Emergent criticality in the Aubry-Andr\'e model with periodic modulation

The Aubry-Andr\'e model describes a system with quasiperiodic lattice modulation. In one dimension the AAH model is known to exhibit a sharp metal to insulator transition at a self-dual critical point at which all the states in the spectrum are critical or multifractal in nature. While such criticality is immediately destroyed by an additional onsite periodic modulation, we show an emergent criticality in the limit of strong periodic modulation strength under proper conditions. The resulting strong-modulation critical phase exhibits multifractal eigenstates and singular continuous spectra, belonging to the universality class of the critical Harper model. Moreover, we reveal that additional periodic potential of period N in the quasiperiodic chain folds the spectrum into N bands with quasiperiodicity being enhanced by a factor of N, producing N numbers of Hofstadter butterflies in each band. Our results reveal a general mechanism for engineering robust criticality and spectral replication in quasiperiodic systems.