Analytic calculation of properties of holographic superconductors

We calculate analytically properties of holographic superconductors in the probe limit. We analyze the range $1/2 < \Delta < 3$, where $\Delta$ is the dimension of the operator that condenses. We obtain the critical temperature in terms of a solution to a certain eigenvalue problem. Near the critical temperature, we apply perturbation theory to determine the temperature dependence of the condensate. In the low temperature limit we show that the condensate diverges as $T^{-\Delta/3}$ for $\Delta < 3/2$ whereas it asymptotes to a constant value for which we provide analytic estimates for $\Delta > 3/2$. We also obtain the frequency dependence of the conductivity by solving analytically the wave equation of electromagnetic perturbations. We show that the real part of the DC conductivity behaves as $e^{-\Delta_g /T}$ and estimate the gap $\Delta_g$ analytically. Our results are in good agreement with numerical results.