Toward a theory of the integer quantum Hall transition: continuum limit of the Chalker-Coddington model

TL;DR

This paper generalizes the Chalker-Coddington network model to N channels, transforming it into a supersymmetric sigma model, and analyzes the critical behavior of the integer quantum Hall transition in the continuum limit.

Contribution

It introduces an exact transformation of the N-channel network model into a lattice field theory, providing a new continuum limit description of the quantum Hall transition.

Findings

The continuum limit yields a supersymmetric Pruisken sigma model with specific couplings.

The N=2 case is shown to be noncritical, indicating no phase transition.

A modified network model is proposed that remains in the quantum Hall universality class.

Abstract

An N-channel generalization of the network model of Chalker and Coddington is considered. The model for N = 1 is known to describe the critical behavior at the plateau transition in systems exhibiting the integer quantum Hall effect. Using a recently discovered equality of integrals, the network model is transformed into a lattice field theory defined over Efetov's sigma model space with unitary symmetry. The transformation is exact for all N, no saddle-point approximation is made, and no massive modes have to be eliminated. The naive continuum limit of the lattice theory is shown to be a supersymmetric version of Pruisken's nonlinear sigma model with couplings sigma_xx = sigma_xy = N/2 at the symmetric point. It follows that the model for N = 2, which describes a spin degenerate Landau level and the random flux problem, is noncritical. On the basis of symmetry considerations and inspection of the Hamiltonian limit, a modified network model is formulated, which still lies in the quantum Hall universality class. The prospects for deformation to a Yang-Baxter integrable vertex model are briefly discussed.