Pair Density Waves from Local Band Geometry.

A band-projection formalism is developed for calculating the superfluid weight in two-dimensional multiorbital superconductors with an orbital-dependent pairing. It is discovered that, in this case, the band geometric superfluid stiffness tensor can be locally nonpositive definite in some regions of the Brillouin zone. When these regions are large enough or include nodal singularities, the total superfluid weight becomes nonpositive definite due to pairing fluctuations, resulting in the transition of a BCS state to a pair density wave (PDW). This geometric BCS-PDW transition is studied in the context of two-orbital superconductors, and proof of the existence of a geometric BCS-PDW transition in a generic topological flat band is established.