Among the extensive studies of fractional quantum anomalous Hall (FQAH) states, there recently has appeared a growing interest in the topological states with coexisting charge density wave (CDW) orders. Such states are referred to as Hall crystals. However, compared to those with integer Hall conductivities, the FQAH crystal (FQAHC) is still elusive, even at the level of a microscopic model. In this work, we numerically study a topological flat-band model on a triangular lattice with spinless fermions. At fractional filling of the Chern band, the nearest-neighbor interaction leads to a commensurate and topologically trivial CDW state. Interestingly, the folded miniband above the CDW gap is nontrivial, and we focus on the doping of it without any projection. A series of (F)QAHC states at (fractional) integer fillings of this miniband is discovered and some FQAHC state might even exist in less “ideal” conditions. The ground-state degeneracies of such (F)QAHC states are enlarged by the CDW degeneracy, and the Hall conductivities—determined by the fillings of the miniband—are different from the fillings of the original Chern band. We also study the thermodynamics of an FQAHC state and find a compressible CDW phase at intermediate temperatures, which might serve as a precursor of a lower-temperature FQAHC phase. Moreover, we numerically demonstrate that such a generic scheme of doping a CDW-folded topological miniband could be applied to bosonic systems, broadening the platforms of Hall-crystal physics and motivating its exploration in quantum moiré and cold-atom systems.

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• Charge density waves
• Fractional quantum Hall effect
• Quantum anomalous Hall effect