TBG IV: Exact Insulator Ground States and Phase Diagram of Twisted Bilayer Graphene

We derive the exact analytic 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)$\times$U(4) symmetry, and we find that the exact ground states at integer filling $-4\le \nu\le 4$ relative to charge neutrality are Chern insulators of Chern numbers $\nu_C=4-|\nu|,2-|\nu|,\cdots,|\nu|-4$, all of which are degenerate. This confirms recent experiments where Chern insulators are found to be competitive low-energy states of TBG. When the chiral-flat limit is reduced to the nonchiral-flat limit which has a U(4) symmetry, we find $\nu=0,\pm2$ has exact ground states of Chern number $0$, while $\nu=\pm1,\pm3$ has perturbative ground states of Chern number $\nu_C=\pm1$, all of which are U(4) ferromagnetic. In the chiral-nonflat limit which has a different U(4) symmetry, different Chern number states are degenerate up to second order perturbations. When reduced to the realistic nonchiral-nonflat case, we find that the perturbative insulator states with Chern number $|\nu_C| 0$, a first-order phase transition for $\nu=\pm1,\pm2$ from Chern number $\nu_C=\text{sgn}(\nu B)(2-|\nu|)$ to $\nu_C=\text{sgn}(\nu B)(4-|\nu|)$ is expected, which agrees with recent experimental observations. Lastly, the TBG Hamiltonian reduces into an extended Hubbard model in the stabilizer code limit.