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Haag duality for Kitaev’s quantum double model for abelian groups

    We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute.

    As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.

    AMSC: 81R15, 46L60, 81T05, 82B20


    • 1. A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303 (2003) 2–30. Crossref, ISI, ADSGoogle Scholar
    • 2. C. Mochon, Anyons from nonsolvable finite groups are sufficient for universal quantum computation, Phys. Rev. A 67 (2003) 022315. Crossref, ISI, ADSGoogle Scholar
    • 3. C. Mochon, Anyon computers with smaller groups, Phys. Rev. A 69 (2004) 032306. Crossref, ISI, ADSGoogle Scholar
    • 4. S. Doplicher, R. Haag and J. E. Roberts, Local observables and particle statistics. I, Comm. Math. Phys. 23 (1971) 199–230. Crossref, ISI, ADSGoogle Scholar
    • 5. S. Doplicher, R. Haag and J. E. Roberts, Local observables and particle statistics. II, Comm. Math. Phys. 35 (1974) 49–85. Crossref, ISI, ADSGoogle Scholar
    • 6. P. Naaijkens, Localized endomorphisms in Kitaev’s toric code on the plane, Rev. Math. Phys. 23 (2011) 347–373. Link, ISI, ADSGoogle Scholar
    • 7. P. Naaijkens, Haag duality and the distal split property for cones in the toric code, Lett. Math. Phys. 101 (2012) 341–354. Crossref, ISI, ADSGoogle Scholar
    • 8. M. A. Rieffel and A. van Daele, The commutation theorem for tensor products of von Neumann algebras, Bull. London Math. Soc. 7 (1975) 257–260. CrossrefGoogle Scholar
    • 9. S. Doplicher and R. Longo, Standard and split inclusions of von Neumann algebras, Invent. Math. 75 (1984) 493–536. Crossref, ISI, ADSGoogle Scholar
    • 10. S. J. Summers and R. Werner, Maximal violation of Bell’s inequalities for algebras of observables in tangent spacetime regions, Ann. Inst. H. Poincaré Phys. Théor. 49 (1988) 215–243. ISIGoogle Scholar
    • 11. D. Buchholz and K. Fredenhagen, Locality and the structure of particle states, Comm. Math. Phys. 84 (1982) 1–54. Crossref, ISI, ADSGoogle Scholar
    • 12. H. Bombin and M. A. Martin-Delgado, Family of non-Abelian Kitaev models on a lattice: Topological condensation and confinement, Phys. Rev. B 78 (2008) 115421. Crossref, ISI, ADSGoogle Scholar
    • 13. V. G. Drinfel’d, Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986) (Amer. Math. Soc., Providence, RI, 1987), pp. 798–820. Google Scholar
    • 14. K. Szlachányi and P. Vecsernyés, Quantum symmetry and braid group statistics in G-spin models, Comm. Math. Phys. 156 (1993) 127–168. Crossref, ISI, ADSGoogle Scholar
    • 15. C. Kassel, Quantum Groups, Graduate Texts in Mathematics, Vol. 155 (Springer-Verlag, New York, 1995). CrossrefGoogle Scholar
    • 16. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2, 2nd edn., Texts and Monographs in Physics (Springer-Verlag, Berlin, 1997). CrossrefGoogle Scholar
    • 17. R. Dijkgraaf, V. Pasquier and P. Roche, Quasi Hopf algebras, group cohomology and orbifold models, Nuclear Phys. B Proc. Suppl. 18B (1991) 60–72; recent advances in field theory (Annecy-le-Vieux, 1990). Google Scholar
    • 18. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. II: Structure and Analysis on Locally Compact Groups; Analysis on Locally Compact Abelian Groups (Springer, 1970). Google Scholar
    • 19. R. Alicki, M. Fannes and M. Horodecki, A statistical mechanics view on Kitaev’s proposal for quantum memories, J. Phys. A 40 (2007) 6451–6467. Crossref, ADSGoogle Scholar
    • 20. P. Naaijkens, Anyons in infinite quantum systems: QFT in d = 2 + 1 and the toric code, Ph.D. thesis, Radboud Universiteit Nijmegen (2012). Google Scholar
    • 21. R. Oeckl, Discrete Gauge Theory: From Lattices to TQFT (Imperial College Press, London, 2005). LinkGoogle Scholar
