Yang-Mills with chemical potential interpretation [1][2][3][4]. On the other hand, according to Teitelboim and Henneaux's mechanism, coupling four-dimensional gravity with antisymmetric gauge field without cosmological constant results into appearance of the cosmological constant as a constant of motion [5], and therefore, the cosmological constant will be a variable. Recently, the cosmological constant was considered as a state-dependant parameter in two dimensional dilaton gravity [6]. It was shown that treating the cosmological constant as a U (1) charge with non-minimal coupling leads to confining electrostatic potential. Larranaga showed that considering the cosmological constant as thermodynamical variable can be extended to use the Smarr formula for inner and outer horizons of BTZ black hole [7].
Recently, there has been an increasing interest in thermodynamical behavior of black holes in asymptotically adS spacetime. This growing interest comes from the fact that using adS/CFT correspondence, one can find answers regarding a conformal field theory of d-dimensions by solving problems that a gravitational field presents in (d + 1)dimensional anti de Sitter spacetime [8]. Hawking and Page, in their pioneering work, showed that the similarity of phase transition between the stable large black hole and thermal gas in adS space can be interpreted as the confinement/deconfinement phase transition in the dual strongly coupled gauge theory [9]. Later, in an interesting article, Witten showed that using adS/CFT correspondence, one can study the thermal phase transition and the interpretation of confinement in gauge theories [10]. On the other hand, one may consider the cosmological constant as a thermodynamical pressure (in order to investigate the phase transition of black holes) to extend the phase space (see [11] for more details) and modify the first law of black holes thermodynamics [12,13]. Another contribution of this consideration is a renewed interpretation for the mass of black holes which from internal energy becomes Enthalpy [1]. This interpretation indicates that the mass of black holes plays more important role in thermodynamical structure of black holes and contains more information regarding phase structure of black holes [14].
For a canonical ensemble with a fixed charge, it was found that there exists a phase transition between small and large black holes. This phase transition behaves very like the gas/liquid phase transition in a Van der Waals system [15]. On the other hand, phase transition of small/large black holes in adS/CFT correspondence may be interpreted as conductor/superconductor regions of the condescend matter systems [16].
Considering the fact that Maxwell theory contains some fundamental problems and nonlinear electromagnetic fields solve some of these shortcomings [17], one is motivated to study different models of nonlinear electrodynamics (NED). One interesting class of these models is Born-Infeld (BI) type which is acquired in the low energy limit of heterotic string theory [18]. Therefore, one is motivated to study these theories (which in this paper we have considered logarithmic [19] and exponential forms [20]) and the nonlinearity effects of electromagnetic field on critical values representing phase transition of black holes.
On the other hand, Einstein gravity is not flawless and has some fundamental problems [21]. Generalization of the Einstein gravity to higher orders of Lovelock gravity is one way to solve some of these problems [22]. Besides, the Lagrangian of Lovelock gravity is obtainable through the use of the low energy effective action of string theory [18]. One can take this fact into account that modification of Einstein gravity may change the conserved quantities of black holes and therefore it is inevitable to see that critical values and phase transition may depend on the choice of gravity model.
In literature, there have been various studies regarding higher orders of Lovelock gravity in presence of different NED [23]. Also, the phase transition and stability conditions of black holes in various gravity models have been studied intensively [24]. These investigations lead to interesting consequences and phenomenologies [25]. In this paper, we consider higher orders of Lovelock gravity in presence of two classes of NED and study their phase structure. We investigate the effects of both nonlinearity of the electrodynamic models and Lovelock parameters on the phase diagrams and the critical values.
