Effective Lagrangians for BCS Superconductors at T=0

TL;DR

This paper derives an effective Lagrangian description of the phase dynamics in BCS superconductors at zero temperature, showing equivalence to a non-linear Schrödinger Lagrangian, and extends it to include amplitude variations.

Contribution

It introduces a novel effective Lagrangian framework for BCS superconductors at T=0, incorporating Galilean invariance and amplitude fluctuations.

Findings

Phase dynamics described by a non-linear Schrödinger Lagrangian.

Amplitude fluctuations lead to coupled non-linear Schrödinger equations.

Provides a unified effective field theory approach for superconductors.

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 non-linear 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 also refer the interested reader to our earlier paper, `Nonlinear Schrodinger equation for superconductors' [cond-mat/9312099], for a different line of derivation