Nearest neighbor tight binding models with an exact mobility edge in one dimension

TL;DR

This paper introduces a deterministic one-dimensional tight binding model with a quasiperiodic potential that exhibits a mobility edge, separating localized and extended states, protected by a duality symmetry, and proposes an experimental realization.

Contribution

It presents the first nearest neighbor tight binding model with a mobility edge in 1D, characterized by a self-duality condition and verified through numerical calculations.

Findings

Identifies a self-dual line defining a mobility edge in the model.

Demonstrates localization-delocalization transition at the mobility edge.

Proposes experimental schemes for realization in optical lattices and photonic waveguides.

Abstract

We investigate localization properties in a family of deterministic (i.e. no disorder) nearest neighbor tight binding models with quasiperiodic onsite modulation. We prove that this family is self-dual under a generalized duality transformation. The self-dual condition for this general model turns out to be a simple closed form function of the model parameters and energy. We introduce the typical density of states as an order parameter for localization in quasiperiodic systems. By direct calculations of the inverse participation ratio and the typical density of states we numerically verify that this self-dual line indeed defines a mobility edge in energy separating localized and extended states. Our model is a first example of a nearest neighbor tight binding model manifesting a mobility edge protected by a duality symmetry. We propose a realistic experimental scheme to realize our results in atomic optical lattices and photonic waveguides.