Lifshitz scaling effects on the holographic p-wave superconductors coupled to nonlinear electrodynamics

We employ gauge/gravity duality to study the effects of Lifshitz scaling on the holographic $p$ wave superconductors in the presence of Born-Infeld (BI) nonlinear electrodynamics. By using the shooting method in the probe limit, we calculate the relation between critical temperature $T_{c}$ and $\rho^{z/d}$ numerically for different values of mass, nonlinear parameter $b$ and Lifshitz critical exponent $z$ in various dimensions. We observe that critical temperature decreases by increasing $b$, $z$ or the mass parameter $m$ which makes conductor/superconductor phase transition harder to form. In addition, we analyze the electrical conductivity and find the behavior of real and imaginary parts as a function of frequency which depend on the model parameters. However, some universal behaviors are seen. For instance at low frequencies, real part of conductivity shows a delta function behavior while the imaginary part has a pole which means that these two parts are connected to each other through Kramers-Kronig relation. The behavior of real part of conductivity in the large frequency regime can be achieved by $Re[\sigma]=\omega^{D-4}$. Furthermore, with increasing the Lifshitz scaling $z$, the energy gap and the minimum values of the real and imaginary parts become unclear.

works [52][53][54]. Moreover, in the framework of condensed matter physics, a dynamical scaling appears near the critical point. In the vicinity of critical point, the scale transformation turns to be [55]: (1) when z = 1, the usual AdS spacetime is obtained. Otherwise, the temporal and the spatial coordinates scale anisotropically. Many investigations have been done regarding the anisotropic superconductors [56]. Nowadays, the applications of superconductors aren't limited to physics. However, the actual use of superconducting devices is limited due to the fact that they must be cooled to low temperatures to become superconductors and hence cannot use for ordinary (lab) temperatures. So by passing time, p-wave and d-wave superconductors as candidates for high temperature superconductors grab a lot of attentions. In some cases p-wave superconductors show anisotropic behavior [3]. In addition, heavy fermion compounds and other materials including high T c superconductors have a metallic phase (dubbed as strange metal) whose properties cannot be explained within the ordinary Landau-Fermi liquid theory. In this phase some quantities exhibit universal behavior such as the resistivity, which is linear in the temperature σ ∼ T . Such universal properties are believed to be the consequence of quantum criticality. At the quantum critical point there is a Lifshitz scaling same as Eq. (1) [57]. Many researches carried out in Lifshitz scaling by using the holographic approach (see e.g. [58][59][60][61]). However, all the previous works about the effects of Lifshitz scaling on holographic p-wave superconductor were done by considering a SU(2) Yang-Mills gauge field in the bulk in the presence of Maxwell electrodynamics [55,58]. In this work, we have explored the effects of Lifshitz scaling on holographic p-wave superconductor by introducing a charged vector field in the bulk. As the consequence of this method all following calculations to analyze the behavior of condensation and conductivity became different from the previous works. However, there are good agreements between our results with the case of m 2 = 0 in [58]. This approach allows us to consider the effects of mass term as well as the spacetime dimension and Lifshitz scaling z, while in the previous method the mass term plays no role and sets to zero. Besides, we can study the effect of conductivity in easier way by turning on only the component δA y = A y e −iωt as perturbation on the black hole background in compared to [58]. Moreover, It is worthful to investigate this model in the presence of nonlinear electrodynamics [55,58]. There exist several types of nonlinear electrodynamics in the literatures, including BI [62], Exponential [63], Logarithmic [64] and Power-Maxwell [13]. Among them, the BI nonlinear electrodynamics is perhaps the most well-known which was proposed to address the problem of the divergence of the electrical field at the position of a point particle. It was later pointed out that BI Lagrangian could be reproduced through string theory, and its action naturally possesses electric-magnetic duality invariance which makes it suitable for describing gauge fields arising from open strings on D-branes [36,49,62,63]. Disclosing the effects of the nonlinear electrodynamics on the behavior of superconductors is of interest both for practical applications and for the study of the fundamental properties of the materials. In any practical electronic application, the designer must know how much power the conductors can handle and at which power level nonlinear effects such as harmonic generation and intermodulation (IM) distortion become appreciable. Therefore, the magnitude and the detailed nature of the nonlinear effects must be measured and understood in order to facilitate widespread application of superconductors in microwave frequency electronics. The effects of nonlinear electrodynamics widely studied in the literatures (see e.g [65][66][67][68][69][70]). Although these investigations extend over many years, new interest in these nonlinear effects has been kindled since the discovery of the high-T, oxide materials [70]. Therefore, it is worthy to provide theoretical approach for prediction of the behavior of real high temperatures superconductors in the presence of nonlinear electrodynamics based on the holographic approach. For example we find out that by increasing the effect of nonlinearity, critical temperature decreases. The effects of nonlinear electrodynamics on the holographic superconductors have been explored widely in the literatures (see e.g. [13,[29][30][31][32][33][34][35][36]49]). Our aim in this work is to investigate the effects of Lifshitz scaling on the holographic p-wave superconductor in arbitrary dimensions. We shall study the phase transition between conductor and superconductor which depends crucially on the parameters m, b and z. In spite of the fact that there is a dynamical exponent, we find out that the condensation has mean-field behavior near the critical temperature which is the same as AdS spacetime. Additionally, the electrical conductivity in gauge/gravity correspondence is achieved by imposing appropriate perturbation on the gauge field. Besides, the conductivity formula, we calculate the behavior of both real and imaginary parts of conductivity as a function of frequency. Although, the obvious differences in graphs based on our choice of m, b and z, they follow same trends in some cases. A good illustration of this is obeying the Kramers-kronig relation by having a delta function and a pole in real and imaginary parts of conductivity. However, the gap frequency which is occurred below the critical temperature, becomes less obvious by enlarging the anisotropy between space and time. The effects of nonlinearity parameter on the conductivity will be clearly indicated via graphs. Our choice of mass in each dimension has a direct outcome on the effect of BI nonlinear electrodynamics on conductivity but generally the gap energy and minimum of conductivity shift toward larger values of frequency by enlarging the nonlinearity effects.
This article is organized as follows. In section II, we analyze the holographic setup via condensation of the vector field in the context of Lifshitz spacetime and in the presence of nonlinear BI electrodynamics. We explore the electrical conductivity of this model in section III. Finally, our outcomes are summarized in section IV.

