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Spin-$s$ Rational $Q$-system (2303.07640v4)

Published 14 Mar 2023 in hep-th, math-ph, math.MP, and nlin.SI

Abstract: Bethe ansatz equations for spin-$s$ Heisenberg spin chain with $s\ge1$ are significantly more difficult to analyze than the spin-$\tfrac{1}{2}$ case, due to the presence of repeated roots. As a result, it is challenging to derive extra conditions for the Bethe roots to be physical and study the related completeness problem. In this paper, we propose the rational $Q$-system for the XXX$_s$ spin chain. Solutions of the proposed $Q$-system give all and only physical solutions of the Bethe ansatz equations required by completeness. This is checked numerically and proved rigorously. The rational $Q$-system is equivalent to the requirement that the solution and the corresponding dual solution of the $TQ$-relation are both polynomials, which we prove rigorously. Based on this analysis, we propose the extra conditions for solutions of the XXX$_s$ Bethe ansatz equations to be physical.

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References (45)
  1. L. D. Faddeev, “How algebraic Bethe ansatz works for integrable model,” in Les Houches School of Physics: Astrophysical Sources of Gravitational Radiation, pp. pp. 149–219. 5, 1996. arXiv:hep-th/9605187.
  2. R. P. Langlands and Y. Saint-Aubin, “Algebro-geometric aspects of the Bethe equations,” in Strings and symmetries, pp. pp. 40–53. 1995.
  3. K. Fabricius and B. M. McCoy, “Bethe’s equation is incomplete for the XXZ model at roots of unity,” J. Statist. Phys. 103 (2001) 647–678, arXiv:cond-mat/0009279.
  4. R. J. Baxter, “Completeness of the Bethe ansatz for the six and eight vertex models,” J. Statist. Phys. 108 (2002) 1–48, arXiv:cond-mat/0111188.
  5. E. Mukhin, V. Tarasov, and A. Varchenko, “Bethe Algebra of Homogeneous XXX Heisenberg Model Has Simple Spectrum,” arXiv e-prints (June, 2007) arXiv:0706.0688, arXiv:0706.0688 [math.QA].
  6. L. V. Avdeev and A. A. Vladimirov, “ON EXCEPTIONAL SOLUTIONS OF THE BETHE ANSATZ EQUATIONS,” Theor. Math. Phys. 69 (1987) 1071.
  7. F. H. L. Essler, V. E. Korepin, and K. Schoutens, “Fine structure of the Bethe ansatz for the spin 1/2 Heisenberg XXX model,” J. Phys. A 25 (1992) 4115–4126.
  8. R. Siddharthan, “Singularities in the Bethe solution of the XXX and XXZ Heisenberg spin chains,” arXiv e-prints (Apr., 1998) cond–mat/9804210, arXiv:cond-mat/9804210 [cond-mat.str-el].
  9. Y. Miao, J. Lamers, and V. Pasquier, “On the Q operator and the spectrum of the XXZ model at root of unity,” SciPost Phys. 11 (2021) 067, arXiv:2012.10224 [cond-mat.stat-mech].
  10. R. J. Baxter, Exactly Solved Models in Statistical Mechanics. Academic Press, 1982.
  11. C. Marboe and D. Volin, “Fast analytic solver of rational Bethe equations,” J. Phys. A 50 no. 20, (2017) 204002, arXiv:1608.06504 [math-ph].
  12. E. Granet and J. L. Jacobsen, “On zero-remainder conditions in the Bethe ansatz,” JHEP 03 (2020) 178, arXiv:1910.07797 [hep-th].
  13. Z. Bajnok, E. Granet, J. L. Jacobsen, and R. I. Nepomechie, “On Generalized Q𝑄Qitalic_Q-systems,” JHEP 03 (2020) 177, arXiv:1910.07805 [hep-th].
  14. R. I. Nepomechie, “Q-systems with boundary parameters,” arXiv:1912.12702 [hep-th].
  15. R. I. Nepomechie, “The Am(1)superscriptsubscript𝐴𝑚1A_{m}^{(1)}italic_A start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT Q-system,” Mod. Phys. Lett. A 35 no. 31, (2020) 2050260, arXiv:2003.06823 [hep-th].
  16. J. Gu, Y. Jiang, and M. Sperling, “Rational Q𝑄Qitalic_Q-systems, Higgsing and Mirror Symmetry,” arXiv:2208.10047 [hep-th].
  17. N. Reshetikhin, “ A method of functional equations in the theory of exactly solvable quantum systems,” Lett. Math. Phys. 7 (1983) 205–213.
  18. I. Krichever, O. Lipan, P. Wiegmann, and A. Zabrodin, “Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations,” Commun. Math. Phys. 188 (1997) 267–304, arXiv:hep-th/9604080.
  19. V. Kazakov, A. S. Sorin, and A. Zabrodin, “Supersymmetric Bethe ansatz and Baxter equations from discrete Hirota dynamics,” Nucl. Phys. B 790 (2008) 345–413, arXiv:hep-th/0703147.
