Advertisement

Kitaev’s Quantum Double Model from a Local Quantum Physics Point of View

  • Pieter Naaijkens
Chapter
Part of the Mathematical Physics Studies book series (MPST)

Abstract

A prominent example of a topologically ordered system is Kitaev’s quantum double model \(\mathcal {D}(G)\) for finite groups G (which in particular includes \(G = \mathbb {Z}_2\), the toric code). We will look at these models from the point of view of local quantum physics. In particular, we will review how in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the different superselection sectors of the model. In this way one finds that the charges are in one-to-one correspondence with the representations of \(\mathcal {D}(G)\), and that they are in fact anyons. Interchanging two of such anyons gives a non-trivial phase, not just a possible sign change. The case of non-abelian groups G is more complicated. We outline how one could use amplimorphisms, that is, morphisms \(\mathfrak {A} \rightarrow M_n(\mathfrak {A})\) to study the superselection structure in that case. Finally, we give a brief overview of applications of topologically ordered systems to the field of quantum computation.

Keywords

Conjugacy Class Hopf Algebra Fusion Rule Tensor Category Fusion Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The author wishes to thank Courtney Brell for helpful comments and discussions and Leander Fiedler for collaboration on [22]. This work is supported by the Dutch Organisation for Scientific Research (NWO) through a Rubicon grant and partly through the EU project QFTCMPS and the cluster of excellence EXC 201 Quantum Engineering and Space-Time Research

