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


SIGMA 9 (2013), 072, 12 pages      arXiv:1309.6165      https://doi.org/10.3842/SIGMA.2013.072

Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz

Samuel Belliard a, b and Nicolas Crampé a, b
a) Laboratoire Charles Coulomb L2C, UMR 5221, CNRS, F-34095 Montpellier, France
b) Laboratoire Charles Coulomb L2C, UMR 5221, Université Montpellier 2, F-34095 Montpellier, France

Received September 29, 2013, in final form November 19, 2013; Published online November 22, 2013

Abstract
We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al. The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the length of the chain. We prove this result for the chains with small length. We obtain also an off-shell equation (i.e. satisfied without the Bethe equations) formally similar to the ones obtained in the periodic case or with diagonal boundaries.

Key words: algebraic Bethe ansatz; integrable spin chain; boundary conditions.

pdf (357 kb)   tex (21 kb)

References

  1. Baseilhac P., Belliard S., The half-infinite XXZ chain in Onsager's approach, Nuclear Phys. B 873 (2013), 550-584, arXiv:1211.6304.
  2. Baseilhac P., Koizumi K., Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory, J. Stat. Mech. Theory Exp. 2007 (2007), P09006, 27 pages, hep-th/0703106.
  3. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press Inc., London, 1982.
  4. Belliard S., Algebraic Bethe ansatz for twisted XXX spin chain, in preparation.
  5. Belliard S., Crampé N., Ragoucy E., Algebraic Bethe ansatz for open XXX model with triangular boundary matrices, Lett. Math. Phys. 103 (2013), 493-506, arXiv:1209.4269.
  6. Belliard S., Pakuliak S., Ragoucy E., Slavnov N.A., The algebraic Bethe ansatz for scalar products in SU(3)-invariant integrable models, J. Stat. Mech. Theory Exp. 2012 (2012), P10017, 25 pages, arXiv:1207.0956.
  7. Bethe H., Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys. 71 (1931), 205-226.
  8. Cao J., Yang W.L., Shi K., Wang Y., Exact solutions and elementary excitations in the XXZ spin chain with unparallel boundary fields, Nuclear Phys. B 663 (2003), 487-519, cond-mat/0212163.
  9. Cao J., Yang W.L., Shi K., Wang Y., Off-diagonal Bethe ansatz and exact solution a topological spin ring, Phys. Rev. Lett. 111 (2013), 137201, 5 pages, arXiv:1305.7328.
  10. Cao J., Yang W.L., Shi K., Wang Y., Off-diagonal Bethe ansatz solution of the XXX spin-chain with arbitrary boundary conditions, Nuclear Phys. B 875 (2013), 152-165, arXiv:1306.1742.
  11. Cao J., Yang W.L., Shi K., Wang Y., Off-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary fields, Nuclear Phys. B 877 (2013), 152-175, arXiv:1307.2023.
  12. Cherednik I.V., Factorizing particles on a half line, and root systems, Theoret. and Math. Phys. 61 (1984), 977-983.
  13. Crampé N., Ragoucy E., Generalized coordinate Bethe ansatz for non-diagonal boundaries, Nuclear Phys. B 858 (2012), 502-512, arXiv:1105.0338.
  14. Crampé N., Ragoucy E., Simon D., Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions, J. Stat. Mech. Theory Exp. 2010 (2010), P11038, 20 pages, arXiv:1009.4119.
  15. Crampé N., Ragoucy E., Simon D., Matrix coordinate Bethe ansatz: applications to XXZ and ASEP models, J. Phys. A: Math. Theor. 44 (2011), 405003, 17 pages, arXiv:1106.4712.
  16. de Vega H.J., González-Ruiz A., Boundary K-matrices for the six vertex and the n(2n−1) An−1 vertex models, J. Phys. A: Math. Gen. 26 (1993), L519-L524, hep-th/9211114.
  17. Faldella S., Kitanine N., Niccoli G., Complete spectrum and scalar products for open spin-1/2 XXZ quantm chains with non-diagonal boundary terms, arXiv:1307.3960.
