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 R Λ of observables localized in Λ and the algebra R Λc 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 Λ 1⊂Λ 2 are two cones whose boundaries are well separated, there is a Type I factor N such that R Λ1 ⊂N ⊂ R Λ2. We demonstrate this by explicitly constructing N. © 2012 The Author(s).