Considering the fact that we are treating black holes as thermodynamical system and interpret the first law of black holes mechanics as the first law of thermodynamics, we expect to see the similar thermodynamical behavior for the black holes and usual thermodynamical systems. Therefore, it is crucial to investigate phase transition and critical values of black holes. Moreover, Lovelock gravity is a generalization of the Einstein gravity and it is a theory that solves some of the Einstein gravity problems. Hence, this generalization also gives a correction to calculated critical values of the Einstein gravity. The thermodynamical behavior of this modification should be reasonable and consistent with thermodynamical concepts. Furthermore, as it was mentioned before, nonlinear electromagnetic fields are introduced to solve shortcomings of Maxwell theory. So it is reasonable to investigate nonlinear effects of NED on the phase transition of black holes. Also, modifications of electrodynamics like gravitational parts must have consistent thermodynamical behavior.
This paper is constructed in the following form. Next section is devoted to introduction to higher orders of Lovelock gravity and conserved quantities. Then, we extend the phase space by considering cosmological constant as thermodynamical pressure and study the Smarr formula of these black holes. After that, we calculate critical values and plot related diagrams for different cases. We give a detailed discussion regarding diagrams, their physical interpretations, and the effects of both NED and gravitational parameters. We finish our paper with closing remarks.

II. SOLUTIONS AND THERMODYNAMIC QUANTITIES OF LOVELOCK GRAVITY
Here, we restrict ourselves to BI types NED models that were introduced by Soleng [19] and Hendi [20] with the following Lagrangians where ENEF and LNEF denote Exponential form of Nonlinear Electromagnetic Field and Logarithmic form of Nonlinear Electromagnetic Field, respectively, and β is nonlinearity parameter. Considering the fact that we are interested in studying the spherically symmetric spacetime, we employ the following metric where dΩ 2 d denotes the standard metric of d-dimensional sphere, S d , with the volume ω d . It is known that the field equations of Einstein gravity in the presence of NED are in the following forms [26] G µν + Λg µν = 1 2 where G µν and Λ are Einstein tensor and cosmological constant, respectively. Einstein gravity in the presence of mentioned NED has been studied in Ref. [26]. Regardless of gravitational sector, one may consider Eqs. (1) and (4) with the metric (2) to obtain the nonzero components of electromagnetic fields with the following explicit forms [26] where q is an integration constant which is proportional to total electric charge of the black hole solutions (with regarding 4π as proportionality constant) Extended phase space thermodynamics and phase diagrams of the Einstein gravity in the presence of NED were investigated in Ref. [27]. So, we consider Gauss-Bonnet (GB) and third order of Lovelock (TOL) gravities and investigate extended phase space thermodynamics and critical behavior of these gravities.

A. GB gravity
At first, we take into account the GB gravity. In order to obtain the field equation of GB gravity, one should add the G GB µν tensor to the left hand side of Eq. (4), in which G GB µν is where L GB = R µνγδ R µνγδ − 4R µν R µν + R 2 and α GB is GB parameter. Considering Eq. (4) with the extra term (6), one can obtain the following solutions [28] g GB (r) = 1 + where in which L W = LambertW ( 4q 2 β 2 r 2d−4 ) and m is an integration constant which is related to the total mass [28] In addition, H and Γ are in the following forms Calculating the Kretschmann scalar, one finds that it diverges at r = 0, so the metric function (7) has an essential singularity at r = 0 [28]. We should note that these solutions may be interpreted as asymptotically adS black holes as those of Einstein case [26]. Now, we take into account the surface gravity interpretation to obtain the Hawking temperature of the mentioned black hole solutions, yielding [28] where and E = E(r)| r=r+ . Since obtained solutions are asymptotically adS, one may obtain the entropy of the black hole solutions by the use of the Gibbs-Duhem relation. After some calculations, one can obtain [28] which confirms that the area law is recovered for α = 0.

B. TOL gravity
Now, we insert the following TOL term, G T OL µν , to the field equation of GB gravity to obtain the solutions of TOL gravity. The tensor G T OL µν may be written as where α T OL and L T OL are, respectively, the coefficient and Lagrangian of TOL gravity Hereafter, we consider the special case to simplify the calculations. It has been shown that the metric function of TOL gravity in the presence of NED can be written as [29] g T OL (r) = 1 + .
