Effects of Backreaction on Power-Maxwell Holographic Superconductors in Gauss-Bonnet Gravity

We analytically and numerically investigate the properties of s-wave holographic superconductors by considering the effects of scalar and gauge fields on the background geometry in five dimensional Einstein-Gauss-Bonnet gravity. We assume the gauge field to be in the form of the Power-Maxwell nonlinear electrodynamics. We employ the Sturm-Liouville eigenvalue problem for analytical calculation of the critical temperature and the shooting method for the numerical investigation. Our numerical and analytical results indicate that higher curvature corrections affect condensation of the holographic superconductors with backreaction. We observe that the backreaction can decrease the critical temperature of the holographic superconductors, while the Power-Maxwell electrodynamics and Gauss-Bonnet coefficient term may increase the critical temperature of the holographic superconductors. We find that the critical exponent has the mean-field value $\beta=1/2$, regardless of the values of Gauss-Bonnet coefficient, backreaction and Power-Maxwell parameters.

dimensional boundary. According to the idea of the holographic superconductors given in [1], in the gravity side, a Maxwell field and a charged scalar field are introduced to describe the U (1) symmetry and the scalar operator in the dual field theory, respectively. This holographic model undergoes a phase transition from black hole with no hair (normal phase/conductor phase) to the case with scalar hair at low temperatures (superconducting phase) [4].
Following [1,2], an overwhelming number of papers have appeared which try to investigate various properties of the holographic superconductors from different perspective [5][6][7][8][9][10][11][12][13]. The studies were also generalized to other gravity theories. In the context of Gauss-Bonnet gravity, the phase transition of the holographic superconductors were explored in [14][15][16][17][18]. The motivation is to study the effects of higher order gravity corrections on the critical temperature of the holographic superconductors. Considering the holographic p-wave and s-wave superconductors in (3 + 1)dimensional boundary field theories, it was shown that when Gauss-Bonnet coefficients become larger the operators on the boundary field theory will be harder to condense [16]. Taking the backreaction of the gauge and scalar field on the background geometry into account, numerical as well as analytical study on the holographic superconductors in five dimensional Einstein-Gauss-Bonnet gravity were carried out in [17]. It was observed that the temperature of the superconductor decreases with increasing the backreaction, although the effect of the Gauss-Bonnet coupling is more subtle: the critical temperature first decreases then increases as the coupling tends towards the Chern-Simons value in a backreaction dependent fashion [17].
In addition to the correction on the gravity side of the action, it is also interesting to consider the corrections to the gauge field on the matter side of the action. In particular, it is interesting to investigate the effects of the nonlinear corrections to the gauge field on the condensation and critical temperature of the holographic superconductors. It was argued that in the Schwarzschild AdS black hole background, the higher nonlinear electrodynamics corrections make the condensation harder [19,20]. When the gauge field is in the form of Born-Infeld nonlinear electrodynamics, analytical study, based on the Sturm-Liouville eigenvalue problem, of holographic superconductors in Einstein [21] and Gauss-Bonnet gravity [23,24] have been carried out. In the background of d-dimensional Schwarzschild AdS black hole, the properties of Power-Maxwell holographic superconductors have been explored in the probe limit [25] and away from the probe limit [26]. In our recent paper [27], we have analytically as well as a numerically studied the holographic s-wave superconductors in Gauss-Bonnet gravity with Power-Maxwell electrodynamics. However, in that work, we did not investigate the effects of backreaction and limited our study to the case where scalar and gauge fields do not have an effect on the background metric. Our purpose in the present work is to disclose the effects of the backreaction on the phase transition and critical temperature of the Power-Maxwell holographic superconductors in Gauss-Bonnet gravity.
The organization of this paper is as follows. In the next section, we provide the basic field equations of Power-Maxwell holographic superconductors in the background of Gauss-Bonnet-AdS black holes by taking into account the backreaction. In section III, based on the Sturm-Liouville eigenvalue problem, we find a relation between the critical temperature and charge density of the backreacting holographic superconductor with Maxwell field in Gauss-Bonnet gravity. In section IV, we extend the study to the case of Power-Maxwell nonlinear electrodynamics. By applying the shooting method, we also compare our analytical calculations with numerical results in this section.
In section V, we calculate the critical exponent and the condensation values of the Power-Maxwell holographic superconductor with backreaction. We finish with conclusion and discussion in section VI.

