×

Topological degeneracy of non-abelian states for dummies. (English) Zbl 1359.82004

Summary: We present a physical construction of degenerate groundstates of the Moore-Read Pfaffian states, which exhibits non-Abelian statistics, on general Riemann surface with genus \(g\). The construction is given by a generalization of the recent argument [the author and T. Senthil, “Fractionalization, topological order, and quasiparticle statistics”, Phys. Rev. Lett. 96, No. 6, Article ID 060601, 4 p. (2006; doi:10.1103/physrevlett.96.060601)] which relates fractionalization and topological order. The nontrivial groundstate degeneracy obtained by N. Read and D. Green [“Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect”, Phys. Rev. B 61, No. 15, 10267–10297, (2000; doi:10.1103/PhysRevB.61.10267)] based on differential geometry is reproduced exactly. Some restrictions on the statistics, due to the fractional charge of the quasiparticle are also discussed. Furthermore, the groundstate degeneracy of the \(p + \mathrm ip\) superconductor in two dimensions, which is closely related to the Pfaffian states, is discussed with a similar construction.

MSC:

82B10 Quantum equilibrium statistical mechanics (general)
81V70 Many-body theory; quantum Hall effect
82D55 Statistical mechanics of superconductors
53C27 Spin and Spin\({}^c\) geometry
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Wen, X.-G.; Niu, Q., Phys. Rev. B, 41, 9377 (1990)
[2] Kitaev, A., Ann. Phys., 303, 2 (2003)
[3] Moore, G.; Read, N., Nucl. Phys., 360, 362 (1991)
[4] Rezayi, E. H.; Haldane, F. D.M., Phys. Rev. Lett., 84, 4685 (2000)
[5] Willett, R. L., Phys. Rev. Lett., 59, 1776 (1987)
[6] Fradkin, E., Nucl. Phys. B, 516, 704 (1998)
[7] Das Sarma, S.; Freedman, M.; Nayak, C., Phys. Rev. Lett., 94, 166802 (2005)
[8] Stern, A.; Halperin, B. I., Phys. Rev. Lett., 96, 016802 (2006)
[9] Bonderson, P.; Kitaev, A.; Shtengel, K., Phys. Rev. Lett., 96, 016803 (2006)
[10] Das Sarma, S.; Nayak, C.; Tewari, S., Phys. Rev. B, 73, 220502 (2006)
[11] S. Tewari, S. Das Sarma, C. Nayak, C. Zhang, P. Zoller,; S. Tewari, S. Das Sarma, C. Nayak, C. Zhang, P. Zoller,
[12] Bravyi, S., Phys. Rev. A, 73, 042312 (2006)
[13] Das Sarma, S.; Freedman, M.; Nayak, C., Phys. Today, 59, 32 (2006), and references therein
[14] Oshikawa, M.; Senthil, T., Phys. Rev. Lett., 96, 060601 (2006)
[15] Wu, Y.-S.; Hatsugai, Y.; Kohmoto, M., Phys. Rev. Lett., 66, 659 (1991)
[16] Sato, M.; Kohmoto, M.; Wu, Y.-S., Phys. Rev. Lett., 97, 010601 (2006)
[17] Read, N.; Green, D., Phys. Rev. B, 61, 10267 (2000)
[18] Nayak, C.; Wilczek, F., Nucl. Phys. B, 516, 704 (1996)
[19] Gurarie, V.; Radzihovsky, L.; Andreev, A. V., Phys. Rev. Lett., 94, 230403 (2005)
[20] (Heinonen, O., Composite Fermions: A Unified View of the Quantum Hall Regime (1998), World Scientific: World Scientific Singapore)
[21] Ivanov, D. A., Phys. Rev. Lett., 86, 268 (2001)
[22] Stern, A.; von Oppen, F.; Mariani, E., Phys. Rev. B, 70, 205338 (2004)
[23] Phys. Rev. Lett., 91, 109901(E) (2003)
[24] Hansson, T. H.; Oganesyan, V.; Sondhi, S. L., Ann. Phys., 313, 497 (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.