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Communications in Mathematical Physics

, Volume 373, Issue 1, pp 219–264 | Cite as

On the Stability of Charges in Infinite Quantum Spin Systems

  • Matthew Cha
  • Pieter NaaijkensEmail author
  • Bruno Nachtergaele
Article
  • 60 Downloads

Abstract

We consider a theory of superselection sectors for infinite quantum spin systems, describing charges that can be approximately localized in cone-like regions. The primary examples we have in mind are the anyons (or charges) in topologically ordered models such as Kitaev’s quantum double models, and perturbations of such models. In order to cover the case of perturbed quantum double models, the Doplicher–Haag–Roberts approach, in which strict localization is assumed, has to be amended. To this end we consider endomorphisms of the observable algebra that are almost localized in cones. Under natural conditions on the reference ground state (which plays a role analogous to the vacuum state in relativistic theories), we obtain a braided tensor \(C^*\)-category describing the sectors. We also introduce a superselection criterion selecting excitations with energy below a threshold. When the threshold energy falls in a gap of the spectrum of the ground state, we prove stability of the entire superselection structure under perturbations that do not close the gap. We apply our results to prove that all essential properties of the anyons in Kitaev’s abelian quantum double models are stable against perturbations.

Notes

Acknowledgements

MC would like to thank the University of Tokyo, for support and hospitality during the summer of 2015 where part of the work on this paper was done. PN thanks Courtney Brell for helpful conversations and suggestions, and Luca Giorgetti and Michael Müger for discussions on “Appendix C”. MC was supported in part by the National Science Foundation under Grant OISE-1515557 and the Japan Society for the Promotion of Science Summer Program 2015. PN has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 657004 and the European Research Council (ERC) Consolidator Grant GAPS (No. 648913). BN and MC were supported in part by the National Science Foundation under Grant DMS-1515850.

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Department of MathematicsUniversity of CaliforniaDavisUSA
  3. 3.JARA Institute for Quantum ComputingRWTH Aachen UniversityAachenGermany
  4. 4.Departamento de Análisis Matemático y Matemática AplicadaUniversidad Complutense de MadridMadridSpain

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