Edge modes in discrete lattices: phase transitions under the three-gap theorem

We present a new paradigm for generating complex structured materials based on the three-gap theorem that unifies and generalizes several key concepts in the study of localized edge states. Our model has both the discretized coupling strengths of the Su–Schrieffer–Heeger (SSH) model and a modulation parameter that can be used to characterize the spectral flow of edge modes and produce images reminiscent of the Hofstadter butterfly. By defining a localization factor associated with each eigenmode, we are able to establish conditions for the existence of localized edge states in finite systems. This allows us to compare their eigenfrequencies with the spectra of the corresponding infinitely periodic problem and characterize the rich pattern of localized edge modes appearing and disappearing (in the sense of becoming delocalized) as the parameters of our three-gap algorithm are varied.