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

SIGMA 13 (2017), 094, 13 pages      arXiv:1710.08490

Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary

Nicolas Crampe
Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, Montpellier, France

Received November 01, 2017, in final form December 06, 2017; Published online December 13, 2017

We solve the XXZ Gaudin model with generic boundary using the modified algebraic Bethe ansatz. The diagonal and triangular cases have been recovered in this general framework. We show that the model for odd or even lengths has two different behaviors. The corresponding Bethe equations are computed for all the cases. For the chain with even length, inhomogeneous Bethe equations are necessary. The higher spin Gaudin models with generic boundary is also treated.

Key words: integrability; algebraic Bethe ansatz; Gaudin models; Bethe equations.

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  1. Avan J., Belliard S., Grosjean N., Pimenta R.A., Modified algebraic Bethe ansatz for XXZ chain on the segment - III: Proof, Nuclear Phys. B 899 (2015), 229-246, arXiv:1506.02147.
  2. Baseilhac P., Belliard S., The half-infinite XXZ chain in Onsager's approach, Nuclear Phys. B 873 (2013), 550-584, arXiv:1211.6304.
  3. Baseilhac P., Belliard S., Crampe N., FRT presentation of the Onsager algebras, arXiv:1709.08555.
  4. 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.
  5. Belliard S., Modified algebraic Bethe ansatz for XXZ chain on the segment - I: Triangular cases, Nuclear Phys. B 892 (2015), 1-20, arXiv:1408.4840.
  6. Belliard S., Crampe N., Heisenberg XXX model with general boundaries: eigenvectors from algebraic Bethe ansatz, SIGMA 9 (2013), 072, 12 pages, arXiv:1309.6165.
  7. Belliard S., Crampe N., Ragoucy E., Algebraic Bethe ansatz for open XXX model with triangular boundary matrices, Lett. Math. Phys. 103 (2013), 493-506, arXiv:1209.4269.
  8. Belliard S., Pimenta R.A., Modified algebraic Bethe ansatz for XXZ chain on the segment—II: General cases, Nuclear Phys. B 894 (2015), 527-552, arXiv:1412.7511.
  9. Belliard S., Pimenta R.A., Slavnov and Gaudin-Korepin formulas for models without ${\rm U}(1)$ symmetry: the twisted XXX chain, SIGMA 11 (2015), 099, 12 pages, arXiv:1506.06550.
  10. Cao J., Lin H.-Q., Shi K.-J., Wang Y., Exact solution of $XXZ$ spin chain with unparallel boundary fields, Nuclear Phys. B 663 (2003), 487-519, cond-mat/0212163.
  11. Cao J., Yang W.-L., Shi K.-J., Wang Y., Off-diagonal Bethe ansatz and exact solution a topological spin ring, Phys. Rev. Lett. 111 (2013), 137201, 5 pages, arXiv:1305.7328.
  12. Cao J., Yang W.-L., Shi K.-J., 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.
  13. Cao J., Yang W.-L., Shi K.-J., Wang Y., Off-diagonal Bethe ansatz solutions of the anisotropic spin-$\frac12$ chains with arbitrary boundary fields, Nuclear Phys. B 877 (2013), 152-175, arXiv:1307.2023.
  14. Cirilo António N., Manojlović N., Nagy Z., Trigonometric $s\ell(2)$ Gaudin model with boundary terms, Rev. Math. Phys. 25 (2013), 1343004, 14 pages, arXiv:1303.2481.
  15. Crampe N., Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries, J. Phys. A: Math. Theor. 48 (2015), 08FT01, 12 pages, arXiv:1411.7954.
  16. Crampe N., Ragoucy E., Generalized coordinate Bethe ansatz for non-diagonal boundaries, Nuclear Phys. B 858 (2012), 502-512, arXiv:1105.0338.
  17. Crampe 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.
  18. Crampe 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.
  19. Faldella S., Kitanine N., Niccoli G., The complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms, J. Stat. Mech. Theory Exp. 2014 (2014), P01011, 35 pages, arXiv:1307.3960.
  20. Faribault A., Tschirhart H., Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields, SciPost Phys. 3 (2017), 009, 24 pages, arXiv:1704.01873.
  21. 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.
  22. Frahm H., Seel A., Wirth T., Separation of variables in the open $XXX$ chain, Nuclear Phys. B 802 (2008), 351-367, arXiv:0803.1776.
  23. 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.
  24. Gaudin M., Diagonalisation d'une classe d'Hamiltoniens de spin, J. Physique 37 (1976), 1089-1098.
  25. Gorissen M., Lazarescu A., Mallick K., Vanderzande C., Exact current statistics of the ASEP with open boundaries, Phys. Rev. Lett. 109 (2012), 170601, 5 pages, arXiv:1207.6879.
  26. Hao K., Yang W.-L., Fan H., Liu S.-Y., Wu K., Yang Z.-Y., Zhang Y.-Z., Determinant representations for scalar products of the XXZ Gaudin model with general boundary terms, Nuclear Phys. B 862 (2012), 835-849, arXiv:1205.0597.
  27. Hikami K., Separation of variables in the BC-type Gaudin magnet, J. Phys. A: Math. Gen. 28 (1995), 4053-4061, solv-int/9506001.
  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. Lazarescu A., Mallick K., An exact formula for the statistics of the current in the TASEP with open boundaries, J. Phys. A: Math. Theor. 44 (2011), 315001, 16 pages, arXiv:1104.5089.
  30. Lazarescu A., Pasquier V., Bethe ansatz and $Q$-operator for the open ASEP, J. Phys. A: Math. Theor. 47 (2014), 295202, 48 pages, arXiv:1403.6963.
  31. Manojlović N., Salom I., Algebraic Bethe ansatz for the XXZ Heisenberg spin chain with triangular boundaries and the corresponding Gaudin model, Nuclear Phys. B 923 (2017), 73-106, arXiv:1705.02235.
  32. Manojlović N., Salom I., Algebraic Bethe ansatz for the trigonometric $s\ell(2)$ Gaudin model with triangular boundary, arXiv:1709.06419.
  33. 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.
  34. 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.
  35. 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.
  36. Nepomechie R.I., An 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.
  37. 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.
  38. 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.
  39. Sklyanin E.K., Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen. 21 (1988), 2375-2389.
  40. Skrypnyk T., Generalized Gaudin spin chains, nonskew symmetric $r$-matrices, and reflection equation algebras, J. Math. Phys. 48 (2007), 113521, 17 pages.
  41. Takhtadzhan L.A., Faddeev L.D., The quantum method for the inverse problem and the Heisenberg $XYZ$ model, Russian Math. Surveys 34 (1979), no. 5, 11-68.
  42. 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.
  43. Yang W.-L., Zhang Y.-Z., Sasaki R., $A_{n-1}$ Gaudin model with open boundaries, Nuclear Phys. B 729 (2005), 594-610, hep-th/0507148.
  44. Zhang X., Li Y.-Y., Cao J., Yang W.-L., Shi K.-J., Wang Y., Retrieve the Bethe states of quantum integrable models solved via the off-diagonal Bethe Ansatz, J. Stat. Mech. Theory Exp. 2015 (2015), P05014, 18 pages, arXiv:1407.5294.

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