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

SIGMA 13 (2017), 015, 17 pages      arXiv:1611.00943      https://doi.org/10.3842/SIGMA.2017.015

Bethe Vectors for Composite Models with $\mathfrak{gl}(2|1)$ and $\mathfrak{gl}(1|2)$ Supersymmetry

Jan Fuksa ab
a) Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia
b) Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Czech Republic

Received November 18, 2016, in final form March 03, 2017; Published online March 13, 2017

Abstract
Supersymmetric composite generalized quantum integrable models solvable by the algebraic Bethe ansatz are studied. Using a coproduct in the bialgebra of monodromy matrix elements and their action on Bethe vectors, formulas for Bethe vectors in the composite models with supersymmetry based on the super-Yangians $Y[\mathfrak{gl}(2|1)]$ and $Y[\mathfrak{gl}(1|2)]$ are derived.

Key words: algebraic Bethe ansatz; composite models.

pdf (421 kb)   tex (24 kb)

References

  1. Babujian H., Karowski M., Zapletal A., Matrix difference equations and a nested Bethe ansatz, J. Phys. A: Math. Gen. 30 (1997), 6425-6450.
  2. Bedürftig G., Frahm H., Thermodynamics of an integrable model for electrons with correlated hopping, J. Phys. A: Math. Gen. 28 (1995), 4453-4468, cond-mat/9504103.
  3. 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.
  4. Belliard S., Pakuliak S., Ragoucy E., Slavnov N.A., Bethe vectors of quantum integrable models with ${\rm GL}(3)$ trigonometric $R$-matrix, SIGMA 9 (2013), 058, 23 pages, arXiv:1304.7602.
  5. Bracken A.J., Gould M.D., Links J.R., Zhang Y.Z., New supersymmetric and exactly solvable model of correlated electrons, Phys. Rev. Lett. 74 (1995), 2768-2771, cond-mat/9410026.
  6. Faddeev L.D., How the algebraic Bethe ansatz works for integrable models, in Symétries Quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 149-219, hep-th/9605187.
  7. Foerster A., Links J., Tonel A.P., Algebraic properties of an integrable $t$-$J$ model with impurities, Nuclear Phys. B 552 (1999), 707-726, cond-mat/9901091.
  8. Frahm H., Doped Heisenberg chains: spin-$S$ generalizations of the supersymmetric $t$-$J$ model, Nuclear Phys. B 559 (1999), 613-636, cond-mat/9904157.
  9. Fuksa J., Slavnov N.A., Form factors of local operators in supersymmetric quantum integrable models, arXiv:1701.05866.
  10. Göhmann F., Korepin V.E., Solution of the quantum inverse problem, J. Phys. A: Math. Gen. 33 (2000), 1199-1220, hep-th/9910253.
  11. Hutsalyuk A., Liashyk A., Pakuliak S.Z., Ragoucy E., Slavnov N.A., Form factors of the monodromy matrix entries in ${\mathfrak{gl}}(2|1)$-invariant integrable models, Nuclear Phys. B 911 (2016), 902-927, arXiv:1607.04978.
  12. Hutsalyuk A., Liashyk A., Pakuliak S.Z., Ragoucy E., Slavnov N.A., Multiple actions of the monodromy matrix in $\mathfrak{gl}(2|1)$-invariant integrable models, SIGMA 12 (2016), 099, 22 pages, arXiv:1605.06419.
  13. Hutsalyuk A., Liashyk A., Pakuliak S.Z., Ragoucy E., Slavnov N.A., Scalar products of Bethe vectors in models with $\mathfrak{gl}(2|1)$ symmetry 1. Super-analog of Reshetikhin formula, J. Phys. A: Math. Theor. 49 (2016), 454005, 28 pages, arXiv:1605.09189.
  14. Hutsalyuk A., Liashyk A., Pakuliak S.Z., Ragoucy E., Slavnov N.A., Scalar products of Bethe vectors in models with $\mathfrak{gl}(2|1)$ symmetry 2. Determinant representation, J. Phys. A: Math. Theor. 50 (2017), 034004, 22 pages, arXiv:1606.03573.
  15. Izergin A.G., Partition function of a six-vertex model in a finite volume, Sov. Phys. Dokl. 32 (1987), 878-879.
  16. Izergin A.G., Korepin V.E., The quantum inverse scattering method approach to correlation functions, Comm. Math. Phys. 94 (1984), 67-92.
  17. 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.
  18. 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.
  19. Korepin V.E., Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982), 391-418.
  20. Korepin V.E., Bogoliubov N.M., Izergin A.G., Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1993.
  21. Kulish P.P., Reshetikhin N.Y., Diagonalisation of ${\rm GL}(N)$ invariant transfer matrices and quantum $N$-wave system (Lee model), J. Phys. A: Math. Gen. 16 (1983), L591-L596.
  22. Kulish P.P., Sklyanin E.K., Solutions of the Yang-Baxter equation, J. Sov. Math. 19 (1982), 1596-1620.
  23. Maillet J.M., Terras V., On the quantum inverse scattering problem, Nuclear Phys. B 575 (2000), 627-644, hep-th/9911030.
  24. Pakuliak S., Ragoucy E., Slavnov N.A., ${\rm GL}(3)$-based quantum integrable composite models. I. Bethe vectors, SIGMA 11 (2015), 063, 20 pages, arXiv:1501.07566.
  25. 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.
  26. Pakuliak S., Ragoucy E., Slavnov N.A., Bethe vectors for models based on the super-Yangian $Y(\mathfrak{gl}(m|n))$, arXiv:1604.02311.
  27. Pfannmüller M.P., Frahm H., Algebraic Bethe ansatz for ${\mathfrak{gl}}(2,1)$ invariant $36$-vertex models, Nuclear Phys. B 479 (1996), 575-593, cond-mat/9604082.
  28. Sklyanin E.K., Takhtadzhyan L.A., Faddeev L.D., Quantum inverse problem method. I, Theoret. and Math. Phys. 40 (1979), 688-706.
  29. Slavnov N.A., The algebraic Bethe ansatz and quantum integrable systems, Russian Math. Surveys 62 (2007), 727-766.
  30. Sutherland B., Further results for the many-body problem in one dimension, Phys. Rev. Lett. 20 (1968), 98-100.
  31. Takhtadzhyan L.A., Faddeev L.D., The quantum method for the inverse problem and the Heisenberg $XYZ$ model, Russian Math. Surveys 34 (1979), no. 5, 11-68.
  32. Varchenko A., Tarasov V., Jackson integral representations of solutions of the quantized Knizhnik-Zamolodchikov equation, St. Petersburg Math. J. 6 (1995), 275-313, hep-th/9311040.
  33. Yang C.N., Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312-1315.

Previous article  Next article   Contents of Volume 13 (2017)