Haag Duality and the Distal Split Property for Cones in the Toric Code

Abstract

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 ${\mathcal{N}}$ .