  We examine tunneling of topological charge between non-Abelian anyons as a
perturbation of the long-range effective theory of a topologically ordered
system. We obtain energy corrections in terms of the anyons' universal
algebraic structure and non-universal tunneling amplitudes. We find that
generic tunneling completely lifts the topological degeneracy of non-Abelian
anyons. This degeneracy splitting is exponentially suppressed for long
distances between anyons, but leaves no topological protection at shorter
distances. We also show that general interactions of anyons can be expressed in
terms of the transfer of topological charge, and thus can be treated
effectively as tunneling interactions.

Quantum Physics;Mesoscale and Nanoscale Physics;Strongly Correlated Electrons