Excitonic insulator and spin-valley superfluid in twisted bilayer transition metal dichalcogenide

Recently striking correlated phenomena emerge in twisted bilayer WSe$_2$ -- including a correlated moir\'e insulating dome at half-filling -- over a range of small twist angles. In this work, we show the half-filling state is an intervalley excitonic insulator, where the electron-hole pairing is strongly enhanced for a certain range of displacement field due to the proximity of a van Hove singularity to the Fermi level. The mean field phase diagram as a function of displacement field, temperature and magnetic field shows a good agreement with the experiment. Furthermore, the excitonic insulator hosts a spin-valley density wave order which can be viewed as a spin superfluid. We propose an all-optical setup to study the superflow of spins in the twisted bilayer transition metal dichalcogenide systems.