Critical behaviour of Lifshitz dilaton black holes

Till now, critical behaviour of Lifshitz black holes, in an extended $P-v$ space, has not been studied, because it is impossible to find an analytical equation of state, $P=P(v,T)$, for an arbitrary Lifshitz exponent $z$. In this paper, we adopt a new approach toward thermodynamic phase space and successfully explore the critical behaviour of $(n+1)$-dimensional Lifshitz dilaton black holes. For this purpose, we write down the equation of state as $Q^s=Q^s(T,\Psi)$ with $\Psi=\left({\partial M}/{\partial Q^{s} }\right)_{S,P}$ is the conjugate of $Q^s$ and construct Smarr relation based on this new phase space as $ M=M(S,Q^{s},P)$, where $s=2p/(2p-1)$ with $p$ is the power of the power-law Maxwell Lagrangian. We justify such a choice mathematically and show that with this new phase space, the system admits the critical behaviour and resembles the Van der Waals fluid system when the cosmological constant (pressure) is treated as a fixed parameter, while the charge of the system varies. We obtain Gibbs free energy of the system and find swallow tail shape in Gibbs diagrams which represents the first order phase transition. Finally, we calculate the critical exponents and show that although thermodynamic quantities depend on the metric parameters such as $z$ , $p$ and $n$, the critical exponents are the same as Van der Walls fluid-gas system. This alternative viewpoint toward phase space of lifshitz dilaton black hole can be understood easily since one can imagine such a change for a given single black hole i.e., acquiring charge which induces the phase transition. Our results further support the viewpoint suggested in \cite{Dehy}.


Introduction
Historically, Maldacena was the first who suggested, two decades ago, the correspondence between gravity in an Anti-de Sitter (AdS) spacetime and the Conformal Field Theory (CFT) living on the boundary of spacetime known as AdS/CFT correspondence [2]. According to Maldacena's conjecture the effects of the string theory in a d-dimensional AdS n+1 × S d−n−1 spacetime can be appeared in the form of a field theory on an ndimensional r-constant brane which is the boundary of AdS n+1 spacetime. This idea has attracted a lot of enthusiasm and has been investigated from various point of view [3]. The metric of the AdS spacetime is given by which is invariant under an isotropic conformal transformation as follows: On the other hand, the application of AdS/CFT is restricted to systems respected isotropic scale invariance and quantum critical systems show scaling symmetry as where z is a dynamical critical exponent and is restricted as z > 1. This parameter shows the degrees of anisotropy between space and time. The Lifshitz spacetime was first introduced in [4,5] as The Lifshitz spacetime is not a vacuum solution of Einstein gravity and so needs matter source. Usually a massive gauge field plays the role of this matter source but it is nearly impossible to obtain an analytic solution for arbitrary z in such models. As shown in Ref. [6] considering a dilaton field, instead of a massive gauge field, can lead to exact analytical solutions in Lifshitz spacetime (see also [7]). Another motivation is that string theory in its low energy limit reduces to Einstein gravity with a scalar dilaton field coupled to gravity and other fields [8].
On the other hand, the studies on the critical behavior of black holes have got a lot of attentions in a wide range of gravity theories. For example, critical behavior of charged AdS black holes has been studied in [9] and the author completed the analogy between Reissner-Nordstrom-AdS black holes with the Van der Walls liquid-gas system, with the same critical exponents. The key assumption is to enlarge the thermodynamic phase space to include the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume [10][11][12][13][14][15]. When the gauge field is the Born-Infeld nonlinear electrodynamics, one needs more extended phase space to introduce a new thermodynamic quantity conjugate to the Born-Infeld parameter which is necessary for consistency of both the first law of thermodynamics and the corresponding Smarr relation [16]. Treating the cosmological constant as a thermodynamic pressure, thermodynamics and P − v criticality of black holes in an extended phase space in the presence of power-Maxwell [17] and exponential nonlinear electrodynamics [18] have been explored. The studies were also generalized to other gravity theories. In this regards, the phase structure of asymptotically AdS black holes with higher curvature corrections such as Gauss-Bonnet [19,20] and Lovelock gravity [21] have also been investigated. The studies were also extended to the rotating black holes, where phase transition and critical behavior of Myers-Perry black holes have been investigated [22]. Other studies on the critical behavior of black hole spacetimes in an extended phase space have been carried out in [23][24][25][26][27].
