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arXiv:cond-mat/9909103

Non-equilibrium, time-dependent effective theory of a weakly coupled superconductor at finite temperature

Tim Evans, D. A. Steer

Published Sep 7, 1999• 0 citations
arXivHTMLPDF

Abstract

We perform a well defined derivative expansion to obtain the time dependent effective theory for a BCS superconductor at finite temperature, using an arbitrary curve in the complex time plane. Our expansion is unique, being free of any problems related to the order in which analytical continuation and the derivative expansion are made, or equivalently to the order in which the zero momentum and zero energy limits are taken. In other words Landau damping terms are present but do not lead to the pathologies found in standard approaches. We discuss the reason for this, suggesting that these standard approaches may be internally inconsistent. The methods presented here are an attempt to overcome those inconsistencies, and finally, our results are compared to others in the literature.

DOI
10.48550/arxiv.cond-mat/9909103

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We perform a well defined derivative expansion to obtain the time dependent effective theory for a BCS superconductor at finite temperature, using an arbitrary curve in the complex time plane. Our expansion is unique, being free of any problems related to the order in which analytical continuation and the derivative expansion are made, or equivalently to the order in which the zero momentum and zero energy limits are taken. In other words Landau damping terms are present but do not lead to the pathologies found in standard approaches. We discuss the reason for this, suggesting that these standard approaches may be internally inconsistent. The methods presented here are an attempt to overcome those inconsistencies, and finally, our results are compared to others in the literature.
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.
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.
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.
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.
In the first part of this review paper, the time-dependent Ginzburg-Landau theory is derived starting from the microscopic BCS model with the help of a derivative expansion. Special attention is paid to two space dimensions, where the entire crossover from the weak-coupling BCS limit to the strong-coupling BEC limit of tightly bound fermion pairs is accessible analytically. The second part deals with the dual approach to the time-independent Ginzburg-Landau theory in three space dimensions. In this approach, the magnetic vortices of a superconductor play the central role, and the superconductor-to-normal phase transition is understood as a proliferation of these vortices.
We derive finite-temperature time-dependent effective actions for the phase of the pairing field, which are appropriate for a two-dimensional electron system with both nonretarded d- and s-wave attraction. As for s-wave pairing, the d-wave effective action contains terms with Landau damping, but their structure appears to be different from the s-wave case due to the fact that the Landau damping is determined by the quasiparticle group velocity vg,{\mathbf{v}}_{g},vg​, which for d-wave pairing does not have the same direction as the noninteracting Fermi velocity vF.{\mathbf{v}}_{F}.vF​. we show that for d-wave pairing the Landau term has a linear low-temperature dependence and in contrast to the s-wave case is important for all finite temperatures. A possible experimental observation of the phase excitations is discussed.