Pairing fluctuations and the superfluid density through the BCS-BEC crossover

TL;DR

This paper derives a comprehensive expression for the superfluid density in a Fermi gas across the BCS-BEC crossover, incorporating pairing fluctuations and finite temperature effects, and separates contributions from fermionic quasiparticles and Bose collective modes.

Contribution

It provides a self-consistent, fluctuation-inclusive formula for superfluid density valid at finite temperatures across the entire BCS-BEC crossover.

Findings

Normal fluid density separates into fermionic and bosonic contributions.

Fermionic contribution matches Landau's formula for BCS superfluids.

Bose contribution reduces to Landau's formula in the BEC limit.

Abstract

We derive an expression for the superfluid density of a uniform two-component Fermi gas through the BCS-BEC crossover in terms of the thermodynamic potential in the presence of an imposed superfluid flow. Treating the pairing fluctuations in a Gaussian approximation following the approach of Nozi\`eres and Schmitt-Rink, we use this definition of $\rho_s$ to obtain an explicit result which is valid at finite temperatures and over the full BCS-BEC crossover. It is crucial that the BCS gap $\Delta$, the chemical potential $\mu$, and $\rho_s$ all include the effect of fluctuations at the same level in a self-consistent manner. We show that the normal fluid density $\rho_n \equiv n - \rho_s$ naturally separates into a sum of contributions from Fermi BCS quasiparticles ($\rho^F_{n}$) and Bose collective modes ($\rho^B_{n}$). The expression for $\rho^F_{n}$ is just Landau's formula for a BCS Fermi superfluid but now calculated over the BCS-BEC crossover. The expression for the Bose contribution $\rho^B_{n}$ is more complicated and only reduces to Landau's formula for a Bose superfluid in the extreme BEC limit, where all the fermions have formed stable Bose pairs and the Bogoliubov excitations of the associated molecular Bose condensate are undamped. In a companion paper, we present numerical calculations of $\rho_s$ using an expression equivalent to the one derived in this paper, over the BCS-BEC crossover, including unitarity, and at finite temperatures.