Stability of superselection sectors

Matthew Cha, Bruno Nachtergaele and I have just uploaded our paper On the stability of charges in infinite quantum spin systems. This is a project we have been thinking about for quite some time. In it we study the superselection sectors (or charges, excitations) in 2D topologically ordered quantum spin systems. Typically these charges are anyons: interchanging two of them is a non-trivial operation. This is also called braiding. Another piece of structure are the fusion rules. They say how two charges can be decomposed into simple or irreducible charges. It is analogous to the decomposition of the tensor product of two representations of a group into irreducibles. Mathematically, typically the excitations are described by a modular tensor category (MTC).

A natural question is then how one can obtain this modular tensor category from a physical description of the system. That is, given a local Hamiltonian. It turns out that this can be done rigorously if one considers the thermodynamic limit, with infinitely many sites. In that case, it is possible to formulate a variant of the Doplicher-Haag-Roberts theory from algebraic quantum field theory in the quantum spin setting. In this setting the charges can be identified with equivalence classes of irreducible representations. These representations satisfy certain conditions with respect to a reference ground state. Typically, the reference state is a pure translation invariant state. One then considers only those irreducible representations that “look like” the reference representation outside of a cone, and moreover, this localization region can be moved around. The set of these representations can then be further analysed to recover the full modular tensor category. This has been worked out for the toric code, and more generally, abelian quantum double models.

The question we ask ourselves in this paper is: what happens if we perturb the system? Because of its topological nature, the superselection sector structure should not change. At least, as long as we stay in the same gapped ground phase. In our paper we provide a rigorous proof of this statement for abelian quantum double models. As a key tool we develop a superselection theory for almost localized charges. In contrast with the analysis for the unperturbed model, in the perturbed model generally the charges can not be exactly localized (here, in a cone-like region extending to infinity), but only up to an exponentially decaying error term. As a consequence, many of the techniques that work in the unperturbed case break down.

The issue can be explained as follows. We consider the toric code as an example. This model has only one translational invariant ground state, which is pure. Denote this by $\omega_0$. With help of the GNS construction, we get a representation $\pi_0$, which we will use as the reference representation. The quasi-local algebra of observables is written $\mathcal{A}$. Now consider a family of local Hamiltonians $s \mapsto H_\Lambda(s)$, where $s=0$ corresponds to the toric code. We assume the assignment is $C^1$ and that the gap does not close. Then by the spectral flow there is an automorphism $\alpha_s \in {Aut}(\mathcal{A})$ such that $\omega_s = \omega_0 \circ \alpha_s$, where $\omega_s$ is a ground state of the perturbed model (in fact, more generally we can consider weak$*$-limits of states whose energy lies below a gap in the spectrum). Hence the new reference representation is $\pi_0 \circ \alpha_s$.

The problem one has to deal with is that $\alpha_s$ is not strictly local, but rather satisfies a Lieb-Robinson bound. This is an issue because in the DHR theory, we consider representations $\pi$ such that for any cone-like region $\Lambda$, we have the following relation:

Or in words, when restricted to observables outside the cone $\Lambda$, the representations look like the reference representation. The problem is that if we compose both sides with $\alpha_s$, this does not need to be true any more, since $\alpha_s$ generally will not map observables localized into cone regions onto observables in cone regions.

In the unperturbed models one can show that representations satisfying the criterion above are equivalent to $\pi_0 \circ \rho$ with an endomorphism $\rho$ of $\mathcal{A}$ that acts non-trivially only on $\mathcal{A}(\Lambda)$ for some cone $\Lambda$. Motivated by the Lieb-Robinson bound for $\alpha_s$, we relax this condition to allow for endomorphisms that are only approximately localized in a cone. To get the tensor category, we study such endomorphisms by looking at (bi-)asymptopia, a concept introduced by Buchholz, Doplicher, Morchio, Roberts and Strocchi. Ultimately, this can be used to show that the abelian quantum double models are indeed stable against local perturbations that do not close the gap.