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

Abstract

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.

Chapter contributed to R. Brunetti, C. Dappiaggi, K. Fredenhagen, J. Yngvason (eds), Advances in Algebraic Quantum Field Theory (Springer 2015). Mainly review

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Cited by in corpus (9)

• The complete set of infinite volume ground states for Kitaev's abelian quantum double models
• The split and approximate split property in 2D systems: stability and absence of superselection sectors
• Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons
• The category of anyon sectors for non-abelian quantum double models
• Classification of the anyon sectors of Kitaev's quantum double model
• Continuous Dependence on the Initial Data in the Kadison Transitivity Theorem and GNS Construction
• Oriented Closed Polyhedral Maps and the Kitaev Model
• $C^{\ast }$-categorical prefactorization algebras for superselection sectors and topological order
• Quantum information theory and Fourier multipliers on quantum groups