II. HOLOGRAPHIC p-WAVE SUPERCONDUCTORS WITH LIFSHITZ SCALING
We consider a (d+1)-dimensional holographic p-wave superconductor living on the boundary of a (d+2)-dimensional Lifshitz black hole in the presence of BI nonlinear electrodynamics, which is described by the following action [38] where m and q are the mass and charge of vector field ρ µ . The metric determinant, the Ricci scalar and the negative cosmological constant are demonstrated by g, R and Λ, respectively. In terms of the radius of Lifshitz spacetime, l, we can formulate the cosmological constant as [58] where z, the Lifshitz scaling, is a dynamical critical exponent standing for the anisotropy between space and time.
Hereafter, for simplicity we set l = 1. The Lagrangian density of the BI nonlinear electrodynamics L(F ) is given by [62] where F = F µν F µν is the Maxwell invariant and b, with dimension of [length] 2 , represents the strength of nonlinearity.
In the Lagrangian of the matter field L m , by using the covariant derivative The last term in the matter Lagrangian shows the strength of interaction between ρ µ and A µ with γ as the magnetic moment in the case with an applied magnetic field which will be ignored in this work. The equations of motion, for the matter fields, can be obtained by varying the action (2) with respect to the gauge field A µ and the vector field ρ µ , where L F = ∂L(F )/∂F . Since we work in the probe limit, the background spacetime is not affected by the vector and gauge fields. Thus, we can write down the metric of (d + 2)-dimensional Lifshitz spacetime as [58] where r + denotes the black hole horizon obeying f (r + ) = 0. We also assume the vector and gauge field has the following form The regularity condition for the gauge field, on the horizon, implies that φ(r + ) = 0 [55]. The Hawking temperature of the black hole, associated with the horizon, is defined [58] Inserting Eqs. (7) and (9) in the field Eqs. (5) and (6), we arrive at ρ ′′ where the prime denotes derivative with respect to r. The linear electrodynamic form of the above equations of motion are recovered in the limiting case where b → 0 [58]. In the remaining part of this paper without loss of generality, we will set r + = 1 and q = 1. At the boundary where r → ∞, the above equations have the asymptotic solutions as According to gauge/gravity duality µ, ρ, ρ x+ and ρ x− play, respectively, the role of the chemical potential, charge density, x-component of the vacuum expectation value of the order parameter J x and source. Since we expect the spontaneous U (1) symmetry breaking so we impose the source free condition i.e. ρ x− = 0. In addition, we follow the Breitenlohner-Freedman (BF) bound for our choice of the mass, Considering the canonical ensemble with fixed ρ, and employing the shooting method, we perform the numerical calculations to derive the relation between critical temperature, T c , and charge density, ρ z/d , for z = 1, 2 in D = d + 2 = 4 and 5 spacetime dimensions. Our results are summarized in table I. We find out that increasing the values of z, m and nonlinearity b, in each dimension, hinders the superconductivity phase by diminishing the critical temperature. Moreover, the trends of condensation J x 1/(1+∆+) as a function of temperature impressed by different values of z, m and b are shown in figures 1 and 2. Based on these graphs, condensation goes down by raising z.

