Duality between two generalized Aubry-Andre models with exact mobility edges

TL;DR

This paper analytically proves the duality between two quasiperiodic models with exact mobility edges, providing insights into their localization properties and potential experimental realization.

Contribution

It establishes a duality relation between two models with exact mobility edges and derives their localization characteristics analytically and numerically.

Findings

Models are mutually dual with exact mobility edges.

Numerical verification confirms the duality and localization properties.

Analytical expressions for mobility edges and localization lengths are obtained.

Abstract

A mobility edge (ME) in energy separating extended from localized states is a central concept in understanding various fundamental phenomena like the metal-insulator transition in disordered systems. In one-dimensional quasiperiodic systems, there exist a few models with exact MEs, and these models are beneficial to provide exact understanding of ME physics. Here we investigate two widely studied models including exact MEs, one with an exponential hopping and one with a special form of incommensurate on-site potential. We analytically prove that the two models are mutually dual, and further give the numerical verification by calculating the inverse participation ratio and Husimi function. The exact MEs of the two models are also obtained by calculating the localization lengths and using the duality relations. Our result may provide insight into realizing and observing exact MEs in both theory and experiment.