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

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 ${\rho }_s$ 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 $\Delta$ and Fermi chemical potential $\mu$, using the Nozières–Schmitt-Rink approximation. This treatment is shown to be necessary for consistent treatment of ${\rho }_s$ 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 ${\rho }_s$. 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 ${\rho }_s$ 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 [Phys. Rev. A 74, 063626 (2006)], we derive an equivalent expression for ${\rho }_s$ from the thermodynamic potential, which exhibits the role of the pairing fluctuations in a more explicit manner.

Figure 1

(a) Off-diagonal mean-field $\Delta$ and (b) Fermi chemical potential $\mu$ in the BCS-BEC crossover. The pairing interaction is measured in terms of the inverse of the two-body scattering length $a_s$, normalized by the Fermi momentum $k_{\mathrm{F}}$. In these panels, the dotted line shows $T_{\mathrm{c}}$ as a function of ${\left(k_{\mathrm{F}}{a}_s\right)}^{-1}$. In the strong-coupling regime, the apparent first-order behavior of the phase transition is an artifact of the NSR Gaussian treatment of pairing fluctuations (see text).

Figure 2

Calculated values of the superfluid order parameter $\Delta$ as a function of temperature. “BCS” labels the weak-coupling BCS limit. The bendover near $T_{\mathrm{c}}$ is an artifact of our NSR Gaussian treatment of fluctuations.

Figure 3

Fermi chemical potential $\mu$ as a function of temperature. For comparison, in the unitarity limit $[{\left(k_{\mathrm{F}}{a}_s\right)}^{-1}=0]$, MC results [12, 21] give $\mu ∕{\epsilon }_{\mathrm{F}}=0.44–0.49$ and an improved version of NSR theory [22] gives $\mu ∕{\epsilon }_{\mathrm{F}}=0.4–0.47$ just below $T_{\mathrm{c}}$.

Figure 4

Calculated superfluid density ${\rho }_s$ in the BCS-BEC crossover. The self-consistent solutions for $\Delta$ and $\mu$ shown in Fig. 1 are used. The dashed line shows $T_{\mathrm{c}}$, where ${\rho }_s$ vanishes.

Figure 5

Superfluid density ${\rho }_s$ as a function of temperature in the BCS region (solid circles), unitarity limit (solid triangles), and BEC regime (open circles). “BCS” labels the mean-field BCS result, given by ${\rho }_s=n-{\rho }_n^F$ with $\mu ={\epsilon }_{\mathrm{F}}$. “BEC” gives ${\rho }_s$ for a dilute Bose gas with $N∕2$ bosons described by the excitation spectrum in Eq. (2.12).

Figure 6

Condensate fraction $N_c$ in the strong-coupling BEC regime. The solid line shows $N_{c0}$ and the dashed line includes the small correction $\delta N_c$ from self-energies due to pairing fluctuations.

Figure 7

Calculated condensate fraction $N_c$ in the BCS-BEC crossover at $T=0$ (solid line). In this figure, as well as in Figs. 8, 9, we only show $N_{c0}$. “MF” shows the condensate fractionin the case when $\Delta$ and $\mu$ are determined in the mean-field results in Eqs. (5.12, 5.13). “BEC” shows the result for a Bose gas described by Eqs. (5.10, 5.11). The solid circles shows recent Monte Carlo results for $N_c$ [12] for comparison.

Figure 8

Condensate fraction $N_c$ as a function of temperature in the BCS-BEC crossover. In this calculation, the self-consistent solutions for $\Delta$ and $\mu$ in Fig. 1 are used.

Figure 9

Condensate fraction $N_c$ as a function of temperature in the BCS regime, unitarity limit, and the BEC regime. The curve “BCS” is the weak-coupling BCS result. The curve labeled “BEC” is the condensate fraction for a Bose superfluid determined by Eq. (2.11). Results are normalized to values at $T=0$.