▎ 摘　　要

We derive the exact insulator ground states of the projected Hamiltonian of magic-angle twisted bilayer graphene (TBG) flat bands with Coulomb interactions in various limits, and study the perturbations away from these limits. We define the (first) chiral limit where the AA stacking hopping is zero, and a flat limit with exactly flat bands. In the chiral-flat limit, the TBG Hamiltonian has a U(4)xU(4) symmetry, and we find that the exact ground states at integer filling -4 0, a first-order phase transition for nu = +/- 1, +/- 2 from Chern number nu(C) = sgn(nu B)(2 - vertical bar nu vertical bar) to nu(C) = sgn(nu B)(4 - vertical bar nu vertical bar) is expected, which agrees with recent experimental observations. Lastly, the TBG Hamiltonian reduces into an extended Hubbard model in the stabilizer code limit.