Localization transition in one dimension using Wegner flow equations

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 $\ensuremath{\alpha}$. We derive the flow equations and the evolution of single-particle operators. The flow equation reveals the delocalized nature of the states for $\ensuremath{\alpha}<\frac{1}{2}$. Additionally, in the regime $\ensuremath{\alpha}>\frac{1}{2}$, 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 $\left(\ensuremath{\alpha}=1\right)$. This method correctly reproduces the critical level-spacing statistics and the fractal dimensionality of the eigenfunctions.