Renormalization Group Analysis of the Hierarchical Anderson Model

Abstract

We apply Feshbach-Krein-Schur renormalization techniques in the hierarchical Anderson model to establish a criterion on the single-site distribution which ensures exponential dynamical localization as well as positive inverse participation ratios and Poisson statistics of eigenvalues. Our criterion applies to all cases of exponentially decaying hierarchical hopping strengths and holds even for spectral dimension $d>2$, which corresponds to the regime of transience of the underlying hierarchical random walk. This challenges recent numerical findings that the spectral dimension is significant as far as the Anderson transition is concerned.

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Cited by in corpus (4)

• Scale-invariant critical dynamics at eigenstate transitions
• Delocalization and continuous spectrum for ultrametric random operators
• A supersymmetric hierarchical model for weakly disordered $3d$ semimetals
• Singular Spectrum and Recent Results on Hierarchical Operators