  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 $-4\le \nu\le 4$ 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
$\nu_C=\pm1$, 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 $\nu_C=0$
($0<|\nu_C|<4-|\nu|$) at integer fillings $\nu$ are fully (partially)
intervalley coherent, while the insulator states with Chern number
$|\nu_C|=4-|\nu|$ are valley polarized. However, for $0<|\nu_C|\le4-|\nu|$, the
fully intervalley coherent states are highly competitive (0.005meV/electron
higher). At nonzero magnetic field $|B|>0$, 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.