References

  1. 1.
    Alicki, R., Fannes, M., Horodecki, M.: A statistical mechanics on Kitaev’s proposal for quantum memories. J. Phys. A 40, 6451–6467 (2007)MathSciNetCrossRefADSzbMATHGoogle Scholar
  2. 2.
    Alicki, R., Fannes, M., Horodecki, M.: On thermalization in Kitaev’s 2D model. J. Phys. A 42, 065303 (2009)MathSciNetCrossRefADSzbMATHGoogle Scholar
  3. 3.
    Bakalov, B., Kirillov Jr, A.: Lectures on Tensor Categories and Modular Functors. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  4. 4.
    Beverland, M.E., König, R., Pastawski, F., Preskill, J., Sijher, S.: Protected gates for topological quantum field theories. Preprint, arXiv:1409.3898 (2014)
  5. 5.
    Bombin, J., Martin-Delgado, M.A.: Family of non-Abelian Kitaev models on a lattice: topological condensation and confinement. Phys. Rev. B. 78, 115421 (2008)CrossRefADSGoogle Scholar
  6. 6.
    Bonderson, P., Freedman, M., Nayak, C.: Measurement-only topological quantum computation via anyonic interferometry. Ann. Phys. 324, 787–826 (2009)MathSciNetCrossRefADSzbMATHGoogle Scholar
  7. 7.
    Bonesteel, N.E., Hormozo, L., Zikos, G., Simon, S.H.: Braid topologies for quantum computation. Phys. Rev. Lett. 95, 140503 (2005)MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    Borchers, H.-J.: A remark on a theorem of B. Misra. Commun. Math. Phys. 4, 315–323 (1967)MathSciNetCrossRefADSzbMATHGoogle Scholar
  9. 9.
    Bratteli, O., Robinson, D.: Operator Algebras and Quantum Statistical Mechanics 1, 2nd edn. Springer, New York (1987)CrossRefzbMATHGoogle Scholar
  10. 10.
    Bratteli, O., Robinson, D.: Operator Algebras and Quantum Statistical Mechanics 2, 2nd edn. Springer, New York (1997)CrossRefzbMATHGoogle Scholar
  11. 11.
    Bravyi, S., Hastings, M.B.: A short proof of stability of topological order under local perturbations. Commun. Math. Phys. 307, 609–627 (2011)MathSciNetCrossRefADSzbMATHGoogle Scholar
  12. 12.
    Bravyi, S., Hastings, M.B., Michalakis, S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010)MathSciNetCrossRefADSzbMATHGoogle Scholar
  13. 13.
    Bravyi, S., Terhal, B.: A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes. New J. Phys. 11, 043029 (2009)CrossRefADSGoogle Scholar
  14. 14.
    Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)MathSciNetCrossRefADSzbMATHGoogle Scholar
  15. 15.
    Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states in gauge field theories. In: Schrader, R., Seiler, R., Uhlenbrock, D. (eds.) Mathematical Problems in Theoretical Physics, pp. 368–371. Springer, Berlin (1982)CrossRefGoogle Scholar
  16. 16.
    Cuntz, J.: Simple \(C^*\)-algebras generated by isometries. Commun. Math. Phys. 57, 173–185 (1977)MathSciNetCrossRefADSzbMATHGoogle Scholar
  17. 17.
    Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi Hopf algebras, group cohomology and orbifold models. Nucl. Phys. B (Proc. Suppl.) 18B, 60–72 (1990)Google Scholar
  18. 18.
    Doplicher, S., Roberts, J.E.: Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys. 131, 51–107 (1990)MathSciNetCrossRefADSzbMATHGoogle Scholar
  19. 19.
    Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics. I. Commun. Math. Phys. 23, 199–230 (1971)MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics. II. Commun. Math. Phys. 35, 49–85 (1974)MathSciNetCrossRefADSGoogle Scholar
  21. 21.
    Feynman, R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fiedler, L., Naaijkens, P.: Haag duality for Kitaev’s quantum double model for abelian groups. Preprint, arXiv:1406.1084 (2014)
  23. 23.
    Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. I. General theory. Commun. Math. Phys. 125, 201–226 (1989)MathSciNetCrossRefADSzbMATHGoogle Scholar
  24. 24.
    Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. II. Geometric aspects and conformal covariance. Rev. Math. Phys. 4(Special Issue), 113–157 (1992)Google Scholar
  25. 25.
    Freedman, M.H.: P/NP, and the quantum field computer. Proc. Natl. Acad. Sci. USA 95, 98–101 (1998)MathSciNetCrossRefADSzbMATHGoogle Scholar
  26. 26.
    Fröhlich, J., Gabbiani, F.: Braid statistics in local quantum theory. Rev. Math. Phys. 2, 251–353 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gabbiani, F., Fröhlich, J.: Operator algebras and Conformal Field Theory. Commun. Math. Phys. 155, 569–640 (1993)CrossRefADSMathSciNetzbMATHGoogle Scholar
  28. 28.
    Haag, R.: Local Quantum Physics: Fields, Particles, Algebras, 2nd edn. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  29. 29.
    Halvorson, H.: Algebraic quantum field theory (with an appendix by M. Müger). In: Butterfield, J., Earman, J. (eds) Philosophy of Physics, pp. 731–922, Elsevier, Amsterdam (2006)Google Scholar
  30. 30.
    Kalmeyer, V., Laughlin, R.B.: Equivalence of the resonating-valence-bond and fractional quantum Hall states. Phys. Rev. Lett. 59, 2095–2098 (1987)CrossRefADSGoogle Scholar
