Abstract
We study the class of \( \mathfrak{o} _{2n+1}\)-invariant quantum integrable models in the framework of the algebraic Bethe ansatz and propose a construction of the \( \mathfrak{o} _{2n+1}\)-invariant Bethe vector in terms of the Drinfeld currents for the Yangian double \( \mathcal{D}Y ( \mathfrak{o} _{2n+1})\). We calculate the action of the monodromy matrix elements on the off-shell Bethe vectors for these models and obtain recurrence relations for these vectors. The action formulas can be used to investigate scalar products of Bethe vectors in \( \mathfrak{o} _{2n+1}\)-invariant models.
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Notes
We recall that the zero modes \(T_{i,j}[0]\) are denoted by \(T_{i,j}\) for \(-n\le i,j\le n\).
In what follows, we omit the superscript in the notation for the monodromy matrix elements \(T^{ \mathbb{K} }_{i,j}(u)\) and always assume expansion (3.8).
References
A. B. Zamolodchikov and Al. B. Zamolodchikov, “Factorized \(S\)-matrices in two dimensions as the exact solutions of certain relativistic quantum field models,” Ann. Phys., 120, 253–291 (1979).
N. Yu. Reshetikhin, “Integrable models of quantum one-dimensional magnets with \(O(n)\) and \(Sp(2k)\) symmetry,” Theor. Math. Phys., 63, 555–569 (1985).
N. Y. Reshetikhin, “Algebraic Bethe ansatz for \(SO(N)\)-invariant transfer matrices,” J. Math. Sci., 54, 940–951 (1991).
M. J. Martins and P. B. Ramos, “The algebraic Bethe ansatz for rational braid-monoid lattice models,” Nucl. Phys. B, 500, 579–620 (1997); arXiv:hep-th/9703023v1 (1997).
V. G. Drinfeld, “Quantum groups,” J. Soviet Math., 41, 898–915 (1988).
V. G. Drinfeld, “A new realization of Yangians and of quantum affine algebras,” Dokl. Math., 36, 212–216 (1988).
J. Ding and I. Frenkel, “Isomorphism of two realizations of quantum affine algebra \(U_q( \mathfrak{gl} (n))\),” Commun. Math. Phys., 156, 277–300 (1993).
N. Jing, F. Yang, and M. Liu, “Yangian doubles of classical types and their vertex representations,” J. Math. Phys., 61, 051704 (2020); arXiv:1810.06484v4 [math.QA] (2018).
N. Jing, M. Liu, and A. Molev, “Isomorphism between the \(R\)-matrix and Drinfeld presentations of Yangian in types \(B\), \(C\), and \(D\),” Commun. Math. Phys., 361, 827–872 (2018); arXiv:1705.08155v3 [math.QA] (2017).
N. Jing, M. Liu, and A. Molev, “Isomorphism between the \(R\)-matrix and Drinfeld presentations of quantum affine algebra: Type \(C\),” J. Math. Phys., 61, 031701 (2020); arXiv:1903.00204v4 [math.QA] (2019).
N. Jing, M. Liu, and A. Molev, “Isomorphism between the \(R\)-matrix and Drinfeld presentations of quantum affine algebra: Types \(B\) and \(D\),” SIGMA, 16, 043 (2020); arXiv:1911.03496v2 [math.QA] (2019).
S. Khoroshkin and S. Pakuliak, “A computation of an universal weight function or the quantum affine algebra \(U_q( \mathfrak{gl} (N))\),” J. Math. Kyoto Univ., 48, 277–321 (2008); arXiv:0711.2819v2 [math.QA] (2011).
S. Pakuliak, E. Ragoucy, and N. A. Slavnov, “Bethe vectors of quantum integrable models based on \(U_q( \widehat { \mathfrak{gl} }(N)\),” J. Phys. A, 47, 105202 (2014); arXiv:1310.3253v1 [math-ph] (2013).
A. A. Hutsalyuk, A. Liashyk, S. Z. Pakulyak, E. Ragoucy, and N. A. Slavnov, “Current presentation for the super-Yangian double \(DY( \mathfrak{gl} (m|n))\) and Bethe vectors,” Russian Math. Surveys, 72, 33–99 (2017); arXiv:1611.09620v2 [math-ph] (2016).
B. Enriquez, S. Khoroshkin, and S. Pakuliak, “Weight functions and Drinfeld currents,” Commun. Math. Phys., 276, 691–725 (2007).
A. Liashyk and S. Z. Pakuliak, “Gauss coordinates vs currents for the Yangian doubles of the classical types,” arXiv:2006.01579v3 [math-ph] (2020).
D. Karakhanyanan and R. Kirschner, “Spinorial \(R\) operator and algebraic Bethe ansatz,” Nucl. Phys. B, 951, 114905 (2020); arXiv:1911.08385v1 [math-ph] (2019).
A. Gerrard and V. Regelskis, “Nested algebraic Bethe ansatz for orthogonal and symplectic open spin chains,” Nucl. Phys. B, 952, 114909 (2020); arXiv:1909.12123v1 [math-ph] (2019).
A. Gerrard and V. Regelskis, “Nested algebraic Bethe ansatz for deformed orthogonal and symplectic spin chains,” Nucl. Phys. B, 956, 115021 (2020); arXiv:1912.11497v2 [math-ph] (2019).
A. N. Liashyk, S. Z. Pakuliak, E. Ragoucy, and N. A. Slavnov, “Bethe vectors for orthogonal integrable models,” Theor. Math. Phys., 201, 1545–1564 (2019); arXiv:1906.03202v1 [math-ph] (2019).
A. Hutsalyuk, A. Liashyk, S. Z. Pakuliak, E. Ragoucy, and N. A. Slavnov, “Actions of the monodromy matrix elements onto \( \mathfrak{gl} (m|n)\)-invariant Bethe vectors,” arXiv:2005.09249v1 [math-ph] (2020).
A. Hutsalyuk, A. Liashyk, S. Z. Pakuliak, E. Ragoucy, and N. A. Slavnov, “Scalar products of Bethe vectors in the models with \( \mathfrak{gl} (m|n)\) symmetry,” Nucl. Phys. B, 923, 277–311 (2017); arXiv:1704.08173v3 [math-ph] (2017).
A. Liashyk, S. Z. Pakuliak, E. Ragoucy, and N. A. Slavnov, “New symmetries of \( \mathfrak{gl} (N)\)-invariant Bethe vectors,” J. Stat. Mech., 2019, 044001 (2019).
S. Belliard, S. Z. Pakuliak, and E. Ragoucy, “Universal Bethe ansatz and scalar products of Bethe vectors,” SIGMA, 6, 094 (2010); arXiv:1012.1455v2 [math-ph] (2010).
Acknowledgments
The authors thank E. Ragoucy and N. A. Slavnov for the fruitful discussions.
Funding
This research was performed at the Skolkovo Institute of Science and Technology under a grant from the Russian Science Foundation (Project No. 19-11-00275).
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Liashyk, A., Pakuliak, S.Z. Algebraic Bethe ansatz for \(\mathfrak o_{2n+1}\)-invariant integrable models. Theor Math Phys 206, 19–39 (2021). https://doi.org/10.1134/S0040577921010025
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DOI: https://doi.org/10.1134/S0040577921010025