Valley Anisotropy in TMD Homobilayer Moires

Hole bands in K-valley TMD homobilayer moires normally have non-zero spin/valley projected Chern numbers. Experimentally, the ground states at half filling are insulating over a wide range of twist angles but are not necessarily valley polarized – as signaled by the presence or absence of a quantized anomalous Hall effect. Here we examine the valley anisotropy of valley-ferromagnetic insulating states by studying an idealized model systems in which the Chern bands in the two valleys are replaced by Landau levels with opposite signs of magnetic field. The XY valley ferromagnet state can be mapped to a vortex-lattice state of Cooper pairs within Landau levels by making a particle-hole transformation in one of the two valleys. We find that the Ising valley ferromagnet is normally the ground state, discuss strategies to stabilize the XY valley ferromagnet, and argue that the fractional quantum spin Hall effect in twist MoTe2 can be explained by XY ground states.