Localization and delocalization in one-dimensional systems with translation-invariant hopping

TL;DR

This paper develops an analytical theory for Anderson localization in one-dimensional systems with translation-invariant, finite-range hopping, revealing conditions for localization and delocalization as hopping range varies.

Contribution

It introduces a new analytical approach to calculate localization length for arbitrary finite-range hopping and explores delocalization possibilities in the infinite-range limit.

Findings

Localization length depends on hopping range

Delocalized states can occur with long-range hopping

Localization criteria can be violated under certain conditions

Abstract

We present a theory of Anderson localization on a one-dimensional lattice with translation-invariant hopping. We find by analytical calculation, the localization length for arbitrary finite-range hopping in the single propagating channel regime. Then by examining the convergence of the localization length, in the limit of infinite hopping range, we revisit the problem of localization criteria in this model and investigate the conditions under which it can be violated. Our results reveal possibilities of having delocalized states by tuning the long-range hopping.