Phase diagram of twisted graphene bilayers near a magic angle*

The way in which interactions in twisted bilayer graphene favor broken symmetry phases is analyzed. The long range electrostatic coupling between electrons gives rise to spin and/or valley polarized states for most fillings. Competition between phases with different polarizations give rise to discontinuous transitions and phase coexistence. Details of the phase diagram are determined by short range interactions, such as the intra-atomic Hubbard repulsion, and also by external effects, such as screening by the environment, strains, and the coupling to the substrate. The effect of acoustic phonons, which overlap with the continuum of electron-hole excitations, will also be discussed.