Superfluid fraction and Leggett bound in a density-modulated strongly interacting Fermi gas at zero temperature

We calculate the superfluid fraction of an interacting Fermi gas, in the presence of a one-dimensional periodic potential of strength $V_0$ and wave vector $q$. Special focus is given to the unitary Fermi gas, characterized by the divergent behavior of the $s$-wave scattering length. Comparison with the Leggett's upper bound ${(\langle n_{1D}\rangle <1/n_{1D}>)}^{-1}$, with $n_{1D}$ the one-dimensional column density, explicitly shows that differently from the case of a dilute interacting Bose gas, the bound significantly overestimates the value of the superfluid fraction, except in the phonon regime of small $q$. Sum rule arguments show that the combined knowledge of the Leggett bound and of the actual value of the superfluid fraction allows for the determination of curvature effects, providing the deviation of the dispersion of the Anderson-Bogoliubov mode from the linear phonon dependence. The comparison with the predictions of the weakly interacting BCS Fermi gas points out the crucial role of two-body interactions. The implications of our predictions on the anisotropic behavior of the sound velocity are also discussed.

Figure 1

Lattice-induced perturbative corrections to the superfluid density, ${\alpha }_S=(1-f_S)E_F^2/V_0^2$, and to the associated Leggett bound, ${\alpha }_S^L=(1-f_S^L)E_F^2/V_0^2$, of a unitary Fermi gas, calculated from the sum-rule approach [see Eqs. (8) and (12)].

Figure 2

Dispersion relation of the Anderson-Bogoliubov collective mode of the unitary Fermi gas. The sum-rule estimates (symbols) in the long-wavelength limit are compared with the full numerical result $\omega ={\omega }_{AB}\left(q\right)$ (black solid line), first discussed in Ref. [13]. Here, $k_F$ is the Fermi momentum and $v_F=k_F/m$ is the Fermi velocity. The sound velocity of the system is $c=0.444v_F$.

Figure 3

Superfluid fraction $f_S$ (blue triangles) and its Leggett bound $f_S^L$ (red circles) for the unitary Fermi gas as a function of the ratio $V_0/E_F$ for (a) $k_{F}d=4$ and (b) $k_{F}d=10$. The green solid line refers to the local density approximation (LDA) at unitarity. In each plot, we also include the corresponding predictions for the weakly interacting BCS regime for $f_S$ (black dotted line), $f_S^L$ (black dashed line), and the LDA (green dashed line).