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"Discrete non-Abelian gauge theories in Josephson-junction arrays and quantum computation", Benoit Doucot, Lev Ioffe, Julien Vidal, 2004

Reviewed October 6, 2023

Citation: Douçot, Benoit, Lev B. Ioffe, and Julien Vidal. "Discrete non-Abelian gauge theories in Josephson-junction arrays and quantum computation." Physical Review B 69.21 (2004): 214501.

Web: https://arxiv.org/abs/cond-mat/0302104

Tags: Pedagogical, Non-abelian-anyons

This paper is a very(early) take on topological quantum computation in the language of gauge theory, which is more grounded than its contemporaries. It considers explicit lattice models with Josephson-junction arrays which are equivalent to discrete non-Abelian gauge theories. These situations host non-abelian anyons which are well described. This paper includes a lovely review of relevant material, and lots of pretty pictures.

Another nice feature of this paper is the surrounding discussion. For instance, in the conclusion the authors raise a very interesting question: "It is not clear whether it is absolutely unavoidable to emulate gauge theory in order to get the topological protection and non-Abelian statistics. What is the most general class of theories that has these properties?". Perhaps one answer to this question is the existence of the Levin-Wen model, based on arbitrary spherical fusion categories. Group-theoretical spherical fusion categories (i.e. Vec_G for some group G) correspond to models emulating gauge theories, but arbitrarily fusion categories don't have such naturally associated gauge groups. Some MTCs come from gauge theories in more subtle ways, like quantum group MTCs. There are MTCs which don't come from quantum groups however, and instead come from more exotic vertex operator algebras.

Another question raised is whether or not Kosterlitz-Thouless type transitions appear in these non-abelian lattice settings. I have no idea. An interesting reference is a paper in which the Kosterlitz-Thouless transition is demonstrated very nicely theoretically in a lattice based model:

> Park, Kyungwha, and David A. Huse. "Superconducting phase with fractional vortices in the frustrated kagom é wire network at f= 1/2." Physical Review B 64.13 (2001): 134522.

Another reason to cite this paper: It is an early example of experimentally-minded people understanding that the dihedral group D_4 is the simplest non-abelian group, not S_3. Their non-abelian models focus entirely on the dihedral groups D_4 and D_5.