  20. V. V. Bazhanov and Z. Tsuboi, “Baxter’s Q-operators for supersymmetric spin chains,” Nucl. Phys. B 805 (2008) 451–516, arXiv:0805.4274 [hep-th].
  21. V. V. Bazhanov, T. Lukowski, C. Meneghelli, and M. Staudacher, “A Shortcut to the Q-Operator,” J. Stat. Mech. 1011 (2010) P11002, arXiv:1005.3261 [hep-th].
  22. V. Kazakov, S. Leurent, and Z. Tsuboi, “Baxter’s Q-operators and operatorial Backlund flow for quantum (super)-spin chains,” Commun. Math. Phys. 311 (2012) 787–814, arXiv:1010.4022 [math-ph].
  23. W. Hao, R. I. Nepomechie, and A. J. Sommese, “Completeness of solutions of Bethe’s equations,” Phys. Rev. E 88 no. 5, (2013) 052113, arXiv:1308.4645 [math-ph].
  24. R. I. Nepomechie and C. Wang, “Algebraic Bethe ansatz for singular solutions,” J. Phys. A 46 (2013) 325002, arXiv:1304.7978 [hep-th].
  25. R. I. Nepomechie and C. Wang, “Twisting singular solutions of Bethe’s equations,” J. Phys. A 47 no. 50, (2014) 505004, arXiv:1409.7382 [math-ph].
  26. G. P. Pronko and Y. G. Stroganov, “Bethe equations ’on the wrong side of equator’,” J. Phys. A 32 (1999) 2333–2340, arXiv:hep-th/9808153.
  27. W. Hao, R. I. Nepomechie, and A. J. Sommese, “Singular solutions, repeated roots and completeness for higher-spin chains,” J. Stat. Mech. 1403 (2014) P03024, arXiv:1312.2982 [math-ph].
  28. N. Kitanine, J. M. Maillet, G. Niccoli, and V. Terras, “On determinant representations of scalar products and form factors in the SoV approach: the XXX case,” J. Phys. A 49 no. 10, (2016) 104002, arXiv:1506.02630 [math-ph].
  29. J. M. Maillet and G. Niccoli, “On quantum separation of variables beyond fundamental representations,” SciPost Phys. 10 no. 2, (2021) 026, arXiv:1903.06618 [math-ph].
  30. Y. He, J. Hou, Y. Liu, and Z. Tan, “ Spin-s𝑠sitalic_s Q𝑄Qitalic_Q-system: twist and open boundaries,” to appear .
  31. A. N. Kirillov, “Combinatorial identities, and completeness of eigenstates of the Heisenberg magnet,” Journal of Soviet Mathematics 30 no. 4, (1985) 2298–2310. https://doi.org/10.1007/BF02105347.
  32. H. Shu, P. Zhao, R.-D. Zhu, and H. Zou, “Bethe-State Counting and the Witten Index,” arXiv:2210.07116 [hep-th].
  33. H. M. Babujian, “EXACT SOLUTION OF THE ISOTROPIC HEISENBERG CHAIN WITH ARBITRARY SPINS: THERMODYNAMICS OF THE MODEL,” Nucl. Phys. B 215 (1983) 317–336.
  34. L. A. Takhtajan, “The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins,” Phys. Lett. A 87 (1982) 479–482.
  35. V. Tarasov, “Completeness of the Bethe Ansatz for the Periodic Isotropic Heisenberg Model,” Rev. Math. Phys. 30 (2018) 1840018.
  36. Y. Jiang and Y. Zhang, “Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE,” JHEP 03 (2018) 087, arXiv:1710.04693 [hep-th].
  37. C. Closset and O. Khlaif, “Twisted indices, Bethe ideals and 3d 𝒩=2𝒩2\mathcal{N}=2caligraphic_N = 2 infrared dualities,” arXiv:2301.10753 [hep-th].
  38. J. Lykke Jacobsen, Y. Jiang, and Y. Zhang, “Torus partition function of the six-vertex model from algebraic geometry,” JHEP 03 (2019) 152, arXiv:1812.00447 [hep-th].
  39. Z. Bajnok, J. L. Jacobsen, Y. Jiang, R. I. Nepomechie, and Y. Zhang, “Cylinder partition function of the 6-vertex model from algebraic geometry,” JHEP 06 (2020) 169, arXiv:2002.09019 [hep-th].
  40. J. Böhm, J. L. Jacobsen, Y. Jiang, and Y. Zhang, “Geometric algebra and algebraic geometry of loop and Potts models,” JHEP 05 (2022) 068, arXiv:2202.02986 [hep-th].
  41. N. A. Nekrasov and S. L. Shatashvili, “Quantum integrability and supersymmetric vacua,” Prog. Theor. Phys. Suppl. 177 (2009) 105–119, arXiv:0901.4748 [hep-th].
  42. N. A. Nekrasov and S. L. Shatashvili, “Supersymmetric vacua and Bethe ansatz,” Nucl. Phys. B Proc. Suppl. 192-193 (2009) 91–112, arXiv:0901.4744 [hep-th].
  43. M. Bullimore, H.-C. Kim, and T. Lukowski, “Expanding the Bethe/Gauge Dictionary,” JHEP 11 (2017) 055, arXiv:1708.00445 [hep-th].
  44. O. Foda and M. Manabe, “Nested coordinate Bethe wavefunctions from the Bethe/gauge correspondence,” JHEP 11 (2019) 036, arXiv:1907.00493 [hep-th].
  45. P. Zhao and H. Zou, “Remarks on 2d Unframed Quiver Gauge Theories,” arXiv:2206.14793 [hep-th].
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