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Topological Eigenfunctions in Two Dimensions

C. Baxter and Rodney Loudon

Abstract

An ion moving in a constant magnetic field is considered as the generic example of two-dimensional topological (Chern-Simons) theory. The eigenfunctions for the generic particle are determined by solving Schrodinger’s equation in the secular and non-secular forms of the theory, corresponding to the inclusion and non-inclusion respectively in the Lagrangian of the particle’s kinetic energy. A notable feature is the self-consistency of the formalism in the limit of the non-secular theory: the eigenfunction remains well-behaved and singlevalued, and the eigenenergy does not diverge. The non-secular eigenfunction is consistent with a half-integral angular momentum spectrum. It is shown that the non-secular regime is unlikely to be achieved by a naive cooling of the ion. A suggestion is made that the non-secular regime might be experimentally accessible through Laguerre-Gaussian optics.