Critical behavior of the Einstein-Maxwell-dilaton black holes has been studied in [28]. When the gauge field is in the form of Born-Infeld [29] and power-Maxwell [30] field, critical behavior of (n + 1)-dimensional dilaton black holes in an extended phase space have been investigated. Taking into account the dilaton field in the presence of logarithmic and exponential forms of nonlinear electrodynamics, and considering the cosmological constant and nonlinear parameter as thermodynamic quantities which can vary, it was shown that indeed there is a complete analogy between the nonlinear dilaton black holes with Van der Waals liquid-gas system [31]. In all mentioned above, one assumes the charge of the black hole as an external fixed parameter and treats the cosmological constant as the pressure of the system which can vary.
In the present work, we would like to investigate the critical behaviour of Lifshitz black holes in Einstein-dilaton gravity in the presence of a power-law Maxwell field. It is worthwhile to mention that the (n + 1)-dimensional Lagrangian in power Maxwell theory is conformally invariant provided p = (n + 1)/4 where p the power of the Lagrangian. Let us first plot a 3-dimensional diagrams for equation of state of Lifshitz black hole to understand the phase behaviour of this system and show its analogy with Van der Walls liquid-gas system (see Fig. 1). A close look at the temperature expression of Lifshitz-   dilaton black holes (see [32] and Eq. (2.23) of the present paper), shows that it is nearly impossible to solve this equation for P (or more precisely for l). Therefore, we cannot have an analytical equation of state, P = P (v, T ), to investigate the critical behavior or calculate critical quantities of Lifshitz black holes. Another way to investigate critical behavior of the black holes is to use the method of Refs. [33,34], but as shown in [1], such a view of thermodynamic conjugate variables (Q and Φ = Q/r + ) which are not mathematically independent can lead to physically irrelevant quantities such as (∂Q/∂Φ) T which is supposed to be a thermodynamic response function, but mathematically ill-defined.
To address this problem, an alternative viewpoint toward thermodynamic phase space of black holes was developed in [1] by treating the cosmological constant as a fixed parameter and considering the charge of the black hole as a thermodynamic variable [35,36]. It was argued that, with fixed cosmological constant, the critical behavior indeed occurs in Q 2 -Ψ plane, where Ψ = 1/2r + is conjugate of Q 2 , and thus the equation of state is written as Q 2 = Q 2 (T, Ψ). We find out that in case of Lifshitz dilaton black holes, the system admits a critical behaviour provided we take the electrodynamics in the form of power-Maxwell field and considering Q s as a thermodynamic variable with Ψ = (∂M /∂Q s ) S,P as its conjugate, where s = 2p/(2p − 1). In this case we can define a new response function which naturally leads to physically relevant quantity. Thus, the equation of state is written in the form of Q s = Q s (T, Ψ) and Smarr relation based on this new phase space as M = M (S, Q s , P ). Clearly, for p = 1, the power Maxwell field reduces to standard Maxwell field and Q s → Q 2 . Following [1], in this approach we keep the cosmological constant (pressure) as a fixed quantity, while the charge of the system can vary.
This paper is outlined as follows. In the next section, we present the action, basic field equations of of Lifshitz dilaton black holes and review thermodynamic properties of this system. In section 3.1, we study the phase structure of the solution and present the modified Smarr relation. In section 3.2, we obtain the equation of state and study the critical behavior of the solutions and compare them with Van der Waals fluid system. We investigate the Gibbs free energy and the critical exponents of the system in sections 3.3 and 3.4, respectively. The last section is devoted to summery and conclusion.

