Optimizing fractional quantum anomalous Hall effect in moiré superlattice of rhombohedralmultilayer graphene

Recently fractional quantum anomalous Hall (FQAH) effect was observed in the moiré system formed by five-layer rhombohedral stacked grapehne aligned with hBN. It was known that the topological band corresponds to the sign of displacement field where electrons are away from the hBN and the superlattice potential is too weak to open the remote gap between the next band. Through Hartree-Fock calculation, we notice that the interaction can open a gap, generating a narrow C=1 Chern band. Next we propose a new moiré setup with n-layer graphene separated with the twisted bilayer graphene (TBG) through a thin hBN. We change the filling of TBG to n_TBG=4 and the inter-layer Coulomb interaction imprints superlattice potentials for all of the n-layers from the density profile of TBG. We find flat C=1 Chern band for n_layer=4,5 without doing even Hartree Fock. We present Hartree Fock and exact diagonalization study of the possibility of FQAH in ths new system.