Haag duality in Kitaev's quantum double models

Our paper on Haag duality in quantum double models has been published in Reviews in Mathematical Physics. This is the first paper of my co-author Leander Fiedler, a PhD student in Hannover with whom I have worked closely together while I was a postdoc there, so this is very exciting. Our paper generalises results first obtained for the toric code (where $G = \mathbb{Z}_2$) to arbitrary abelian groups $G$. In particular, we prove that Haag duality holds in the translational invariant ground state and uncover the superselection structure of the model.

So what exactly is Haag duality? We consider Kitaev’s quantum double model in two dimensions in the thermodynamic limit. For simplicity, we assume that we have a square lattice, with at each edge between neighbouring vertices a $\vert G\vert$-dimensional Hilbert space. This leads in a natural way to a $C^*$-algebra $\mathfrak{A}$ of observables, called the quasi-local algebra.¹ This is very general. The physical input comes from specifying dynamics for the system in terms of local Hamiltonians, in our case those first introduced by Kitaev.² The precise form is not important here: the main point is that this allows us to select ground states for these dynamics. In the present case, there are many ground states, but for our purposes there is a preferred one: there is a unique translational invariant ground state (which is pure).³

Once we have selected a state $\omega_0$ on $\mathfrak{A}$ we can obtain a representation $\pi_0$ on a Hilbert space $\mathcal{H}$, such that the state $\omega_0$ is implemented by a vector $\Omega \in \mathcal{H}$. This can be done via the GNS construction. Now consider a “cone region” of the system: take a point in the lattice and two semi-infinite lines emanating from them. We say that, roughly speaking, the sites within the two lines form a cone $\Lambda$.⁴ Associated to the cone is the $C^*$-algebra $\mathfrak{A}(\Lambda)$ of quasi-local observables localised in that cone. The algebra of all quasi-local observables outside the cone is denoted by $\mathfrak{A}(\Lambda)$. We can then define two new algebras:

where the prime means taking the commutant in $\mathfrak{B}(\mathcal{H})$. By taking the double commutant we obtain von Neumann algebras. It should be noted that these algebras strongly depend on the choice of representation $\pi_0$. By locality, the property that observables localised in disjoint regions commute, it is easy to see that $\mathcal{R}_\Lambda \subset \mathcal{R}_{\Lambda^c}'$. Haag duality then is the claim that the reverse inclusion also holds.

The proof relies on a good understanding of how the Hilbert space can be build by acting with local observables on the ground state vector $\Omega$. It turns out that we can label a convenient orthonormal basis by looking at “charge configurations” of anyons. These configurations can be obtained by acting with local operators associated to paths on the lattice on the ground state vector. A good understanding of the commutation properties of these operators is essential. In this case, these relations have a nice algebraic structure stemming from the representation theory of Hopf algebras. It allows us to look at the subspace of all charge configurations in the cone $\Lambda$ (and trivial charge outside), and see how $\mathcal{R}_\Lambda$ and $\mathcal{R}_{\Lambda^c}'$ act when restricted to this subspace. By proving a density result, it is possible to show that they are each others commutants. This result then extends to the whole Hilbert space.

Haag duality is mainly used as a technical tool in the study of the superselection sectors of the model. For a short introduction of the main ideas behind this construction, which originates from algebraic quantum field theory, see here (or the arXiv version). In essence, it allows us to replace representations of $\mathfrak{A}$, which satisfy a physically reasonable selection criterion,⁵ by endomorphisms of $\mathfrak{A}$. These endomorphisms have a much richer structure than the representations. For example, two endomorphisms can be composed. Physically speaking this is the same as “fusion”: bringing two charges together and see how they combine.⁶ By studying these endomorphisms we can uncover all the relevant physical properties of the anyonic charges in the model. As expected, they are described by the representation theory of the quantum double of the group $G$.

Footnotes