Twisted bilayer graphene. VI. An exact diagonalization study at nonzero integer filling

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