Thermodynamics of Lifshitz dilaton black holes
In this section we are going to review the solutions of charged Lifshitz black holes with power Maxwell field [32], with emphasizing on their thermodynamic properties. The (n+1)dimensional action of Einstein-dilaton gravity in the presence of a power Maxwell electromagnetic and two linear Maxwell fields can be written as where R is the Ricci scalar on manifold M, Φ is the dilaton field, λ 1 and λ i are constants.
, with A µ and (B i ) µ are the electromagnetic potentials. Varying the action (2.1) with respect to the metric g µν , the dilaton field Φ, electromagnetic potentials A µ and (B i ) µ , lead to the following field equations [32] We assume the line element of the higher-dimensional asymptotic Lifshitz spacetime has the following form [32] where dΩ 2 n−1 is an (n − 1)-dimensional hypersurface with constant curvature (n − 1)(n − 2)k and volume ω n−1 . Following the method of [32], one can find the solutions of the field equations (2.2)-(2.5) as (2.14) It was argued in [32] that p and z are restricted as for 1/2 < p ≤ n/2, all z(≥ 1) values are allowed, Using the modified BY formalism [37], one can calculate the mass of the solution per unit volume ω n−1 as [32] where the mass parameter m can be written in term of the horizon radius r + by using the fact that f (r + ) = 0. We find m(r + ) = r z+n−1 One can also calculate the charge of the black hole by applying the Gauss law where n µ and u ν are the unit spacelike and timelike normals to the hypersurface of radius r given as Using (2.18), we obtain the charge per unit volume ω n−1 as The electric potential U , measured at infinity with respect to horizon is defined by where χ = p∂ t is the null generator of the horizon. Using (2.9), we can obtain electric potential The entropy of the black holes can be calculated by using the area law of the entropy which is applied to almost all kinds of black holes in Einstein gravity including dilaton black holes. Thus, the entropy of our solutions per unit volume ω n−1 is The Hawking temperature can also be obtained as As one can see from expression (2.23) it is nearly impossible to solve this equation for P (or more precisely for l) and write an analytical equation of state, P = P (v, T ) for an arbitrary Lifshitz exponent z. This implies that, for the Lifshitz dialton black holes, one cannot investigate the critical behavior of the system through an extended P − v phase space by treating the cosmological constant (pressure) as a thermodynamic variable. However, as we shall see in the next section, it is quite possible to investigate the critical behaviour of this system through a new Q s − Ψ phase space and show its similarity with Van der Waals fluid system.

Phase structure
It is now generally accepted that charged black holes in AdS spaces allow critical behavior similar to the Van der Waals fluid system, provided one treats the cosmological constant as a thermodynamic variable (pressure) in an extended phase space [9]. Also it has been shown in [1] that there is deeper connection between charged AdS black holes and Van der Waals fluid system. Indeed, it was argued that similar behavior can be found without extending the phase space [1] by even keeping the cosmological constant as a fixed parameter. The key assumption in this picture, is to treat the square of the charge of black hole, Q 2 , as a thermodynamic variable instead of charge Q [1]. Besides, the equation of state has been written as Q 2 = Q 2 (T, Ψ) where Ψ = 1/v (conjugate of Q 2 ) is the inverse of the specific volume. With this new picture, the authors completed analogy of charged AdS black holes with Van der Waals fluid system with exactly the same critical exponents. In this section, we would like to consider Lifshitz dilaton black holes with power-law Maxwell field and investigate the critical behavior as well as analogy with Van der Walls fluid for this system.
The usual first law of thermodynamics in an extended phase space is in the from of dM = T dS + V dP + U dQ.
It was shown in [1] that by replacing term U dQ in the first law with term ΨdQ 2 , the system allows critical behavior similar to the Van der Waals fluid system. First of all, let us review the motivation of this selection. The well-known thermodynamic quantities of AdS black holes are given [1,9] and the usual Smarr formula is [1,9] M = 2 (T S − V P ) + QU.
