Superfluid weight in the isolated band limit within the generalized random phase approximation

The superfluid weight of a generic lattice model with attractive Hubbard interaction is computed analytically in the isolated band limit within the generalized random phase approximation. Time-reversal symmetry, spin rotational symmetry, and the uniform pairing condition are assumed. It is found that the relation obtained in [https://link.aps.org/doi/10.1103/PhysRevB.106.014518] between the superfluid weight in the flat band limit and the so-called minimal quantum metric is valid even at the level of the generalized random phase approximation. For an isolated, but not necessarily flat, band it is found that the correction to the superfluid weight obtained from the generalized random phase approximation $D_{\rm s}^{(1)} = D_{\rm s,c}^{(1)}+D_{\rm s,g}^{(1)}$ is also the sum of a conventional contribution $D_{\rm s,c}^{(1)}$ and a geometric contribution $D_{\rm s,g}^{(1)}$, as in the case of the known mean-field result $D_{\rm s}^{(0)}=D_{\rm s,c}^{(0)}+D_{\rm s,g}^{(0)}$, in which the geometric term $D_{\rm s,g}^{(0)}$ is a weighted average of the quantum metric. The conventional contribution is geometry independent, that is independent of the orbital positions, while it is possible to find a preferred, or natural, set of orbital positions such that $D_{\rm s,g}^{(1)}=0$. Useful analytic expressions are derived for both the natural orbital positions and the minimal quantum metric, including its extension to bands that are not necessarily flat. Finally, using some simple examples, it is argued that the natural orbital positions may lead to a more refined classification of the topological properties of the band structure.