Using exact diagonalization, we study the projected Hamiltonian with Coulomb interaction in the 8 flat bands of first magic angle twisted bilayer graphene. Employing the U(4) (U(4)$times$U(4)) symmetries in the nonchiral (chiral) flat band limit, we reduced the Hilbert space to an extent which allows for study around $ u=pm 3,pm2,pm1$ 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 $pm1$ excitations found in [arXiv:2009.14200] are the lowest charge excitations up to system sizes $8times8$ (for restricted Hilbert space) in the chiral-flat limit. We also find that the Flat Metric Condition (FMC) used in [arXiv:2009.11301,2009.11872,2009.12376,2009.13530,2009.14200] for obtaining a series of exact ground-states and excitations holds in a large parameter space. For $ u=-3$, the ground state is the spin and valley polarized Chern insulator with $ u_C=pm1$ at $w_0/w_1lesssim0.9$ (0.3) with (without) FMC. At $ u=-2$, we can only numerically access the valley polarized sector, and we find a spin ferromagnetic phase when $w_0/w_1gtrsim0.5t$ where $tin[0,1]$ is the factor of rescaling of the actual TBG bandwidth, and a spin singlet phase otherwise, confirming the perturbative calculation [arXiv:2009.13530]. The analytic FMC ground state is, however, predicted in the intervalley coherent sector which we cannot access [arXiv:2009.13530]. For $ u=-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 $ u=-3$ suggests a competition among (nematic) metal, momentum $M_M$ ($pi$) stripe and $K_M$-CDW orders at large $w_0/w_1$.
تحميل البحث