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From string nets to nonabelions. (English) Zbl 1196.82072

Summary: 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 \(SO(3)_{3} \times SO(3)_{3}\) doubled Chern-Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wave function, we describe the properties of this phase. Our discussion is informed by mappings of string net wave functions to the chromatic polynomial and the Potts model.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05C10 Planar graphs; geometric and topological aspects of graph theory
81S99 General quantum mechanics and problems of quantization
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References:

[1] Levin M., Wen X.G.: Phys. Rev. B. 71, 045110 (2005)
[2] Fendley P., Fradkin E.: Phys. Rev. B. 72, 024412 (2005)
[3] Moore G., Read N.: Nucl. Phys. B 360, 362 (1991)
[4] Morf R.H.: Phys. Rev. Lett. 80, 1505 (1998)
[5] Rezayi E.H., Haldane F.D.M.: Phys. Rev. Lett. 84, 4685 (2000)
[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)
[7] Freedman, M., Nayak, C., Shtengel, K.: http://arXiv.org/list/cond-mat/0309120 , 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)
[8] Chayes J.T., Chayes L., Kivelson S.A.: Commun. Math. Phys. 123, 53 (1989)
[9] Tutte W.T.: On Chromatic Polynomials and the Golden Ratio. J. Combin. Th., Ser. B 9, 289–296 (1970) · Zbl 0209.55001
[10] Cardy, J.: In: Les Houches 1988, Fields, Strings and Critical Phenomena. Brezin, E., Zinn-Justin, J. (eds.) London: Elsevier, 1989
[11] Saleur, H.: Nucl. Phys. B 360, 219 (1991); Pasquier, V., Saleur, H.: Nucl. Phys. B, 330, 523 (1990)
[12] Baxter R.: J. Phys. A: Math. Gen. 13, L61–L70 (1980)
[13] Freedman, M., Larsen, M., Wang, Z.: http://arXiv.org/list/quant-ph/0001108 , 2000
[14] Levin M., Wen X.G.: Phys. Rev. Lett. 96, 110405 (2006)
[15] Witten E.: Commun. Math. Phys. 121, 351 (1989) · Zbl 0667.57005
[16] Turaev V.G.: Quantum Invariants of Knots and 3-Manifolds. Berlin-New York, Walter de Gruyter (1994) · Zbl 0812.57003
[17] Fortuin C.M., Kasteleyn P.W.: Physica 57, 536 (1972)
[18] Kitaev A.: Ann. Phys. 303, 2 (2003) · Zbl 1012.81006
[19] Trebst S. et al.: Phys. Rev. Lett. 98, 070602 (2007)
[20] Kauffman L.H., Lins S.L.: Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds. Princeton, NJ, Princeton, University Press (1994) · Zbl 0821.57003
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