An Exact Diagonalization Study of Twisted Bilayer Graphene at Integer Fillings: Phase Diagram*

We study the projected Hamiltonian with Coulomb interaction in the 8 flat bands of first magic angle twisted bilayer graphene using exact diagonalization. Employing the U(4) (U(4)xU(4)) symmetries in the nonchiral (chiral) flat band limit, we reduced the Hilbert space to an extent which allows for study around ν=±3, ±2, ±1 fillings. In the first chiral limit, we find that the ground states at these fillings are extremely well-described by Chern insulator Slater determinants. We also find that the Flat Metric Condition (FMC) for obtaining a series of exact ground states holds in a large parameter space. For ν=-3, the ground state is the spin and valley polarized Chern insulator with ν_C=±1 at w_0/w_1≤0.3 (0.9) with (without) FMC. At ν=-2, we can only numerically access the valley polarized sector, and we find a spin ferromagnetic-singlet phase transition, confirming the perturbative calculation.