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The Quantum Double Model with Boundary: Condensations and Symmetries

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Abstract

Associated to every finite group, Kitaev has defined the quantum double model for every orientable surface without boundary. In this paper, we define boundaries for this model and characterize condensations; that is, we find all quasi-particle excitations (anyons) which disappear when they move to the boundary. We then consider two phases of the quantum double model corresponding to two groups with a domain wall between them, and study the tunneling of anyons from one phase to the other. Using this framework we discuss the necessary and sufficient conditions when two different groups give the same anyon types. As an application we show that in the quantum double model for S 3 (the permutation group over three letters) there is a chargeon and a fluxion which are not distinguishable. This group is indeed a special case of groups of the form of the semidirect product of the additive and multiplicative groups of a finite field, for all of which we prove a similar symmetry.

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Authors and Affiliations

  1. Institute for Quantum Information, California Institute of Technology, MC 305-16, 1200 E. California Blvd., Pasadena, CA, 91125-1600, USA

    Salman Beigi

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran

    Salman Beigi

  3. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Peter W. Shor & Daniel Whalen

Authors
  1. Salman Beigi
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  2. Peter W. Shor
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  3. Daniel Whalen
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Correspondence to Salman Beigi.

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Communicated by Y. Kawahigashi

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Beigi, S., Shor, P.W. & Whalen, D. The Quantum Double Model with Boundary: Condensations and Symmetries. Commun. Math. Phys. 306, 663–694 (2011). https://doi.org/10.1007/s00220-011-1294-x

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  • DOI: https://doi.org/10.1007/s00220-011-1294-x

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