Orbital Chern Insulator at $ν=-2$ in Twisted MoTe$_{2}$

In twisted MoTe$_{2}$, latest transport measurement has reported observation of quantum anomalous Hall effect at hole filling $\nu=-1$, which undergoes a topological phase transition to a trivial ferromagnet as layer hybridization gets suppressed by interlayer bias $D$. Here we show that this underlies the existence of an orbital Chern insulating state with gate ($D$) switchable sign in an antiferromagtic spin background at hole filling $\nu=-2$. From momentum-space Hartree Fock calculations, we find this state has a topological phase diagram complementary to that of the $\nu=-1$ one: by sweeping $D$ from negative to positive, the Chern number of this $\nu=-2$ state can be switched between $+1$, $0$, and $-1$, accompanied by a sign change of a sizable orbital magnetization. In range of $D$ where this antiferronagnet is the ground state, the orbital magnetization allows magnetic field initialization of the spin antiferromagnetic order and the Chern number.