Communications in Mathematical Physics

, Volume 357, Issue 1, pp 125–157 | Cite as

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

  • Matthew Cha
  • Pieter Naaijkens
  • Bruno Nachtergaele


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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  1. 1.
    Alicki R., Fannes M., Horodecki M.: A statistical mechanics view on Kitaev’s proposal of quantum memories. J. Phys. A 40(24), 6451–6467 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Araki H., Matsui T.: Ground states of the XY-model. Commun. Math. Phys. 101, 213–245 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arovas D., Schrieffer J.R., Wilczek F.: Fractional statistics and the quantum Hall effect. Phys. Rev. Lett. 53, 722–723 (1984)ADSCrossRefGoogle Scholar
  4. 4.
    Bachmann, S.: Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons. Rev. Math. Phys. 29, 1750018 (2017)Google Scholar
  5. 5.
    Bachmann S., Michalakis S., Nachtergaele B., Sims R.: Automorphic equicalence within gapped phases of quantum lattice systems. Commun. Math. Phys. 309, 835–871 (2012)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Bachmann S., Ogata Y.: C 1-classification of gapped parent Hamiltonians of quantum spin chains. Commun. Math. Phys. 338, 1011–1042 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bais, F.A., van Driel, P., De Wild Propitius, M.: Anyons in discrete gauge theories with Chern–Simons terms. Nucl. Phys. B. 393, 547–570 (1993)Google Scholar
  8. 8.
    Bakalov, B., Kirillov, A., Jr.: Lectures on Tensor Categories and Modular Functors (University Lecture Series 21). American Mathematical Society, Providence, RI (2001)Google Scholar
  9. 9.
    Beigi S., Shor P.W., Whalen D.: The quantum double model with boundary: condensations and symmetries. Commun. Math. Phys. 306, 663–694 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bombin H., Martin-Delgado M.A.: A family of non-abelian Kitaev models on the lattice: topological condensation and confinement. Phys. Rev. B. 78, 115421 (2008)ADSCrossRefGoogle Scholar
  11. 11.
    Bonderson P., Shtengel K., Slingerland J.K.: Interferometry of non-abelian anyons. Ann. Phys. 323, 2709–2755 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Brandão F.G.S.L., Horodecki M.: Exponential decay of correlations implies area law. Commun. Math. Phys. 333, 761 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bratteli O., Kishimoto A., Robinson D.: Ground states of infinite quantum spin systems. Commun. Math. Phys. 64, 41–48 (1978)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Bratteli O., Robinson D.W.: Operator algebras and quantum statistical mechanics 1 and 2. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  15. 15.
    Bravyi S., Hastings M., Michalakis S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bravyi, S., Kitaev, A.: Quantum codes on a lattice with boundary (1998). arXiv:quant-ph/9811052v1
  17. 17.
    Chen X., Gu Z.-C., Wen X.-G.: Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. Phys. Rev. B 84, 155138 (2011)Google Scholar
  18. 18.
    Dijkgraaf R., Pasquier V., Roche P.: Quasi Hopf algebras, group cohomology and orbifold models. Nucl. Phys. B. Proc. Suppl. 18, 60–72 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Doplicher S., Haag R., Roberts J.E.: Local observables and particle statistics. I. Commun. Math. Phys. 23, 199–230 (1971)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Doplicher S., Haag R., Roberts J.E.: Local observables and particle statistics. II. Commun. Math. Phys. 35, 49–85 (1974)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Fannes M., Nachtergaele B., Werner R.F.: Finitely correlated states of quantum spin chains. Commun. Math. Phys. 144, 443–490 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fannes M., Werner R.F.: Boundary conditions for quantum lattice systems. Helv. Phys. Acta 68, 635–657 (1995)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Fiedler L., Naaijkens P.: Haag duality for Kitaev’s quantum double model for abelian groups. Rev. Math. Phys. 27, 1550021 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fredenhagen K., Rehren K.-H., Schroer B.: Superselection sectors with braid group statistics and exchange algebras. Commun. Math. Phys. 125, 201–226 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Freedman M.: P/NP, and the quantum field computer. Proc. Natl. Acad. Sci. USA 95, 98–101 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Freedman M., Meyer D.A.: Projective plane and planar quantum codes. Found. Comput. Math. 1, 325–332 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Fröhlich J., Gabbiani F.: Braid statistics in local quantum theory. Rev. Math. Phys. 2, 251–353 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gottstein, C.T., Werner, R.F.: Ground states of the q-deformed Heisenberg ferromagnet (1995). arXiv:cond-mat/9501123
  29. 29.
    Haag R.: Local Quantum Physics: Fields, Particles, Algebras, Texts and Monographs in Physics. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  30. 30.
    Haag R., Hugenholtz N.M., Winnink M.: On the equilibrium states in quantum statistical mechanics. Commun. Math. Phys. 5, 215–236 (1967)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Haah J.: An invariant of topologically ordered states under local unitary transformations. Commun. Math. Phys. 342, 771–801 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hastings M.B.: An area law for one-dimensional quantum systems. J. Stat. Mech. 2007, P08024 (2007)MathSciNetGoogle Scholar
  33. 33.
    Hastings M.B., Koma T.: Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265, 781–804 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kitaev A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kitaev A.: Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Koma T., Nachtergaele B.: The complete set of ground states of the ferromagnetic XXZ chains. Adv. Theor. Math. Phys. 2, 533–558 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Matsui T.: On ground states of the one-dimensional ferromagnetic XXZ chain. Lett. Math. Phys. 37, 397–403 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Michalakis S., Zwolak J.P.: Stability of frustration-free Hamiltonians. Commun. Math. Phys. 322, 277–302 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Moore G., Read N.: Nonabelions in the fractional quantum Hall effect. Nucl. Phys. B 360, 362–396 (1990)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Naaijkens P.: Localized endomorphisms in Kitaev’s toric code on the plane. Rev. Math. Phys. 23, 347–373 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Naaijkens, P.: Kitaev’s quantum double model from a local quantum physics point of view. In: Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J. (eds.) Advances in Algebraic Quantum Field Theory, pp. 365–395. Springer, Berlin (2015)Google Scholar
  42. 42.
    Nachtergaele B., Ogata Y., Sims R.: Propagation of correlations in quantum lattice systems. J. Stat. Phys. 124, 1–13 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Nachtergaele B., Sims R.: Lieb–Robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 265, 119–130 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Ogata Y.: A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization I. Commun. Math. Phys. 348, 847–895 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Ogata Y.: A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization II. Commun. Math. Phys. 348, 897–957 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Ogata Y.: A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization III. Commun. Math. Phys. 352, 1205–1263 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. I: Functional Analysis, Revised and Enlarged edition. Academic Press (1980)Google Scholar
  48. 48.
    Szlachányi K., Vecsernyés P.: Quantum symmetry and braid group statistics in G-spin models. Commun. Math. Phys. 156, 127–168 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Wen X.-G.: Vacuum degeneracy of chiral spin states in compactified space. Phys. Rev. Lett. B 40, 7387–7390 (1989)ADSCrossRefGoogle Scholar
  50. 50.
    Wilczek F.: Fractional Statistics and Anyon Superconductivity. 2nd edn.World Scientific Publishing Co., Inc., Teaneck (1990)CrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, DavisDavisUSA
  2. 2.JARA Institute for Quantum InformationRWTH Aachen UniversityAachenGermany

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