Communications in Mathematical Physics

, Volume 287, Issue 3, pp 805–827 | Cite as

From String Nets to Nonabelions

  • Lukasz Fidkowski
  • Michael Freedman
  • Chetan Nayak
  • Kevin Walker
  • Zhenghan Wang
Open Access


We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for the system to be in the DFib (Doubled Fibonacci) topological phase, that is, be described at low energy by an SO(3)3 × SO(3)3 doubled Chern-Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wavefunction, we describe the properties of this phase. Our discussion is informed by mappings of string net wavefunctions to the chromatic polynomial and the Potts model.


Hilbert Space Golden Ratio Topological Phase Chromatic Polynomial Trivalent Graph 
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We would like to thank Paul Fendley and Eduardo Fradkin for useful discussions. C.N. would like to acknowledge the support of the NSF under grant no. DMR-0411800 and the ARO under grant W911NF-04-1-0236 (C.N.). This research has been supported by the NSF under grants DMR-0130388 and DMR-0354772 (Z.W.). L.F. would like to acknowledge the support of the NSF under grant no. PHY -0244728.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Levin M., Wen X.G.: Phys. Rev. B. 71, 045110 (2005)CrossRefADSGoogle Scholar
  2. 2.
    Fendley P., Fradkin E.: Phys. Rev. B. 72, 024412 (2005)CrossRefADSGoogle Scholar
  3. 3.
    Moore G., Read N.: Nucl. Phys. B 360, 362 (1991)CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Morf R.H.: Phys. Rev. Lett. 80, 1505 (1998)CrossRefGoogle Scholar
  5. 5.
    Rezayi E.H., Haldane F.D.M.: Phys. Rev. Lett. 84, 4685 (2000)CrossRefADSGoogle Scholar
  6. 6.
    Moessner, R., Sondhi, S.L.: Phys. Rev. Lett 86, 1881 (2001); Nayak, C., Shtengel, K.: Phys. Rev. B 64, 064422 (2001); Balents, L. et al.: Phys. Rev. B 65, 224412 (2002); Ioffe, L.B. et al.: Nature 415, 503 (2002); Motrunich, O.I., Senthil, T.: Phys. Rev. Lett. 89, 277004 (2002)Google Scholar
  7. 7.
    Freedman, M., Nayak, C., Shtengel, K.:, 2003; Freedman, M., Nayak, C., Shtengel, K.: Phys. Rev. Lett. 94, 066401 (2005); Freedman, M., Nayak, C., Shtengel, K.: Phys. Rev. Lett. 94, 147205 (2005)Google Scholar
  8. 8.
    Chayes J.T., Chayes L., Kivelson S.A.: Commun. Math. Phys. 123, 53 (1989)CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Tutte W.T.: On Chromatic Polynomials and the Golden Ratio. J. Combin. Th., Ser. B 9, 289–296 (1970)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cardy, J.: In: Les Houches 1988, Fields, Strings and Critical Phenomena. Brezin, E., Zinn-Justin, J. (eds.) London: Elsevier, 1989Google Scholar
  11. 11.
    Saleur, H.: Nucl. Phys. B 360, 219 (1991); Pasquier, V., Saleur, H.: Nucl. Phys. B, 330, 523 (1990)Google Scholar
  12. 12.
    Baxter R.: J. Phys. A: Math. Gen. 13, L61–L70 (1980)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Freedman, M., Larsen, M., Wang, Z.:, 2000
  14. 14.
    Levin M., Wen X.G.: Phys. Rev. Lett. 96, 110405 (2006)CrossRefADSGoogle Scholar
  15. 15.
    Witten E.: Commun. Math. Phys. 121, 351 (1989)MATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Turaev V.G.: Quantum Invariants of Knots and 3-Manifolds. Berlin-New York, Walter de Gruyter (1994)MATHGoogle Scholar
  17. 17.
    Fortuin C.M., Kasteleyn P.W.: Physica 57, 536 (1972)CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Kitaev A.: Ann. Phys. 303, 2 (2003)MATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Trebst S. et al.: Phys. Rev. Lett. 98, 070602 (2007)CrossRefADSGoogle Scholar
  20. 20.
    Kauffman L.H., Lins S.L.: Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds. Princeton, NJ, Princeton, University Press (1994)MATHGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Lukasz Fidkowski
    • 1
    • 4
  • Michael Freedman
    • 1
  • Chetan Nayak
    • 1
    • 2
  • Kevin Walker
    • 1
  • Zhenghan Wang
    • 1
    • 3
  1. 1.Microsoft Station QUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Department of Physics and AstronomyUniversity of CaliforniaLos AngelesUSA
  3. 3.Department of MathematicsIndiana UniversityBloomingtonUSA
  4. 4.Department of PhysicsStanford UniversityStanfordUSA

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