The geometric and thermodynamic properties of the asymptotically adS black holes have been studied before [29]. The finite mass of these solutions is the same as that of Einstein gravity, where m can be obtained as a function of r + from the metric function of TOL gravity. The Hawking temperature and the entropy of the TOL solutions can be calculated as [29] and

III. EXTENDED PHASE SPACE AND PHASE DIAGRAMS
As we mentioned in introduction, there are some motivations to view the cosmological constant as a variable. In addition, there are various theories in which some physical constants (such as gauge coupling constants, Newton constant, Lovelock coefficients and BI parameter) are not fixed but dynamical ones. In that case, it is logical to consider the variation of these parameters into the first law of black hole thermodynamics [30].
In order to investigate the phase structure of these classes of gravities, we employ the approach in which the cosmological constant is a thermodynamical variable (pressure) with the following relation This consideration could be justified due to the fact that in quantum context, fundamental fixed parameters could vary. As one can see the conjugating thermodynamical variable to this assumption (cosmological constant as thermodynamical pressure) will be volume where in literature the derived volume for different types of black holes are in agreement with the topology of the spacetime [14,24,27]. In order to calculate the volume of these thermodynamical systems we use the following relation Also, we should consider the effects of cosmological constant in the first law of thermodynamics and extend our phase space. With doing so the total finite mass of the black hole will play the role of Enthalpy and hence the corresponding Gibbs free energy will be in form of Using the mentioned comments, one can obtain the volume with the following form which is consistent with topological structure of spherical symmetric spacetime. Equation (22) was obtained in Einstein gravity [27] which indicates that although considering Lovelock gravity modifies the metric function and some conserved quantities of the black hole, it does not change the volume of the black hole. In addition, it was shown that the Smarr formula may be extended to Lovelock gravity as well as nonlinear theories of electrodynamics [31,32]. Geometrical techniques (scaling argument) were used to derive an extension of the first law and its related modified Smarr formula. The result includes variations in the cosmological constant, Lovelock coefficients and also nonlinearity parameter. In our case, Lovelock gravity in the presence of NED, M should be the function of entropy, pressure, charge, Lovelock parameters and BI coupling coefficient [31]. Regarding the previous section, we find that those thermodynamic quantities satisfy the following differential form where we have achieved T , and one can obtain ∂M ∂α 3 S,Q,P,α2,β , B = ∂M ∂β S,Q,P,α2,α3 .
Using the redefinition of α 2 and α 3 with respect to the single parameter, α, we can rewrite A ′ 1 dα 2 + A ′ 2 dα 3 as a single differential form Moreover, by scaling argument, we can obtain the generalized Smarr relation for our asymptotically adS solutions in the extended phase space , Next step will be calculating critical values. Due to relation between volume and radius of the black hole, we use horizon radius (specific volume) in order to investigate the critical behavior of these systems [24]. In order to do so, we use the method in which critical values are obtained through the use of P − r + diagrams. At first, we use the following relations to obtain the proper equations for critical radius For the economical reasons we will not bring obtained relations for calculating critical horizon radius. We employ numerical method for calculating critical values which result into following diagrams for different classes of Lovelock gravity. We present various tables in order to plot P − r + , T − r + and G − T and study the effects of gravitational parameter which is presented by α and NED parameter which is presented by β. In this paper, we have considered two classes of NED. Studying different phase diagrams for these NED shows that they have similar behavior. Therefore, for economical reason, we will regard only LNEF branch and calculate related critical values of this NED model.
It will be constructive to give a short description regarding to different phase diagrams and the information they contain before presenting tables and phase diagrams. G − T diagrams are representing energy level of different states that phase transition takes place between them and shows the changes in energy level of before and after phase transition states. The characteristic swallowtail that is seen in these diagrams shows the process that we know as phase transition. It also gives interesting information regarding temperature of critical points. For T − r + plot, it contains information regarding critical temperature and horizon radius in which phase transition takes place. Also, it gives some insight about single state regions which in our case is small/large black holes. Finally, studying P − r + plot gives us information regarding the behavior of pressure as function of horizon radius, critical pressure and critical horizon radius (volume) of phase transition. One of the reasons for studying these diagrams is the similarity between phase structure of black holes and the Van der Waals thermodynamical systems.