  31. 31.
    Kassel, C.: Quantum Groups. Springer, New York (1995)CrossRefzbMATHGoogle Scholar
  32. 32.
    Kawahigashi, Y., Longo, R., Müger, M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001)CrossRefADSMathSciNetzbMATHGoogle Scholar
  33. 33.
    Kay, A., Colbeck, R.: Quantum self-correcting stabilizer codes. Preprint arXiv:0810.3557 (2008)
  34. 34.
    Keyl, M., Matsui, T., Schlingemann, D., Werner, R.F.: Entanglement, Haag-duality and type properties of infinite quantum chains. Rev. Math. Phys. 18, 935–970 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kitaev, A.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003)MathSciNetCrossRefADSzbMATHGoogle Scholar
  36. 36.
    Landon-Cardinal, O., Poulin, D.: Local topological order inhibits thermal stability in 2D. Phys. Rev. Lett. 110, 090502 (2013)CrossRefADSGoogle Scholar
  37. 37.
    Matsui, T.: The split property and the symemtry breaking of the quantum spin chain. Commun. Math. Phys. 218, 393–416 (2001)MathSciNetCrossRefADSzbMATHGoogle Scholar
  38. 38.
    Mourik, V., Zuo, K., Frolov, S.M., Plissard, S.R., Bakkers, E.P.A.M., Kouwenhoven, L.P.: Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices. Science 336, 1003–1007 (2012)CrossRefADSGoogle Scholar
  39. 39.
    Müger, M.: On the structure of modular categories. Proc. London Math. Soc. 87, 291–308 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Müger, M.: Tensor categories: a selective guided tour. Rev. Unión Mat. Argentina 51, 95–163 (2010)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Mochon, C.: Anyons from nonsolvable finite groups are sufficient for universal quantum computation. Phys. Rev. A 67, 022315 (2003)CrossRefADSGoogle Scholar
  42. 42.
    Mochon, C.: Anyon computers with smaller groups. Phys. Rev. A 69, 032306 (2004)CrossRefADSGoogle Scholar
  43. 43.
    Naaijkens, P.: Localized endomorphisms in Kitaev’s toric code on the plane. Rev. Math. Phys. 23, 347–373 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Naaijkens, P.: Anyons in infinite quantum systems: QFT in \(d=2+1\) and the Toric Code. Ph.D. thesis, Radboud Universiteit Nijmegen (2012)Google Scholar
  45. 45.
    Naaijkens, P.: Haag duality and the distal split property for cones in the toric code. Lett. Math. Phys. 101, 341–354 (2012)MathSciNetCrossRefADSzbMATHGoogle Scholar
  46. 46.
    Naaijkens, P.: Kosaki-Longo index and classification of charges in 2D quantum spin models. J. Math. Phys. 54, 081901 (2013)MathSciNetCrossRefADSzbMATHGoogle Scholar
  47. 47.
    Nachtergaele, B., Sims, R.: Lieb-Robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 265, 119–130 (2006)MathSciNetCrossRefADSzbMATHGoogle Scholar
  48. 48.
    Nayak, C., Simon, S.H., Stern, A., Freedman, M., Das Sarma, S.: Non-abelian anyons and topological quantum computation. Rev. Modern Phys. 80, 1083–1159 (2008)Google Scholar
  49. 49.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  50. 50.
    Nill, F., Szlachányi, K.: Quantum chains of Hopf algebras with quantum double cosymmetry. Commun. Math. Phys. 187, 159–200 (1997)CrossRefADSMathSciNetzbMATHGoogle Scholar
  51. 51.
    Oeckl, R.: Discrete Gauge Theory: From Lattices to TQFT. Imperial College Press, London (2005)CrossRefzbMATHGoogle Scholar
  52. 52.
    Ostrik, V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8, 177–206 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Rehren, K.-H.: Braid group statistics and their superselection rules. In: Kastler, D. (ed.) The Algebraic Theory of Superselection Sectors, pp. 333–355. World Scientific Publishing, River Edge (1990)Google Scholar
  54. 54.
    Rieffel, M., Van Daele, A.: The commutation theorem for tensor products of von Neumann algebras. Bull. London Math. Soc. 7, 257–260 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Szlachányi, K., Vecsernyés, P.: Quantum symmetry and braid group statistics in \(G\)-spin models. Comm. Math. Phys. 156, 127–168 (1993)MathSciNetCrossRefADSzbMATHGoogle Scholar
  56. 56.
    Turaev, V.G.: Quantum Invariants of Knots and 3-manifolds. Walter de Gruyter & Co., Berlin (1994)zbMATHGoogle Scholar
  57. 57.
    Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B. 300, 360–376 (1988)MathSciNetCrossRefADSzbMATHGoogle Scholar
  58. 58.
    Wang, Z.: Topological quantum computation. Conference Board of the Mathematical Sciences, Washington, DC (2010)Google Scholar
  59. 59.
    Wen, X.-G.: Vacuum degeneracy of chiral spin states in compactified space. Phys. Rev. B 40, 7387–7390 (1989)CrossRefADSGoogle Scholar
  60. 60.
    Wick, G.C., Wightman, A.S., Wigner, E.P.: The intrinsic parity of elementary particles. Phys. Rev. Lett. 88, 101–105 (1952)MathSciNetADSzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikLeibniz Universität HannoverHannoverGermany

Personalised recommendations