A plane wave study on the localized-extended transitions in the one-dimensional incommensurate systems

TL;DR

This paper introduces a plane wave framework to analyze the localized-extended transition in one-dimensional incommensurate systems, revealing a fundamental limit on localization length and connecting theoretical predictions with numerical results.

Contribution

It develops a scattering-based approach to understand the transition mechanism and identifies a maximum localization length, contrasting with Anderson localization.

Findings

Existence of an upper limit on localization length for localized states

Connection between incommensurate potentials and transition mechanisms

Numerical validation of theoretical predictions

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 in the ultracold atom and photonic crystal. We formulate a propagator based scattering picture for the transition at the ground state and single particle mobility edge, in which the deeper connection between the incommensurate potentials, eigenstate compositions and transition mechanism is revealed. We further show that there exists a upper limit of localization length for all localized eigenstates, leading to an fundamental difference to the Anderson localization. Numerical calculations are presented alongside the analysis to justify our statements. The theoretical analysis and numerical methods can also be generalized to systems in higher dimensions, with different potentials or beyond the single particle regime, which would benefit the future studies in the related fields.