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

SIGMA 11 (2015), 063, 20 pages      arXiv:1501.07566      https://doi.org/10.3842/SIGMA.2015.063

${\rm GL}(3)$-Based Quantum Integrable Composite Models. I. Bethe Vectors

Stanislav Pakuliak ^abc, Eric Ragoucy ^d and Nikita A. Slavnov ^e
a) Institute of Theoretical & Experimental Physics, 117259 Moscow, Russia
^b) Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow reg., Russia
^c) Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Moscow reg., Russia
^d) Laboratoire de Physique Théorique LAPTH, CNRS and Université de Savoie, BP 110, 74941 Annecy-le-Vieux Cedex, France
^e) Steklov Mathematical Institute, Moscow, Russia

Received February 18, 2015, in final form July 22, 2015; Published online July 31, 2015

Abstract
We consider a composite generalized quantum integrable model solvable by the nested algebraic Bethe ansatz. Using explicit formulas of the action of the monodromy matrix elements onto Bethe vectors in the ${\rm GL}(3)$-based quantum integrable models we prove a formula for the Bethe vectors of composite model. We show that this representation is a particular case of general coproduct property of the weight functions (Bethe vectors) found in the theory of the deformed Knizhnik-Zamolodchikov equation.

Key words: Bethe ansatz; quantum affine algebras, composite models.

pdf (439 kb)   tex (24 kb)

References

1. Belliard S., Pakuliak S., Ragoucy E., Slavnov N.A., Bethe vectors of ${\rm GL}(3)$-invariant integrable models, J. Stat. Mech. Theory Exp. 2013 (2013), P02020, 24 pages, arXiv:1210.0768.
2. Enriquez B., Khoroshkin S., Pakuliak S., Weight functions and Drinfeld currents, Comm. Math. Phys. 276 (2007), 691-725, math.QA/0610398.
3. Izergin A.G., Partition function of a six-vertex model in a finite volume, Sov. Phys. Dokl. 32 (1987), 878-879.
4. Izergin A.G., Korepin V.E., The quantum inverse scattering method approach to correlation functions, Comm. Math. Phys. 94 (1984), 67-92.
5. Izergin A.G., Korepin V.E., Reshetikhin N.Yu., Correlation functions in a one-dimensional Bose gas, J. Phys. A: Math. Gen. 20 (1987), 4799-4822.
6. Khoroshkin S., Pakuliak S., A computation of universal weight function for quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$, J. Math. Kyoto Univ. 48 (2008), 277-321, arXiv:0711.2819.
7. Khoroshkin S., Pakuliak S., Tarasov V., Off-shell Bethe vectors and Drinfeld currents, J. Geom. Phys. 57 (2007), 1713-1732, math.QA/0610517.
8. Kitanine N., Kozlowski K., Maillet J.M., Slavnov N.A., Terras V., On correlation functions of integrable models associated with the six-vertex $R$-matrix, J. Stat. Mech. Theory Exp. 2007 (2007), P01022, 17 pages, hep-th/0611142.
9. Kitanine N., Maillet J.M., Terras V., Correlation functions of the $XXZ$ Heisenberg spin-${1\over2}$ chain in a magnetic field, Nuclear Phys. B 567 (2000), 554-582, math-ph/9907019.
10. Korepin V.E., Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982), 391-418.
11. Pakuliak S., Ragoucy E., Slavnov N.A., Bethe vectors of quantum integrable models based on $U_q(\widehat{\mathfrak{gl}}_N)$, J. Phys. A: Math. Theor. 47 (2014), 105202, 16 pages, arXiv:1310.3253.
12. Pakuliak S., Ragoucy E., Slavnov N.A., Zero modes method and form factors in quantum integrable models, Nuclear Phys. B 893 (2015), 459-481, arXiv:1412.6037.
13. Pakuliak S., Ragoucy E., Slavnov N.A., Form factors of local operators in a one-dimensional two-component Bose gas, arXiv:1503.00546.
14. Slavnov N.A., Calculation of scalar products of wave functions and form-factors in the framework of the algebraic Bethe ansatz, Theoret. and Math. Phys. 79 (1989), 502-508.
15. Slavnov N.A., The algebraic Bethe ansatz and quantum integrable systems, Russian Math. Surveys 62 (2007), 727-766.
16. Varchenko A.N., Tarasov V.O., Jackson integral representations for solutions of the Knizhnik-Zamolodchikov quantum equation, St. Petersburg Math. J. 6 (1994), 275-313, hep-th/9311040.