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