Quantum Network Models and Classical Localization Problems

TL;DR

This paper reviews quantum network models related to the spin quantum Hall transition and explores their connection to classical localization problems via mappings to classical random walks and percolation theory.

Contribution

It establishes a link between quantum network models and classical localization, providing rigorous insights into classical localization through percolation in various dimensions.

Findings

Quantum network models describe the spin quantum Hall transition.

Many observables can be mapped to classical random walks in random environments.

Rigorous results are obtained for classical localization via percolation theory.

Abstract

A review is given of quantum network models in class C which, on a suitable 2d lattice, describe the spin quantum Hall plateau transition. On a general class of graphs, however, many observables of such models can be mapped to those of a classical walk in a random environment, thus relating questions of quantum and classical localization. In many cases it is possible to make rigorous statements about the latter through the relation to associated percolation problems, in both two and three dimensions.