Geometric and Conventional Contributions to Superfluid Weight in the Minimal Models for Superconducting Flat Bands Induced by Doping

The density functional theory calculations and tight-binding models for the doped lead apatite (and other materials in the same symmetry group) support flat bands, which could be susceptible to the emergence of various correlated phases including superconductivity. We develop a theory for the geometric contribution of the superfluid weight arising from the momentum-space topology of the Bloch wave functions of these flat bands, and we compare our results to the paradigmatic case of \(s\)-wave superconductivity on an isolated topological flat band. We show that, in contrast to the standard paradigm of flat-band superconductivity, there does not exist any lower bound for the superfluid weight in these models. Moreover, although the nontrivial quantum geometries of the normal-state bands are the same when the superconductivity appears in the ferromagnetic and paramagnetic phases, the emerging superconducting phases have very different superfluid weights. In the case of superconductivity appearing on the spin-polarized bands, the superfluid weight varies a lot as a function of model parameters. On the other hand, if the superconductivity emerges in the paramagnetic phase, the superfluid weight is robustly large and it contains a significant geometric component. Abstract Published by the Jagiellonian University 2026 authors