We analytically as well as numerically study the properties of one-dimensional holographic pwave superconductors in the presence of backreaction. We employ the Sturm-Liouville eigenvalue problem for the analytical calculation and the shooting method for the numerical investigations. We apply the AdS3/CFT2 correspondence and determine the relation between the critical temperature Tc and the chemical potential µ for different values of mass m of charged spin-1 field ρµ and backreacting parameters. We observe that the data of both analytical and numerical studies are in good agreement. We find out that increasing the backreaction as well as the mass parameter cause the greater values for Tc/µ. Therefore, it makes the condensation harder to form. In addition, the analytical and numerical approaches show that the value of the critical exponent β is 1/2 which is the same as in the mean field theory. Moreover, both methods confirm the exhibition of a second order phase transition.

We investigate the holographic p-wave superconductors in the presence of the higher order corrections on the gravity as well as on the gauge field side. On the gravity side, we add the Gauss-Bonnet curvature correction terms, while on the gauge field side we take the nonlinear Lagrangian in the form F + bF 2 , where F is the Maxwell Lagrangian and b indicates the strength of nonlinearity. We employ the shooting method for the numerical calculations in order to obtain the ratio of the critical temperature Tc over ρ 1/(d−2) . We observe that by increasing the values of the mass and the nonlinear parameters the critical temperature decreases and thus the condensation becomes harder to form. In addition, the stronger Gauss-Bonnet parameter α hinders the superconducting phase in Gauss-Bonnet gravity. Furthermore, we calculate the electrical conductivity based on the holographic setup. The real and imaginary parts are related to each other by Kramers-Kronig relation which indicates a delta function and pole in low frequency regime, respectively. However, at enough large frequencies the trend of real part can be interpreted by Re[σ] = ω (d−4) . Moreover, in holographic model the ratio ωg/Tc is always much larger than the BCS value 3.5 due to the strong coupling of holographic superconductors. In both gravity kinds, decreasing the temperature or increasing the effect of nonlinearity shifts the gap frequency toward larger values. Besides, the gap frequency is occurred at larger values by enlarging the Gauss-Bonnet parameter. In general, the behavior of conductivity depends on the choice of the mass, the nonlinear and the Gauss-Bonnet parameters.

We investigate analytically as well as numerically the effects of nonlinear Born-Infeld (BI) electrodynamics on the properties of (1 + 1)-dimensional holographic p-wave superconductor in the context of gauge/gravity duality. We consider the case in which the gauge and vector fields backreact on the background geometry. We apply the Sturm-Liouville eigenvalue problem for the analytical approach as well as the shooting method for the numerical calculations. In both methods, we find out the relation between critical temperature Tc and chemical potential µ and show that both approaches are in good agreement with each other. We find that if one strengthen the effect of backreaction as well as nonlinearity, the critical temperature decreases which means that the condensation is harder to form. We also explore the conductivity of the one-dimensional holographic p-wave superconductor for different values of b and T /Tc. We find out that the real and imaginary parts of the conductivity have different behaviors in higher dimensions. The effects of different values of temperature is more apparent for larger values of nonlinearity parameter. In addition, for the fixed value of T /Tc by increasing the effect of nonlinearity we observe larger values for Drude-like peak in real part of conductivity and deeper minimum for imaginary part.

We analytically as well as numerically study the effects of Born-Infeld nonlinear electrodynamics on the properties of (1 + 1)-dimensional s-wave holographic superconductors. We relax the probe limit and further assume the scalar and gauge fields to affect the background spacetime. We thus explore the effects of backreaction on the condensation of the scalar hair. For the analytical method, we employ the Sturm-Liouville eigenvalue problem, and for the numerical method, we employ the shooting method. We show that these methods are powerful enough to analyze the critical temperature and phase transition of the one-dimensional holographic superconductor. We find that increasing the backreaction as well as the nonlinearity makes the condensation harder to form. In addition, this one-dimensional holographic superconductor faces a second order phase transition and the critical exponent has the mean field value β = 1/2.

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