Localization transition in one dimension using Wegner flow equations

TL;DR

This paper applies Wegner flow equations to a disordered one-dimensional model with power-law hoppings, revealing a localization transition at a critical decay exponent and accurately capturing critical properties.

Contribution

It introduces a novel application of flow equations combined with a strong-bond RG approach to analyze localization transitions in a disordered 1D system with power-law decaying hoppings.

Findings

States are delocalized for <1/2

Critical point at =1 is identified

Method reproduces level-spacing statistics and eigenfunction fractality

Abstract

The flow equation method was proposed by Wegner as a technique for studying interacting systems in one dimension. Here, we apply this method to a disordered one dimensional model with power-law decaying hoppings. This model presents a transition as function of the decaying exponent $\alpha$. We derive the flow equations, and the evolution of single-particle operators. The flow equation reveals the delocalized nature of the states for $\alpha<1/2$. Additionally, in the regime, $\alpha>1/2$, we present a strong-bond renormalization group structure based on iterating the three-site clusters, where we solve the flow equations perturbatively. This renormalization group approach allows us to probe the critical point $\left(\alpha=1\right)$. This method correctly reproduces the critical level-spacing statistics, and the fractal dimensionality of the eigenfunctions.