Localization transition in one dimension using Wegner flow equations

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 ([alpha] = 1). This method correctly reproduces the critical level-spacing statistics and the fractal dimensionality of the eigenfunctions.