Current and charge distributions of the fractional quantum Hall liquids with edges

TL;DR

This paper develops an effective Chern-Simons theory to analyze charge and current distributions in fractional quantum Hall liquids with edges, revealing a single non-localized mode and resolving edge singularity issues.

Contribution

It introduces a unified approach to study edge and bulk properties in fractional quantum Hall states and provides exact solutions demonstrating non-localized edge modes.

Findings

Charge and current distributions lack diverging singularities.

Only a single non-localized mode exists even in hierarchical states.

Edge modes are not exponentially localized near edges.

Abstract

An effective Chern-Simons theory for the quantum Hall states with edges is studied by treating the edge and bulk properties in a unified fashion. An exact steady-state solution is obtained for a half-plane geometry using the Wiener-Hopf method. For a Hall bar with finite width, it is proved that the charge and current distributions do not have a diverging singularity. It is shown that there exists only a single mode even for the hierarchical states, and the mode is not localized exponentially near the edges. Thus this result differs from the edge picture in which electrons are treated as strictly one dimensional chiral Luttinger liquids.