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

Abstract

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{sub 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{ge}0.4T{sub 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 Schr{umlt o}dinger equation by giving the pair mass a small temperature-dependent positive imaginary part. {copyright} {ital 1997} {ital The American Physical Society}