About this Journal Submit a Manuscript Table of Contents 
Advances in Mathematical Physics
Volume 2010 (2010), Article ID 671039, 18 pages
doi:10.1155/2010/671039
Research Article

Microscopic Description of 2D Topological Phases, Duality, and 3D State Sums

Zoltán Kádár,1 Annalisa Marzuoli,1,2 and Mario Rasetti1,3

1Institute for Scientific Interchange Foundation, Villa Gualino, Viale Settimio Severo 75, 10131 Torino, Italy
2Dipartimento di Fisica Nucleare e Teorica, Istituto Nazionale di Fisica Nucleare, Universita degli Studi di Pavia, Sezione di Pavia, via A. Bassi 6, 27100 Pavia, Italy
3Dipartimento di Fisica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received 7 September 2009; Accepted 23 January 2010

Academic Editor: Debashish Goswami

Copyright © 2010 Zoltán Kádár et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. Witten, “2+1-dimensional gravity as an exactly soluble system,” Nuclear Physics B, vol. 311, no. 1, pp. 46–78, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  2. M. Atiyah, “Topological quantum field theories,” IHÉS Publications Mathématiques, no. 68, pp. 175–186, 1988. View at Zentralblatt MATH · View at MathSciNet
  3. X.-G. Wen, “Topological orders and edge excitations in fractional quantum Hall states,” Advances in Physics, vol. 44, no. 5, pp. 405–473, 1995. View at Scopus
  4. A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics, vol. 303, no. 1, pp. 2–30, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. E. Rowell, R. Stong, and Z. Wang, “On classification of modular tensor categories,” Communications in Mathematical Physics, vol. 292, no. 2, pp. 343–389, 2009. View at Publisher · View at Google Scholar · View at Scopus
  6. R. K. Kaul, T. R. Govindarajan, and P. Ramadevi, “Schwarz type topological quantum field theories,” in Encyclopedia of Mathematical Physics, Elsevier, Amsterdam, The Netherlands, 2005.
  7. M. A. Levin and X.-G. Wen, “String-net condensation: a physical mechanism for topological phases,” Physical Review B, vol. 71, no. 4, Article ID 045110, 21 pages, 2005. View at Publisher · View at Google Scholar
  8. Z. Kádár, A. Marzuoli, and M. Rasetti, “Braiding and entanglement in spin networks: a combinatorial approach to topological phases,” International Journal of Quantum Information, vol. 7, supplement, pp. 195–203, 2009. View at Publisher · View at Google Scholar · View at Scopus
  9. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Reviews of Modern Physics, vol. 80, no. 3, pp. 1083–1159, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  10. A. Marzuoli and M. Rasetti, “Computing spin networks,” Annals of Physics, vol. 318, no. 2, pp. 345–407, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. M. H. Freedman, “Quantum computation and the localization of modular functors,” Foundations of Computational Mathematics, vol. 1, no. 2, pp. 183–204, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. M. Aguado and G. Vidal, “Entanglement renormalization and topological order,” Physical Review Letters, vol. 100, no. 7, Article ID 070404, 4 pages, 2008. View at Publisher · View at Google Scholar
  13. R. König, B. W. Reichardt, and G. Vidal, “Exact entanglement renormalization for string-net models,” Physical Review B, vol. 79, no. 19, Article ID 195123, 6 pages, 2009. View at Publisher · View at Google Scholar
  14. O. Buerschaper, M. Aguado, and G. Vidal, “Explicit tensor network representation for the ground states of string-net models,” Physical Review B, vol. 79, no. 8, Article ID 085119, 5 pages, 2009. View at Publisher · View at Google Scholar
