Kitaev's quantum double model from a local quantum physics point of view

• Main Author: Naaijkens, Pieter
• Format: Journal Article
• Language: English
• Published: 28-08-2015
• Subjects: Mathematics - Mathematical Physics; Physics - Mathematical Physics; Physics - Quantum Physics
• DOI: 10.48550/arxiv.1508.07170

Advances in Algebraic Quantum Field Theory, pp 365-395 (Springer
2015) A prominent example of a topologically ordered system is Kitaev's quantum
double model $\mathcal{D}(G)$ for finite groups $G$ (which in particular
includes $G = \mathbb{Z}_2$, the toric code). We will look at these models from
the point of view of local quantum physics. In particular, we will review how
in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the
different superselection sectors of the model. In this way one finds that the
charges are in one-to-one correspondence with the representations of
$\mathcal{D}(G)$, and that they are in fact anyons. Interchanging two of such
anyons gives a non-trivial phase, not just a possible sign change. The case of
non-abelian groups $G$ is more complicated. We outline how one could use
amplimorphisms, that is, morphisms $A \to M_n(A)$ to study the superselection
structure in that case. Finally, we give a brief overview of applications of
topologically ordered systems to the field of quantum computation.