Annales Henri Poincaré

, Volume 20, Issue 2, pp 339–392 | Cite as

Nested Algebraic Bethe Ansatz for Open Spin Chains with Even Twisted Yangian Symmetry

  • Allan Gerrard
  • Niall MacKay
  • Vidas RegelskisEmail author


We present a nested algebraic Bethe ansatz for a one-dimensional open spin chain whose boundary quantum spaces are irreducible \(\mathfrak {so}_{2n}\)- or \(\mathfrak {sp}_{2n}\)-representations, and the monodromy matrix satisfies the defining relations of the Olshanskii twisted Yangian \(Y^\pm (\mathfrak {gl}_{2n})\). We use a generalization of the Bethe ansatz introduced by De Vega and Karowski which allows us to relate the spectral problem of a \(\mathfrak {so}_{2n}\)- or \(\mathfrak {sp}_{2n}\)-symmetric open spin chain to that of a \(\mathfrak {gl}_{n}\)-symmetric periodic spin chain. We explicitly derive the structure of the Bethe vectors and the nested Bethe equations.


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The authors thank Samuel Belliard, Nicolas Crampé, Nicolas Guay, Bart Vlaar and Curtis Wendlandt for useful discussions and the anonymous referee for comments and suggestions. V.R. was in part supported by the UK EPSRC under the Grant EP/K031805/1 and by the European Social Fund, Grant Number 09.3.3-LMT-K-712-02-0017. A.G. was supported by an EPSRC PhD studentship.


