Interacting phase diagram of twisted bilayer MoTe${}_2$ in magnetic field

We study electron-electron interaction induced states of twisted bilayer MoTe${}_2$ in an out-of-plane magnetic field $B\widehat{\mathbf{z}}$ near one hole per moiré unit cell filling. The 3D phase diagram showing the evolution of competing phases with $B$, interaction strength and an out-of-plane electric field is presented at electron fillings that follow the Diophantine equation along Chern number $-\text{sign}\left(B\right)$ line, that is pointing away from the charge neutral filling, where we find prominent Chern insulators consistent with the experiments. We also explain the experimental absence of prominent Chern insulators along the Chern number $+\text{sign}\left(B\right)$ line.