Intertwined fractional quantum anomalous Hall states and charge density waves

Abstract: 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 $\sigma_H = - 2/3$ to a trivial CDW insulator with $\sigma_H = 0$, and more interestingly, to a QAH-crystal with $\sigma_H= -1$.