Two-body correlations and the superfluid fraction for nonuniform systems

We extend the one-body phase function upper bound on the superfluid fraction f s in a periodic solid (a spatially ordered supersolid) to include two-body phase correlations. The one-body current density is no longer proportional to the gradient of the one-body phase times the one-body density, but rather it becomes $\vec{j}(\vec{r}_{1})=\rho_{1}(\vec{r}_{1})\frac{\hbar}{m}\vec{\nabla }_{1}\phi_{1}(\vec{r}_{1})+\frac{1}{N}\int d\vec{r}_{2}\rho_{2}(\vec{r}_{1},\vec{r}_{2})\frac{\hbar }{m}\vec{\nabla}_{1}\phi_{2}(\vec{r}_{1},\vec{r}_{2})$ . This expression therefore depends also on two-body correlation functions. The equations that simultaneously determine the one-body and two-body phase functions require a knowledge of one-, two-, and three-body correlation functions. The approach can also be extended to disordered solids. Fluids, with two-body densities and two-body phase functions that are translationally invariant, cannot take advantage of this additional degree of freedom to lower their energy.