  We present a theory on the quantum phase diagram of AB-stacked
MoTe$_2$/WSe$_2$ using a self-consistent Hartree-Fock calculation performed in
the plane-wave basis, motivated by the observation of topological states in
this system. At filling factor $\nu=2$ (two holes per moir\'e unit cell),
Coulomb interaction can stabilize a $\mathbb{Z}_2$ topological insulator by
opening a charge gap. At $\nu=1$, the interaction induces three classes of
competing states, spin density wave states, an in-plane ferromagnetic state,
and a valley polarized state, which undergo first-order phase transitions tuned
by an out-of-plane displacement field. The valley polarized state becomes a
Chern insulator for certain displacement fields. Moreover, we predict a
topological charge density wave forming a honeycomb lattice with ferromagnetism
at $\nu=2/3$. Future directions on this versatile system hosting a rich set of
quantum phases are discussed.