II. BACKREACTING GAUSS-BONNET HOLOGRAPHIC SUPERCONDUCTORS
To study a (3 + 1)-dimensional holographic superconductor, we begin with a (4 + 1)-dimensional action of Einstein-Gauss-Bonnet-AdS gravity which is coupled to Power-Maxwell field and a charged scalar field, where κ 2 = 8πG 5 with G 5 is the 5-dimensional gravitational constant, Λ = −6/l 2 is the negative cosmological constant, where l is the AdS radius of spacetime, and α is the Gauss-Bonnet coefficient.
Here, R and R µν and R µνσρ are, respectively, Ricci scalar, Ricci tensor and Riemann curvature tensor. F µν is the electromagnetic field tensor and q is the power parameter of the Power-Maxwell field. ψ is complex scalar field with the charge e and the mass m, and A is the gauge field. Also, b is coupling constant and due to positivity of energy density has sign (−1) q+1 [28,29]. For latter convenience we shall take b = (−1/2) q+1 . With this choice, the Power-Maxwell Lagrangian will reduce to the Maxwell Lagrangian in the limit q = 1.
It is easy to check that by re-scaling ψ →ψ/e, φ →φ/e and b →be 2q−2 , a factor 1/e 2 will appear in front of matter part of action (1). Thus, the probe limit can be deduced when κ 2 /e 2 → 0.
In order to take the backreaction into account, in this paper, we keep κ 2 /e 2 finite and for simplicity we set e as unity.
Taking the backreaction into account, the plane-symmetric black hole solution with an asymptotically AdS behavior in 5-dimensional spacetime may be written where is the effective AdS radius of the spacetime. The ratio of l eff /l can be smaller than unity for α > 0, while for α < 0 it is obvious that l eff /l is larger than unity.
Superconductivity phase transition is dual to formation of charged matter field in the bulk, and for occurrence this phase transition in bulk, one needs to prevent the charged matter field to falls into the black hole, thus we expect greater curvature of spacetime in bulk make condensation harder which corresponds to the positive values of α. Also for α < 0, we shall see that the scalar field can be formed easier, means at higher temperature.
The Hawking temperature of black hole is given by where r + is the black hole horizon and the prime denotes derivative with respect to r. We choose the electromagnetic gauge potential and scalar field as Without lose of generality, we can take φ(r) and ψ(r) real. The equation of motions can be obtained by varying action (1) with respect to the metric and matter fields. We find: In order to solve the above field equations, we need appropriate boundary conditions both on the horizon r + , which is defined by f (r + )=0, and on the AdS boundary where r → ∞. On the horizon, the regularity condition imposes and thus from Eqs. (8) and (9) we have Since our solutions are asymptotically AdS, thus as r → ∞, we have where µ and ρ are, respectively, chemical potential and charge density of the CFT boundary, and ∆ ± is defined as According to the AdS/CFT correspondence, ψ ± =< O ± >, where O ± is the dual operator to the scalar field with the conformal dimension ∆ ± . We have the freedom to impose boundary conditions such that either ψ − or ψ + vanish. We prefer to keep fixed ∆ ± while we vary α, thus we setm 2 = m 2 l 2 eff . For example, form 2 = −3, we have ∆ + = 3 for all values of parameter α. It is important to note that, unlike other known electrodynamics, the boundary condition for the gauge field φ(r) given in Eq. (13), depends on the power parameter q. Using boundary condition (13) and the fact that φ should be finite as r → ∞, we require that (4 − 2q)/(2q − 1) > 0 which restricts q to ranges as 1/2 < q < 2.
It is easier to work in the dimensionless variable, z = r + /r, instead of variable r. Under this transformation, equations of motion (6)-(9) become Here the prime indicates the derivative with respect to the new coordinate z which ranges in the interval [0, 1], where z = 0 and z = 1 correspond to the boundary and horizon, respectively. Since near the critical point the expectation value of scalar operator (< O ± >) is small, we can select it as an expansion parameter where i = ±. Using the fact that ǫ ≪, we can expand f and χ around the Gauss-Bonnet AdS spacetime as Note that since we are interested in solution in which condensation is small, ψ and φ can also be expanded as We further assume the chemical potential is expanded as [30], where δµ 2 > 0. Thus near the critical point for the order parameter as the function of chemical potential we have It is obvious when µ → µ 0 , the order parameter approaches zero which indicate phase transition point. Thus phase transition occurs at the critical value µ c = µ 0 . Let us note that the order parameter grows with exponent 1/2 which is the universal result from the Ginzburg-Landau mean field theory.
