Electrically tuned topology and magnetism in twisted bilayer MoTe$_2$ at $\nu_h=1$

We present a theoretical study of an interaction-driven quantum phase diagram of twisted bilayer MoTe$_2$ at hole filling factor $\nu_h=1$ as a function of twist angle $\theta$ and layer potential difference $V_z$, where $V_z$ is generated by an applied out-of-plane electric field. At $V_z=0$, the phase diagram includes quantum anomalous Hall insulators in the intermediate $\theta$ regime and topologically trivial multiferroic states with coexisting ferroelectricity and magnetism in both small and large $\theta$ regimes. There can be two transitions from the quantum anomalous Hall insulator phase to topologically trivial out-of-plane ferromagnetic phase, and finally to in-plane 120$^\circ$ antiferromagnetic phase as $|V_z|$ increases, or a single transition without the intervening ferromagnetic phase. We show explicitly that the spin vector chirality of various 120$^\circ$ antiferromagnetic states can be electrically switched. We discuss the connection between the experimentally measured Curie-Weiss temperature and the low-temperature magnetic order based on an effective Heisenberg model with magnetic anisotropy.