In what follows, we present various tables to investigate the effects of electrodynamics and gravity models on the critical values of phase transition. We also plot P − r + , T − r + and G − T diagrams for GB and TOL gravities and interpret them. It is notable to mention that considering the metric function of GB gravity, one finds that there is an upper limit for GB parameter to have a real solution.

IV. DISCUSSION ON THE RESULTS OF DIAGRAMS
As one can confirm, higher orders of Lovelock gravity modify the phase diagrams and critical values of volume, pressure and temperature (see tables 1 − 6 and related Figs. of 1 -6). It is clear that considering higher orders of Lovelock gravity leads to different kinds of thermodynamical systems which was also evident through calculated conserved quantities.
In this paper, we consider thermodynamical effects of Lovelock gravity up to GB and TOL, separately. It is evident that critical temperature and pressure are decreasing functions of α and also orders of Lovelock gravity whereas the critical volume is an increasing function of these two factors (see Figs. 8,9,11 and 12). On the other hand, the energy gap between two phases increases drastically by transforming from lower order of Lovelock gravity to higher one (see Figs. 8,9,11 and 12 right panels) or by increasing α for each order (see Figs. 8 and 9 right panels). In addition, the length of subcritical isobars increases which means that the single phase region of small/large black holes decreases (see Figs. 9 and 12 middle panels). Also, phase transitions region is an increasing function of α and order of Lovelock gravity (see Figs. 9 and 12 left panels).
To conclude, plotted figures show that the highest critical temperature and pressure, and the lowest critical volume and energy gap belong to Einstein gravity. On the other hand, system needs higher temperature to have phase transition in Einstein gravity. Whereas the critical temperature decreases by considering higher orders of Lovelock gravity or increasing Lovelock coefficients. Considering the fact that increasing Lovelock parameters and/or adding higher orders of Lovelock gravity increase the power of gravity, one may say that in stronger gravitational regimes, phase transitions take place in lower temperature.
As for the effects of NED, we find the following results. It is evident that the critical temperature in which swallow tail is formed (see Figs. 7, 11 and 12 right panels) and critical pressure (see Figs. 7, 11 and 12 left panels) are decreasing functions of nonlinearity parameter whereas the critical volume is an increasing function. In addition, the length of subcritical isobars is an increasing function of nonlinearity parameter. This means that single phase region of small/large black holes is a decreasing function of β (see Figs. 7, 11 and 12 middle panels). Therefore, for higher values of nonlinearity parameter (weak nonlinearity strength), black holes need to absorb less mass in order to have phase transition. Another issue that must be taken into account is the fact that as β increases, the gap between isobars decreases whereas for small β this gap is greater.
Due to the fact that we take into account BI type models, for large values of nonlinearity parameter, they will lead to Maxwell theory. Obtained results show that the lowest critical temperature and pressure and highest critical volume belong to Maxwell theory. On the other hand, one can conclude that the power of the nonlinearity causes the system to need higher critical temperature to have a phase transition. This effect is opposite of what was observed for gravity.
In addition, the energy gap between two phases, critical temperature and Gibbs free energy are increasing functions of dimensions (Fig. 10). In other words, for higher dimensions, system needs to have more energy for having phase transition. It is worthwhile to mention that subcritical isobars (also critical region) are increasing functions of dimensions (see Figs. 10 middle panels). Whereas critical volume is a decreasing function of dimensions. Finally, the ratio Pcvc Tc is a decreasing (an increasing) function of α (β).

