Pair-density-wave and chiral superconductivity in twisted bilayer transition-metal-dichalcogenides

We theoretically explore possible unconventional superconductivity induced by weak repulsive interactions in twisted bilayer TMD (e.g. WSe$_2$) in the presence of an out-of-plane electric field. Using renormalization group (RG) analysis, we show that topological chiral superconducting states with Chern number $\mathcal{N}=1,2,4$ (namely $p+ip$, $d+id$, $g+ig$) appear over a large parameter region with moire filling factor around $n=1$. At some special values of applied electric field and in the presence of a weak out-of-plane Zeeman field, spin-polarized pair density wave (PDW) superconductivity can emerge. This spin-polarized PDW state can be probed by experiments such as spin-polarized STM measuring spin-resolved pairing gap and quasi-particle interference. Moreover, the spin-polarized PDW could lead to spin-polarized superconducting diode effect.