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

SIGMA 19 (2023), 046, 47 pages      arXiv:2212.09696      https://doi.org/10.3842/SIGMA.2023.046

Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via $U_{\mathfrak{q}}\mathfrak{sl}_2$ Symmetry

Dmitry Chernyak ab, Azat M. Gainutdinov c, Jesper Lykke Jacobsen abd and Hubert Saleur be
a) Laboratoire de Physique de l'École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, 75005 Paris, France
b) Institut de Physique Théorique, Paris Saclay, CEA, CNRS, 91191 Gif-sur-Yvette, France
c) Institut Denis Poisson, CNRS, Université de Tours, Parc de Grandmont, 37200 Tours, France
d) Sorbonne Université, École Normale Supérieure, CNRS, Laboratoire de Physique (LPENS), 75005 Paris, France
e) USC Physics and Astronomy Department, Los Angeles Ca 90089, USA

Received January 26, 2023, in final form July 04, 2023; Published online July 16, 2023

Abstract
We derive by the traditional algebraic Bethe ansatz method the Bethe equations for the general open XXZ spin chain with non-diagonal boundary terms under the Nepomechie constraint [J. Phys. A 37 (2004), 433-440, arXiv:hep-th/0304092]. The technical difficulties due to the breaking of $\mathsf{U}(1)$ symmetry and the absence of a reference state are overcome by an algebraic construction where the two-boundary Temperley-Lieb Hamiltonian is realised in a new $U_{\mathfrak{q}}\mathfrak{sl}_2$-invariant spin chain involving infinite-dimensional Verma modules on the edges [J. High Energy Phys. 2022 (2022), no. 11, 016, 64 pages, arXiv:2207.12772]. The equivalence of the two Hamiltonians is established by proving Schur-Weyl duality between $U_{\mathfrak{q}}\mathfrak{sl}_2$ and the two-boundary Temperley-Lieb algebra. In this framework, the Nepomechie condition turns out to have a simple algebraic interpretation in terms of quantum group fusion rules.

Key words: quantum integrable models; non-diagonal K-matrices; Verma modules; Temperley-Lieb algebras.

