d-Wave pair density wave superconductivity in a two-orbital model

Motivated by exploring superconductivity in multi-orbital systems, we study two orbital models of spinful fermions representing ($p_x,p_y$) or ($d_{xz}, d_{yz})$ orbitals on the square lattice. For minimal interorbital $t$-$J$ or $t$-$V$ on-site interactions, a random phase approximation uncovers regimes of instability towards incommensurate $d_{xy}$ pair density wave ($d$-PDW) superconductivity with driven by interband pairing. We study the competition of PDW order with uniform nodal $d_{xy}$ pairing states and magnetic and charge density wave (CDW) instabilities. At strong coupling, we derive an effective hard-core Cooper pair Hamiltonian which we study using a bosonic Gutzwiller ansatz to reveal a period-$2$ PDW over a wide range of fillings as well as a checkerboard CDW at quarter-filling. Our results apply to correlated multi-orbital materials with quasi-1D bands, Hubbard models on the square-octagon lattice, and atomic fermions in $p$-orbitals. Our work highlights the role of the orbital content and multiband Fermi surfaces in stabilizing interband PDW states.