Topology and Fractional Quantum Hall Effect

TL;DR

This paper links the fractional quantum Hall effect to topological vector bundles, showing how the filling factors relate to topological invariants and demonstrating stability and boundary condition independence in the large particle limit.

Contribution

It explicitly constructs vector bundles from Laughlin wave functions and proves their topological properties relate to observed fractional filling factors.

Findings

Filling factor $ u$ equals the ratio of the first Chern number to the bundle rank

Stable vector bundles correspond to experimentally observed fractional fillings

Curvature fluctuations vanish as particle number approaches infinity

Abstract

Starting from Laughlin type wave functions with generalized periodic boundary conditions describing the degenerate groundstate of a quantum Hall system we explictly construct $r$ dimensional vector bundles. It turns out that the filling factor $\nu$ is given by the topological quantity $c_1 \over r$ where $c_1$ is the first chern number of these vector bundles. In addition, we managed to proof that under physical natural assumptions the stable vector bundles correspond to the experimentally dominating series of measured fractional filling factors $\nu = {n \over 2pn\pm 1}$. Most remarkably, due to the very special form of the Laughlin wave functions the fluctuations of the curvature of these vector bundles converge to zero in the limit of infinitely many particles which shows a new mathematical property. Physically, this means that in this limit the Hall conductivity is independent of the boundary conditions which is very important for the observabilty of the effect. Finally we discuss the relation of this result to a theorem of Donaldson.