Non-equilibrium, time-dependent effective theory of a weakly coupled superconductor at finite temperature

TL;DR

This paper develops a unique, well-defined derivative expansion to derive a time-dependent effective theory for a weakly coupled BCS superconductor at finite temperature, avoiding common analytical issues.

Contribution

It introduces a novel derivative expansion method that is free from standard analytical continuation and limit order problems in superconductor effective theories.

Findings

Landau damping terms are included without causing pathologies.

The method overcomes inconsistencies in standard approaches.

Results are compared favorably with existing literature.

Abstract

We perform a well defined derivative expansion to obtain the time dependent effective theory for a BCS superconductor at finite temperature, using an arbitrary curve in the complex time plane. Our expansion is unique, being free of any problems related to the order in which analytical continuation and the derivative expansion are made, or equivalently to the order in which the zero momentum and zero energy limits are taken. In other words Landau damping terms are present but do not lead to the pathologies found in standard approaches. We discuss the reason for this, suggesting that these standard approaches may be internally inconsistent. The methods presented here are an attempt to overcome those inconsistencies, and finally, our results are compared to others in the literature.