III. ELECTRIC CONDUCTIVITY
In this section, we are going to investigate the effects of Lifshitz scaling and nonlinear parameter on the electric conductivity of holographic p-wave superconductor. For this purpose, we apply appropriate electromagnetic perturbation by turning on δA y = A y e −iωt on the black hole background which acts as boundary electrical current in holographic setup [58]. So, we have In the Maxwell limit Eq. (16), except a factor 2 in the last term which is originated from different approaches to calculate conductivity, turns to corresponding equation in Ref. [58]. The above equation has the asymptotic behavior as which admits the following solution for A y , where A (0) , A (1) and Ω are constant parameters. Furthermore, by considering z = 1, Eqs. (17) and (18) have the same form as in AdS case [54]. Based on gauge/gravity duality, the electrical current is given by in which the on-shell bulk action S o.s by using equation (19) is defined by The electrical conductivity in a corresponding framework is [6] σ yy = J y E y , E y = −∂ t δA y .
Employing Eqs. (19), (20) and (21) and using appropriate counterterms, based on the re-normalization method to remove the divergency, the electrical conductivity is obtained as [71] σ yy = For z = 1, we obtain the same equations as in the AdS background [54]. In order to follow our research, we imply an ingoing wave boundary condition near the horizon as where by Taylor expansion of equation (17) around the horizon, coefficients a and b are obtained. The behavior of real and imaginary parts of conductivity as a function of ω/T are shown in Figs. 3-6. The conductivity along the y direction in Lifshitz holographic p-wave superconductors is the same as σ xx in s-types [58]. Although the figures follow different trends but in all cases the behavior of real part of conductivity in large frequency regime can be predicted by a power law function as Re[σ] = ω D−4 similar to [54]. The real and imaginary parts of conductivity follow the Kramers-Kronig relation. Thus, we observe the appearance of the delta function and pole, respectively. By increasing the Lifshitz scalaing z, the gap energy becomes unclear like the minimum of the imaginary part. However, in some cases, we observe that by decreasing the temperature [55] for strong BI nonlinear electrodynamics, the gap and minimum in real and imaginary parts of conductivity appear which is obvious in D = 5 with m 2 = 0 and b = 0.04. When z = 1, below the critical temperature, the superconducting gap appears and becomes deeper and sharper by diminishing the temperature which yields to larger values of ω g which makes the conductor/superconductor phase transition harder to form because it can be interpreted as the energy needs to break the fermion pairs. With the use of figures 7-8 which are graphed in T = 0.3T c , the value ω g = 8T c is generally achieved for z = 1 which differs from the predicted value 3.5 of BCS theory. This difference is originated from the fact that holographic superconductors are strongly coupled systems and because of this character, they are expected to be suitable for description the high temperature superconductors [52]. In addition, larger values of nonlinearity parameter shifts the maximum and minimum parts of conductivity toward larger values but the effects of nonlinearity on the value of energy gap depends on our choice of mass which implies the fact that the value of ω g /T c is characterized by our selection of mass m, nonlinearity b and dynamical critical exponent z in each dimension.

IV. SUMMARY AND CONCLUSION
In this work by employing the gauge/gravity duality, we have studied the holographic p-wave superconductors with Lifshitz scaling in the presence of BI nonlinear electrodynamics. We applied the shooting method to calculate the equations of motion and analyze the behavior of the condensation as a function of temperature numerically. We found the relation between critical temperature T c and ρ z/d for different values of mass m, nonlinearity effect b and Lifshitz scaling z in 4D and 5D spacetime. Based on our results, we observe that the temperature decreases with increasing each of three parameters m, z and b, which means that superconductivity phase faces with more difficulties to occur. The condensation behavior in Lifshitz scaling is similar to AdS spacetime by obeying the mean field trend in the vicinity of critical point. Increasing the anisotropy between space and time, diminishes the condensation value. After that, by applying a suitable perturbation on black hole background as δA y = A y e −iωt , we investigated the effects of Lifshitz scaling on the electrical conductivity of the holographic p-wave superconductors and plot the behavior of real and imaginary parts of conductivity as a function of frequency. The plotted Figures are different with each other but they follow some universal behaviors. For instance, in large frequencies, we can predict the behavior of real part of conductivity as Re[σ] = ω D−4 . In addition, the real and imaginary parts of conductivity are related to each other via the Kramers-Kronig relation. Actually, the real part shows a delta function behavior and the imaginary part has a pole at zero frequency. At low frequencies with z = 1, real and imaginary parts of conductivity show a gap energy and minimum which shift toward larger frequencies by diminishing temperature. In this regime, increasing the Lifshitz  critical exponent z, makes the gap energy and minimum unclear. However in some cases, they were occurred by going down temperatures and raising the nonlinearity effect. Our choice of mass in each dimension has a direct outcome on the effect of BI nonlinear electrodynamics on conductivity but generally the gap energy and minimum of conductivity shift toward larger values of frequency by enlarging the nonlinearity effect. In the limiting case where z = 1, the ratio ω g /T c ≃ 8 is obtained generally which is larger that the BCS value because of the strong coupling between the pairs.