Thermodynamics of charged rotating dilaton black branes coupled to logarithmic nonlinear electrodynamics

We construct a new class of charged rotating black brane solutions in the presence of logarithmic nonlinear electrodynamics with complete set of the rotation parameters in arbitrary dimensions. The topology of the horizon of these rotating black branes are flat, while, due to the presence of the dilaton field the asymptotic behaviour of them are neither flat nor (anti)-de Sitter [(A)dS]. We investigate the physical properties of the solutions. The mass and angular momentum of the spacetime are obtained by using the counterterm method inspired by AdS/CFT correspondence. We derive temperature, electric potential and entropy associated with the horizon and check the validity of the first law of thermodynamics on the black brane horizon. We study thermal stability of the solutions in both canonical and grand canonical ensemble and disclose the effects of the rotation parameter, nonlinearity of electrodynamics and dilaton field on the thermal stability conditions. We find the solutions are thermally stable for $\alpha<1$, while for $\alpha>1$ the solutions may encounter an unstable phase where $\alpha$ is dilaton-electromagnetic coupling constant.

the first time in [27]. Although this type of nonlinear electrodynamics has no direct relation with superstring theory, it can be motivated from different sides. First, it can be regarded as a toy model showing that certain nonlinear field theories can produce particle-like solutions that can realize the limiting curvature hypothesis in cosmological theories [27]. Second, the behavior of logarithmic electrodynamics and BI electrodynamics Lagrangians are the same for large values of nonlinear parameter β. Third, from gauge/gravity duality point of view, the ratio of holographic viscosity to entropy density is not affected by nonlinear terms rise from a nonlinear electrodynamics in contrast with gravitational corrections [28]. Fourth, the values of important parameters of a holographic superconductor system such as critical temperature and order parameter are firmly sensitive to choice of electrodynamics [29,30]. Other studies on the nonlinear electrodynamics have been carried out in [31][32][33][34][35][36][37][38].
The above pointed out motivations are convincingly enough to satisfy one to seek for the effects of logarithmic electrodynamics on the solutions. Till now, exact rotating solutions of logarithmic electrodynamics in the context of dilaton gravity has not been constructed. In this paper, we would like to construct the rotating dilaton black branes in the presence of logarithmic nonlinear electrodynamics and investigate their thermodynamics as well as their thermal stability.
The outline of this paper is as follows. In the next section, we present the basic field equations. In section III, we construct the rotating dilaton black branes with a complete set of rotation parameters in all higher dimensions and investigate their properties. In section IV, we study thermodynamics of the spacetime, by calculating the conserved and thermodynamic quantities. In section V, we perform a stability analysis and show that the dilaton creates an unstable phase for the solutions. The last section is devoted to conclusions and discussions.

II. BASIC FIELD EQUATIONS
We consider an n-dimensional action in which gravity is coupled to a dilaton and a nonlinear electrodynamic field where the Lagrangian of the logarithmic nonlinear electrodynamics coupled to the dilaton field (LNd) is chosen in the following form L(F, Φ) = −8β 2 e 4αΦ/(n−2) ln 1 + e −8αΦ/(n−2)) F 2 8β 2 . (2) In action (1), R is the Ricci scalar curvature, Φ is the dilaton field, and V (Φ) is a potential for Φ. The dilaton parameter α determines the strength of coupling of the scalar and LNd fields, is the electromagnetic tensor field, A µ is the vector potential, and β is the nonlinear parameter with dimension of mass. The last term in Eq. (1) is the Gibbons-Hawking boundary term which is chosen such that the variational principle is well-defined. The manifold M has metric g µν and covariant derivative ∇ µ . Θ is the trace of the extrinsic curvature Θ ab of any boundary(ies) ∂M of the manifold M, with induced metric(s) h ab . In this paper, we consider the action (1) with a Liouville type potential, where Λ is a constant which may be referred to as the cosmological constant, since in the absence of the dilaton field (Φ = 0) the action (1) reduces to the action of Einstein gravity in the presence of nonlinear electrodynamics with cosmological constant. For later convenience, we redefine it as Λ = −(n − 1)(n − 2)/2l 2 , where l is a constant with dimension of length. The series expansion of (2) for large β, leads to For latter convenience we rewrite where we have defined By varying the action (1) with respect to the gravitational field g µν , the dilaton field Φ and the gauge field A µ . We find In the limiting case where β → ∞, we have L(Y ) = Y . In this case the system of field equations (4)-(6) restore the well-known equations of EMd gravity [10][11][12][13][14], as expected.