  15. A. Hamma, R. Ionicioiu, and P. Zanardi, “Bipartite entanglement and entropic boundary law in lattice spin systems,” Physical Review A, vol. 71, no. 2, Article ID 022315, 10 pages, 2005. View at Publisher · View at Google Scholar
  16. M. Levin and X.-G. Wen, “Detecting topological order in a ground state wave function,” Physical Review Letters, vol. 96, no. 11, Article ID 110405, 4 pages, 2006. View at Publisher · View at Google Scholar
  17. J. Kogut and L. Susskind, “Hamiltonian formulation of Wilson's lattice gauge theories,” Physical Review D, vol. 11, no. 2, pp. 395–408, 1975. View at Publisher · View at Google Scholar · View at Scopus
  18. C. Rovelli and L. Smolin, “Spin networks and quantum gravity,” Physical Review D, vol. 52, no. 10, pp. 5743–5759, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  19. C. Rovelli, Quantum Gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, Mass, USA, 2004. View at MathSciNet
  20. Z. Kádár, “Polygon model from first-order gravity,” Classical and Quantum Gravity, vol. 22, no. 5, pp. 809–823, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. G. Ponzano and T. Regge, “Semiclassical limit of Racah coefficients,” in Spectroscopic and Group Theoretical Methods in Physics, F. Bloch, et al., Ed., pp. 1–58, North-Holland, Amsterdam, The Netherlands, 1968.
  22. L. Freidel and K. Krasnov, “Spin foam models and the classical action principle,” Advances in Theoretical and Mathematical Physics, vol. 2, no. 6, pp. 1183–1247, 1998. View at Zentralblatt MATH · View at MathSciNet
  23. D. Oriti, Spin foam models of quantum spacetime, Ph.D. thesis, University of Cambridge.
  24. V. G. Turaev and O. Y. Viro, “State sum invariants of 3-manifolds and quantum 6j-symbols,” Topology, vol. 31, no. 4, pp. 865–902, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  25. V. G. Drinfel'd, “Quantum groups,” in Proceedings of the International Congress of Mathematicians—ICM (Berkeley 1986), pp. 798–820, American Mathematical Society, Providence, RI, USA, 1987.
  26. A. Beliakova and B. Durhuus, “Topological quantum field theory and invariants of graphs for quantum groups,” Communications in Mathematical Physics, vol. 167, no. 2, pp. 395–429, 1995. View at Zentralblatt MATH · View at MathSciNet
  27. V. Turaev, “Quantum invariants of links and 3-valent graphs in 3-manifolds,” Publications Mathematiques de L'IHES, no. 77, pp. 121–171, 1993. View at MathSciNet
  28. O. Buerschaper and M. Aguado, “Mapping Kitaev's quantum double lattice models to Levin and Wen's string-net models,” Physical Review B, vol. 80, no. 15, 2009. View at Publisher · View at Google Scholar
  29. J. C. Baez, “Spin networks in gauge theory,” Advances in Mathematics, vol. 117, no. 2, pp. 253–272, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. E. Witten, “Gauge theories and integrable lattice models,” Nuclear Physics B, vol. 322, no. 3, pp. 629–697, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  31. S. Garnerone, A. Marzuoli, and M. Rasetti, “Quantum automata, braid group and link polynomials,” Quantum Information & Computation, vol. 7, no. 5-6, pp. 479–503, 2007. View at Zentralblatt MATH · View at MathSciNet
  32. S. Garnerone, A. Marzuoli, and M. Rasetti, “Quantum geometry and quantum algorithms,” Journal of Physics A, vol. 40, no. 12, pp. 3047–3066, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. S. Garnerone, A. Marzuoli, and M. Rasetti, “Efficient quantum processing of 3-manifolds topological invariants,” Advances in Theoretical and Mathematical Physics. In press.
  34. T. Ohtsuki, “Problems on invariants of knots and 3-manifolds,” in Invariants of Knots and 3-Manifolds, vol. 4 of Geometry and Topology Monographs, pp. 377–572, Geom. Topol. Publ., Coventry, UK, 2002. View at Zentralblatt MATH · View at MathSciNet