▎ 摘　　要

Using exact diagonalization, we study the projected Hamiltonian with the Coulomb interaction in the eight flat bands of first magic angle twisted bilayer graphene. Employing the U(4) [U(4) x U(4)] symmetries in the nonchiral (chiral) flat band limit, we reduced the Hilbert space to an extent that allows for study around nu = +/- 3, +/- 2, +/- 1 fillings. In the first chiral limit w(0)/w(1) = 0, where w(0) (w(1)) is the AA (AB) stacking hopping, we find that the ground states at these fillings are extremely well-described by Slater determinants in a so-called Chern basis, and the exactly solvable charge +/- 1 excitations found in Bernevig et al. [Phys. Rev. B 103, 205415 (2021)] are the lowest charge excitations up to system sizes 8 x 8 (for restricted Hilbert space) in the chiral-flat limit. We also find that the flat metric condition (FMC) used by Bernevig et al. [Phys. Rev. B 103, 205411 (2021)], Song et al. [Phys. Rev. B 103, 205412 (2021)], Bernevig et al. [Phys. Rev. B 103, 205413 (2021)], Lian et al. [Phys. Rev. B 103, 205414 (2021)], and Bernevig et al. [Phys. Rev. B 103, 205415 (2021)] for obtaining a series of exact ground states and excitations holds in a large parameter space. For nu = -3, the ground state is the spin and valley polarized Chern insulator with nu(C) = +/- 1 at w(0)/w(1) less than or similar to 0.9 (0.3) with (without) FMC. At nu = -2, we can only numerically access the valley polarized sector, and we find a spin ferromagnetic phase when w(0)/w(1) greater than or similar to 0.5t where t is an element of [0, 1] is the factor of rescaling of the actual TBG bandwidth, and a spin singlet phase otherwise, confirming the perturbative calculation [Lian. et al., Phys. Rev. B 103, 205414 (2021), Bultinck et al., Phys. Rev. X 10, 031034 (2020)]. The analytic FMC ground state is, however, predicted in the intervalley coherent sector which we cannot access [Lian et al., Phys. Rev. B 103, 205414 (2021), Bultinck et al., Phys. Rev. X 10, 031034 (2020)]. For nu = -3 with/without FMC, when w(0)/w(1) is large, the finite-size gap Delta to the neutral excitations vanishes, leading to phase transitions. Further analysis of the ground state momentum sectors at nu = -3 suggests a competition among (nematic) metal, momentum M-M (pi) stripe and K-M-CDW orders at large w(0)/w(1).