Superfluid density and condensate fraction in the BCS-BEC crossover regime at finite temperatures

Abstract

The superfluid density is a fundamental quantity describing the response to a rotation as well as in two-fluid collisional hydrodynamics. We present extensive calculations of the superfluid density ρ s in the BCS-BEC crossover regime of a uniform superfluid Fermi gas at finite temperatures.We include strong-coupling or fluctuation effects on these quantities within a Gaussian approximation. We also incorporate the same fluctuation effects into the BCS single-particle excitations described by the superfluid order parameter ∆ and Fermi chemical potential µ, using the Nozières and Schmitt-Rink (NSR) approximation. This treatment is shown to be necessary for consistent treatment of ρ s over the entire BCS-BEC crossover. We also calculate the condensate fraction N c as a function of the temperature, a quantity which is quite different from the superfluid density ρ s . We show that the mean-field expression for the condensate fraction N c is a good approximation even in the strong-coupling BEC regime. Our numerical results show how ρ s and N c depend on temperature, from the weak-coupling BCS region to the BEC region of tightly-bound Cooper pair molecules. In a companion paper by the authors (cond-mat/0609187), we derive an equivalent expression for ρ s from the thermodynamic potential, which exhibits the role of the pairing fluctuations in a more explicit manner.

Variational theory of two-fluid hydrodynamic modes at unitarity

“…The NSR-type Gaussian fluctuation theory, on the other hand, suffers from a re-entrance problem close to T c . This problem first appears around (k F a s ) −1 = −0.5 on the BCS side and persists into the BEC side of unitarity [19]. This spurious first-order phase transition [19] is due to the NSR Gaussian treatment of pairing fluctuations used to calculate ∆ 0 and µ self-consistently.…”

“…This problem first appears around (k F a s ) −1 = −0.5 on the BCS side and persists into the BEC side of unitarity [19]. This spurious first-order phase transition [19] is due to the NSR Gaussian treatment of pairing fluctuations used to calculate ∆ 0 and µ self-consistently. The problem is equivalent to one that arises in a self-consistent calculation of the condensate density and chemical potential close to T c in Bose gases using the Bogoliubov-Popov approximation (for further discussion and references, see p. 34 of Shi and Griffin [31]).…”

“…[12,17,18,19] includes the contributions from the BCS Fermi excitations plus the bosonic pairing fluctuations. As discussed in Refs.…”

“…The original NSR data from Ref. [19] is denoted by the blue circles (the data points used in the curve fitting are given by the filled circles).…”

“…[19] and a scaled BCS mean-field superfluid density. A superfluid with a two-component order parameter (and a bosonic fluctuation spectrum) undergoes a second order phase transition with a superfluid density that varies as ρ s ∝ (T c − T ) 2/3 close to the transition temperature, independent of the interaction strength [32].…”