A. FINITE ACTION IN CANONICAL AND GRAND-CANONICAL ENSEMBLES
In general, the total action I given in Eq. (1) is divergent when evaluated on a solution. One way of dealing with the divergences of the action is adding some counterterms to the action (1). The counterterms should contain a part which removes the divergence of the gravity part of the action and a part for dealing with the divergence of the matter action. Since the horizon of our solution is flat, the counterterm which removes the divergence of the gravity part should be proportional to √ −h. The counterterm for the matter part of the action in the presence of the dilaton is given by where l eff is given by (13) and I deriv is a collection of terms involving derivatives of the boundary fields that could involve the curvature tensor constructed from the boundary metric. Since in our case the boundary is flat so I deriv is zero on the boundary. The variation of the total action (I tot = I + I ct ) about the solutions of the equations of motion is where Equation (8) shows that the variation of the total action with respect to A a will only give the equation of motion of the nonlinear massless field A a provided the variation is at fixed nonlinear massless gauge potential on the boundary. Thus, the total action, I tot = I + I ct , given in Eq. (8) is appropriate for the grand-canonical ensemble, where δA a = 0 on the boundary. But in the canonical ensemble, where the electric charge −e −4αΦ/(n−2) ∂ Y L(Y )n a F ab is fixed on the boundary, the appropriate action is The last term in Eq. (10) is the generalization of the boundary term introduced by Hawking for linear electromagnetic field [39] and the results of [40,41] for the nonlinear Lifshitz black holes to the exponential nonlinear gauge field coupled to the dilaton field. Thus, both in canonical and grand-canonical ensemble, the variation of total action about the solutions of the field equations is That is, the nonlinear gauge field is absent in the variation of the total action both in canonical and grand-canonical ensembles.
In order to obtain the conserved charges of the spacetime, we use the counterterm method [42,43] inspired by (A)dS/CFT correspondence. For asymptotically AdS solutions this method works very well [43]. However, in our paper we have the scalar dilaton field with a Liouville potential. It was argued that the presence of Liouville-type dilaton potential, which is regarded as the generalization of the cosmological constant, changes the asymptotic behavior of the solutions to be neither asymptotically flat nor (A)dS. It has been shown that no dilaton dS or AdS black hole solution exists with the presence of only one Liouville-type dilaton potential [9]. But, as in the case of asymptotically AdS spacetimes, according to the domain-wall/QFT (quantum field theory) correspondence [44], there may be a suitable counterterm for the stress-energy tensor which removes the divergences. In this paper, we deal with the spacetimes with zero curvature boundary [R abcd (h) = 0], and therefore the counterterm for the stress-energy tensor should be proportional to h ab . We find the finite stress-energy tensor in n-dimensional Einstein-dilaton gravity with Liouville-type in the form [38] where l eff is given by In the particular case α = 0, the effective l 2 eff of Eq. (13) reduces to l 2 = −(n − 1)(n − 2)/2Λ of the AdS spacetimes. The first two terms in Eq. (12) is the variation of the action (1) with respect to h ab , and the last term is the counterterm which removes the divergences. One may note that the counterterm has the same form as in the case of asymptotically AdS solutions with zero curvature boundary, where l is replaced by l eff . If we choose the Killing vector field ξ on spacelike surface B in ∂M with metric σ ij , then the quasilocal conserved quantities may be obtained from the following relation [38] where σ is the determinant of the boundary metric σ ij and n a is the unit normal vector on the boundary B. In our case, the boundary B has two Killing vector fields timelike (∂/∂t) and rotational (∂/∂ϕ). The corresponding conserved charges are the quasilocal mass and angular momentum may be obtained as

