Conserving Approximation of Pairing theories in Fermionic superfluid phase

TL;DR

This paper extends a conserving approximation framework to the superfluid phase of fermionic gases, incorporating order parameter fluctuations to better understand transport phenomena like viscosity.

Contribution

It develops a conserving approximation for the $G_0G$ t-matrix theory in the superfluid phase, including order parameter fluctuations and Goldstone mode contributions.

Findings

The approximation satisfies the stress tensor Ward identity in the superfluid phase.

Inclusion of Goldstone mode fluctuations is essential for conservation laws.

Results are relevant for understanding viscosity in fermionic superfluids.

Abstract

Respecting the conservation laws of momentum and energy in a many body theory is very important for understanding the transport phenomena. The previous conserving approximation requires that the self-energy of a single particle can be written as a functional derivative of a full dressed Green's function. This condition can not be satisfied in the $G_0G$ t-matrix or pair fluctuation theory which emphasizes the fermion pairing with a stronger than the Bardeen-Cooper-Schrieffer (BCS) attraction. In the previous work\cite{stressWI}, we have shown that when the temperature is above the superfluid transition temperature $T_c$, the $G_0G$ t-matrix theory can be put into a form that satisfies the stress tensor Ward identity (WI) or local form of conservation laws by introducing a new type of vertex correction. In this paper, we will extend the above conservation approximation to the superfluid phase in the BCS mean field level. To establish the stress tensor WI, we have to include the fluctuation of the order parameter or the contribution from the Goldstone mode. The result will be useful for understanding the transport properties such as the behavior of the viscosity of Fermionic gases in the superfluid phases.