Dual hidden landscapes in Anderson localization on discrete lattices

TL;DR

This paper reveals that in 1D discrete lattices, low and high energy Anderson localized modes are governed by two distinct dual landscapes, improving understanding and estimation of localization properties across the energy spectrum.

Contribution

It introduces the concept of dual landscapes for low and high energy modes in discrete lattices, extending the landscape theory to these systems.

Findings

High energy modes are governed by a dual landscape derived from the low energy landscape.

Dual landscapes accurately estimate localization length across energies, especially at weak disorder.

Localization subregions are characterized by these dual landscapes in 1D discrete lattices.

Abstract

The localization subregions of stationary waves in continuous disordered media have been recently demonstrated to be governed by a hidden landscape that is the solution of a Dirichlet problem expressed with the wave operator. In this theory, the strength of Anderson localization confinement is determined by this landscape, and continuously decreases as the energy increases. However, this picture has to be changed in discrete lattices in which the eigenmodes close to the edge of the first Brillouin zone are as localized as the low energy ones. Here we show that in a 1D discrete lattice, the localization of low and high energy modes is governed by two different landscapes, the high energy landscape being the solution of a dual Dirichlet problem deduced from the low energy one using the symmetries of the Hamiltonian. We illustrate this feature using the one-dimensional tight-binding Hamiltonian with random on-site potentials as a prototype model. Moreover we show that, besides unveiling the subregions of Anderson localization, these dual landscapes also provide an accurate overal estimate of the localization length over the energy spectrum, especially in the weak disorder regime.