  1. 1.
    Avan, J., Doikou, A., Karaiskos, N.: The \({sl}(N)\) twisted Yangian: bulk-boundary scattering and defects. J. Stat. Mech. P05024 (2015). arXiv:1412.6480
  2. 2.
    Arnaudon, D., Avan, J., Crampé, N., Doikou, A., Frappat, L., Ragoucy, E.: General boundary conditions for the \({\mathfrak{s}}{\mathfrak{l}}(N)\) and \({\mathfrak{s}}{\mathfrak{l}}(M|N)\) open spin chains. J. Stat. Mech. P08005 (2004). arXiv:math-ph/0406021
  3. 3.
    Arnaudon, D., Crampe, N., Doikou, A., Frappat, L., Ragoucy, E.: Analytical Bethe Ansatz for open spin chains with soliton non preserving boundary conditions. Int. J. Mod. Phys. A 21, 1537 (2006). arXiv:math-ph/0503014 ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Arnaudon, D., Crampe, N., Doikou, A., Frappat, L., Ragoucy, E.: Spectrum and Bethe ansatz equations for the \(U_{q}(gl(N))\) closed and open spin chains in any representation. Ann. H. Poincaré 7, 1217 (2006). arXiv:math-ph/0512037 CrossRefzbMATHGoogle Scholar
  5. 5.
    Babichenko, A., Regelskis, V.: On boundary fusion and functional relations in the Baxterized affine Hecke algebra. J. Math. Phys. 55, 043503 (2014). arXiv:1305.1941 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Batchelor, M.T., Baxter, R.J., O’Rourke, M.J., Yung, C.M.: Exact solution and interfacial tension of the six-vertex model with anti-periodic boundary conditions. J. Phys. A 28, 2759–2770 (1995). arXiv:hep-th/9502040 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Belliard, S., Ragoucy, E.: The nested Bethe ansatz for ‘all’ closed spin chains. J. Phys. A 41, 295202 (2008). arXiv:0804.2822 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Belliard, S., Ragoucy, E.: Nested Bethe ansatz for ‘all’ open spin chains with diagonal boundary conditions. J. Phys. A 42, 205203 (2009). arXiv:0902.0321 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Doikou, A.: Quantum spin chain with “soliton non-preserving” boundary conditions. J. Phys. A 33, 8797–8808 (2000). arXiv:hep-th/0006197 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    De Vega, H.J., Karowski, M.: Exact Bethe ansatz solution of O(2N) symmetric theories. Nuc. Phys. B 280, 225–254 (1987)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Fan, H.: Bethe ansatz for the Izergin–Korepin model. Nucl. Phys. B 488, 409–425 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Faddeev, L.D., Takhtajan, L.A.: The quantum method of the inverse problem and the Heisenberg XYZ model. Russ. Math. Surv. 34, 11–60 (1979)Google Scholar
  13. 13.
    Frappat, L., Khoroshkin, S., Pakuliak, S., Ragoucy, E.: Bethe ansatz for the universal weight function. Ann. Henri Poincaré 10, 513–548 (2009). arXiv:0810.3135 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Galleas, W.: Functional relations from the Yang-Baxter algebra: eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions. Nucl. Phys. B 790(3), 524–542 (2008). arXiv:0708.0009
  15. 15.
    Gombor, T., Palla, L.: Algebraic Bethe Ansatz for O(2N) sigma models with integrable diagonal boundaries. JHEP 02, 158 (2016). arXiv:1511.03107
  16. 16.
    Guang-Liang, L., Kang-Jie, S., Rui-Hong, Y.: Algebraic Bethe Ansatz Solution to \(C_N\) Vertex Model with Open Boundary Conditions. Commun. Theor. Phys. 44(1), 89–98 (2005)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Guay, N., Regelskis, V.: Twisted Yangians for symmetric pairs of types B, C. D. Math. Z. 284, 131 (2016). arXiv:1407.5247 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hutsalyuk, A., Liashyk, A., Pakuliak, S. Z., Ragoucy, E., Slavnov, N. A.: Scalar products of Bethe vectors in the models with \({\mathfrak{g}}{\mathfrak{l}}(m|n)\) symmetry. Nucl. Phys. B 923, 277–311 (2017). arXiv:1704.08173
  19. 19.
    Hutsalyuk, A., Liashyk, A., Pakuliak, S. Z., Ragoucy, E., Slavnov, N. A.: Scalar products and norm of Bethe vectors for integrable models based on \(U_q({\hat{\mathfrak{g}\mathfrak{l}}}_n)\). SciPost Phys. 4, 006 (2018). arXiv:1711.03867
  20. 20.
    Izergin, A.G., Korepin, V.E.: The quantum inverse scattering method approach to correlation functions. Commun. Math. Phys. 94, 67–92 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jing, N., Liu, M., Molev, A.: Isomorphism between the R-matrix and Drinfeld presentations of Yangian in types B, C and D. arXiv:1705.08155
  22. 22.
    Kitanine, N., Kozlowski, K., Maillet, J.-M., Slavnov, N. A., Terras, V.: A form factor approach to the asymptotic behavior of correlation functions. J. Stat. Mech. P12010 (2011), arXiv:1110.0803 [hep-th]
  23. 23.
    Kitanine, N., Kozlowski, K., Maillet, J.-M., Slavnov, N. A., Terras, V.: Form factor approach to dynamical correlation functions in critical models. J. Stat. Mech. P09001 (2012). arXiv:1206.2630
  24. 24.
    Kitanine, N., Maillet, J.-M., Slavnov, N.A., Terras, V.: Master equation for spin-spin correlation functions of the XXZ chain. Nucl. Phys. B 712, 600–622 (2005). arXiv:hep-th/0406190 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kitanine, N., Maillet, J.-M., Terras, V.: Form factors of the XXZ Heisenberg spin-\(1/2\) finite chain. Nucl. Phys. B 554, 647–678 (1999). arXiv:math-ph/9807020 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Korepin, V.E.: Calculation of norms of Bethe wave functions. Commun. Math. Phys. 86(3), 391–418 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kulish, P.P., Reshetikhin, NYu.: Diagonalisation of GL(N) invariant transfer matrices and quantum N-wave system (Lee model). J. Phys. A: Math. Gen. 16, 591–596 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Molev, A., Nazarov, M., Olshanskii, G.: Yangians and classical Lie algebras. Russ. Math. Surv. 51(2), 205–282 (1996). arXiv:hep-th/9409025 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Molev, A.: Finite-dimensional irreducible representations of twisted Yangians. J. Math. Phys. 39, 5559–5600 (1998). arXiv:q-alg/9711022
  30. 30.
    Molev, A.: Irreducibility criterion for tensor products of Yangian evaluation modules. Duke Math. J. 112, 307–341 (2002). arXiv:math/0009183 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Molev, A.: Yangians and Classical Lie Algebras. Mathematical Surveys and Monographs, vol. 143. American Mathematical Society, Providence (2007)Google Scholar
  32. 32.
    Olshanskii, G.: Twisted Yangians and infinite-dimensional classical Lie algebras. Quantum groups (Leningrad, 1990), pp. 104–119. Lecture Notes in Math., vol. 1510. Springer, Berlin (1992)Google Scholar
  33. 33.
    Pakuliak, S., Ragoucy, E., Slavnov, N.: Bethe vectors of quantum integrable models based on \(U_q({\hat{{\mathfrak{gl}}}}_n)\). J. Phys. A 47, 105202 (2014). arXiv:1310.3253 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Pakuliak, S., Ragoucy, E., Slavnov, N.: Bethe vectors for models based on the super-Yangian \(Y(\mathfrak{g}\mathfrak{l}(m|n))\). J. Integrable Syst. 2, 1–31 (2017). arXiv:1604.02311
  35. 35.
    Pakuliak, S., Ragoucy, E., Slavnov, N.: Nested Algebraic Bethe Ansatz in integrable models: recent results. arXiv:1803.00103
  36. 36.
    Reshetikhin, NYu.: Algebraic Bethe ansatz for SO(N)-invariant transfer matrices. J. Sov. Math. 54, 940 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sklyanin, E.K.: Boundary conditions for integrable quantum systems. J. Phys. A 21, 2375 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Sklyanin, E.K., Takhtadzhyan, L.A., Faddeev, L.D.: Quantum inverse problem method. I. Theor. Math. Phys. 40(2), 688–706 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Slavnov, N.A.: Calculation of scalar products of wave functions and form factors in the framework of the algebraic Bethe ansatz. Theor. Math. Phys. 79, 502–508 (1989)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Slavnov, N.A.: The algebraic Bethe ansatz and quantum integrable systems. Russ. Math. Surv. 62, 727 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Tarasov, V.A.: Algebraic Bethe ansatz for the Izergin-Korepin R matrix. Theor. Math. Phys. 56, 793 (1988)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Tarasov, V., Varchenko, A.: Combinatorial formulae for nested Bethe vectors. SIGMA 9, 048 (2013). arXiv:math/0702277 MathSciNetzbMATHGoogle Scholar
  43. 43.
    Wang, Y., Yang, W.-Li, Cao, J., Shi, K.: Off-Diagonal Bethe Ansatz for Exactly Solvable Models. Springer 2015Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of YorkYorkUK
  2. 2.Institute of Theoretical Physics and AstronomyVilnius UniversityVilniusLithuania

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