Haag duality and the distal split property for cones in the toric code

TL;DR

This paper proves Haag duality and the distal split property for cones in the toric code model, establishing important algebraic structures and their explicit construction in this quantum many-body system.

Contribution

It establishes Haag duality and the distal split property for cones in the toric code, with explicit construction of the Type I factor N.

Findings

Haag duality holds for cones in the toric code

The distal split property is valid for well-separated cones

Explicit construction of the Type I factor N

Abstract

We prove that Haag duality holds for cones in the toric code model. That is, for a cone Lambda, the algebra R_Lambda of observables localized in Lambda and the algebra R_{Lambda^c} of observables localized in the complement Lambda^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 N such that R_{Lambda_1} \subset N \subset R_{Lambda_2}. We demonstrate this by explicitly constructing N.