On the stability of charges in infinite quantum spin systems

Abstract: We consider a theory of superselection sectors for infinite quantum spin systems, describing charges that can be approximately localized in cone-like regions. The primary examples we have in mind are the anyons (or charges) in topologically ordered models such as Kitaev's quantum double models and perturbations of such models. In order to cover the case of perturbed quantum double models, the Doplicher-Haag-Roberts approach, in which strict localization is assumed, has to be amended. To this end we consider endomorphisms of the observable algebra that are almost localized in cones. Under natural conditions on the reference ground state (which plays a role analogous to the vacuum state in relativistic theories), we obtain a braided tensor $C^*$-category describing the sectors. We also introduce a superselection criterion selecting excitations with energy below a threshold. When the threshold energy falls in a gap of the spectrum of the ground state, we prove stability of the entire superselection structure under perturbations that do not close the gap. We apply our results to prove that all essential properties of the anyons in Kitaev's abelian quantum double models are stable against perturbations.