Interacting Quantum Topologies and the Quantum Hall Effect

TL;DR

This paper presents a gauge covariant formulation of the integer quantum Hall effect using noncommutative geometry, revealing twisted statistics and interactions between matter and gauge fields based on Moyal algebras.

Contribution

It introduces a novel noncommutative geometric framework for IQHE, incorporating twisted coproducts and statistics, and models interactions via algebras with different deformation parameters.

Findings

Gauge covariant formulation of IQHE using A_theta(R^2)

Identification of twisted statistics in many-particle sectors

Explicit construction of twisted two-particle Laughlin wave functions

Abstract

The algebra of observables of planar electrons subject to a constant background magnetic field B is given by A_theta(R^2) x A_theta(R^2) the product of two mutually commuting Moyal algebras. It describes the free Hamiltonian and the guiding centre coordinates. We argue that A_theta(R^2) itself furnishes a representation space for the actions of these two Moyal algebras, and suggest physical arguments for this choice of the representation space. We give the proper setup to couple the matter fields based on A_theta(R^2) to electromagnetic fields which are described by the abelian commutative gauge group G_c(U(1)), i.e. gauge fields based on A_0(R^2). This enables us to give a manifestly gauge covariant formulation of integer quantum Hall effect (IQHE). Thus, we can view IQHE as an elementary example of interacting quantum topologies, where matter and gauge fields based on algebras A_theta^prime with different theta^prime appear. Two-particle wave functions in this approach are based on A_theta(R^2) x A_theta(R^2). We find that the full symmetry group in IQHE, which is the semi-direct product SO(2) \ltimes G_c(U(1)) acts on this tensor product using the twisted coproduct Delta_theta. Consequently, as we show, many particle sectors of each Landau level have twisted statistics. As an example, we find the twisted two particle Laughlin wave functions.