Letters in Mathematical Physics

, Volume 101, Issue 3, pp 341–354

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

Authors

    • Institute for Mathematics, Astrophysics and Particle PhysicsRadboud University Nijmegen
    • Institut für Theoretische PhysikLeibniz Universität Hannover
Open AccessArticle

DOI: 10.1007/s11005-012-0572-7

Cite this article as:
Naaijkens, P. Lett Math Phys (2012) 101: 341. doi:10.1007/s11005-012-0572-7

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

Mathematics Subject Classification (2010)

81R15 (46L60, 81T05, 82B20)

Keywords

Haag duality distal split property toric code

Copyright information

© The Author(s) 2012