V. CONCLUSIONS
In this paper, we have considered both GB and TOL gravities in presence of two classes of BI type NED and studied their phase diagrams. We have considered cosmological constant as pressure and its conjugating quantity as thermodynamical volume. The obtained volume for these cases was consistent with topological structure of black holes and what was obtained previously [27]. It was shown that although both higher orders of Lovelock gravity and NED modify thermodynamical quantities but the volume of the black holes in these cases is independent of these two modifications and only depends on the topology of the solutions. By employing numerical method, we have calculated critical thermodynamical values for different cases and studied the effects of gravitational and nonlinear electromagnetic field parameters on these critical values. It was shown that black holes under consideration have similar behavior as Van der Waals liquid-gas system.
We found that critical temperature and pressure were decreasing functions of orders and/or coefficients of Lovelock gravity and critical volume and energy gap were increasing functions of them. In other words, black holes with higher orders of Lovelock gravity go under phase transition and acquire stable state with lower temperature comparing to Einstein case. On the other hand, the length of the subcritical isobars and region of the phase transition were increasing functions of the orders and/or coefficients of Lovelock gravity. Orders of the Lovelock gravity are denoting different powers of the curvature scalar. From what we have obtained, one can argue that the critical temperature, pressure and the region of the small/large black holes are decreasing functions of the power of the curvature scalar. While, the critical volume, subcritical isobars and region of the phase transition are increasing functions of it. Therefore, the power of the curvature scalar indeed has a crucial role in variation of the critical values. It is also regarded that considering the higher orders of Lovelock theory, increases the power of gravity.
It was shown that critical temperature and pressure were decreasing functions of β whereas the critical volume is an increasing function of it. In comparison between BI type and Maxwell electrodynamics, it was found that the lowest critical pressure and temperature and the largest critical volume belong to linear (Maxwell) theory.
The gravitational and electromagnetic fields have opposite effects on critical pressure and also phase transition. According to these results, one can say that phase transition is fundamentally related to both gravity and electrody- namics and their powers. As the power of gravitational field (electromagnetic field) increases (decreases) the critical temperature decreases (increases). Interaction between the gravitational and electrodynamic sectors of charged black hole may be found through the metric function. In addition, investigations of the phase diagrams confirm that they are weakening each others effects.
As for dimensionality, we found that as it increases, the energy gap, critical temperature and critical pressure increase. The universal ratio of Pcvc Tc was a decreasing function of β and increasing function of dimensions and Lovelock parameter.
For higher orders of Lovelock gravity we have higher value of entropy (α > 0) which indicates that the thermodynamical system that the gravity describe contains higher value of disorder. In addition, considering higher orders of Lovelock gravity will cause the black holes to have higher degree of complexity in their geometrical structure. If one considers the complexity of the geometrical structure as a disorder measurement of the system, it is logical to expect to see higher value of entropy for higher order of Lovelock gravity. On the other hand, higher value of entropy means that our systems will have phase transition in lower critical temperature. That is the result that we have derived through our numerical calculations.
It was shown that [32] GB gravity in the presence of Maxwell field has no phase transition for arbitrary electric charge in higher than 5-dimensions. While we found that adjusting the nonlinearity parameter, β, there is phase transition for various values of q and d ≥ 5. In other words, the nonlinearity parameter of electrodynamics has modified both electrodynamics and thermodynamical behavior of a black hole system.
Due to the opposite effects of gravitational power in Lovelock gravity and the power of nonlinearity in electrodynamics, it is constructive to find the dominant effect for various domains of thermodynamical systems. Also, one can study whether these two different fields for certain values of α and β, cancel each other effects or not.
Considering the effects of Hawking radiation, we expect to see different behavior for higher orders of Lovelock gravity. This indicates that in order to investigate black holes evaporation and phase transition of black holes, one could take both effects simultaneously. Another interesting issue is studying the connection between complexity of the spacetime (topological structure of black hole) and entropy of the system and the interpretation of entropy as geometrical property. We leave these problems for the future work.