Correlated insulating phases of twisted bilayer graphene at commensurate filling fractions: a Hartree-Fock study

Motivated by the recently observed insulating states in twisted bilayer graphene, we study the nature of the correlated insulating phases of the twisted bilayer graphene at commensurate filling fractions. We use the continuum model and project the Coulomb interaction onto the flat bands to study the ground states by using a Hartree-Fock approximation. In the absence of the hexagonal boron nitride substrate, the ground states are the intervalley coherence states at charge neutrality (filling $\nu$ = 0, or four electrons per moir\'e cell) and at $\nu$ = -1/4 and -1/2 (three and two electrons per cell, respectively) and the $C_2\mathcal{T}$ symmetry-broken state at $\nu$= -3/4 (one electron per cell). The hexagonal boron nitride substrate drives the ground states at all $\nu$ into $C_2\mathcal{T}$ symmetry broken-states. Our results provide good reference points for further study of the rich correlated physics in the twisted bilayer graphene.