Inclusion of Landau damping in a time-dependent effective theory for a weak-coupling superconductor at finite temperature
I. J. R. Aitchison, D. J. Lee
DOI 10.1103/PhysRevB.56.8303 · Physical Review B
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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=0.6Tc. 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>~0.4Tc. 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 Schrödinger equation by giving the pair mass a small temperature-dependent positive imaginary part.
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