III. ROTATING DILATON BLACK BRANES IN HIGHER DIMENSIONS
In this section, we would like to construct the rotating black brane solutions of the field equations (4)-(6) with k rotation parameters. The number of independent rotation parameters for an n-dimensional localized object is equal to the number of Casimir operators, which is [(n − 1)/2] ≡ k, where [x] is the integer part of x [45]. The metric of n-dimensional rotating solution with cylindrical or toroidal horizons and k rotation parameters can be written as [46,47] where a i 's are k rotation parameters. There are two unknown functions f (r) and R(r) in the above metric which should be determined by solving the field equations. The range of the angular coordinates are 0 ≤ φ i ≤ 2π and dX 2 is the Euclidean metric on the (n − k − 2)-dimensional submanifold with volume Σ n−k−2 .
First of all, we integrate the electromagnetic field equation (6). The result is where q, is an integration constant related to the electric charge of the brane. When β → ∞, F tr reduces to the electric field of n-dimensional black brane of Einstein-Maxwell-dilaton gravity [45] In order to solve the system of equations (4) and (5) for three unknown functions f (r), R(r) and Φ(r), we make the ansatz [45] R(r) = e 2αΦ/(n−2) .
In order to justify this choice for the metric function R(r), let us note that R(r) is indeed added to the metric (17) in order to increase the degrees of freedom for obtaining solutions in the presence of the dilaton field. Choosing R(r) in the form of Eq. (21), is an ansatz. However, it is chosen such that in the absence of the dilaton field Φ = 0, we have R(r) = 1, as expected. With this ansatz, we are able to solve the field equation, analytically.
Substituting (21), the electromagnetic fields (18)- (19) and the metric (17) into the field equations (4) and (5), one can obtain the following solutions where c and b are constant of integration. We find that these solutions will fully satisfy the system of equations (4) and (5) provided we choose c = 0. Note that b has the dimension of [Length] to make the argument of logarithmic function dimensionless. In the above solutions γ = 2α 2 /(1 + α 2 ), and In the above expression, m appears as an integration constant and is related to the mass of the black hole. The integration of Eq. (22) can be performed using the MATHEMATICA software. The resulting solution can be written , z) is the hypergeometric function [48]. It is worth mentioning that the solutions are ill-defined for α = √ n − 1. We expect that for large β, the function f (r) reduces to the n-dimensional charged rotating dilaton black brane solutions given in Ref. [45]. Indeed, if we expand Eq. (25) for large β, we arrive at Setting α = γ = 0 in (26), we reach The last term in the right hand side of the above expression is the leading nonlinear correction to the AdS black brane with dilaton field. In the absence of a nontrivial dilaton (α = γ = 0), the above solutions reduce to the asymptotically AdS charged rotating black brane solutions of Einstein gravity in the presence of EN electrodynamics [33]. Finally, in the limit β 2 → ∞ and α = 0 = γ, the solution given by Eq. (26) has the form of the asymptotically AdS black brane solutions [46,49]. Figs. (1) and (2) depict the behavior of f (r) given by Eq. (25) for different α's and β's respectively.
A. Asymptotic behavior of the spacetime Next, we study the geometry of this spacetime. For this purpose, we first seek for the curvature singularities in the presence of dilaton and nonlinear electrodynamic fields. It is a matter of calculation to show that the Ricci scalar and the Kretschmann invariant behave as which indicate that there is an essential singularity at r = 0. In order to study the asymptotic behavior of the solutions, we expand the metric function f (r) for r → ∞ limit. We find Let us note that in the absence of the dilaton field (α = 0 = γ), the metric function becomes which describes an asymptotically AdS (Λ < 0) or dS (Λ > 0) spacetimes. However, as one can see from Eq. (29), in the presence of the dilaton field, the asymptotic behavior is neither flat nor (A)dS. For example, taking α = √ 2, n = 6 and b = 1, we have Clearly, the metric function (31) is neither flat nor (A)dS. This is consistent with the argument given in [9], which states that no dilaton dS or AdS black hole solution exists with the presence of only one or two Liouville-type dilaton potential. It is important to note that this asymptotic behavior is not due to the nonlinear nature of the electrodynamic field, since as r → ∞ the effects of the nonlinearity disappear. Besides, from the dilaton field (23) we see that as r → ∞, the dilaton field does not vanishes, while in case of asymptotic flat or (A)dS we expect to have lim r−→∞ Φ(r) = 0. Indeed, by solving the field equation (5) we find however, the system of equation (4)-(6) will be fully satisfied provided we choose c = 0. From the above arguments we conclude that the asymptotic behavior of the obtained solutions is neither flat nor (A)dS.

