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

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.