Finite Temperature Time-Dependent Effective Theory For The Goldstone Field In A BCS-Type Superfluid

TL;DR

This paper develops a finite temperature effective theory for the Goldstone field in a BCS superfluid, extending the zero-temperature model to include Landau damping effects and analyzing the resulting fluid components.

Contribution

It introduces a local time-dependent non-linear Schrödinger Lagrangian for the Goldstone field at finite temperature, incorporating Landau damping and distinguishing superfluid and normal fluid components.

Findings

Effective theory can be written as a local TDNLSL neglecting Landau damping.

Landau damping introduces non-local terms destroying Galilean invariance.

Retarded θ-propagator can be modeled with two poles indicating damping.

Abstract

We extend to finite temperature the time-dependent effective theory for the Goldstone field (the phase of the pair field) $ \theta $ which is appropriate for a superfluid containing one species of fermions with s-wave interactions, described by the BCS Lagrangian. We show that, when Landau damping is neglected, the effective theory can be written as a local time-dependent non-linear Schr\"{o}dinger Lagrangian (TDNLSL) which preserves the Galilean invariance of the zero temperature effective theory and is identified with the superfluid component. We then calculate the relevant Landau terms which are non-local and which destroy the Galilean invariance. We show that the retarded $\theta$-propagator (in momentum space) can be well represented by two poles in the lower-half frequency plane, describing damping with a predicted temperature, frequency and momentum dependence. It is argued that the real parts of the Landau terms can be approximately interpreted as contributing to the normal fluid component.