IV. THERMODYNAMICS OF BLACK BRANES
It is easy to show that the metric given by (17) and (25) has both Killing and event horizons [45]. The Killing horizon is a null surface whose null generators are tangent to a Killing field. It is easy to see that the Killing vector is the null generator of the event horizon, where Ω i is the ith component of angular velocity of the outer horizon which may be obtained by analytic continuation of the metric. The Hawking temperature and the angular velocities of the outer event horizon can be obtained as where η + = η(r = r + ) and we have used f (r + ) = 0 for deleting m. For large β, we can expand T + and arrive at the temperature of the higher dimensional black branes in EMd gravity [45] T The mass and angular momentum of the black branes (α < √ n − 1) can be calculated through the use of Eqs. (15) and (16). Denoting the volume of the hypersurface boundary at constant t and r by V n−2 = (2π) k Σ n−k−2 , the mass and angular momentum per unit volume V n−2 of the black branes can be obtained as Note that, in order to avoid repeating the factor V n−2 , we calculate, in this paper, the mass M and extensive quantities such as angular momentum J i , entropy S and charge Q appearing in first law of thermodynamics per unit volume. For the static case where a i = 0 (Ξ = 1), the angular momentum per unit volume vanishes, and therefore a i 's are the rotational parameters of the black branes. Black hole entropy typically satisfies the so called area law of the entropy [50]. This near universal law applies to almost all kinds of black holes and black branes in Einstein gravity [51]. It is easy to show that the entropy per unit volume V n−2 of the black brane can be written as The electric charge per unit volume V n−1 can be found by calculating the flux of the electric field at infinity, yielding where the volume is replaced by an arbitrary spacelike hypersurface Σ (partial Cauchy surface) with boundary ∂Σ.
In addition, the volume element on Σ is a non-spacelike covector (1-form) dS ν and dS µν is the area element of ∂Σ. We should note that for linear Maxwell case (β −→ ∞), one obtains ∂ Y L(Y ) = −1.
The electric potential U , measured at infinity with respect to the horizon, is defined by where χ is the null generator of the horizon given by Eq. (33). One can easily show that the vector potential A µ corresponding to the electromagnetic tensor (18) and (19) can be written as where 3 F 2 is the hypergeometric function and we have set the constant of integration equal to zero. Therefore, the electric potential may be obtained as Now, we are in a position to verify the first law of thermodynamics. In order to do this, we obtain the mass M as a function of extensive quantities S, J and Q. Using the expression for the mass, the angular momenta, the entropy, and the charge given in Eqs. (37), (38), (39), (40) and the fact that f (r + ) = 0, one can obtain a Smarr-type formula as where J = k i J i 2 , and Z = Ξ 2 is the positive real root of the following equation where ζ = π 2 Q 2 / S 2 β 2 . We can regard the parameters S, J, and Q as a complete set of extensive parameters for the mass M (S, J, Q) and define the intensive parameters conjugate to S, J and Q. This parameters are, respectively, the temperature, the angular velocities, and the electric potential, which are defined as Numerical calculations show that the intensive quantities calculated by Eq. (46) coincide with Eqs. (34), (35) and (44). Thus, these thermodynamics quantities satisfy the first law of thermodynamics

