Equality of bulk and edge Hall conductance revisited

TL;DR

This paper provides a new proof demonstrating the equality of bulk and edge Hall conductance in the quantum Hall effect, emphasizing the stability of an index as a flux tube crosses the boundary.

Contribution

It introduces an alternative proof based on a generalized index of operator pairs, extending previous K-theory approaches.

Findings

Bulk and edge conductances are equal under broad conditions.

The proof relies on the stability of a generalized index during flux tube movement.

The approach offers a new perspective on quantum Hall conductance invariance.

Abstract

The integral quantum Hall effect can be explained either as resulting from bulk or edge currents (or, as it occurs in real samples, as a combination of both). This leads to different definitions of Hall conductance, which agree under appropriate hypotheses, as shown by Schulz-Baldes et al. by means of K-theory. We propose an alternative proof based on a generalization of the index of a pair of projections to more general operators. The equality of conductances is an expression of the stability of that index as a flux tube is moved from within the bulk across the boundary of a sample.