Effective theory for the Goldstone field in the BCS-BEC crossover at T=0

TL;DR

This paper derives a comprehensive effective Lagrangian for the Goldstone mode in a superfluid Fermi gas across the BCS-BEC crossover at zero temperature, including next-to-leading order terms and their dependence on interaction regimes.

Contribution

It provides the most general form of the effective Lagrangian at next-to-leading order, with explicit coefficient functions calculated at one-loop level across the entire crossover.

Findings

Identifies an invariant combination of higher spatial gradients in the unitary limit.

Calculates coefficient functions analytically using elliptic integrals.

Analyzes behavior of coefficients in BCS, BEC, and unitary regimes.

Abstract

We perform a detailed study of the effective Lagrangian for the Goldstone mode of a superfluid Fermi gas at zero temperature in the whole BCS-BEC crossover. By using a derivative expansion of the response functions, we derive the most general form of this Lagrangian at the next to leading order in the momentum expansion in terms of four coefficient functions. This involves the elimination of all the higher order time derivatives by careful use of the leading order field equations. In the infinite scattering length limit where conformal invariance is realized, we show that the effective Lagrangian must contain an unnoticed invariant combination of higher spatial gradients of the Goldstone mode, while explicit couplings to spatial gradients of the trapping potential are absent. Across the whole crossover, we determine all the coefficient functions at the one-loop level, taking into account the dependence of the gap parameter on the chemical potential in the mean-field approximation. These results are analytically expressed in terms of elliptic integrals of the first and second kind. We discuss the form of these coefficients in the extreme BCS and BEC regimes and around the unitary limit, and compare with recent work by other authors.