Multi-Channel Luttinger Liquids at the Edge of Quantum Hall Systems

TL;DR

This paper rigorously analyzes the edge transport in quantum Hall systems, showing that their large-scale behavior is described by a multi-channel Luttinger model, with universal conductance determined by edge mode chiralities.

Contribution

It provides a rigorous proof that the edge conductance in quantum Hall systems is universal and equals the sum of chiralities, using renormalization group methods and lattice Ward identities.

Findings

Edge conductance is universal and equals the sum of chiralities.

Large-scale edge correlations are described by a multi-channel Luttinger model.

Backscattering effects are fully incorporated in the analysis.

Abstract

We consider the edge transport properties of a generic class of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the large-scale behavior of the edge correlation functions is effectively described by the multi-channel Luttinger model. In particular, we prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. The proof is based on rigorous renormalization group methods, that allow to fully take into account the effect of backscattering at the edge. Universality arises as a consequence of the integrability of the emergent multi-channel Luttinger liquid combined with lattice Ward identities for the microscopic $2d$ theory.