It is clear that the charge of black hole appears as Q s in the above equations. This motivates us to choose the new thermodynamic variable as Q s which can simplify all calculations. Besides, from Eq. (3.10) we see that any other choice except Q s , make the equation of state so complicated which cannot be solved analytically to investigate it's critical behavior and critical quantities. While selecting Q s as a new variable causes to a simple and solvable equation of state. Fortunately, in the limit of p = z = 1 and n = 3, Q s −→ Q 2 i.e., our result reduces to the one of [1]. As we will show in the next sections, the system allows the critical behavior similar to the Van der Walls fluid with fixed cosmological constant by replacing U dQ with ΨdQ s . Thus, we write down the first law in the form dM = T dS + V dP + ΨdQ s , (3.13) where T = (∂M /∂S) P,Q s , V = (∂M /∂P ) S,Q s and the conjugate of Q s is (3.14) while η = 2 pz + n − 4 p − z + 1. When p = z = 1 and in 3-dimensions, the above definition for Ψ reduces to Ψ = 1/(2r + ) [1] . In this new picture we can write the Smarr formula for the charged Lifshitz dilaton black hole as It is worth noting that we have replaced the usual ΦdQ term in the first law with ΨdQ s . The extended phase space associated with P = −Λ/(8π) is still the same. Using (3.11) and (3.14), straightforward calculations shows QU = ΨQ s and so Smarr equation

Equation of state
Using Eq. (3.14) and treating pressure or more precisely l as a fixed parameter, Eq. (2.23) can be written as In order to compare the critical behavior of the system with Van der Waals gas, one may plot isotherm diagrams Q s − Ψ, which are displayed in Figs. 2-4. As we know second order phase transition occurs in the point with following conditions: Solving Eqs. (3.19) yield the coordinates of the critical point as
(b) ρc − p for b = 1, n = 3, q = 1 and z = 1. As we see ρc is positive in the range 1/2 < p < 3/2. Following the new definition ρ c = Q s T c Ψ c [1], the energy density of Lifshitz black hole at the critical point is Since ρ c should be positive quantity, we can determine the range of the parameters which satisfy this condition. In the general case it is difficult to calculate it, but using diagram (5), show that ρ c is positive provided Eq. (2.15) is satisfied. For n = 3, ρ c is independent of the value of l. As we expect when n = 3, z = 1 = p, our results reduces to those of RN-AdS black holes [1] .
To see how the critical quantities change with p and z, one may plot the Figs. 6-8. It is clear that in the limit of z = 2, ρ c and Q s c are equal to zero, while Ψ c and T c goes to infinity. Indeed T c = 0 at z = 2, while for z > 2, T c may have an imaginary value.

Gibbs free energy
Gibbs free energy is one of the most important item which can help us to study phase transition of a thermodynamical system. As we know, there is no phase transition when Gibbs free energy is a continuous function. Any discontinuity in Gibbs free energy, known as a zero order phase transition. Also, first-order phase transition occurs when the Gibbs free energy is continuous, but its first derivative with respect to the temperature and pressure is discontinuous. At first, we calculate the Gibbs free energy of Lifshitz dilaton black hole. Then, we try to plot Gibbs diagrams to find out more details about phase transition of the    system. We associate the energy of the system with the Gibbs free energy G = M − T S [10]. The Gibbs free energy can be obtained as In the limiting case where p = z = 1 and n = 3, the Gibbs free energy reduces to [1]    where r + = r + (T, Q 2 ). We have plotted the Gibbs energy diagrams in Figs. 9-11. These diagrams have been shifted for more clarity. The swallowtail behavior of Figs. 9-11 show that a first order phase transition occurs in the system.

