Strong coupling phases of partially filled twisted bilayer graphene narrow bands

We identify the many-body insulating states favored by the Coulomb interactions projected onto the microscopically constructed Wannier basis of the four narrow bands of the twisted bilayer graphene near the ``magic angle''. The non-local interaction terms are novel and their form can be traced to the topologically non-trivial nature of the narrow bands. In the strong coupling limit and at the filling of two elecrons/holes per moire unit cell, the ground state can be found exactly; it is SU(4) ferromagnetic with exactly one electron/hole on each honeycomb lattice site. The kinetic terms break the SU(4) symmetry and select the ground state in which the two valleys with opposite spins are equally mixed. Although such an insulating state is not quite a spin singlet, the magnetic moment per particle vanishes in the thermodynamic limit and therefore its Zeeman coupling to an external magnetic field also vanishes at linear order. We also find gapped extended excited states with a gap that decreases in an external magnetic field. Finally, we propose that an insulating stripe ferromagnetic phase could be the ground state for the filling of one electron/hole per unit cell, with a gap that does not decrease in an external magnetic field.