Structure factors and quantum geometry in multiband BCS superconductors

We consider multiband BCS superconductors that exhibit time-reversal symmetry and uniform pairing, and analyze their dynamic density and spin structure factors using linear-response theory within the mean-field BCS-BEC crossover framework at zero temperature. Our results for the multi-orbital Hubbard model satisfy the associated f-sum rules in several limits. In particular, in the strong-coupling limit, they coincide with those of a weakly-interacting Bose gas of Cooper pairs, where the low-energy collective Goldstone modes serve as Bogoliubov phonons. We further reveal that the quantum-geometric origin of the low-energy structure factors, along with related observables such as the superfluid-weight tensor and the effective-mass tensor of Cooper pairs, can be traced all the way back to the effective-mass theorem for Bloch bands in this limit. As an illustration, we investigate the pyrochlore-Hubbard model numerically and demonstrate that the Goldstone modes are the only relevant collective degrees of freedom in the flat-band regime.