Excitonic density wave and spin-valley superfluid in bilayer transition metal dichalcogenide

Artificial moir\'e superlattices in 2d van der Waals heterostructure is a new venue for realizing and controlling correlated electronic phenomena. Recently, twisted bilayer WSe$_2$ emerged as a new robust moir\'e system hosting a correlated insulator at moir\'e half-filling over a range of twist angle. In this work, we present a theory of this insulating state as an excitonic density wave due to intervalley electron-hole pairing. We show that exciton condensation is strongly enhanced by a van Hove singularity near the Fermi level. Our theory explains the remarkable sensitivity of the insulating gap to the vertical electric field. In contrast, the gap is weakly reduced by a perpendicular magnetic field, with quadratic dependence at low field. The different responses to electric and magnetic field can be understood in terms of pair-breaking versus non-pair-breaking effects in a BCS analog of the system. We further predict superfluid spin transport in this electrical insulator, which can be detected by optical spin injection and spatial-temporal imaging.