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The Complete Set of Infinite Volume Ground States for Kitaev’s Abelian Quantum Double Models

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Abstract

We study the set of infinite volume ground states of Kitaev’s quantum double model on \({\mathbb{Z}^2}\) for an arbitrary finite abelian group G. It is known that these models have a unique frustration-free ground state. Here we drop the requirement of frustration freeness, and classify the full set of ground states. We show that the set of ground states decomposes into \({|G|^2}\) different charged sectors, corresponding to the different types of abelian anyons (also known as superselection sectors). In particular, all pure ground states are equivalent to ground states that can be interpreted as describing a single excitation. Our proof proceeds by showing that each ground state can be obtained as the weak* limit of finite volume ground states of the quantum double model with suitable boundary terms. The boundary terms allow for states that represent a pair of excitations, with one excitation in the bulk and one pinned to the boundary, to be included in the ground state space.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, CA, 95616, USA

    Matthew Cha, Pieter Naaijkens & Bruno Nachtergaele

  2. JARA Institute for Quantum Information, RWTH Aachen University, Otto-Blumenthal-Straße 20, 52074, Aachen, Germany

    Pieter Naaijkens

Authors
  1. Matthew Cha
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  2. Pieter Naaijkens
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  3. Bruno Nachtergaele
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Correspondence to Pieter Naaijkens.

Additional information

Communicated by D. Buchholz, K. Fredenhagen, Y. Kawahigashi

Dedicated to the memory of Rudolf Haag

2017 by the authors. This article may be reproduced in its entirety for non-commercial purposes.

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Cha, M., Naaijkens, P. & Nachtergaele, B. The Complete Set of Infinite Volume Ground States for Kitaev’s Abelian Quantum Double Models. Commun. Math. Phys. 357, 125–157 (2018). https://doi.org/10.1007/s00220-017-2989-4

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