1/4 is the new 1/2: Interaction-induced Unification of Quantum Anomalous and Spin Hall Effects

Abstract

We introduce interactions into two general models for quantum spin Hall physics. Although the traditional picture is that such physics appears when the two lower spinful bands are occupied, that is, half-filling, we show using determinantal quantum Monte Carlo as well as from an exactly solvable model that in the presence of strong interactions, the quarter-filled state instead exhibits the quantum spin Hall effect at high temperature. A topological Mott insulator is the underlying cause. The peak in the spin susceptibility is consistent with a possible ferromagnetic state at $T=0$. The onset of such magnetism would convert the quantum spin Hall to a quantum anomalous Hall effect. We argue that it is the consistency with the Lieb-Schultz-Mattis theorem\cite{lsm1,lsm2} for interacting systems with an odd number of charges per unit cell that underlies the emergence of the quantum anomalous Hall effect as a low-temperature symmetry-broken phase of the quantum spin Hall effect. While such a symmetry-broken phase typically is accompanied by a gap, we find that the interaction strength must exceed a critical value for the gap to form using quantum Monte Carlo dynamical cluster approximation simulations. Hence, we predict that topology can obtain in a gapless phase but only in the presence of interactions in dispersive bands. These results are applied to recent experiments on moir\'e systems and shown to be consistent with valley-coherent quantum anomalous Hall physics.Comment: Figure 4e,f added as well as a referenc