  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.
