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The superfluid weight of a generic lattice model with attractive Hubbardinteraction is computed analytically in the isolated band limit within thegeneralized random phase approximation. Time-reversal symmetry, spin rotationalsymmetry, and the uniform pairing condition are assumed. It is found that therelation 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 minimalquantum metric is valid even at the level of the generalized random phaseapproximation. For an isolated, but not necessarily flat, band it is found thatthe correction to the superfluid weight obtained from the generalized randomphase approximation $D_{\rm s}^{(1)} = D_{\rm s,c}^{(1)}+D_{\rm s,g}^{(1)}$ isalso the sum of a conventional contribution $D_{\rm s,c}^{(1)}$ and a geometriccontribution $D_{\rm s,g}^{(1)}$, 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 geometricterm $D_{\rm s,g}^{(0)}$ is a weighted average of the quantum metric. Theconventional contribution is geometry independent, that is independent of theorbital positions, while it is possible to find a preferred, or natural, set oforbital positions such that $D_{\rm s,g}^{(1)}=0$. Useful analytic expressionsare derived for both the natural orbital positions and the minimal quantummetric, including its extension to bands that are not necessarily flat.Finally, using some simple examples, it is argued that the natural orbitalpositions may lead to a more refined classification of the topologicalproperties of the band structure.

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