Pair distribution functions of a superfluid spin-1/2 Fermi gas with contact interactions in the linearized time-dependent BCS theory

We show that the minimal mean-field theory to use for calculating the pair distribution functions $g_{\sigma\sigma'}(\vec{r},\vec{r}\,')$ of a spatially homogeneous, unpolarized spin-1/2 superfluid Fermi gas is not the ordinary static BCS theory, but the linearized time-dependent BCS theory implemented via the fluctuation-dissipation theorem. Indeed, the former completely ignores the acoustic excitation branch - the phonons - of the superfluid, while the latter explicitly takes it into account, as well as the quantum fluctuations induced by the broken-pair continuum. Unlike the first, the second theory (i) reflects the effect of these collective excitations on the system's equation of state, including at zero temperature, (ii) allows the function $g_{\uparrow\downarrow}(\vec{r},\vec{r}\,')$ to go at sufficiently large distances strictly below its asymptotic value $(\rho/2)^2$ where $\rho$ is the gas density, as expected according to the quantum hydrodynamics of Landau and Khalatnikov at low temperatures, and (iii) predicts in the function $g_{\uparrow\uparrow}(\vec{r},\vec{r}\,')$ at short distances subdominant contributions $|\vec{r}-\vec{r}\,'|^2\ln|\vec{r}-\vec{r}\,'|$ in 3D and $|\vec{r}-\vec{r}\,'|^2\ln(-\ln|\vec{r}-\vec{r}\,'|)$ in 2D, alongside the dominant contributions $|\vec{r}-\vec{r}\,'|$ in 3D and $|\vec{r}-\vec{r}\,'|^2\ln|\vec{r}-\vec{r}\,'|$ in 2D already present in static BCS theory but with a lower coefficient. This discussion is relevant to the recent theoretical work of Obeso-Jureidini and Romero-Rochin, and to the ongoing experiments on cold atomic gases at ENS and MIT.