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

SIGMA 14 (2018), 054, 18 pages      arXiv:1804.00597      https://doi.org/10.3842/SIGMA.2018.054

Modified Algebraic Bethe Ansatz: Twisted XXX Case

Samuel Belliard ab, Nikita A. Slavnov c and Benoit Vallet b
a) Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Energies, LPTHE, F-75005, Paris, France
b) Institut de Physique Théorique, DSM, CEA, URA2306 CNRS Saclay, F-91191, Gif-sur-Yvette, France
c) Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Received April 09, 2018, in final form May 28, 2018; Published online June 07, 2018

Abstract
We prove the modified algebraic Bethe Ansatz characterization of the spectral problem for the closed XXX Heisenberg spin chain with an arbitrary twist and arbitrary positive (half)-integer spin at each site of the chain. We provide two basis to characterize the spectral problem and two families of inhomogeneous Baxter T-Q equations. The two families satisfy an inhomogeneous quantum Wronskian equation.

Key words: integrable spin chain; algebraic Bethe ansatz; Baxter T-Q equation; quantum Wronskian equation.

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References

  1. Avan J., Belliard S., Grosjean N., Pimenta R.A., Modified algebraic Bethe ansatz for XXZ chain on the segment - III - Proof, Nuclear Phys. B 899 (2015), 229-246, arXiv:1506.02147.
  2. Belliard S., Modified algebraic Bethe ansatz for XXZ chain on the segment - I: Triangular cases, Nuclear Phys. B 892 (2015), 1-20, arXiv:1408.4840.
  3. Belliard S., Modified algebraic Bethe ansatz for XXZ chain on the segment - II - General cases, Nuclear Phys. B 894 (2015), 527-552, arXiv:1412.7511.
  4. Belliard S., Crampé N., Heisenberg XXX model with general boundaries: eigenvectors from algebraic Bethe ansatz, SIGMA 9 (2013), 072, 12 pages, arXiv:1209.4269.
  5. Belliard S., Crampé N., Ragoucy E., The scattering matrix for a general ${\rm gl}(2)$ spin chain, J. Stat. Mech. Theory Exp. 2009 (2009), P12003, 29 pages, arXiv:0909.1520.
  6. 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.
  7. Belliard S., Pimenta R.A., Slavnov and Gaudin-Korepin formulas for models without ${\rm U}(1)$ symmetry: the twisted XXX chain, SIGMA 11 (2015), 099, 12 pages, arXiv:1506.06550.
  8. Belliard S., Slavnov N.A., A note on $\mathfrak{gl}_2$-invariant Bethe vectors, J. High Energy Phys. 2018 (2018), no. 4, 031, 15 pages, arXiv:1802.07576.
  9. Cao J., Yang W.L., Shi K., Wang Y., Off-diagonal Bethe ansatz and exact solution a topological spin ring, Phys. Rev. Lett. 111 (2013), 137201, 5 pages, arXiv:1305.7328.
  10. Cao J., Yang W.-L., Shi K., Wang Y., Off-diagonal Bethe ansatz solution of the XXX spin chain with arbitrary boundary conditions, Nuclear Phys. B 875 (2013), 152-165, arXiv:1306.1742.
  11. Crampé N., Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries, J. Phys. A: Math. Theor. 48 (2015), 08FT01, 12 pages, arXiv:1411.7954.
  12. Crampé N., Algebraic Bethe ansatz for the XXZ Gaudin models with generic boundary, SIGMA 13 (2017), 094, 13 pages, arXiv:1710.08490.
  13. Derkachev S.E., The $R$-matrix factorization, $Q$-operator, and variable separation in the case of the XXX spin chain with the ${\rm SL}(2,{\mathbb C})$ symmetry group, Theoret. and Math. Phys. 169 (2011), 1539-1550.
  14. Drinfel'd V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
  15. Faddeev L.D., Sklyanin E.K., Takhtajan L.A., Quantum inverse problem. I, Theoret. and Math. Phys. 40 (1979), 688-706.
  16. Faddeev L.D., Takhtadzhyan L.A., Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model, J. Sov. Math. 24 (1984), 241-267.
  17. Fioravanti D., Nepomechie R.I., An inhomogeneous Lax representation for the Hirota equation, J. Phys. A: Math. Theor. 50 (2017), 054001, 14 pages, arXiv:1609.06761.
  18. Gromov N., Levkovich-Maslyuk F., Sizov G., New construction of eigenstates and separation of variables for ${\rm SU}(N)$ quantum spin chains, J. High Energy Phys. 2017 (2017), no. 9, 111, 40 pages, arXiv:1610.08032.
  19. Hao W., Nepomechie R.I., Sommese A.J., Completeness of solutions of Bethe's equations, Phys. Rev. E 88 (2013), 052113, 8 pages, arXiv:1308.4645.
  20. Jiang Y., Zhang Y., Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE, J. High Energy Phys. 2018 (2018), no. 3, 087, 40 pages, arXiv:1710.04693.
  21. Marboe C., Volin D., Fast analytic solver of rational Bethe equations, J. Phys. A: Math. Theor. 50 (2017), 204002, 14 pages, arXiv:1608.06504.
  22. Molev A.I., Yangians and their applications, in Handbook of Algebra, Handb. Algebr., Vol. 3, Elsevier/North-Holland, Amsterdam, 2003, 907-959, math.QA/0211288.
  23. Nepomechie R.I., An inhomogeneous $T$-$Q$ equation for the open XXX chain with general boundary terms: completeness and arbitrary spin, J. Phys. A: Math. Theor. 46 (2013), 442002, 7 pages, arXiv:1307.5049.
  24. Niccoli G., Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables, J. Math. Phys. 54 (2013), 053516, 20 pages, arXiv:1206.2418.
  25. Pronko G.P., Stroganov Y.G., Bethe equations ''on the wrong side of the equator'', J. Phys. A: Math. Gen. 32 (1999), 2333-2340, hep-th/9808153.
  26. Ribeiro G.A.P., Martins M.J., Galleas W., Integrable ${\rm SU}(N)$ vertex models with general toroidal boundary conditions, Nuclear Phys. B 675 (2003), 567-583, nlin.SI/0308011.
  27. Sklyanin E.K., Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen. 21 (1988), 2375-2389.
  28. Sklyanin E.K., Quantum inverse scattering method. Selected topics, in Quantum Group and Quantum Integrable Systems, Nankai Lectures Math. Phys., World Sci. Publ., River Edge, NJ, 1992, 63-97, \mboxhep-th/9211111.
  29. Wang Y., Yang W.-L., Cao J., Shi K., Off-diagonal Bethe ansatz for exactly solvable models, Springer, Heidelberg, 2015.
  30. Zhang X., Li Y.-Y., Cao J., Yang W.-L., Shi K., Wang Y., Bethe states of the XXZ spin-$\frac12$ chain with arbitrary boundary fields, Nuclear Phys. B 893 (2015), 70-88, arXiv:1412.6905.
  31. Zhang X., Li Y.-Y., Cao J., Yang W.-L., Shi K., Wang Y., Retrieve the Bethe states of quantum integrable models solved via the off-diagonal Bethe Ansatz, J. Stat. Mech. Theory Exp. (2015), P05014, 18 pages, arXiv:1407.5294.

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