Heun's equation and analytic structure of the Gap in Holographic superconductivity

TL;DR

This paper introduces a new method to analyze the critical temperature and AC conductivity in holographic superconductors, revealing regimes where the critical temperature is ill-defined and providing analytical expressions that match numerical results.

Contribution

The paper develops a novel analytical approach to study the critical temperature and conductivity in holographic superconductors, especially in regimes lacking a well-defined critical temperature.

Findings

Critical temperature is ill-defined for 1/2 ≤ Δ < 1.

Derived analytical expression for AC conductivity at Δ=2.

Analytical results agree with numerical simulations.

Abstract

We present the new method to calculate the critical temperature as a function of $\Delta$, conformal dimension of the cooper operator. We find that, in the regime $1/2\leq \Delta <1$ where the AC conductivity does not show a gap, the critical temperature is not well defined. We also got expression of AC conductivity for $\Delta=2$, which agrees with numerical result in the probe approximation.