Interaction-induced Quantum Anomalous and Spin Hall Mott Insulators: beyond Z_2 topology

We introduce interactions into two general models for quantum spin Hall physics. Although the traditional picture is that such physics appears at half-filling, we show that in the presence of strong interactions, the quarter-filled state instead exhibits the quantum spin Hall effect with spin Chern number $C_s=1$. A topological Mott insulator is the underlying cause that lies outside the standard $mathbb Z_2$ topological classification. An intermediate interacting regime emerges that exhibits a quantum anomalous Hall effect when the lower band is `flat'. This state transitions to the quantum spin Hall effect once the interactions are sufficiently large. We show that this intermediate regime is consistent with the simultaneous observation in transition metal dichalcogenide moir{'e} materials of a quantum anomalous Hall phase at quarter filling and a quantum spin Hall effect at half-filling. The valley coherence seen in such moir{'e} systems also matches expectations from a simple bi-layer extension of our results.