Fractional quantum anomalous Hall effects in rhombohedral multilayer graphene in the moir\'eless limit and in Coulomb imprinted superlattice

The standard theoretical framework for fractional quantum anomalous Hall effect (FQAH) assumes an isolated flat Chern band in the single particle level. In this paper we challenges this paradigm for the FQAH recently observed in the pentalayer rhombohedral stacked graphene aligned with hexagon boron nitride (hBN). We show that the external moir\'e superlattice potential is simply a perturbation in a model with continuous translation symmetry. Through Hartree Fock calculation, we find that interaction opens a sizable remote band gap, resulting an isolated narrow $C=1$ Chern band at filling $\nu=1$. From exact diagonalization (ED) we identify FQAH phases at various fillings. But they exist also in the calculations without any external moir\'e potential. We suggest that the QAH insulator at $\nu=1$ should be viewed as an interaction driven topological Wigner crystal with QAH effect, which is then pinned by a small moir\'e potential. The $C=1$ QAH crystal is robust with a crystal period around $10\mathrm{nm}$ in 4-layer, 5-layer, 6-layer and 7-layer graphene systems. Our work suggests a new direction to exploring the interplay of topology and FQAH with spontaneous crystal formation in the vanishing moir\'e potential limit. We also propose a new system to generate and control both honeycomb and triangular moir\'e superlattice potential through Coulomb interaction from another control layer, which can stabilize or suppress the QAH crystal depending on the density of the control layer.