Abstract
We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing \( \mathfrak{g}{\mathfrak{l}}_3 \)-invariant R-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided the system of Bethe equations is fulfilled.
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References
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, New Construction of Eigenstates and Separation of Variables for SU(N) Quantum Spin Chains, JHEP 09 (2017) 111 [arXiv:1610.08032] [INSPIRE].
H. Bethe, On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain, Z. Phys. 71 (1931) 205 [INSPIRE].
C.-N. Yang, Some exact results for the many body problems in one dimension with repulsive delta function interaction, Phys. Rev. Lett. 19 (1967) 1312 [INSPIRE].
B. Sutherland, Further Results for the Many-Body Problem in One Dimension, Phys. Rev. Lett. 20 (1968) 98 [INSPIRE].
B. Sutherland, A General Model for Multicomponent Quantum Systems, Phys. Rev. B 12 (1975) 3795 [INSPIRE].
L.D. Faddeev, E.K. Sklyanin and L.A. Takhtajan, The Quantum Inverse Problem Method. 1 Theor. Math. Phys. 40 (1980) 688 [INSPIRE].
L.A. Takhtajan and L.D. Faddeev, The Quantum method of the inverse problem and the Heisenberg XYZ model, Russ. Math. Surveys 34 (1979) 11 [Usp. Math. Nauk 34 (1979) 13] [INSPIRE].
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge (1993).
L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in Relativistic gravitation and gravitational radiation. Proceedings, School of Physics, Les Houches, France, September 26-October 6, 1995, pp. 149-219 [hep-th/9605187] [INSPIRE].
P.P. Kulish, Classical and quantum inverse problem method and generalized Bethe ansatz, Physica D 3 (1981) 246.
P.P. Kulish and N.Yu. Reshetikhin, GL 3 -invariant solutions of the Yang-Baxter equation and associated quantum systems, J. Sov. Math. 34 (1982) 1948 [Zap. Nauchn. Sem. POMI. 120 (1982) 92].
P.P. Kulish and N.Yu. Reshetikhin, Diagonalization of GL(N) invariant transfer matrices and quantum N -wave system (Lee model), J. Phys. A 16 (1983) L591 [INSPIRE].
V. Tarasov and A. Varchenko, Jackson integral representations for solutions of the quantized Knizhnik-Zamolodchikov equation, St. Petersburg Math. J. 6 (1995) 275 [Algebra i Analiz 6 (1994) 90] [hep-th/9311040] [INSPIRE].
V. Tarasov and A. Varchenko, Asymptotic solutions to the quantized Knizhnik-Zamolodchikov equation and Bethe vectors, hep-th/9406060 [INSPIRE].
V. Tarasov and A. Varchenko, Combinatorial formulae for nested Bethe vectors, SIGMA 9 (2013) 048 [math/0702277].
S. Belliard and É. Ragoucy, Nested Bethe ansatz for ‘all’ closed spin chains, J. Phys. A 41 (2008) 295202 [arXiv:0804.2822] [INSPIRE].
S. Pakuliak and S. Khoroshkin, The weight function for the quantum affine algebra \( {U}_q\left({\widehat{\mathfrak{sl}}}_3\right) \), Theor. Math. Phys. 145 (2005) 1373 [math/0610433].
S. Khoroshkin, S. Pakuliak and V. Tarasov, Off-shell Bethe vectors and Drinfeld currents, J. Geom. Phys. 57 (2007) 1713 [math/0610517].
S. Khoroshkin and S. Pakuliak, A computation of universal weight function for quantum affine algebra U q (gl N ), J. Math. Kyoto Univ. 48 (2008) 277 [arXiv:0711.2819].
L. Frappat, S. Khoroshkin, S. Pakuliak and E. Ragoucy, Bethe Ansatz for the Universal Weight Function, Ann. Henri Poincaré 10 (2009) 513 [arXiv:0810.3135].
