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Twisted Bilayer Graphene IV. Exact Insulator Ground States and Phase Diagram

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)$\times$U(4) symmetry, and we find that the exact ground states at integer filling $-4\le ν\le 4$ relative to charge neutrality are Chern insulators of Chern numbers $ν_C=4-|ν|,2-|ν|,\cdots,|ν|-4$, 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 $ν=0,\pm2$ has exact ground states of Chern number $0$, while $ν=\pm1,\pm3$ has perturbative ground states of Chern number $ν_C=\pm1$, which are U(4) ferromagnetic. In the chiral-nonflat limit with a different U(4) symmetry, different Chern number states are degenerate up to second order perturbations. In the realistic nonchiral-nonflat case, we find that the perturbative insulator states with Chern number $ν_C=0$ ($0<|ν_C|<4-|ν|$) at integer fillings $ν$ are fully (partially) intervalley coherent, while the insulator states with Chern number $|ν_C|=4-|ν|$ are valley polarized. However, for $0<|ν_C|\le4-|ν|$, the fully intervalley coherent states are highly competitive (0.005meV/electron higher). At nonzero magnetic field $|B|>0$, a first-order phase transition for $ν=\pm1,\pm2$ from Chern number $ν_C=\text{sgn}(νB)(2-|ν|)$ to $ν_C=\text{sgn}(νB)(4-|ν|)$ is expected, which agrees with recent experimental observations. Lastly, the TBG Hamiltonian reduces into an extended Hubbard model in the stabilizer code limit.