    • 22. J. J. Bisognano and E. H. Wichmann, On the duality condition for quantum fields, J. Math. Phys. 17 (1976) 303–321. Crossref, ISI, ADSGoogle Scholar
    • 23. D. Buchholz, G. Mack and I. Todorov, Localized automorphisms of the U(1)-current algebra on the circle: an instructive example, in The Algebraic Theory of Superselection Sectors (Palermo, 1989) (World Sci. Publ., River Edge, NJ, 1990), pp. 356–378. Google Scholar
    • 24. M. Takesaki, Theory of Operator Algebra I, Encyclopaedia of Mathematical Sciences, Vol. 124 (Springer Berlin/Heidelberg, 2002). Google Scholar
    • 25. D. Buchholz, Product states for local algebras, Comm. Math. Phys. 36 (1974) 287–304. Crossref, ISI, ADSGoogle Scholar
    • 26. D. Buchholz, S. Doplicher and R. Longo, On Noether’s theorem in quantum field theory, Ann. Phys. 170 (1986) 1–17. Crossref, ISI, ADSGoogle Scholar
    • 27. P. Naaijkens, Kosaki-Longo index and classification of charges in 2D quantum spin models, J. Math. Phys. 54 (2013) 081901. Crossref, ISIGoogle Scholar
    • 28. R. Werner, Local preparability of states and the split property in quantum field theory, Lett. Math. Phys. 13 (1987) 325–329. Crossref, ISI, ADSGoogle Scholar
    • 29. R. Haag, Local Quantum Physicsc: Fields, Particles, Algebras, 2nd edn., Texts and Monographs in Physics (Springer-Verlag, Berlin, 1996). CrossrefGoogle Scholar
    • 30. J. C. A. Barata and F. Nill, Electrically and magnetically charged states and particles in the (2 + 1)-dimensional ZN-Higgs gauge model, Comm. Math. Phys. 171 (1995) 27–86. Crossref, ISI, ADSGoogle Scholar
    • 31. K. Fredenhagen and M. Marcu, Charged states in Z2 gauge theories, Comm. Math. Phys. 92 (1983) 81–119. Crossref, ISI, ADSGoogle Scholar
    • 32. B. Bakalov and A. Kirillov, Jr., Lectures on Tensor Categories and Modular Functors, University Lecture Series, Vol. 21 (American Mathematical Society, Providence, RI, 2001). Google Scholar
    • 33. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321 (2006) 2–111. Crossref, ISI, ADSGoogle Scholar
    • 34. Z. Wang, Topological Quantum Computation, CBMS Regional Conference Series in Mathematics, Vol. 112, Published for the Conference Board of the Mathematical Sciences (Amer. Math. Soc., 2010). Google Scholar
    • 35. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd edn., Texts and Monographs in Physics (Springer-Verlag, New York, 1987). CrossrefGoogle Scholar
    • 36. J. Fröhlich and F. Gabbiani, Braid statistics in local quantum theory, Rev. Math. Phys. 2 (1990) 251–353. LinkGoogle Scholar
    • 37. K. Fredenhagen, K.-H. Rehren and B. Schroer, Superselection sectors with braid group statistics and exchange algebras. I. General theory, Comm. Math. Phys. 125 (1989) 201–226. Crossref, ISI, ADSGoogle Scholar
    • 38. H. Halvorson, Algebraic quantum field theory, in Philosophy of Physics, eds. J. ButterfieldJ. Earman (Elsevier, 2006), pp. 731–922. Google Scholar
    • 39. E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nuclear Phys. B 300 (1988) 360–376. Crossref, ISI, ADSGoogle Scholar
    • 40. K.-H. Rehren, Braid group statistics and their superselection rules, in The Algebraic Theory of Superselection Sectors (Palermo, 1989), ed. D. Kastler (World Sci. Publ., River Edge, NJ, 1990), pp. 333–355. Google Scholar
    • 41. K.-H. Rehren, Markov traces as characters for local algebras, Nuclear Phys. B Proc. Suppl. 18 (1991) 259–268; Recent advances in field theory (Annecy-le-Vieux, 1990). Google Scholar
    • 42. S. Beigi, P. W. Shor and D. Whalen, The quantum double model with boundary: Condensations and symmetries, Comm. Math. Phys. 306 (2011) 663–694. Crossref, ISI, ADSGoogle Scholar
    • 43. F. Nill and K. Szlachányi, Quantum chains of Hopf algebras with quantum double cosymmetry, Comm. Math. Phys. 187 (1997) 159–200. Crossref, ISI, ADSGoogle Scholar
    Published: 17 November 2015