Localization in one-dimensional incommensurate lattices beyond the Aubry-André model

Abstract

Localization properties of particles in one-dimensional incommensurate lattices without interaction are investigated with models beyond the tight-binding Aubry-Andr\'e (AA) model. Based on a tight-binding ${t}_{1}\text{\ensuremath{-}}{t}_{2}$ model with finite next-nearest-neighbor hopping ${t}_{2}$, we find the localization properties qualitatively different from those of the AA model, signaled by the appearance of mobility edges. We then further go beyond the tight-binding assumption and directly study the system based on the more fundamental single-particle Schr\"odinger equation. With this approach, we also observe the presence of mobility edges and localization properties dependent on incommensuration.