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We analyze the phase diagram of twisted graphene bilayers near a magic angle.We consider the effect of the long range Coulomb interaction, treated withinthe self consistent Hartree-Fock approximation, and we study arbitrary bandfillings. We find a rich phase diagram, with different broken symmetry phases,although tehy do not show necessarily a gap at the Fermi energy. There are nontrivial effects of the electrostatic potential on the shape and the gaps of thebands in the broken symmetry phases. The results suggest that the nonsuperconducting broken symmetry phases observed experimentally are induced bythe long range exchange interaction.

Band structure and insulating states driven by the Coulomb interaction in twisted bilayer graphene