Auxiliary field formalism for dilute fermionic atom gases with tunable interactions

TL;DR

This paper introduces an auxiliary field formalism for dilute spin-1/2 fermionic gases with tunable interactions, unifying the BCS-BEC crossover description and connecting to existing theories at zero and finite temperatures.

Contribution

It develops a new auxiliary field formalism for fermionic gases that parallels recent Bose gas theories, enabling systematic improvements and a unified understanding of the BCS-BEC crossover.

Findings

LOAF reproduces BCS equations at zero temperature

LOAF aligns with established finite-temperature theories

Framework allows systematic corrections beyond mean-field

Abstract

We develop the auxiliary field formalism corresponding to a dilute system of spin-1/2 fermions. This theory represents the Fermi counterpart of the BEC theory developed recently by F. Cooper et al. [Phys. Rev. Lett. 105, 240402 (2010)] to describe a dilute gas of Bose particles. Assuming tunable interactions, this formalism is appropriate for the study of the crossover from the regime of Bardeen-Cooper-Schriffer (BCS) pairing to the regime of Bose-Einstein condensation (BEC) in ultracold fermionic atom gases. We show that when applied to the Fermi case at zero temperature, the leading-order auxiliary field (LOAF) approximation gives the same equations as those obtained in the standard BCS variational picture. At finite temperature, LOAF leads to the theory discussed by by Sa de Melo, Randeria, and Engelbrecht [Phys. Rev. Lett. 71, 3202(1993); Phys. Rev. B 55, 15153(1997)]. As such, LOAF provides a unified framework to study the interacting Fermi gas. The mean-field results discussed here can be systematically improved upon by calculating the one-particle irreducible (1-PI) action corrections, order by order.