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

Abstract

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)×U(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 w0/w1 = 0 where w0 (w1) 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 × 8 (for restricted Hilbert space) in the chiral-flat limit. We also find that the Flat Metric Condition (FMC) used in 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 ν = −3, the ground state is the spin and valley polarized Chern insulator with νC = ±1 at w0/w1 . 0.9 (0.3) with (without) FMC. At ν = −2, we can only numerically access the valley polarized sector, and we find a spin ferromagnetic phase when w0/w1 & 0.5t where t ∈ [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 ν = −3 with/without FMC, when w0/w1 is large, the finite-size gap ∆ to the neutral excitations vanishes, leading to phase transitions. Further analysis of the ground state momentum sectors at ν = −3 suggests a competition among (nematic) metal, momentum MM (π) stripe and KM -CDW orders at large w0/w1.