Perturbative criteria for Anderson localization in long-ranged 1D tight-binding models

TL;DR

This paper introduces a perturbative scaling method to determine Anderson localization criteria in one-dimensional long-range tight-binding models, revealing a transition at a critical decay exponent.

Contribution

It presents a novel perturbative approach to analyze localization in 1D models with power-law decaying interactions, confirming previous RG results.

Findings

For $oldsymbol{ ext{α} = 1}$, all states are extended at weak disorder.

For $oldsymbol{ ext{α} > 1}$, all states are localized regardless of disorder strength.

The approach aligns with earlier renormalization group findings.

Abstract

We develop an alternative scaling approach to determine the criteria for Anderson localization in one-dimensional tight-binding models with random site energies having a bandwidth that decays as a power law in space, $H_{ij} \propto |i - j|^{-\alpha}$. At the first order in perturbation theory the scale dependence of the exchange-narrowed energy of the disorder is compared to the energy level spacing of the ideal system to establish whether or not the disorder has a perturbative effect on the Bloch states. We find that at $\alpha =1$, the perturbative condition is satisfied and for sufficiently weak disorder strength all states are extended. For $\alpha > 1$, all states are localized for arbitrary disorder strength, in agreement with the earlier renormalization group treatment by Levitov.