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Motivated by the recently observed insulating states in twisted bilayergraphene, we study the nature of the correlated insulating phases of thetwisted bilayer graphene at commensurate filling fractions. We use thecontinuum model and project the Coulomb interaction onto the flat bands tostudy the ground states by using a Hartree-Fock approximation. In the absenceof the hexagonal boron nitride substrate, the ground states are the intervalleycoherence states at charge neutrality (filling $\nu$ = 0, or four electrons permoir\'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 theground states at all $\nu$ into $C_2\mathcal{T}$ symmetry broken-states. Ourresults provide good reference points for further study of the rich correlatedphysics in the twisted bilayer graphene.

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