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

SIGMA 6 (2010), 080, 9 pages      arXiv:1005.4603      https://doi.org/10.3842/SIGMA.2010.080
Contribution to the Proceedings of the International Workshop “Recent Advances in Quantum Integrable Systems”

Quantum Integrable 1D anyonic Models: Construction through Braided Yang-Baxter Equation

Anjan Kundu
Theory Group & CAMCS, Saha Institute of Nuclear Physics, Calcutta, India

Received May 25, 2010, in final form October 03, 2010; Published online October 09, 2010

Abstract
Applying braided Yang-Baxter equation quantum integrable and Bethe ansatz solvable 1D anyonic lattice and field models are constructed. Along with known models we discover novel lattice anyonic and q-anyonic models as well as nonlinear Schrödinger equation (NLS) and the derivative NLS quantum field models involving anyonic operators, N-particle sectors of which yield the well known anyon gases, interacting through δ and derivative δ-function potentials.

Key words: nonultralocal model; braided YBE; quantum integrability; 1D anyonic and q-anyonic lattice models; anyonic NLS and derivative NLS field models; algebraic Bethe ansatz.

pdf (203 kb)   ps (139 kb)   tex (14 kb)

References

  1. Wilczek F., Magnetic flux, angular momentum and statistics, Phys. Rev. Lett. 48 (1982), 1144-1146.
  2. Camino F.F., Zhou W., Goldman V.J., Realization of a Laughlin quasiparticle interferometer: observation of fractional statistics, Phys. Rev. B 72 (2005), 075342, 8 pages, cond-mat/0502406.
    Kim E., Lawer M., Vishweshwara S., Fradkin E., Signature of fractional statistics in noise experiments in quantum Hall fluids, Phys. Rev. Lett. 95 (2005), 176402, 4 pages, cond-mat/0507428.
  3. Kitaev A.Yu., Fault-tolerant quantum computation by anyons, Ann. Physics 303 (2003), 2-30, quant-ph/9707021.
    Kitaev A.Yu., Anyons in an exactly solved model and beyond, Ann. Physics 321 (2006), 2-111, cond-mat/0506438.
    Trebst S., Troyer M., Wang Z., Ludwig A.W., A short introduction to Fibonacci anyon models, Progr. Theoret. Phys. Suppl. 176 (2008), 384-407, arXiv:0902.3275.
    Nayak C., Simon S.H., Stern A., Freedman M., Das Sarma S., Non-abelian anyons and topological quantum computation, Rev. Modern Phys. 80 (2008), 1083-1159, arXiv:0707.1889.
  4. Pâtu O.I., Korepin V.E., Averin D.V., Correlation functions of one-dimensional Lieb-Liniger anyons, J. Phys. A: Math. Theor. 40 (2007), 14963-14984, arXiv:0707.4520.
  5. Kundu A., Exact solution of double δ-function Bose gas through an interacting anyon gas, Phys. Rev. Lett. 83 (1999), 1275-1278, hep-th/9811247.
  6. Batchelor M.T., Guan X.-W., Kundu A., One-dimensional anyons with competing δ-function and derivative δ-function potentials, J. Phys. A: Math. Theor. 41 (2008), 352002, 8 pages, arXiv:0805.1770.
  7. Batchelor M.T., Foerster A., Guan X.-W., Links J., Zhou H.-Q., The quantum inverse scattering method with anyonic grading, J. Phys. A: Math. Theor. 41 (2008), 465201, 13 pages, arXiv:0807.3197.
  8. Batchelor M.T., Guan X.-W., Oelkers N., One-dimensional interacting anyon gas: low energy properties and Haldane exclusion statistics, Phys. Rev. Lett. 96 (2006), 210402, 4 pages, cond-mat/0603643.
    Girardeau M.D., Anyon-fermion mapping and applications to ultracold gasses in tight waveguides, Phys. Rev. Lett. 97 (2006), 100402, 4 pages, cond-mat/0604357.
    Averin D.V., Nesteroff J.A., Coulomb blockade of anyons in quantum antidots, Phys. Rev. Lett. 99 (2007), 096801, 4 pages, arXiv:0704.0439.
    Batchelor M.T., Guan X.-W., He J.-S., The Bethe ansatz for one-dimensional interacting anyons, J. Stat. Mech. Theory Exp. 2007 (2007), P03007, 19 pages, cond-mat/0611450.
    Calabrese P., Mintchev M., Correlation functions of one-dimensionalanyonic fluids, Phys. Rev. B 75 (2007), 233104, 4 pages, cond-mat/0703117.
    Santachiara R., Increasing of entangement entropy from pure to random quantum critical chains, J. Stat. Mech. Theory Exp. 2006 (2006), L06002, 8 pages, cond-mat/0602527.
    Pâtu O.I., Korepin V.E., Averin D.V., One-dimensional impenetrable anyons in thermal equillibrium. I. Anyonic generalizations of Lenard's formula, J. Phys. A: Math. Theor. 41 (2008), 145006, 15 pages, arXiv:0801.4397.
    Pâtu O.I., Korepin V.E., Averin D.V., One-dimensional impenetrable anyons in thermal equillibrium. II. Determinant represenation for the dynamic correlation functions, J. Phys. A: Math. Theor. 41 (2008), 255205, 19 pages, arXiv:0803.0750.
    del Campo A., Fermionization and bosonization of expanding one-dimensional anyonic fluids, Phys. Rev. A 78 (2008), 045602, 4 pages, arXiv:0805.3786.
  9. Keilmann T., Lanzmich S., McCulloch I., Roncaglia M., Statistically induced phase transitions: turning bosons smoothly via anyons into fermions, arXiv:1009.2036.
  10. Lieb E.H., Liniger W., Exact analysis of an interacting Bose gas. I. The general solution and the ground state, Phys. Rev. 130 (1963), 1605-1616.
  11. Shnirman A.G., Malomed B.A., Ben-Jacob E., Nonperturbative studies of a quantum higher order nonlinear Schrödinger model using the Bethe ansatz, Phys. Rev. A 50 (1994), 3453-3463.
  12. Hlavatý L., Kundu A., Quantum integrability of non-ultralocal models through Baxterisation of quantised braided algebra, Internat. J. Modern Phys. A 11 (1996), 2143-2165, hep-th/9406215.
  13. Kundu A., Exact Bethe ansatz solution of non-ultralocal quantum mKdV model, Modern Phys. Lett. A 10 (1995), 2955-2966, hep-th/9510131.
  14. Faddeev L.D., Quantum completely integrable models in field theory, in Mathematical Physics Reviews, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., Harwood Academic, Chur, 1980, Vol. 1, 107-155.
  15. Kundu A., Exact solution of anyonic NLS quantum field model, in preparation.
    Kundu A., Novel nonultralocal quantum group and exactly solvable anyonic DNLS quantum field model, in preparation.
  16. Korepin V.E., Bogoliubov N.M., Izergin A.G., Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1993.

Previous article   Next article   Contents of Volume 6 (2010)