Kitaev’s Quantum Double Model from a Local Quantum Physics Point of View

  • Pieter NaaijkensEmail author
Part of the Mathematical Physics Studies book series (MPST)


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 \(\mathfrak {A} \rightarrow M_n(\mathfrak {A})\) 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.


Conjugacy Class Hopf Algebra Fusion Rule Tensor Category Fusion Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The author wishes to thank Courtney Brell for helpful comments and discussions and Leander Fiedler for collaboration on [22]. This work is supported by the Dutch Organisation for Scientific Research (NWO) through a Rubicon grant and partly through the EU project QFTCMPS and the cluster of excellence EXC 201 Quantum Engineering and Space-Time Research


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikLeibniz Universität HannoverHannoverGermany

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