Letters in Mathematical Physics

, Volume 101, Issue 3, pp 341–354 | Cite as

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

  • Pieter NaaijkensEmail author
Open Access


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) 


Haag duality distal split property toric code 



This research is funded by the Netherlands Organisation for Scientific Research (NWO) grant no. 613.000.608. The author wishes to thank Klaas Landsman, Michael Müger and Reinhard Werner for valuable feedback on the manuscript.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


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Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Institute for Mathematics, Astrophysics and Particle PhysicsRadboud University NijmegenNijmegenThe Netherlands
  2. 2.Institut für Theoretische PhysikLeibniz Universität HannoverHannoverGermany

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