Finite-temperature time-dependent effective theory for the Goldstone field in a BCS-type superfluid

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 nonlinear Schrodinger Lagrangian 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 that are nonlocal and that destroy the Galilean invariance. We show that the retarded B 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.