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

2008/10/24 by Akhanjee, Shimul
#Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences

paper · doi:10.48550/arxiv.0810.4532

Abstract

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 ∝ |i - j|^-α. 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 α=1, the perturbative condition is satisfied and for sufficiently weak disorder strength all states are extended. For α> 1, all states are localized for arbitrary disorder strength, in agreement with the earlier renormalization group treatment by Levitov.

Related

• Direct Scaling Analysis of localization in disordered systems. I. Single-particle models
• Anderson Localisation for periodically driven systems
• Weak-disorder expansion for localization lengths of quasi-1D systems
• The scaling behaviour of localised and extended states in one-dimensional tight-binding models with disorder
• Anderson Localization for Time Periodic Random Schrödinger operators