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Citation: Naaijkens, Pieter. "Haag duality and the distal split property for cones in the toric code." Letters in Mathematical Physics 101.3 (2012): 341-354.
This paper proves Haag duality for the toric code. This property was first motivated and conjectured in
> Naaijkens, Pieter. "Localized endomorphisms in Kitaev's toric code on the plane." Reviews in Mathematical Physics 23.04 (2011): 347-373.and later generalized to arbitrary finite abelian groups in
> Fiedler, Leander, and Pieter Naaijkens. "Haag duality for Kitaev's quantum double model for abelian groups." Reviews in Mathematical Physics 27.09 (2015): 1550021.
Haag duality is a general "you're not allowed to commute with everybody around you" condition. That is, Haag duality for a subspace of some space says that any operator that commutes with every operator localized on a subspace must itself be localized in the compliment of that subspace.
This paper proves Haag duality for cones. By cone, one literally means cone. We are working with the toric code, which we visualize as the Z^2 lattice. On this lattice one can choose a base point and two rays coming out of it, which will form a literal geometric cone. Every operator which commutes with every operator localized in this cone will be localized in the compliment.
The "distal split" property says that if one cone contains another and their boundaries are sufficiently far apart (sufficiently = they can still be pretty close), then there is a type I subfactor living between the spaces of operators localized in these two cones. This type I subfactor gives a sort of of concrete "+epsilon" separation between the two operator spaces. This paper and its successor seem to have a lot of very interesting mathematics, with some highly non-trivial ingredients at play.