In the next two sections we solve the field equations (15) At zeroth order for the expansion parameter, Eq. (16) may be written as which is the equation of motion of the electromagnetic field in the Maxwell theory and has solution φ 0 (z) = µ 0 1 − z 2 with µ 0 = ρ/r 2 + . At the critical point, we have µ 0 = µ c = ρ/r 2 +c , where r +c is the radius of the horizon at the phase transition point. Therefore, solution of φ 0 (z) at the critical point may be written as Inserting back this solution into Eq. (18), we find the metric function at the zeroth order: where we have used the fact that on the horizon f 0 (1) = 0, and we have defined a new function g(z) for convenience. We note that f 0 (z) restores the metric function of Gauss-Bonnet-AdS gravity in the probe limit as κ → 0.
At the first order approximation, the asymptotic AdS boundary conditions for ψ can be expressed as Near the boundary z = 0, we introduce trial function F (z) with boundary condition F (0) = 1 and F ′ (0) = 0. Substituting Eq. (30) into (15) we arrive at We can convert Eq. (31) into the Standard Sturm-Liouville equation, namely where , According to the Sturm-Liouville eigenvalue problem, ζ 2 can be obtained via In order to determine T (z) we need to solve equation where p(z) is Since α is small, we can expand the above expression for p(z) and keep terms up to O(α 2 ). Then we put the result in Eq. (35) and obtain the following solution for T (z) For small backreaction parameter, κ, the explicit expressions for T (z), Q(z) and P (z) up to second order terms of α and κ, are given by where s = 3α 2 + α + 1 and hereafter we set l = 1 for simplicity. In order to use Sturm-Liouville eigenvalue problem, we will use iteration method in the rest of this section. We take κ = κ n ∆κ where ∆κ = κ n+1 − κ n is step size of iterative procedure and we choose ∆κ = 0.05. Using the fact that and taking κ −1 = ζ| κ −1 = 0, we obtain the minimum eigenvalue of Eq. (32). We also take the trial function F (z) = 1 − az 2 . For example form 2 = −3, α = 0.05 and κ = 0, we have which attains its minimum ζ 2 min = 19.9456 for a = 0.7147. In the second iteration, we take κ = 0.05 and ζ 2 | κ 0 = 19.9456 in calculation of integrals in Eq. (34), and therefore for ζ 2 κ 1 , we get which has the minimum value ζ 2 min = 19.7936 at a = 0.7119. In the Table I we summarize our results for ζ min and a with different values of Gauss-Bonnet coupling parameter α, backreaction parameter κ and reduced mass of scalar fieldm 2 .
Combining Eqs. (4), (12), (27) and using definition of ζ, we obtain the following expression for the critical temperature We apply the iterative procedure to obtain critical temperature for different values of α, κ and m 2 . In table II we summarize critical temperature of phase transition of holographic superconductor in Maxwell electrodynamics for ∆ + obtained analytically from Sturm-Liouville method. For comparison, we also provide numerical results which we obtain by using shooting method. In this numerical method we solve Eq. (15) with φ(z) and f (z) given in Eqs. (27) and (28). Then we find the critical charge density ρ which satisfy the boundary condition ψ − = 0 in z → 0. We obtain discrete values of critical ρ which had this situation. Due to the stability condition [32], we chose the lowest value of ρ c and by using dimensionless quantity T 3 /ρ we calculated critical temperature of the phase transition for different values of Gauss-Bonnet parameter and backreaction parameter.

