Inclusion of Landau damping in a time-dependent effective theory for a weak-coupling superconductor at finite temperature

The effective propagator for the Goldstone mode (phase of the order parameter) is calculated for a neutral BCS system in the long-wavelength/low-frequency limit, with inclusion of Landau damping terms, for temperatures between T = 0 and T = 0.6T(c). The Landau terms are first evaluated numerically; and then accurate closed-form expressions are found for them. The resulting propagator is shown to be well approximated by the product of two simple poles at complex energy, corresponding to a damped mode with linear (and T-dependent) dispersion for both the real and imaginary parts. Damping is only significant for T greater than or equal to 0.4T(c). By considering the Fourier transform of the inverse of this pole-dominated propagator, an effective local equation of motion for the phase degree of freedom is obtained, which includes a specific damping term. The damping may be phenomenologically included in the equivalent time-dependent nonlinear Schrodinger equation by giving the pair mass a small temperature-dependent positive imaginary part.