Role of band geometry on the transition from BCS to a pair density wave state

The superfluid weight in multi-band superconductors contains a geometric contribution that is proportional to the quantum metric of the band for a uniform pairing of the orbitals. Here, we develop a band-projection formalism to calculate additional geometric contributions to the superfluid density in the presence of a non-trivial orbital-dependent pairing order parameter. We find that this band geometric contribution to the stiffness tensor can be locally non-positive-definite in some regions of the Brillouin zone. When these regions are large enough or include nodal zeroes, the total superfluid weight becomes non-positive-definite, indicating that the BCS $d{q}=0$ state is no longer stable. We identify this instability as a transition from the BCS state to a pair density wave, where the Cooper pairs have the same nonzero center-of-mass momentum $d{q}$. This pair density wave is stabilized by band geometry instead of a Zeeman field, so has no de-pairing in the momentum space.