We have developed the recent investigations on the second-order phase transition in the holographic superconductor using the probe limit for a nonlinear Maxwell field strength coupled to a massless scalar field. By analytical methods, based on the variational Sturm-Liouville minimization technique, we study the effects of the spacetime dimension and the nonlinearity parameter on the critical temperature and the scalar condensation of the dual operators on the boundary. Further, as a motivated result, we analytically deduce the DC conductivity in the low and zero temperatures regime. Especially in the zero temperature limit and in two dimensional toy model, we thoroughly compute the conductivity analytically. Our work clarifies more features of the holographic superconductors both in different space dimensions and on the effect of the nonlinearity in Maxwell's strength field. 1. Introduction In the recent years, using the holographic picture of the world, the AdS/CFT (anti de Sitter/conformal field theory) correspondence [1–3] has been applied to study some strongly correlated systems in condensed matter physics, especially for strongly coupled systems with the scale-invariance. Particularly, people studied the low temperature, quantum critical systems near critical point (see, e.g., [4, 5] and references therein). The critical phenomena, which happen here, is a second-order phase transition from normal phase to the superconducting phase, in which below a specific temperature , the DC conductivity becomes infinite. Such second-order phase transitions happen in the high-temperature superconductors and can be described very well by the AdS/CFT dictionary [6, 7]. From the classical and phenomenological point of view, superconductivity, in the high-temperature type II superconductors, modeled by a phenomenological Landau-Ginzburg Lagrangian. This Lagrangian contains a general complex value scalar field , plays the role of a condensate in a superconductive phase. Basically, to have a scalar condensation in the boundary quantum field theory using CFT on the boundary of the bulk, Hartnoll et al. [8] introduced a abelian gauge field and a typical conformal coupled charged complex scalar field in the bulk black hole background. The conformal mass is above the Breitenlohner-Freedman (BF) bound [9]. To solve the negative mass problem, finally Gubser [10] showed that the vector potential modifies the mass term of scalar field and we have a possibility to have hairy black holes in some parts of the parameter space. The full description of the superconductivity in the
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