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Motivated by the recent experimental breakthrough on the observation of thefractional quantum anomalous Hall (FQAH) effects in semiconductor and graphenemoir\'{e} materials, we explore the rich physics associated with thecoexistence of FQAH effect and the charge density wave (CDW) order thatspontaneously breaks the translation symmetry. We refer to a state with bothproperties as "FQAH-crystal". We show that the interplay between FQAH effectand CDW can lead to a rich phase diagram including multiple topological phasesand topological quantum phase transitions at the same moir\'e filling. Inparticular, we demonstrate the possibility of direct quantum phase transitionsfrom a FQAH-crystal with Hall conductivity $\sigma_H = - 2/3$ to a trivial CDWinsulator with $\sigma_H = 0$, and more interestingly, to a QAH-crystal with$\sigma_H= -1$.

Intertwined fractional quantum anomalous Hall states and charge density waves