  18. Frahm H., Grelik J.H., Seel A., Wirth T., Functional Bethe ansatz methods for the open XXX chain, J. Phys. A: Math. Theor. 44 (2011), 015001, 19 pages, arXiv:1009.1081.
  19. Frahm H., Seel A., Wirth T., Separation of variables in the open XXX chain, Nuclear Phys. B 802 (2008), 351-367, arXiv:0803.1776.
  20. Galleas W., Functional relations from the Yang-Baxter algebra: eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions, Nuclear Phys. B 790 (2008), 524-542, arXiv:0708.0009.
  21. Gaudin M., La fonction d'onde de Bethe, Masson, Paris, 1983.
  22. Ghoshal S., Zamolodchikov A., Boundary S matrix and boundary state in two-dimensional integrable quantum field theory, Internat. J. Modern Phys. A 9 (1994), 3841-3885, hep-th/930600.
  23. Gorissen M., Lazarescu A., Mallick K., Vanderzande C., Exact current statistics of the asymmetric simple exclusion process with open boundaries, Phys. Rev. Lett. 109 (2012), 170601, 5 pages, arXiv:1207.6879.
  24. Karaiskos N., Grabinski A.M., Frahm H., Bethe ansatz solution of the small polaron with nondiagonal boundary terms, J. Stat. Mech. Theory Exp. 2013 (2013), P07009, 19 pages, arXiv:1304.2659.
  25. Kitanine N., Kozlowski K.K., Maillet J.M., Niccoli G., Slavnov N.A., Terras V., Correlation functions of the open XXZ chain. I, J. Stat. Mech. Theory Exp. 2007 (2007), P10009, 37 pages, arXiv:0707.1995.
  26. Kitanine N., Kozlowski K.K., Maillet J.M., Niccoli G., Slavnov N.A., Terras V., Correlation functions of the open XXZ chain. II, J. Stat. Mech. Theory Exp. 2008 (2008), P07010, 33 pages, arXiv:0803.3305.
  27. Kitanine N., Maillet J.M., Terras V., Form factors of the XXZ Heisenberg spin-1/2 finite chain, Nuclear Phys. B 554 (1999), 647-678, math-ph/9807020.
  28. Lazarescu A., Matrix ansatz for the fluctuations of the current in the ASEP with open boundaries, J. Phys. A: Math. Theor. 46 (2013), 145003, 21 pages, arXiv:1212.3366.
  29. Melo C.S., Ribeiro G.A.P., Martins M.J., Bethe ansatz for the XXX-S chain with non-diagonal open boundaries, Nuclear Phys. B 711 (2005), 565-603, nlin.SI/0411038.
  30. Murgan R., Nepomechie R.I., Bethe ansatz derived from the functional relations of the open XXZ chain for new special cases, J. Stat. Mech. Theory Exp. 2005 (2005), P05007, 12 pages, hep-th/0504124.
  31. Nepomechie R.I., Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms, J. Phys. A: Math. Gen. 37 (2004), 433-440, hep-th/0304092.
  32. Nepomechie R.I., Inhomogeneous T-Q equation for the open XXX chain with general boundary terms: completeness and arbitrary spin, J. Phys. A: Math. Theor. 46 (2013), 442002, 7 pages, arXiv:1307.5049.
  33. Niccoli G., Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: complete spectrum and matrix elements of some quasi-local operators, J. Stat. Mech. Theory Exp. 2012 (2012), P10025, 42 pages, arXiv:1206.0646.
  34. Pimenta R.A., Lima-Santos A., Algebraic Bethe ansatz for the six vertex model with upper triangular K-matrices, J. Phys. A: Math. Theor. 46 (2013), 455002, 13 pages, arXiv:1308.4446.
  35. Sklyanin E.K., Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen. 21 (1988), 2375-2389.
  36. Sklyanin E.K., Takhtadzhyan L.A., Faddeev L.D., Quantum inverse problem method. I, Theoret. and Math. Phys. 40 (1979), 688-706.
  37. Tsuchiya O., Determinant formula for the six-vertex model with reflecting end, J. Math. Phys. 39 (1998), 5946-5951, solv-int/9804010.
  38. Yang C.N., Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312-1315.
  39. Yang W.L., Zhang Y.Z., On the second reference state and complete eigenstates of the open XXZ chain, J. High Energy Phys. 2007 (2007), no. 4, 044, 11 pages, hep-th/0703222.

Previous article  Next article   Contents of Volume 9 (2013)