Correcting inconsistencies in the conventional superfluid path integral scheme

TL;DR

This paper proposes a modified path integral scheme for fermionic superfluids that ensures gauge invariance and satisfies the compressibility sum rule by adjusting the saddle point condition in the presence of external fields.

Contribution

It introduces a new saddle point condition in the path integral approach to achieve gauge invariance and sum rule consistency in superfluid electrodynamics and thermodynamics.

Findings

The scheme recovers the gauge-invariant BCS electrodynamic response.

It ensures the compressibility sum rule is satisfied at the BCS level.

The approach can be extended to higher order fluctuation theories.

Abstract

In this paper we show how to redress a shortcoming of the path integral scheme for fermionic superfluids and superconductors. This approach is built around a simultaneous calculation of electrodynamics and thermodynamics. An important sum rule, the compressibility sum rule, fails to be satisfied in the usual calculation of the electromagnetic and thermodynamic response at the Gaussian fluctuation level. Here we present a path integral scheme to address this inconsistency. Specifically, at the leading order we argue that the superconducting gap should be calculated using a different saddle point condition modified by the presence of an external vector potential. This leads to the well known gauge-invariant BCS electrodynamic response and is associated with the usual (mean field) expression for thermodynamics. In this way the compressibility sum rule is satisfied at the BCS level. Moreover, this scheme can be readily extended to address arbitrary higher order fluctuation theories. At any level this approach will lead to a gauge invariant and compressibility sum rule consistent treatment of electrodynamics and thermodynamics.