Exact and perturbative insulator states and excitations in twisted bilayer graphene*

We study the low energy states of the flat-band projected Hamiltonian of magic angle twisted bilayer graphene with Coulomb interactions. By treating the nonchiral interaction and the kinetic energy as perturbations, we analytically derive the exact or perturbative strongly interacting (Chern) insulator ground states at all integer fillings |\nu|<4. We find the ground state has Chern number mod(\nu,2), and is intervalley coherent (valley polarized) at fillings |\nu|=0,1,2 (|\nu|=3). We further show that the insulators have analytically computable charge 0, +-1,+-2 excitations. We show that the Goldstone modes of the ground states are quadratic. Furthermore, from the charge 1 and 2 excitations, we prove the absence of Cooper pairing from the Coulomb interaction in the flat band limit, which suggests other superconductivity mechanisms.