Localization properties of a one-dimensional tight-binding model with non-random long-range inter-site interactions

TL;DR

This paper investigates the localization-delocalization transition in a one-dimensional tight-binding model with long-range interactions and uncorrelated disorder, revealing that delocalized states form a null measure set at the band edge in the thermodynamic limit.

Contribution

It provides an analytical and numerical analysis of the mobility edge in a long-range hopping model, showing that delocalized states are of null measure at the transition point.

Findings

Delocalized states tend to form a null measure set at the band edge.

The critical disorder magnitude depends on the interaction exponent.

The mobility edge approaches the band edge in the thermodynamic limit.

Abstract

We perform both analytical and numerical studies of the one-dimensional tight-binding Hamiltonian with stochastic uncorrelated on-site energies and non-fluctuating long-range hopping integrals . It was argued recently [A. Rodriguez at al., J. Phys. A: Math. Gen. 33, L161 (2000)] that this model reveals a localization-delocalization transition with respect to the disorder magnitude provided . The transition occurs at one of the band edges (the upper one for and the lower one for). The states at the other band edge are always localized, which hints on the existence of a single mobility edge. We analyze the mobility edge and show that, although the number of delocalized states tends to infinity, they form a set of null measure in the thermodynamic limit, i.e. the mobility edge tends to the band edge. The critical magnitude of disorder for the band edge states is computed versus the interaction exponent by making use of the conjecture on the universality of the normalized participation number distribution at transition.