Kagome chiral spin liquid in transition metal dichalcogenide moir\'e bilayers

At $n=3/4$ filling of the moir\'e flat band, transition metal dichalcogenide moir\'e bilayers will develop kagome charge order. We derive an effective spin model for the resulting localized spins and find that its further neighbor spin interactions can be much less suppressed than the corresponding electron hopping strength. Using density matrix renormalization group simulations, we study its phase diagram and, for realistic model parameters relevant for WSe$_2$/WS$_2$, we show that this material can realize the exotic chiral spin liquid phase and the highly debated kagome spin liquid. Our work thus demonstrates that the frustration and strong interactions present in TMD heterobilayers provide an exciting platform to study spin liquid physics.