Is Hall Conductance in Hall Bar Geometry a Topological Invariant?

TL;DR

This paper explores the topological nature of Hall conductance in realistic geometries, showing it is invariant under finite size effects, unlike some related quantum field theory coefficients.

Contribution

It demonstrates that Hall conductance remains a topological invariant without finite size correction in realistic geometries, contrasting with finite corrections in related quantum field coefficients.

Findings

Hall conductance has no finite size correction in quantum Hall regime.

Coefficient of induced Chern-Simons term shows small finite size correction.

Hall conductance is confirmed as a topological invariant in realistic conditions.

Abstract

A deep connection between the Hall conductance in realistic situation and a topological invariant is pointed out based on von-Neumann lattice representation in which Landau level electrons have minimum spatial extensions. We show that the Hall conductance has no finite size correction in quantum Hall regime, but a coefficient of induced Chern-Simons term in QED$_3$ has a small finite size correction, although both of them are similar topological invariant.