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

Abstract

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) [$\mathrm{U}(4)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(4)$] symmetries in the nonchiral (chiral) flat band limit, we reduced the Hilbert space to an extent that allows for study around $\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}3,\ifmmode\pm\else\textpm\fi{}2,\ifmmode\pm\else\textpm\fi{}1$ 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 $\ifmmode\pm\else\textpm\fi{}1$ excitations found in Bernevig et al. [Phys. Rev. B 103, 205415 (2021)] are the lowest charge excitations up to system sizes $8\ifmmode\times\else\texttimes\fi{}8$ (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 $\ensuremath{\nu}=\ensuremath{-}3$, the ground state is the spin and valley polarized Chern insulator with ${\ensuremath{\nu}}_{C}=\ifmmode\pm\else\textpm\fi{}1$ at ${w}_{0}/{w}_{1}\ensuremath{\lesssim}0.9$ (0.3) with (without) FMC. At $\ensuremath{\nu}=\ensuremath{-}2$, we can only numerically access the valley polarized sector, and we find a spin ferromagnetic phase when ${w}_{0}/{w}_{1}\ensuremath{\gtrsim}0.5t$ where $t\ensuremath{\in}[0,1]$ 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 $\ensuremath{\nu}=\ensuremath{-}3$ with/without FMC, when ${w}_{0}/{w}_{1}$ is large, the finite-size gap $\mathrm{\ensuremath{\Delta}}$ to the neutral excitations vanishes, leading to phase transitions. Further analysis of the ground state momentum sectors at $\ensuremath{\nu}=\ensuremath{-}3$ suggests a competition among (nematic) metal, momentum ${M}_{M}$ ($\ensuremath{\pi}$) stripe and ${K}_{M}$-CDW orders at large ${w}_{0}/{w}_{1}$.

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