Particle-vortex dynamics in noncommutative space

TL;DR

This paper investigates the quantum behavior of a charged particle in a magnetic field and vortex within noncommutative space, revealing how noncommutativity alters eigenstates and connects to anyon physics.

Contribution

It introduces two noncommutative Hamiltonian extensions and constructs their eigenfunctions, showing how noncommutativity modifies the spectrum compared to ordinary space.

Findings

Eigenfunctions and eigenvalues are derived for both noncommutative extensions.

Half of the states in noncommutative space correspond to those in ordinary space.

The results connect to the physics of anyons in the limit as noncommutativity vanishes.

Abstract

We study the problem of a charged particle in the presence of a uniform magnetic field plus a vortex in noncommutative planar space considering the two possible non-commutative extensions of the corresponding Hamiltonian, namely the ``fundamental'' and the ``antifundamental'' representations. Using a Fock space formalism we construct eigenfunctions and eigenvalues finding in each case half of the states existing in the ordinary space case. In the limit of $\theta \to 0$ we recover the two classes of states found in ordinary space, relevant for the study of anyon physics.