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The experimental discoveries of fractional quantum anomalous Hall effectsunder zero magnetic fields in both transition metal dichalcogenide andpentalayer graphene moir\'e superlattices have aroused significant researchinterest. In this work, we theoretically study the fractional quantum anomalousHall states (also known as fractional Chern insulator states) in pentalayergraphene moir\'e superlattice. Starting from the highest energy scale($\sim\!2\,$eV) of the continuum model, we first construct a renormalizedlow-energy model that applies to a lower cutoff $\sim\!0.15\,$eV usingrenormalization group approach. Then, we study the ground states of therenormalized low-energy model at filling 1 under Hartree-Fock approximation inthe 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 withChern number 1 and 0, respectively. We continue to explore the interactingground 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 1/3with respect to charge neutrality). Setting $\epsilon_r=5$, ourexact-diagonalization calculations suggest that the system stays in fractionalChern 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 FCIstate is obtained at 1/3 electron filling. We have also obtainedcomposite-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 experimentalobservations.

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