Orbital Chern insulator at $\nu =-2$ in twisted ${\mathrm{MoTe}}_2$

In twisted ${\mathrm{MoTe}}_2$, the latest transport measurement has reported observation of the 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 antiferromagnetic 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 the range of $D$ where this antiferromagnet is the ground state, the orbital magnetization allows magnetic field initialization of the spin antiferromagnetic order and the Chern number.