Exact results for anyons on a torus

Abstract

We analyse the quantum mechanics of a system of non-relativistic anyons on a torus, the statistics of which is described by an abelian Chern-Simons field theory with an arbitrary rational coupling constant. We construct the hamiltonian and the momentum operator, and show that by properly taking into account the topological components of the field they satisfy usual commutation relations. The structure of the Hilbert space for the general case is then described and we determine also exact eigenfunctions in some cases. By introducing a constant external magnetic field orthogonal to the surface and an electric field tangent to it, we are able to study the Hall effect for the anyons and their resulting motion on the torus, and we derive the exact eigenfunctions for the ground state at a given momentum for those configurations relevant to the fractional Hall effect hierarchy.