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)xU(4) symmetry, and we find that the exact ground states at integer filling -4 <= nu <= 4 relative to charge neutrality are Chern insulators of Chern numbers nu(C) = 4 - vertical bar nu vertical bar, 2 - vertical bar nu vertical bar, ..., vertical bar nu vertical bar - 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, +/- 2 has exact ground states of Chern number 0, while nu = +/- 1, +/- 3 has perturbative ground states of Chern number nu(C) = +/- 1, 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 < vertical bar nu(C)vertical bar < 4 - vertical bar nu vertical bar) at integer fillings. are fully (partially) intervalley coherent, while the insulator states with Chern number vertical bar nu(C)vertical bar = 4 - vertical bar nu vertical bar are valley polarized. However, for 0 < vertical bar nu(C)vertical bar <= 4 - vertical bar nu vertical bar, the fully intervalley coherent states are highly competitive (0.005 meV/electron higher). At nonzero magnetic field vertical bar B vertical bar > 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.

Automated summary

This study investigates the exact insulator states and Chern insulators in different limits of the projected Hamiltonian for magic-angle twisted bilayer graphene flat bands with Coulomb interactions. The results show competitive low-energy states of TBG, with different Chern number states being degenerate in certain limits. Transition behaviors are also observed in the presence of magnetic fields. The TBG Hamiltonian converges into an extended Hubbard model in the stabilizer code limit.