Critical exponents
The behavior of the physical quantities in the vicinity of critical point can be characterized by the critical exponents. Following the approach of [16], one can calculate the critical exponents α ′ , β ′ , γ ′ and δ ′ for the phase transition of charged Lifshitz black holes in the presence of power-Maxwell field. To obtain the critical exponents, we define the reduced  thermodynamic variables as Since the critical exponents should be studied near the critical point, we write the reduced variables in the form T r = 1 + t and ψ r = 1 + φ, indicating deviation from the critical point.
One may expand Eq. (3.16) near the critical point as To calculate the critical exponent α ′ , we consider the entropy S given in Eq. (2.22) as a function of T and ψ. Using Eq. (3.14) we have Obviously, this is independent of T and therefore the specific heat vanishes, C ψ = T (∂S/∂T ) ψ = 0. Since the exponent α ′ governs the behavior of the specific heat at fixed ψ C v ∝ |t| α ′ , hence the exponent α ′ = 0. Differentiating Eq. (3.28) at a fixed t < 0 with respect to φ, we get Now, we apply the Maxwell's equal area law [38]. Denoting the variable φ for small and large black holes with φ s and φ l , respectively, we obtain Equation (3.31) leads to the unique non-trivial solution which gives the order parameter as Thus, the exponent β ′ which describes the behaviour of the order parameter near the critical point is β ′ = 1/2. To calculate the exponent γ ′ , one may determine the behavior of the following function near the critical point Differentiating Eq. (3.28) with respect to φ, near the critical point may be written as Finally, the shape of the critical isotherm t = 0 is given by Eq. (3.28). We find

Summery and conclusions
The critical behaviour of the Lifshitz dilaton black hole in an extended phase space, where the cosmological constant is treated as the thermodynamical pressure, cannot be studied due to the complicated form of the solution. Indeed, it can be seen from Eq. (2.23) that it is almost impossible to solve this equation for P (or more precisely for l). Therefore, we cannot have an analytical equation of state, P = P (v, T ), to investigate P − v critically of the system. Also, investigating the phase space of the system in U − Q plan leads to physically irrelevant quantities which are mathematically ill-defined [1]. Here, we address, for the first time, the critical behavior of an (n + 1)-dimensional dilaton Lifshitz black hole in the presence of a power-law Maxwell field via an alternative phase space developed in [1]. We have treated the cosmological constant as a fixed parameter and the charge of the system as thermodynamical variable. It was argued in [1] that without extension the phase space and by keeping the cosmological constant (pressure) as a fixed quantity instead of the charge of the system, it is quite possible to have critical behaviour similar to those of Van der Walls system provided one take the equation of state of the form Q 2 = Q 2 (T, ψ) where Ψ = 1/2r + is conjugate of Q 2 .
In this work, we disclosed that in order to investigate critical behaviour of Lifshitz black holes with power Maxwell field, we should modify the method developed in [1] by considering Q s as a thermodynamic variable and write down the equation of state in the form of Q s = Q s (T, ψ), where s = 2p/(2p − 1) with p is the power of the power-Maxwell Lagrangian. In this approach we keep the cosmological constant (pressure) as a fixed quantity and treat the charge of the black hole as a thermodynamic variable. This is in contrast to the extended phase space of [9], where the charge is fixed and the cosmological constant is treated as a thermodynamic variable. The isotherm diagrams Q s − Ψ show the complete analogy between our system and Van der Walls liquid-gas system. Also the swallow tail behavior of the Gibbs free energy represented a first order phase transition is occurring in the system. Furthermore, we calculated the critical quantities such as T c , ρ c , P c and Ψ c , at the critical point which depend on the metric parameters. Finally, we obtained the critical exponents of the system and found out that they are universal and exactly the same as Van der Walls fluid system. Indeed, our study shows that the approach here is powerful to investigate the critical behaviour of Lifshitz black holes with power Maxwell field. It could help to extract critical exponents of the system without extending the phase space, which is useful in studying the thermodynamical properties of the black holes. We expect to confirm that this approach is viable and can be applied for other gravity theories such as Gauss-Bonnet and Lovelock gravity.