▎ 摘　　要

We study the atomic and electronic structures of the commensurate double moire superlattices in fully relaxed twisted bilayer graphene (TBG) nearly aligned with the hexagonal boron nitride (BN). The single-particle effective Hamiltonian ((H) over cap (0)) taking into account the relaxation effect and the full moire Hamiltonian introduced by BN has been built for TBG/BN. The mean-field (MF) band structures of the self-consistent Hartree-Fock (SCHF) ground states at a different number (nu) of filled flat bands relative to the charge neutrality point (CNP) are obtained based on (H) over cap (0) in the plane-wave-like basis. The single-particle flat bands in TBG/BN become separated by the opened gap at CNP due to the symmetry breaking in (H) over cap (0). We find that the broken C-2 symmetry in (H) over cap (0) mainly originates from the intralayer inversion-asymmetric structural deformation in the graphene layer adjacent to BN, which introduces spatially nonuniform modifications of the intralayer Hamiltonian. The gapped flat bands have finite Chern numbers. For TBG/BN with the magic twist angle, the SCHF ground states with vertical bar nu vertical bar = 1-3 are all insulating with narrow MF gaps. When the flat conduction bands are filled, the gap at nu = 1 is smaller than that at nu = 3, suggesting that the nontrivial topological properties associated with the flat Chern bands are more likely to be observed at nu = 3. This is similar for negative v with empty valence bands. The dependence of the electronic structure of TBG/BN on positive v is roughly consistent with recent experimental observations.