arXiv:1406.1084 · Reviews in Mathematical Physics

Haag duality for Kitaev’s quantum double model for abelian groups

Leander Fiedler, Pieter Naaijkens

Published Oct 1, 2015• 34 citations

Abstract

We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.

DOI: 10.1142/s0129055x1550021x

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arXiv:1406.1084 · Reviews in Mathematical Physics

Haag duality for Kitaev’s quantum double model for abelian groups

Leander Fiedler, Pieter Naaijkens

Published Oct 1, 2015• 34 citations

arXiv HTML PDF

Abstract

DOI: 10.1142/s0129055x1550021x

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We prove that Haag duality holds for cones in the toric code model. That is, for a cone Λ, the algebra

{\mathcal{R}_{\Lambda}}

of observables localized in Λ and the algebra

{\mathcal{R}_{\Lambda^c}}

of observables localized in the complement Λ c generate each other’s commutant as von Neumann algebras. Moreover, we show that the distal split property holds: if

{\Lambda_1 \subset \Lambda_2}

are two cones whose boundaries are well separated, there is a Type I factor

{\mathcal{N}}

such that

{\mathcal{R}_{\Lambda_1} \subset \mathcal{N} \subset \mathcal{R}_{\Lambda_2}}

. We demonstrate this by explicitly constructing

{\mathcal{N}}

\textbf{Rep}_f {{\mathcal {D}}}(G)

Rep f D ( G ) , the representation category of the quantum double of G . This establishes for the first time the full DHR structure for a class of 2d quantum lattice models with non-abelian anyons.

G

. We identify the boundary nets of algebras with fusion categorical nets associated to

(\mathsf{Hilb}(G),\mathbb{C}[G])

(\mathsf{Rep}(G),\mathbb{C}^G)

depending on whether the boundary cut is rough or smooth respectively. This allows us to make connections to work of Ogata on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial

G

-symmetry protected topological phase (

G

-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3+1)D Quantum Double model associated to an abelian group.