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arXiv:cond-mat/9411047 · Physical review. B, Condensed matter

Effective Lagrangians for BCS superconductors atT=0

I. J. R. Aitchison, Ping Ao, D. J. Thouless, X.-M. Zhu

Published Mar 1, 1995• 92 citations
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Abstract

We show that the low-frequency, long-wavelength dynamics of the phase of the pair field for a BCS-type s-wave superconductor at T=0 is equivalent to that of a time-dependent nonlinear Schr\"odinger Lagrangian (TDNLSL), when terms required by Galilean invariance are included. If the modulus of the pair field is also allowed to vary, the system is equivalent to two coupled TDNLSL's.

DOI
10.1103/physrevb.51.6531

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We show that the low-frequency, long-wavelength dynamics of the phase of the pair field for a BCS-type s-wave superconductor at T=0 is equivalent to that of a time-dependent nonlinear Schr"odinger Lagrangian (TDNLSL), when terms required by Galilean invariance are included. If the modulus of the pair field is also allowed to vary, the system is equivalent to two coupled TDNLSL's.
We extend to finite temperature the time-dependent effective theory for the Goldstone field (the phase of the pair field) $\ensuremath{\theta}$ 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 Schr\"odinger 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 $\ensuremath{\theta}$ 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.
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=0T=0T=0 and T=0.6Tc.{T=0.6T}_{c}.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 TTT-dependent) dispersion for both the real and imaginary parts. Damping is only significant for T> 0.4Tc.T>~{0.4T}_{c}.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 Schr\"odinger equation by giving the pair mass a small temperature-dependent positive imaginary part.
A standard perturbative expansion around the mean-field solution is used to derive the low-energy effective action for superconductors at T=0.T=0.T=0. Taking into account the density fluctuations at the outset we get the effective action where the density $\ensuremath{\rho}$ is the conjugated momentum to the phase $\ensuremath{\theta}$ of the order parameter. In the hydrodynamic regime, the dynamics of the superconductor is described by a time-dependent nonlinear Schr\"odinger (TDNLS) equation for the field $\ensuremath{\Psi}(x)=\sqrt{\ensuremath{\rho}/2}{e}^{i\ensuremath{\theta}}.$ The evolution of the density fluctuations in the crossover from weak-coupling (BCS) to strong-coupling (Bose condensation of localized pairs) superconductivity is discussed for the attractive Hubbard model. In the bosonic limit, the TDNLS equation reduces to the Gross-Pitaevskii equation for the order parameter, as in the standard description of superfluidity. The conditions under which a phase-only action can be derived in the presence of a long-range interaction to describe the physics of the superconductivity of ``bad metals'' are discussed.
Using a functional approach to the Keldysh formalism we develop a time-dependent Ginzburg-Landau equation for the long-wavelength and low-frequency dynamics of a weak-coupling superconductor, which is also valid in situations that deviate significantly from thermal equilibrium. The theory takes the interactions between quasiparticles and Cooper pairs into account exactly and is explicitly Galilean-invariant at zero temperature. However, the temporal and spatial dependence of the order parameter is assumed to be such that the effects of Landau damping can be neglected. We also consider the screening properties of the superconductor and show that the inclusion of the polarizability of the electron gas is necessary for a complete description.