Renormalization Group Analysis of the Hierarchical Anderson Model

Abstract

Abstract. We apply Feshbach-Krein-Schur renormalization techniques in the hierarchical Anderson model to establish a criterion on the singlesite 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.Mathematics Subject Classification (2010). 47B80, 82B44.

Multifractality of wave functions on a Cayley tree: From root to leaves

“…61, there exists a number of models that show phases (rather than isolated points as in the case of conventional Anderson transitions) with multifractal eigenstates. These include, in particular, power-law random banded matrices 2 , small-world networks 74,75 , Rosenzweig-Porter model [76][77][78] , and random Levy matrices 79 . It would be interesting to see whether there are deeper connections between all these models (or, at least, some of them).…”

A Supersymmetric Hierarchical Model for Weakly Disordered 3d Semimetals

“…See also [10,11] for a recent extension of this result to SUSY ϕ 4 4 , relevant for the study of the weakly self-avoiding walk, and [12] for a detailed discussion of the hierarchical approximation of the model. Hierarchical models have also been considered in the context of random Schrödinger operators [20,53,54,65,73,74]; see also [64,66] for discussions about the connection with the Anderson localization/delocalization transition. There, the model is defined on a one-dimensional lattice, and the range of the hierarchical hopping is tuned to fix the effective dimension of the system.…”

“…Hierarchical models have also been considered in the context of random Schrödinger operators [ 20 , 53 , 54 , 65 , 73 , 74 ]; see also [ 64 , 66 ] for discussions about the connection with the Anderson localization/delocalization transition. There, the model is defined on a one-dimensional lattice, and the range of the hierarchical hopping is tuned to fix the effective dimension of the system.…”

“…There, the model is defined on a one-dimensional lattice, and the range of the hierarchical hopping is tuned to fix the effective dimension of the system. The works [ 53 , 54 , 65 , 73 , 74 ] prove that, as long as the hopping is summable, the model is in the localized phase.…”

A supersymmetric hierarchical model for weakly disordered $3d$ semimetals

“…Hierarchical models have also been considered in the context of random Schrödinger operators [20,64,50,51,71,72], see also [65,63] for discussions about the connection with the Anderson localization/delocalization transition. There, the model is defined on a one-dimensional lattice, and the range of the hierarchical hopping is tuned to fix the effective dimension of the system.…”