Simultaneous quantization of edge and bulk Hall conductivity

TL;DR

This paper demonstrates the quantization of both edge and bulk Hall conductivities in disordered systems, establishing their equality through K-theoretic methods and discussing experimental implications.

Contribution

It provides a rigorous proof of the simultaneous quantization of edge and bulk Hall conductivities and their equality in disordered quantum Hall systems.

Findings

Edge Hall conductivity is an integer multiple of e^2/h.

Edge and bulk Hall conductivities are equal due to K-theoretic arguments.

In experiments, only a small fraction of current is carried by edge states.

Abstract

The edge Hall conductivity is shown to be an integer multiple of $e^2/h$ which is almost surely independent of the choice of the disordered configuration. Its equality to the bulk Hall conductivity given by the Kubo-Chern formula follows from K-theoretic arguments. This leads to quantization of the Hall conductance for any redistribution of the current in the sample. It is argued that in experiments at most a few percent of the total current can be carried by edge states.