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References

  1. Ang M., Sun X., Integrability of the conformal loop ensemble, arXiv:2107.01788.
  2. 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.
  3. Bajnok Z., Equivalences between spin models induced by defects, J. Stat. Mech. Theory Exp. 2006 (2006), P06010, 13 pages, arXiv:hep-th/0601107.
  4. Bajnok Z., Granet E., Jacobsen J.L., Nepomechie R.I., On generalized $Q$-systems, High Energy Phys. J 2020 (2020), no. 3, 177, 27 pages, arXiv:1910.07805.
  5. 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, arXiv:hep-th/0703106.
  6. 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.
  7. Belliard S., Crampé N., Heisenberg XXX model with general boundaries: eigenvectors from algebraic Bethe ansatz, SIGMA 9 (2013), 072, 12 pages, arXiv:1309.6165.
  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. Boos H., Göhmann F., Klümper A., Nirov K.S., Razumov A.V., Exercises with the universal $R$-matrix, J. Phys. A 43 (2010), 415208, 35 pages.
  10. Cao J., Lin H., Shi K., Wang Y., Exact solution of $XXZ$ spin chain with unparallel boundary fields, Nuclear Phys. B 663 (2003), 487-519, arXiv:cond-mat/0212163.
  11. Chari V., Pressley A., Quantum affine algebras, Comm. Math. Phys. 142 (1991), 261-283.
  12. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
  13. Chernyak D., Leurent S., Volin D., Completeness of Wronskian Bethe equations for rational $\mathfrak{gl}_{m|n}$ spin chains, Comm. Math. Phys. 391 (2022), 969-1045, arXiv:2004.02865.
  14. Chernyak D., Gainutdinov A.M., Saleur H., $U_{\mathfrak q}\mathfrak{sl}_2$-invariant non-compact boundary conditions for the XXZ spin chain, J. High Energy Phys. 2022 (2022), no. 11, 016, 64 pages, arXiv:2207.12772.
  15. Crampé N., Poulain d'Andecy L., Baxterisation of the fused Hecke algebra and $R$-matrices with $\mathfrak{gl}(N)$-symmetry, Lett. Math. Phys. 111 (2021), 92, 21 pages, arXiv:2004.05035.
  16. 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.
  17. Daugherty Z., Ram A., Calibrated representations of two boundary Temperley-Lieb algebras, arXiv:2009.02812.
  18. Daugherty Z., Ram A., Two boundary Hecke algebras and combinatorics of type C, arXiv:1804.10296.
  19. de Gier J., Nichols A., The two-boundary Temperley-Lieb algebra, J. Algebra 321 (2009), 1132-1167, arXiv:math.RT/0703338.
  20. de Gier J., Pyatov P., Bethe ansatz for the Temperley-Lieb loop model with open boundaries, J. Stat. Mech. Theory Exp. 2004 (2004), 002, 27 pages, arXiv:hep-th/0312235.
  21. de Vega H.J., Gonz'alez-Ruiz A., Boundary $K$-matrices for the six vertex and the $n(2n-1) A_{n-1}$ vertex models, J. Phys. A 26 (1993), L519-L524, arXiv:hep-th/9211114.
  22. de Vega H.J., Gonz'alez-Ruiz A., Boundary $K$-matrices for the $XYZ$, $XXZ$ and $XXX$ spin chains, J. Phys. A 27 (1994), 6129-6137, arXiv:hep-th/9306089.
  23. Delius G.W., Nepomechie R.I., Solutions of the boundary Yang-Baxter equation for arbitrary spin, J. Phys. A 35 (2002), L341-L348, arXiv:hep-th/0204076.
  24. Doikou A., Martin P.P., Hecke algebraic approach to the reflection equation for spin chains, J. Phys. A 36 (2003), 2203-2226, arXiv:hep-th/0206076.
  25. Drinfel'd V.G., Hopf algebras and the quantum Yang-Baxter equation, Sov. Math., Dokl. 32 (1985), 256-258.
  26. Drinfel'd V.G., Quantum groups, J. Sov. Math. 41 (1988), 898-915.
  27. Dubail J., Conditions aux bords dans des théories conformes non unitaires, Ph.D. Thesis, Université Paris Sud - Paris XI, 2010.
  28. Dubail J., Jacobsen J.L., Saleur H., Conformal two-boundary loop model on the annulus, Nuclear Phys. B 813 (2009), 430-459, arXiv:0812.2746.
  29. Dubail J., Jacobsen J.L., Saleur H., Exact solution of the anisotropic special transition in the $O(n)$ model in two dimensions, Phys. Rev. Lett. 103 (2009), 145701, 4 pages, arXiv:0909.2949.
  30. Dubail J., Jacobsen J.L., Saleur H., Conformal boundary conditions in the critical ${\mathcal O}(n)$ model and dilute loop models, Nuclear Phys. B 827 (2010), 457-502, arXiv:0905.1382.
  31. Filali G., Kitanine N., Spin chains with non-diagonal boundaries and trigonometric SOS model with reflecting end, SIGMA 7 (2011), 012, 22 pages, arXiv:1011.0660.
  32. Gainutdinov A.M., Nepomechie R.I., Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity, Nuclear Phys. B 909 (2016), 796-839, arXiv:1603.09249.
  33. Galleas W., Martins M.J., Solution of the ${\rm SU}(N)$ vertex model with non-diagonal open boundaries, Phys. Lett. A 335 (2005), 167-174, arXiv:nlin.SI/0407027.
