# Haag duality for Kitaev's quantum double model for abelian groups

June 04, 2014

We prove Haag duality for conelike regions in the ground state representation
of Kitaev's quantum double model for finite abelian groups. This property says
that if an observable commutes with all observables localised outside the cone
region, it actually is an element of the von Neumann algebra generated by the
local observables inside the cone. This strengthens locality, which says that
observables localised in disjoint regions commute.
As an application we consider the superselection structure of the quantum
double model for abelian groups on an infinite lattice in the spirit of the
Doplicher-Haag-Roberts program in algebraic quantum field theory. We find that,
as is the case for the toric code model on an infinite lattice, the
superselection structure is given by the category of irreducible
representations of the quantum double.

Keywords:

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