An analytic study on the excited states of holographic superconductors

TL;DR

This paper introduces an analytic Sturm-Liouville method to study excited states in holographic superconductors, showing good agreement with numerical results and confirming the second-order nature of phase transitions.

Contribution

It develops a general analytic approach for excited states in holographic superconductors using Sturm-Liouville eigenvalue problems, extending previous numerical studies.

Findings

Excited states have lower critical temperatures than ground states.

The difference in critical chemical potential between states is about 5.

Phase transition remains second order for excited states.

Abstract

Based on the Sturm-Liouville eigenvalue problem, we develop a general analytic technique to investigate the excited states of the holographic superconductors. By including more higher order terms in the expansion of the trial function, we observe that the analytic results agree well with the numeric data, which indicates that the Sturm-Liouville method is very powerful to study the holographic superconductors even if we consider the excited states. For both the holographic s-wave and p-wave models, we find that the excited state has a lower critical temperature than the corresponding ground state and the difference of the dimensionless critical chemical potential between the consecutive states is around 5. Moreover, we analytically confirm that the holographic superconductor phase transition with the excited states belongs to the second order, which can be used to back up the numerical findings for both s-wave and p-wave superconductors.