  34. Goodman F.M., Wenzl H., The Temperley-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334.
  35. Granet E., Jacobsen J.L., On zero-remainder conditions in the Bethe ansatz, J. High Energy Phys. 2020 (2020), no. 3, 178, 14 pages, arXiv:1910.07797.
  36. Ikhlef Y., Jacobsen J.L., Saleur H., Three-point functions in $c\leq 1$ Liouville theory and conformal loop ensembles, Phys. Rev. Lett. 116 (2016), 130601, 5 pages, arXiv:1509.03538.
  37. Inami T., Odake S., Zhang Y.Z., Reflection $K$-matrices of the $19$-vertex model and $XXZ$ spin-$1$ chain with general boundary terms, Nuclear Phys. B 470 (1996), 419-432, arXiv:hep-th/9601049.
  38. Jacobsen J.L., Saleur H., Combinatorial aspects of boundary loop models, J. Stat. Mech. Theory Exp. 2008.
  39. Jacobsen J.L., Saleur H., Conformal boundary loop models, Nuclear Phys. B 788 (2008), 137-166, arXiv:math-ph/0611078.
  40. Jimbo M., A $q$-difference analogue of $U({\mathfrak g})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69.
  41. Jimbo M., A $q$-analogue of $U({\mathfrak g}{\mathfrak l}(N+1))$, Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247-252.
  42. Jones V.F.R., Baxterization, Internat. J. Modern Phys. A 6 (1991), 2035-2043.
  43. Kassel C., Quantum groups, Grad. Texts in Math., Vol. 155, Springer, New York, 1995.
  44. Kitanine N., Maillet J.M., Niccoli G., Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from separation of variables, J. Stat. Mech. Theory Exp. (2014), P05015, 30 pages, arXiv:1401.4901.
  45. Kupiainen A., Rhodes R., Vargas V., Integrability of Liouville theory: proof of the DOZZ formula, Ann. of Math. 191 (2020), 81-166, arXiv:1707.08785.
  46. Mangazeev V.V., Lu X., Boundary matrices for the higher spin six vertex model, Nuclear Phys. B 945 (2019), 114665, 21 pages, arXiv:1903.00274.
  47. Martin P., Saleur H., The blob algebra and the periodic Temperley-Lieb algebra, Lett. Math. Phys. 30 (1994), 189-206, arXiv:hep-th/9302094.
  48. Martin P.P., On Schur-Weyl duality, $A_n$ Hecke algebras and quantum ${\rm sl}(N)$ on $\bigotimes^{n+1}{\bf C}^N$, in Infinite analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., Vol. 16, World Sci. Publ., River Edge, NJ, 1992, 645-673.
  49. Martin P.P., McAnally D.S., On commutants, dual pairs and nonsemisimple algebras from statistical mechanics, in Infinite analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., Vol. 16, World Sci. Publ., River Edge, NJ, 1992, 675-705.
  50. Mukhin E., Tarasov V., Varchenko A., Bethe algebra of homogeneous $XXX$ Heisenberg model has simple spectrum, Comm. Math. Phys. 288 (2009), 1-42, arXiv:0706.0688.
  51. Mukhin E., Tarasov V., Varchenko A., Spaces of quasi-exponentials and representations of the Yangian $Y(\mathfrak{gl}_N)$, Transform. Groups 19 (2014), 861-885, arXiv:1303.1578.
  52. Naǐmark M.A., Decomposition of a tensor product of irreducible representations of the proper Lorentz group into irreducible representations. I. The case of a tensor product of representations of the fundamental series, Trudy Moskov. Mat. Obvsč. 8 (1959), 121-153.
  53. Nepomechie R.I., Functional relations and Bethe ansatz for the $XXZ$ chain, J. Stat. Phys. 111 (2003), 1363-1376, arXiv:hep-th/0211001.
  54. Nepomechie R.I., Bethe ansatz solution of the open $XXZ$ chain with nondiagonal boundary terms, J. Phys. A 37 (2004), 433-440, arXiv:hep-th/0304092.
  55. Nepomechie R.I., Ravanini F., Completeness of the Bethe ansatz solution of the open $XXZ$ chain with nondiagonal boundary terms, J. Phys. A 36 (2003), 11391-11401, arXiv:hep-th/0307095.
  56. Pasquier V., Saleur H., Common structures between finite systems and conformal field theories through quantum groups, Nuclear Phys. B 330 (1990), 523-556.
  57. Simon D., Construction of a coordinate Bethe ansatz for the asymmetric simple exclusion process with open boundaries, J. Stat. Mech. Theory Exp. 2009 (2009), P07017, 28 pages, arXiv:0903.4968.
  58. Sklyanin E.K., Boundary conditions for integrable quantum systems, J. Phys. A 21 (1988), 2375-2389.
  59. Temperley H.N.V., Lieb E.H., Relations between the ''percolation'' and ''colouring'' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ''percolation'' problem, Proc. Roy. Soc. London Ser. A 322 (1971), 251-280.
  60. Wang Y., Yang W.-L., Wang Y., Shi K., Off-diagonal Bethe ansatz for exactly solvable models, Springer, Berlin, 2015.

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