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.

We study the set of infinite volume ground states of Kitaev's quantum double model on Z 2 for an arbitrary finite abelian group G. It is known that these models have a unique frustration-free ground state. Here we drop the requirement of frustration freeness, and classify the full set of ground states. We show that the set of ground states decomposes into |G| 2 different charged sectors, corresponding to the different types of abelian anyons (also known as superselection sectors). In particular, all pure ground states are equivalent to ground states that can be interpreted as describing a single excitation. Our proof proceeds by showing that each ground state can be obtained as the weak * limit of finite volume ground states of the quantum double model with suitable boundary terms. The boundary terms allow for states that represent a pair of excitations, with one excitation in the bulk and one pinned to the boundary, to be included in the ground state space. arXiv:1608.04449v2 [math-ph] 22 Dec 2016 the complete the set of ground states for a given model, and proving that it indeed is the complete set, however, has been solved only in a few cases.In this work, we study quantum double models for abelian groups, in their implementation as quantum spin Hamiltonians with short-range interactions as defined by Kitaev [34]. The simplest example is the toric code model, which corresponds to the choice G = Z 2 . The abelian quantum double model is particularly interesting because it has all of the characteristic features of topologically ordered systems, while at the same time being simple enough to be tackled directly. The main features of the model are: it is exactly solvable in the sense that the Hamiltonian can be explicitly diagonalized; the dimension of the space of ground states of the models defined on a compact orientable surface is a topological invariant and corresponds to the number of flat G-connections on the lattice (up to conjugation); there is a spectral gap above the ground state; the elementary excitations correspond to quasi-particles with braid statistics, see [4] for a rigorous treatment of these results.The focus of this paper is the set of infinite volume ground states of the model for finite abelian groups. Although the quantum double models are exactly solvable in finite volume, much less is known about the thermodynamic limit. The first results in this direction are due to Alicki, Fannes and Hordecki [1]. They showed that in the case G = Z 2 , also known as the toric code, there is a unique frustration-free ground state, which coincides with the translation invariant ground state. This uniqueness property is not general [28], but is related to topological order in the ground state. The difficulty of solving the full ground state problem can be understood as follows. If δ is the derivation generating the dynamics, one has to find all states ω on the quasi-local algebra A of observables that satisfy ω(A * δ(A)) ≥ 0 for all A in the domain of δ. It is possible to construct ground states as weak * ...

Part 1. Preliminaries Chapter 2. Infinite quantum spin systems 2.1. Dynamics 2.2. Ground states 2.3. Gapped ground state phases 2.4. Stability of frustration-free Hamiltonians Chapter 3. Kitaev's quantum double models 3.1. The model 3.2. Topological order 3.3. Infinite system Part 2. Results Chapter 4. The complete set of infinite volume ground states for Kitaev's abelian quantum double models 4.1. Excitations and superselection sectors 4.2. Single excitation ground states 4.3. The complete set of ground states Chapter 5. On the stability of charges in infinite quantum spin systems -ii-5.1. Superselection sectors 5.2. Stability of the superselection structure 5.3. Stability of anyons in Kitaev's abelian quantum double models Bibliography -iii-