Twisted bilayer graphene. IV. Exact insulator ground states and phase diagram

We derive the exact insulator ground states of projected Hamiltonian magic-angle twisted bilayer graphene (TBG) flat bands with Coulomb interactions in various limits, and study perturbations away from these limits. define (first) chiral limit where AA stacking hopping is zero, a exactly bands. In chiral-flat limit, TBG has U(4)$\times$U(4) symmetry, we find that at integer filling $-4\le \nu\le 4$ relative to charge neutrality are Chern insulators numbers $\nu_C=4-|\nu|,2-|\nu|,\cdots,|\nu|-4$, all which degenerate. This confirms recent experiments found be competitive low-energy TBG. When reduced nonchiral-flat U(4) $\nu=0,\pm2$ number $0$, while $\nu=\pm1,\pm3$ perturbative $\nu_C=\pm1$, ferromagnetic. chiral-nonflat different degenerate up second order perturbations. realistic nonchiral-nonflat case, $\nu_C=0$ ($0<|\nu_C|<4-|\nu|$) fillings $\nu$ fully (partially) intervalley coherent, $|\nu_C|=4-|\nu|$ valley polarized. However, for $0<|\nu_C|\le4-|\nu|$, coherent highly (0.005meV/electron higher). At nonzero magnetic field $|B|>0$, first-order phase transition $\nu=\pm1,\pm2$ $\nu_C=\text{sgn}(\nu B)(2-|\nu|)$ B)(4-|\nu|)$ expected, agrees experimental observations. Lastly, reduces into an extended Hubbard model stabilizer code limit.