, Volume 101, Issue 3, pp 341-354,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 17 Jun 2012

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


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 \({\mathcal{N}}\) .