IV. CRITICAL TEMPERATURE OF GB HOLOGRAPHIC SUPERCONDUCTOR WITH POWER-MAXWELL FIELD
In this section we investigate the behavior of holographic superconductor for the general case q = 1 away from probe limit in the Gauss-Bonnet gravity. Just like previous section, we need solution of Eqs. (16), (17) and (18) in order to solve (15). Using expansion (20)- (23) and at the zeroth order of small parameter ǫ, one can easily check that φ 0 and g have the following solution where g(z) = f 0 (z)/r 2 + . Expanding the above expression for g(z) up to O(κ 4 ) and O(α 2 ), one gets One may substitute Eq. (30) and Eq. (45) into Eq. (15) and get an expression for F (z), and then converting it to the Sturm-Liouville equation form (32), resulting in: Again, using Eq. (32), with trial function F (z) = 1 − az 2 , we obtain the minimum eigenvalue ζ 2 min for the Power-Maxwell electrodynamic case. For example, with q = 3/4, α = 0.1, κ = 0.05, and m 2 = −3 and using iterative procedure, we get Varying ζ 2 κ 1 with respect to a to find minimum value of ζ 2 , we obtain ζ 2 min = 9.50679 at a = 0.5675. Also for the case q = 5/4, α = −0.19, κ = 0.1 andm 2 = 0 we obtain ζ 2 κ 2 = 461.3339a 2 − 968.8766a + 593.0286 a 2 − 3.2657a + 3.0859 , which attains its minimum ζ min = 98.9682 at a = 0.8909. Then, we find the critical temperature from Eqs. (4), (12) and (45) as Clearly, T c depends on the Power-Maxwell parameter q, Gauss-Bonnet parameter α and backreaction parameter κ. In Fig. 1, we present reduced critical temperature of phase transition for a (3 + 1)-dimensional holographic superconductor as a function of q with different values of κ and α. For simplicity, we focus on the boundary condition which ψ − = 0, and as an example, we takẽ m 2 = −3 in these figure.
In Fig. 1(a) we fix the backreaction parameter to κ = 0.05 in order to investigate behavior of critical temperature as a function of power parameter q for three allowed value of Gauss-Bonnet parameter. It clearly indicates that for any values of α, by decreasing q, superconductor phase is more accessible. Also, we find out that in the presence of backreaction of the matter fields on the metric, increasing Gauss-Bonnet parameter α makes condensation harder and and thus the critical temperature of the phase transition decreases. It is interesting that decreasing α from zero to negative values in the allowed range can cause the phase transition to superconductor phase easier for any values of the power parameter q.
We also provide Fig. 1(b) by fixing the Gauss-Bonnet parameter to α = 0.1 for studying the behavior of reduced critical temperature in terms of the power parameter q for different values of the backreaction parameter κ. From this figure we see that for any values of q, by increasing the backreaction of the matter fields on the background geometry, which is corresponding to decreasing the charge of the scalar field, the phase transition is made harder in the Einstein-Gauss-Bonnet gravity.
We mention that in the allowed range of the power parameter, there exist some un-physical regimes in which critical temperature becomes negative. For example, by increasing backreaction parameter to greater values, we may obtain negative T c which means for some values of the power parameter we do not have phase transition if complex field charge is less than some critical charge.
Here we disregard these regimes and work in regimes with positive temperatures.
Finally, we present table III to compare the results of critical temperature from analytical  and arrive at where g(z) is defined as in Eq. (47). Near the critical point, T c ≈ T 0 , and inspired by Eq. (45), we assume that Eq. (54) has the following solution where Substituting Eq. (55) into (54) and keeping terms up to < O i > 2 , we reach where η = 2∆ i − 4q + 5 − 4q 2q − 1 (2 − 2q).
This is a differential equation for Ξ(z) independent of r + , r +c and < O i >. Therefore Ξ(z) in any z has a value independent of T , T c and order parameter < O i >.
The boundary condition for φ given by Eq. (13), in the z coordinate, can be rewritten as It is reliable while z ≈ 0, independent of temperature and order parameter. Also near the critical temperature where ψ is small, Eq. (45) may be expressed as Since it is valid for all values of z, we can equate the above expression with Eq. (59) for z → 0 to find µ = ρ Using the analytical Sturm-Liouvill method, we have calculated the proportional constant between the critical temperature and the charge density for all allowed values of the power parameter q, different values of the Gauss-Bonnet coupling constant α, and backreaction parameter κ. We realized that decreasing q from Maxwell case (q = 1) to it's lower bound (q = 1/2) increases the critical temperature, regardless of the values of α and κ. Besides, for a fixed values of q and κ, critical temperature increases with decreasing the Gauss-Bonnet coefficient α. This means that, increasing q and α will decrease the critical condensation of the scalar field and make it harder to form. Also, we observed that taking backreaction into account, decreases the critical temperature regardless of the values of the other parameters. We have confirmed these analytical results by providing the numerical calculations based on the shooting method. Finally, our investigation of critical exponent indicates that the critical exponent β of the superconducting phase transition for the five dimensional Power-Maxwell holographic superconductor with backreaction has the mean field value 1/2 which seems to be a universal constant.