A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, E. Ragoucy and N.A. Slavnov, Current presentation for the double super-Yangian DY \( \left(\mathfrak{g}{\mathfrak{l}}_3\left(m\Big|n\right)\right) \) and Bethe vectors, Russ. Math. Surv. 72 (2017) 33, [arXiv:1611.09020].
E.K. Sklyanin, Functional Bethe Ansatz, in Integrable and Superintegrable Theories B. Kupershmidt ed., World Scientific, Singapore (1990), pp. 8-33.
E.K. Sklyanin, Quantum inverse scattering method. Selected topics, hep-th/9211111 [INSPIRE].
E.K. Sklyanin, Separation of variables — new trends, Prog. Theor. Phys. Suppl. 118 (1995) 35 [solv-int/9504001] [INSPIRE].
S. Belliard, S. Pakuliak, É. Ragoucy and N.A. Slavnov, Bethe vectors of GL(3)-invariant integrable models, J. Stat. Mech. 1302 (2013) P02020 [arXiv:1210.0768] [INSPIRE].
A. Molev, M. Nazarov and G. Olshansky, Yangians and classical Lie algebras, Russ. Math. Surveys 51 (1996) 205 [hep-th/9409025] [INSPIRE].
A. Molev, Yangians and Classical Lie Algebras, Mathematical Surveys and Monographs, vol. 143, American Mathematical Society, Providence, RI (2007).
A.G. Izergin and V.E. Korepin, A Lattice model related to the nonlinear Schrödinger equation, Dokl. Akad. Nauk Ser. Fiz. 259 (1981) 76 [INSPIRE].
P.P. Kulish and E.K. Sklyanin, Quantum spectral transform method. Recent developments, in Integrable Quantum Field Theories, Lect. Notes Phys. 151 (1982) 61 [INSPIRE].
M. Nazarov and V. Tarasov, Representations of Yangians with Gelfand-Zetlin bases, J. Reine Angew. Math. 496 (1998) 181 [q-alg/9502008].
V.E. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys. 86 (1982) 391 [INSPIRE].
A.G. Izergin, Partition function of the six-vertex model in a finite volume, Sov. Phys. Dokl. 32 (1987) 878 [Dokl. Akad. Nauk SSSR 297 (1987) 331].
A. Gorsky, A. Zabrodin and A. Zotov, Spectrum of Quantum Transfer Matrices via Classical Many-Body Systems, JHEP 01 (2014) 070 [arXiv:1310.6958] [INSPIRE].
A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, ��. Ragoucy and N.A. Slavnov, Scalar products of Bethe vectors in models with \( \mathfrak{g}{\mathfrak{l}}_3\left(2\Big|1\right) \) symmetry 2. Determinant representation, J. Phys. A 50 (2017) 034004 [arXiv:1606.03573] [INSPIRE].
E.H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas. 1. The General solution and the ground state, Phys. Rev. 130 (1963) 1605 [INSPIRE].
E.H. Lieb, Exact Analysis of an Interacting Bose Gas. 2. The Excitation Spectrum, Phys. Rev. 130 (1963) 1616 [INSPIRE].
N.A. Slavnov, One-dimensional two-component Bose gas and the algebraic Bethe ansatz, Theor. Math. Phys. 183 (2015) 800 [arXiv:1502.06749].
S. Belliard and N.A. Slavnov, A note on \( \mathfrak{g}{\mathfrak{l}}_2 \) -invariant Bethe vectors, JHEP 04 (2018) 031 [arXiv:1802.07576] [INSPIRE].
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Liashyk, A., Slavnov, N.A. On Bethe vectors in \( \mathfrak{g}{\mathfrak{l}}_3 \)-invariant integrable models. J. High Energ. Phys. 2018, 18 (2018). https://doi.org/10.1007/JHEP06(2018)018
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DOI: https://doi.org/10.1007/JHEP06(2018)018