Chiral Anomaly and Effective Field Theory for the Quantum Hall Liquid with Edges

TL;DR

This paper develops an effective field theory for quantum Hall liquids with edges, clarifying the role of chiral edge currents and their relation to the Hall conductance, especially in topologically non-trivial surfaces.

Contribution

It introduces a gauge-invariant effective action incorporating edge effects and distinguishes the roles of edge currents and Hall currents in quantized conductance.

Findings

Chiral edge current is necessary for gauge invariance.

Hall current is irrelevant to the chiral edge current.

Quantized Hall conductance occurs only when Hall current does not flow at the edge.

Abstract

Under general assumptions, we present a low-energy effective action for the quantum Hall state when edges exist. It is shown that the chiral edge current is necessary to make the effective action to be gauge invariant. However the chiral edge current is irrelevant to the Hall current. The exactly quantized value of $\sigma_{xy}$ is observed only when the Hall current does not flow at the edge region. Our effective theory is applicable to the quantum Hall liquid on a surface with non-trivial topology and physical meanings of the topology are discussed.