Continuous transition and gapless roton inside fractional quantum anomalous Hall states

Collective excitations are crucial for understanding exotic phases and phase transitions in quantum many-body systems. In this work, we numerically demonstrate the microscopic realization of a transition from a translationally invariant fractional quantum anomalous Hall (FQAH) state to a spontaneously broken-translation-symmetry FQAH state, driven by softening of the magnetoroton mode via isotropic interactions in a topological flat-band model. At the critical point, the neutral excitation gap closes at finite momentum, while the charge gap remains stable. This transition shares similarities with Landau-level quantum Hall crystals and nematics but exhibits unique features. Criticality analysis shows that this non-trivial transition belongs to the Ising universality class. Such spontaneous translation symmetry breaking within a topologically ordered FQAH state could serve as a universal mechanism across various systems, with potential implications for quantum moiré materials and cold-atom systems.