Pair density wave, FFLO, and re-entrance in multi-orbital superconductors

In a spin-singlet superconductor, finite-momentum superconductivity, known as the FFLO phase, arises as a field-driven instability of the uniform state. In contrast, finite-momentum spin-triplet pairing can not be stabilized in the same way due to the different pair-breaking effects of the field on a triplet state. However, finite-momentum pairing in zero field, a pair-density wave, has been conjectured to be present in a variety of correlated materials of interest. Nonetheless, this phenomenon remains relatively less explored. Here we study how such a state can arise in a correlated multi-orbital system in the presence of the magnetic field and spin-orbit coupling. We show how the pair density wave and FFLO phases, as well as field-induced and re-entrant phases for large fields arise in correlated metals. The application to several correlated materials of interest is briefly discussed.