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Symmetry breaking in the double moiré superlattices of relaxed twisted bilayer graphene on hexagonal boron nitride

We study the atomic and electronic structures of the commensurate double moiré superlattices in fully relaxed twisted bilayer graphene (TBG) nearly aligned with the hexagonal boron nitride (BN). The single-particle effective Hamiltonian ($\hat{H}^0$) taking into account the relaxation effect and the full moiré 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 different number ($ν$) of filled flat bands relative to the charge neutrality point (CNP) are obtained based on $\hat{H}^0$ 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 $\hat{H}^0$. We find that the broken $C_2$ symmetry in $\hat{H}^0$ mainly originates from the intralayer inversion-asymmetric structural deformation in the graphene layer adjacent to BN, which introduces spatially non-uniform 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 $|ν|$ = 1-3 are all insulating with narrow MF gaps. When the flat conduction bands are filled, the gap at $ν$ = 1 is smaller than that at $ν$ = 3, suggesting that the nontrivial topological properties associated with the flat Chern bands are more likely to be observed at $ν= 3$. This is similar for negative $ν$ with empty valence bands. The dependence of the electronic structure of TBG/BN on positive $ν$ is roughly consistent with recent experimental observations.