Theory of fractional Chern insulator states in pentalayer graphene moir\'e superlattice

The experimental discoveries of fractional quantum anomalous Hall effects under zero magnetic fields in both transition metal dichalcogenide and pentalayer graphene moir\'e superlattices have aroused significant research interest. In this work, we theoretically study the fractional quantum anomalous Hall states (also known as fractional Chern insulator states) in pentalayer graphene moir\'e superlattice. Starting from the highest energy scale ($\sim\!2\,$eV) of the continuum model, we first construct a renormalized low-energy model that applies to a lower cutoff $\sim\!0.15\,$eV using renormalization group approach. Then, we study the ground states of the renormalized low-energy model at filling 1 under Hartree-Fock approximation in the presence of tunable but self-consistently screened displacement field $D$ with several experimentally relevant background dielectric constant $\epsilon_r$. Two competing Hartree-Fock states are obtained at filling 1, which give rise to two types of topologically distinct isolated flat bands with Chern number 1 and 0, respectively. We continue to explore the interacting ground states of the two types of isolated flat bands at hole dopings of 1/3, 2/5, 3/5, and 2/3 (corresponding electron fillings of 2/3, 3/5, 2/5, and 1/3 with respect to charge neutrality). Setting $\epsilon_r=5$, our exact-diagonalization calculations suggest that the system stays in fractional Chern insulator (FCI) state at 2/3 electron filling when $0.9\,\textrm{V/nm}\leq\!D\!\leq 0.92\,\textrm{V/nm}$; while no robust FCI state is obtained at 1/3 electron filling. We have also obtained composite-fermion type FCI ground states at 3/5 electron filling within $0.9\,\textrm{V/nm}\leq\! D \!\leq\!0.95\,\textrm{V/nm}$ and $\epsilon_r=5$. These numerical results are quantitatively consistent with experimental observations.