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 multiorbital 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.

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• BCS-BEC crossover
• Geometric & topological phases
• Multiband superconductivity
• Sum rules
• Hubbard model