Towards a theory of the integer quantum Hall transition: from the nonlinear sigma model to superspin chains

TL;DR

This paper develops a supersymmetric lattice model for the integer quantum Hall transition, linking nonlinear sigma models to superspin chains, and explores the critical behavior and conformal invariance at the transition point.

Contribution

It introduces a superspin chain Hamiltonian derived from the nonlinear sigma model, providing a new framework to analyze the quantum Hall transition and its critical properties.

Findings

The superspin chain resembles an isotropic antiferromagnet with spin 1/2.

The model relates conductance coefficients to superspin correlation functions.

The approach suggests the ratio of magnetic length to potential correlation length is irrelevant at the transition.

Abstract

A careful study of the supersymmetric version of Pruisken's nonlinear sigma model for the integer quantum Hall effect is presented. The lattice regularized model is cast in Hamiltonian form by taking the anisotropic limit and interpreting the topological density as an alternating sum of Wess-Zumino terms. It is argued that the relevant large-scale physics of the model is preserved by projection of the quantum Hamiltonian on its sector of degenerate strong-coupling ground states. For values of the Hall conductivity close to e^2/2h (mod e^2/h), where a delocalization transition occurs, this yields the Hamiltonian of a quantum superspin chain which is closely related to an anisotropic version of the Chalker-Coddington model. The relation implies that the ratio of magnetic length over potential correlation length is an irrelevant parameter at the transition. The superspin chain resembles a 1d isotropic antiferromagnet with spin 1/2. It has an alternating structure which however permits an invariance under translation by one site. The conductance coefficients of a quantum Hall system with N small contacts translate into N-superspin correlation functions which are governed by conformal invariance. The superspin formalism provides a framework for studying the crossover from classical to quantum percolation. It does not however encompass the frequency-dependent correlations of wave amplitudes at criticality.