Finite-temperature time-dependent effective theory for the Goldstone fieldin 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 f or a superfluid containing one species of fermions with s-wave interactions , described by the BCS Lagrangian. We show that, when Landau damping is neg lected, the effective theory can be written as a local time-dependent nonli near Schrodinger Lagrangian which preserves the Galilean invariance of the zero-temperature effective theory and is identified with the superfluid com ponent. We then calculate the relevant Landau terms that are nonlocal and t hat 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, frequenc y, 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.