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

Naaijkens, Pieter 2015. Kitaev's quantum double model from a local quantum physics point of view. In: Brunetti, Romeo, Dappiaggi, Claudio, Fredenhagen, Klaus and Yngvason, Jakob eds. Advances in Algebraic Quantum Field Theory, Mathematical Physics Studies, pp. 365-395. (10.1007/978-3-319-21353-8_9)

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Abstract

A prominent example of a topologically ordered system is Kitaev’s quantum double model D(G) for finite groups G (which in particular includes G=Z2 , 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 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→Mn(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.

Item Type: Book Section
Status: Published
Schools: Mathematics
ISSN: 0921-3767
Last Modified: 14 Feb 2020 16:45
URI: http://orca.cf.ac.uk/id/eprint/129634

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