V. THERMAL STABILITY OF THE BLACK BRANES IN CANONICAL AND GRAND-CANONICAL ENSEMBLES
In this section, we intend to investigate thermal stability of our nonlinearly charged rotating black brane solutions in both canonical and grand-canonical ensembles. We know that the entropy of a thermally stable system is at local maximum. The aim of thermal stability analysis is to find the situations under which the system is stable thermally i.e. its entropy is a local maximum. Therefore, the stability of charged rotating black brane is studied in terms of entropy S(M, Q, J). However, the thermal stability can also be discussed in terms of internal energy. When the entropy is at local maximum, the internal energy is at local minimum. Hence, we can equivalently analyse thermal stability in terms of Legendre transformation of entropy namely internal energy M (S, Q, J). This analysis is commonly done by studying the determinant of the Hessian matrix of M (S, Q, J) with respect to its extensive variables X i , H M XiXj = ∂ 2 M/∂X i ∂X j [53,54]. The positivity of H M XiXj shows that the system is thermally stable. The number of thermodynamic variables depends on the ensemble in which the system is studied. For instance, in canonical ensemble where the charge and angular momenta are fixed, the entropy is the only variable and consequently H M XiXj reduces to (∂ 2 M/∂S 2 ) Q,J . Thus, in this ensemble, the positivity of (∂ 2 M/∂S 2 ) Q,J is sufficient to ensure the thermal stability of course in the ranges the temperature T is positive as well. In grand-canonical ensemble Q and J are no longer fixed.
Since the presence of charge does not change stable solutions to unstable ones [49], we first study thermal stability for uncharged case i.e. q → 0. In this case and Since Ξ 2 ≥ 1, both Eq. (48) and H M SJ are positive for α ≤ 1, therefore the uncharged rotating solutions are stable in both canonical and grand-canonical ensembles provided α ≤ 1. For this case, the temperature is also always positive as one can see from (36). As pointed out before, the charge cannot change thermal stability and therefore we always have thermally stable rotating black brane solutions for α ≤ 1. This fact is illustrated in Figs. (3) and (4) for different values of r + . The positivity of temperature for them is shown in Fig. (5). For different q's, Figs. (6) and (7) show that charge does not affect the thermal stability and therefore charged solutions are still stable for α ≤ 1. The positivity of T for mentioned parameters in Figs. (6) and (7) is shown in Fig. (8). Now, we discuss the stability for nonlinearly charged rotating black brane solutions for α > 1. One can see from (49) that α = 1 is the root of H M SJ . Numerical investigations show that α = 1 is the root of determinant of Hessian matrix in charged case too. Also, for α > 1, H M SQJ is always negative as H M SJ obviously is (see (49)). Therefore, we have unstable solutions for α > 1 in grand-canonical ensemble. Figs. (4) and (7) illustrate this fact. However, in canonical ensemble we have both stable and unstable solutions for α > 1. Figures (3) and (6) show that there is an α max (> 1) that we have stable solutions for values lower than it (note that α < √ n − 1; see sentences above (37)). There is also a r + max that for r + > r + max solutions are unstable (see Fig. (9)). The behavior of (∂ 2 M/∂S 2 ) Q,J in terms of q and β are depicted in Figs. (10) and (11) respectively. These figures show that there are q min and β min that for values greater than them black branes are thermally stable.

VI. CONCLUSIONS AND DISCUSSIONS
In this paper, we studied the higher dimensional action in the context of dilaton gravity and in the presence of the logarithmic nonlinear electrodynamics. By varying the action, we found the field equations of this theory. Then, we constructed a new class of charged, rotating black brane solutions, with k = [(n − 1)/2] rotation parameters, in an arbitrary dimension. We found that the presence of the dilaton field changes the asymptotic behavior of the obtained solutions to be neither flat nor (A)dS. We presented the suitable counterterm which remove the divergences of the action in the presence of the dilaton field. In the absence of a non-trivial dilaton (α = γ = 0), these solutions reduce to the asymptotically AdS charged rotating black brane solutions of Einstein theory in the presence of logarithmic nonlinear electrodynamics [33]. When β −→ ∞, these solutions reduce to the charged rotating dilaton black brane solutions given in Ref. [45]. We also calculated the conserved and thermodynamic quantities of the spacetime such as mass, angular momentum, temperature, entropy and electric potential and checked that the first law of thermodynamics holds on the black brane horizon. Then, we explored thermal stability of the nonlinearly charged rotating black brane solutions in both canonical and grand-canonical ensembles. We found that in both ensembles the solutions are thermally stable for α ≤ 1, while for α > 1 the solutions are always thermally unstable in the grand-canonical ensemble where α is the dilatonelectromagnetic coupling constant. In the canonical ensemble, however, we can have both stable and unstable solutions for α > 1. We found that, in this ensemble, there is an α max > 1 for which the solutions are thermally stable provided α < α max . The pointed out results implies that the thermal stability is ensemble-dependent and α influences the stability under thermal perturbations. These results are expectable since different ensembles allow different sets of quantities to be variable and a thermally stable system is one which is stable under varying variable quantities. On the other hand, values of conserved and thermodynamic quantities depend on values of parameters such as α and therefore the fact that dilaton-electromagnetic coupling has direct effect on thermal stability of the system seems natural.  It is notable to mention that in this paper, we only constructed the charged rotating dilaton black branes of nonlinear electrodynamics with flat horizon. One can try to construct the rotating dilaton black holes of this theory with curved horizon. One can laso investigate the thermodynamic geometry of these solutions. The latter issue is now under investigation and the results will be presented elsewhere.

Conflict of Interests
The author declares no conflict of interests for the present paper.