Mean-field study of pair density wave order in multiband models

One of the indispensable ingredients for pair density wave (PDW) superconductivity is the presence of an attractive pairing interaction, i.e. attraction in the BCS channel, at finite momentum. Here we show that in multiband systems, this condition can be met with straightforward electron-density interactions. The electron-density interaction, when projected to the band basis, acquires form factors with nontrivial momentum dependence and thereby exhibits a potential tendency to a finite-momentum pairing instability. By applying a mean-field analysis to two simple multiband models, we find that PDW order can indeed become the leading instability over a wide range of interaction parameters. Moreover, the condition for the transition from a uniform superconductor to a PDW superconductor is shown via a simple quantum geometric argument.