Coexistence of extended and localized states in one-dimensional systems

TL;DR

This paper introduces a class of one-dimensional tight-binding models with position-dependent electron mass that exhibit controllable transitions between localized and extended states, providing new insights into mobility edges in lower dimensions.

Contribution

It demonstrates how to identify and control mobility edge transitions in 1D models with exact solutions, advancing understanding of localization phenomena in lower-dimensional systems.

Findings

Controlled transition between localized and extended states across energy spectrum

Mathematically exact solutions for extreme parameter conditions

Precise determination of mobility edge energy in intermediate regime

Abstract

Mobility edge transitions from localized to extended states have been observed in two and three dimensional systems, for which sound theoretical explanations have also been derived. One-dimensional lattice models have failed to predict their emergence, offering no clues on how to actually probe this phenomenon in lower dimensions. This work reports results for a class of tight-binding models with electron-mass position dependence, for which localized-extended wave function transitions can be identified. We show that it is possible to control the density of localized and extended states by tuning the transition-related parameter for a continuous range of energy values. Mathematically exact results for extended or localized states are derived in two extreme conditions of this parameter, as well as an exact energy value for the mobility edge transition in the intermediate regime. Our framework provides a clear point of view on the phenomena and can also be harnessed for setting up experiments to probe to precisely evaluate the associated mobility edges using state-of-the-art technology.