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


SIGMA 18 (2022), 036, 20 pages      arXiv:2104.04651      https://doi.org/10.3842/SIGMA.2022.036

A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain. II. The Polynomials $p_n$

Linnea Hietala ab
a) Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden
b) Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96 Gothenburg, Sweden

Received August 06, 2021, in final form April 29, 2022; Published online May 15, 2022

Abstract
By specializing the parameters in the partition function of the 8VSOS model with domain wall boundary conditions and diagonal reflecting end, we find connections between the three-color model and certain polynomials $p_n(z)$, which are conjectured to be equal to certain polynomials of Bazhanov and Mangazeev, appearing in the eigenvectors of the Hamiltonian of the supersymmetric XYZ spin chain. This article is a continuation of a previous paper where we investigated the related polynomials $q_n(z)$, also conjectured to be equal to polynomials of Bazhanov and Mangazeev, appearing in the eigenvectors of the supersymmetric XYZ spin chain.

Key words: eight-vertex SOS model; domain wall boundary conditions; reflecting end; three-color model; XYZ spin chain; polynomials; positive coefficients.

pdf (460 kb)   tex (26 kb)  

References

  1. Baxter R.J., Three-colorings of the square lattice: a hard squares model, J. Math. Phys. 11 (1970), 3116-3124.
  2. Baxter R.J., Partition function of the eight-vertex lattice model, Ann. Physics 70 (1972), 193-228.
  3. Baxter R.J., Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain I. Some fundamental eigenvectors, Ann. Physics 76 (1973), 1-24.
  4. Bazhanov V.V., Mangazeev V.V., Eight-vertex model and non-stationary Lamé equation, J. Phys. A: Math. Gen. 38 (2005), L145-L153, arXiv:hep-th/0411094.
  5. Bazhanov V.V., Mangazeev V.V., The eight-vertex model and Painlevé VI, J. Phys. A: Math. Gen. 39 (2006), 12235-12243, arXiv:hep-th/0602122.
  6. Brasseur S., Hagendorf C., Sum rules for the supersymmetric eight-vertex model, J. Stat. Mech. Theory Exp. 2021 (2021), 023102, 45 pages, arXiv:2009.14077.
  7. Filali G., Elliptic dynamical reflection algebra and partition function of SOS model with reflecting end, J. Geom. Phys. 61 (2011), 1789-1796, arXiv:1012.0516.
  8. Hagendorf C., Fendley P., The eight-vertex model and lattice supersymmetry, J. Stat. Phys. 146 (2012), 1122-1155, arXiv:1109.4090.
  9. Hietala L., A combinatorial description of certain polynomials related to the XYZ spin chain, SIGMA 16 (2020), 101, 26 pages, arXiv:2004.09924.
  10. Izergin A.G., Partition function of the six-vertex model in a finite volume, Soviet Phys. Dokl. 32 (1987), 878-879.
  11. Izergin A.G., Coker D.A., Korepin V.E., Determinant formula for the six-vertex model, J. Phys. A: Math. Gen. 25 (1992), 4315-4334.
  12. Korepin V.E., Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982), 391-418.
  13. Kuperberg G., Another proof of the alternating-sign matrix conjecture, Int. Math. Res. Not. 1996 (1996), 139-150, arXiv:math.CO/9712207.
  14. Kuperberg G., Symmetry classes of alternating-sign matrices under one roof, Ann. of Math. 156 (2002), 835-866, arXiv:math.CO/0008184.
  15. Lieb E.H., Residual entropy of square ice, Phys. Rev. 162 (1967), 162-172.
  16. Mangazeev V.V., Bazhanov V.V., The eight-vertex model and Painlevé VI equation II: eigenvector results, J. Phys. A: Math. Theor. 43 (2010), 085206, 16 pages, arXiv:0912.2163.
  17. Mills W.H., Robbins D.P., Rumsey Jr. H., Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359.
  18. Razumov A.V., Stroganov Yu.G., Spin chains and combinatorics, J. Phys. A: Math. Gen. 34 (2001), 3185-3190, arXiv:cond-mat/0012141.
  19. Razumov A.V., Stroganov Yu.G., Refined enumerations of some symmetry classes of alternating-sign matrices, Theoret. and Math. Phys. 141 (2004), 1609-1630, arXiv:math-ph/0312071.
  20. Razumov A.V., Stroganov Yu.G., A possible combinatorial point for the XYZ spin chain, Theoret. and Math. Phys. 164 (2010), 977-991, arXiv:0911.5030.
  21. Rosengren H., An Izergin-Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices, Adv. in Appl. Math. 43 (2009), 137-155, arXiv:0801.1229.
  22. Rosengren H., The three-colour model with domain wall boundary conditions, Adv. in Appl. Math. 46 (2011), 481-535, arXiv:0911.0561.
  23. Rosengren H., Special polynomials related to the supersymmetric eight-vertex model. I. Behaviour at cusps, arXiv:1305.0666.
  24. Rosengren H., Special polynomials related to the supersymmetric eight-vertex model: a summary, Comm. Math. Phys. 340 (2015), 1143-1170, arXiv:1503.02833.
  25. Sutherland B., Exact solution of a two-dimensional model for hydrogen-bonded crystals, Phys. Rev. Lett. 19 (1967), 103-104.
  26. Tsuchiya O., Determinant formula for the six-vertex model with reflecting end, J. Math. Phys. 39 (1998), 5946-5951, arXiv:solv-int/9804010.
  27. Wang Y.-S., Boundary spontaneous polarization in the six-vertex model with a reflecting boundary, J. Phys. A: Math. Gen. 36 (2003), 4007-4013.
  28. Zeilberger D., Proof of the alternating sign matrix conjecture, Electron. J. Combin. 3 (1996), R13, 84 pages, arXiv:math.CO/9407211.
  29. Zinn-Justin P., Sum rule for the eight-vertex model on its combinatorial line, in Symmetries, Integrable Systems and Representations, Springer Proc. Math. Stat., Vol. 40, Springer, Heidelberg, 2013, 599-637, arXiv:1202.4420.

Previous article  Next article  Contents of Volume 18 (2022)