Edge and Bulk of the Fractional Quantum Hall Liquids

TL;DR

This paper develops a unified Chern-Simons theory for Abelian quantum Hall states, deriving edge modes and quantized Hall conductance under boundary conditions that ensure gauge invariance and current confinement.

Contribution

It introduces a boundary condition-based effective theory that unifies edge and bulk properties and establishes a criterion for the coupling matrix in quantum Hall liquids.

Findings

Hall conductance is always quantized.

Edge modes are derived from gauge invariance.

The coupling matrix condition relates to filling factors.

Abstract

An effective Chern-Simons theory for the Abelian quantum Hall states with edges is proposed to study the edge and bulk properties in a unified fashion. We impose a condition that the currents do not flow outside the sample. With this boundary condition, the action remains gauge invariant and the edge modes are naturally derived. We find that the integer coupling matrix $K$ should satisfy the condition $\sum_I(K^{-1})_{IJ} = \nu/m$ ($\nu$: filling of Landau levels, $m$: the number of gauge fields ) for the quantum Hall liquids. Then the Hall conductance is always quantized irrespective of the detailed dynamics or the randomness at the edge.