Intertwined fractional quantum anomalous Hall states and charge density waves

Motivated by the recent experimental breakthrough on the observation of the fractional quantum anomalous Hall (FQAH) effects in semiconductor and graphene moir\'e materials, we explore the rich physics associated with the coexistence of FQAH effect and the charge density wave (CDW) order that spontaneously breaks the translation symmetry. We refer to a state with both properties as ``FQAH crystal.'' We show that the interplay between FQAH effect and CDW can lead to a rich phase diagram including multiple topological phases and topological quantum phase transitions at the same moir\'e filling. In particular, we demonstrate the possibility of direct quantum phase transitions from a FQAH crystal with Hall conductivity σH=-2/3 to a trivial CDW insulator with σH=0 and, more interestingly, to a QAH crystal with σH=-1.