Plane wave study on the localized-extended transition in the one-dimensional incommensurate systems

Abstract Based on our recently proposed plane wave framework, we theoretically study the localized-extended transition in the one dimensional incommensurate systems with cosine type of potentials, which are in close connection to many recent experiments of the ultracold atoms and photonic crystals. With the aid of numerical calculations, we formulate a propagator based scattering picture to explain the transition at the ground state and single particle mobility edge, in which deeper connections among the incommensurate potentials, eigenstate compositions and transition mechanism are revealed. We further show that there exists an upper limit of localization length for all localized eigenstates, leading to a fundamental difference to the Anderson localization. The scattering picture provides unique insights on the localization mechanism in the incommensurate systems, and can be generalized to study the more complex phenomena in higher dimensions, with different potentials or beyond the single particle regime.