  We consider a generic Hamiltonian that is suitable for describing a uniform
BCS superfluid on a lattice with a two-point basis, and study its collective
excitations at zero temperature. For this purpose, we first derive an
effective-Gaussian action for the pairing fluctuations, and then extract the
low-energy dispersion relations for the in-phase Goldstone and out-of-phase
Leggett modes along with the corresponding amplitude (i.e., the so-called
Higgs) ones. We find that while the Goldstone mode is gapless at zero momentum
and propagating in general, the Leggett mode becomes undamped only with
sufficiently strong interactions. Furthermore, we show that, in addition to the
conventional contribution that is controlled by the energy of the Bloch bands,
the velocity of the Goldstone mode has a geometric contribution that is
governed by the quantum metric tensor of the Bloch states. Our results suggest
that the latter contribution dominates the velocity when the former one becomes
negligible for a narrow- or a flat-band.
