Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 048, 55 pages      arXiv:2302.08027      https://doi.org/10.3842/SIGMA.2024.048

Oriented Closed Polyhedral Maps and the Kitaev Model

Kornél Szlachányi
Wigner Research Centre for Physics, Budapest, Hungary

Received April 07, 2023, in final form May 14, 2024; Published online June 08, 2024

Abstract
A kind of combinatorial map, called arrow presentation, is proposed to encode the data of the oriented closed polyhedral complexes $\Sigma$ on which the Hopf algebraic Kitaev model lives. We develop a theory of arrow presentations which underlines the role of the dual of the double $\mathcal{D}(\Sigma)^*$ of $\Sigma$ as being the Schreier coset graph of the arrow presentation, explains the ribbon structure behind curves on $\mathcal{D}(\Sigma)^*$ and facilitates computation of holonomy with values in the algebra of the Kitaev model. In this way, we can prove ribbon operator identities for arbitrary f.d. C$^*$-Hopf algebras and arbitrary oriented closed polyhedral maps. By means of a combinatorial notion of homotopy designed specially for ribbon curves, we can rigorously formulate ''topological invariance'' of states created by ribbon operators.

Key words: Hopf algebra; polyhedral map; quantum double; ribbon operator; topological invariance.

pdf (3114 kb)   tex (2230 kb)  

References

  1. Bombin H., Martin-Delgado M., A family of non-Abelian Kitaev models on a lattice: topological condensation and confinement, Phys. Rev. B 78 (2008), 115421, 28 pages, arXiv:0712.0190.
  2. Buchholz D., Fredenhagen K., Locality and the structure of particle states, Comm. Math. Phys. 84 (1982), 1-54.
  3. Buerschaper O., Mombelli J.M., Christandl M., Aguado M., A hierarchy of topological tensor network states, J. Math. Phys. 54 (2013), 012201, 46 pages, arXiv:1007.5283.
  4. Cha M., Naaijkens P., Nachtergaele B., On the stability of charges in infinite quantum spin systems, Comm. Math. Phys. 373 (2020), 219-264, arXiv:1804.03203.
  5. Cowtan A., Majid S., Quantum double aspects of surface code models, J. Math. Phys. 63 (2022), 042202, 49 pages, arXiv:2107.04411.
  6. Doplicher S., Haag R., Roberts J.E., Fields, observables and gauge transformations. I, Comm. Math. Phys. 13 (1969), 1-23.
  7. Doplicher S., Haag R., Roberts J.E., Fields, observables and gauge transformations. II, Comm. Math. Phys. 15 (1969), 173-200.
  8. Doplicher S., Haag R., Roberts J.E., Local observables and particle statistics. I, Comm. Math. Phys. 23 (1971), 199-230.
  9. Doplicher S., Haag R., Roberts J.E., Local observables and particle statistics. II, Comm. Math. Phys. 35 (1974), 49-85.
  10. Hirmer A.K., Meusburger C., Categorical generalizations of quantum double models, arXiv:2306.05950.
  11. Jones G., Singerman D., Maps, hypermaps and triangle groups, in The Grothendieck Theory of Dessins D'enfants (Luminy, 1993), London Math. Soc. Lecture Note Ser., Vol. 200, Cambridge University Press, Cambridge, 1994, 115-145.
  12. Kitaev A.Yu., Fault-tolerant quantum computation by anyons, Ann. Physics 303 (2003), 2-30, arXiv:quant-ph/9707021.
  13. Lando S.K., Zvonkin A.K., Graphs on surfaces and their applications, Encyclopaedia Math. Sci., Vol. 141, Springer, Berlin, 2004.
  14. Larson R.G., Radford D.E., Semisimple cosemisimple Hopf algebras, Amer. J. Math. 110 (1988), 187-195.
  15. Meusburger C., Kitaev lattice models as a Hopf algebra gauge theory, Comm. Math. Phys. 353 (2017), 413-468, arXiv:1607.01144.
  16. Meusburger C., Wise D.K., Hopf algebra gauge theory on a ribbon graph, Rev. Math. Phys. 33 (2021), 2150016, 93 pages, arXiv:1512.03966.
  17. Montgomery S., Hopf algebras and their actions on rings, CBMS Reg. Conf. Ser. Math., Vol. 82, American Mathematical Society, Providence, RI, 1993.
  18. Naaijkens P., Localized endomorphisms in Kitaev's toric code on the plane, Rev. Math. Phys. 23 (2011), 347-373, arXiv:1012.3857.
  19. Naaijkens P., Kitaev's quantum double model from a local quantum physics point of view, in Advances in Algebraic Quantum Field Theory, Math. Phys. Stud., Springer, Cham, 2015, 365-395, arXiv:1508.07170.
  20. Sweedler M.E., Hopf algebras, Math. Lecture Note Ser., W.A. Benjamin, Inc., New York, 1969.
  21. White A.T., Graphs of groups on surfaces. Interactions and models, North-Holland Math. Stud., Vol. 188, North-Holland Publishing Co., Amsterdam, 2001.
  22. Yan B., Chen P., Cui S.X., Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras, J. Phys. A 55 (2022), 185201, 34 pages, arXiv:2105.08202.

Previous article  Next article  Contents of Volume 20 (2024)