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


SIGMA 14 (2018), 064, 23 pages      arXiv:1801.09635      https://doi.org/10.3842/SIGMA.2018.064
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

The Functional Method for the Domain-Wall Partition Function

Jules Lamers
Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Göteborg, Sweden

Received January 30, 2018, in final form June 17, 2018; Published online June 26, 2018

Abstract
We review the (algebraic-)functional method devised by Galleas and further developed by Galleas and the author. We first explain the method using the simplest example: the computation of the partition function for the six-vertex model with domain-wall boundary conditions. At the heart of the method lies a linear functional equation for the partition function. After deriving this equation we outline its analysis. The result is a closed expression in the form of a symmetrized sum - or, equivalently, multiple-integral formula - that can be rewritten to recover Izergin's determinant. Special attention is paid to the relation with other approaches. In particular we show that the Korepin-Izergin approach can be recovered within the functional method. We comment on the functional method's range of applicability, and review how it is adapted to the technically more involved example of the elliptic solid-on-solid model with domain walls and a reflecting end. We present a new formula for the partition function of the latter, which was expressed as a determinant by Tsuchiya-Filali-Kitanine. Our result takes the form of a `crossing-symmetrized' sum with $2^L$ terms featuring the elliptic domain-wall partition function, which appears to be new also in the limiting case of the six-vertex model. Further taking the rational limit we recover the expression obtained by Frassek using the boundary perimeter Bethe ansatz.

Key words: six-vertex model; solid-on-solid model; reflecting end; functional equations.

pdf (593 kb)   tex (230 kb)

References

  1. Baxter R.J., Perimeter Bethe ansatz, J. Phys. A: Math. Gen. 20 (1987), 2557-2567.
  2. Bogoliubov N.M., Pronko A.G., Zvonarev M.B., Boundary correlation functions of the six-vertex model, J. Phys. A: Math. Gen. 35 (2002), 5525-5541, math-ph/0203025.
  3. Cherednik I.V., Factorizing particles on a half line, and root systems, Theoret. and Math. Phys. 61 (1984), 977-983.
  4. Filali G., Dynamical reflection algebra and associated boundary integrable models, Ph.D. Thesis, Université de Cergy Pontoise, 2011, available at https://tel.archives-ouvertes.fr/tel-00664076.
  5. 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.
  6. Filali G., Kitanine N., The partition function of the trigonometric SOS model with a reflecting end, J. Stat. Mech. Theory Exp. 2010 (2010), L06001, 11 pages, Erratum, J. Stat. Mech. Theory Exp. 2010 (2010), E07002, arXiv:1004.1015.
  7. Frassek R., Boundary perimeter Bethe ansatz, J. Phys. A: Math. Theor. 50 (2017), 265202, 19 pages, arXiv:1703.10842.
  8. Galleas W., Functional relations for the six-vertex model with domain wall boundary conditions, J. Stat. Mech. Theory Exp. 2010 (2010), P06008, 15 pages, arXiv:1002.1623.
  9. Galleas W., A new representation for the partition function of the six-vertex model with domain wall boundaries, J. Stat. Mech. Theory Exp. 2011 (2011), P01013, 12 pages, arXiv:1010.5059.
  10. Galleas W., Multiple integral representation for the trigonometric SOS model with domain wall boundaries, Nuclear Phys. B 858 (2012), 117-141, arXiv:1111.6683.
  11. Galleas W., Functional relations and the Yang-Baxter algebra, J. Phys. Conf. Ser. 474 (2013), 012020, 19 pages, arXiv:1312.6816.
  12. Galleas W., Refined functional relations for the elliptic SOS model, Nuclear Phys. B 867 (2013), 855-871, arXiv:1207.5283.
  13. Galleas W., Scalar product of Bethe vectors from functional equations, Comm. Math. Phys. 329 (2014), 141-167, arXiv:1211.7342.
  14. Galleas W., Off-shell scalar products for the $XXZ$ spin chain with open boundaries, Nuclear Phys. B 893 (2015), 346-375, arXiv:1412.5389.
  15. Galleas W., Elliptic solid-on-solid model's partition function as a single determinant, Phys. Rev. E 94 (2016), 010102, 5 pages, arXiv:1604.01223.
  16. Galleas W., New differential equations in the six-vertex model, J. Stat. Mech. Theory Exp. 2016 (2016), 033106, 13 pages, arXiv:1508.04690.
  17. Galleas W., On the elliptic $\mathfrak{gl}_2$ solid-on-solid model: functional relations and determinants, arXiv:1606.06144.
  18. Galleas W., Continuous representations of scalar products of Bethe vectors, J. Math. Phys. 58 (2017), 083504, 15 pages, arXiv:1607.08524.
  19. Galleas W., Six-vertex model and non-linear differential equations I. Spectral problem, arXiv:1705.03408.
  20. Galleas W., Lamers J., Reflection algebra and functional equations, Nuclear Phys. B 886 (2014), 1003-1028, arXiv:1405.4281.
  21. Garbali A., The domain wall partition function for the Izergin-Korepin nineteen-vertex model at a root of unity, J. Stat. Mech. Theory Exp. 2016 (2016), 033112, 19 pages, arXiv:1411.2903.
  22. Göhmann F., Korepin V.E., Solution of the quantum inverse problem, J. Phys. A: Math. Gen. 33 (2000), 1199-1220.
  23. Izergin A.G., Partition function of a six-vertex model in a finite volume, Sov. Phys. Dokl. 32 (1987), 878-879.
  24. 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.
  25. Kitanine N., Kozlowski K.K., Maillet J.M., Niccoli G., Slavnov N.A., Terras V., Correlation functions of the open $XXZ$ chain. I, J. Stat. Mech. Theory Exp. 2007 (2007), P10009, 37 pages, arXiv:0707.1995.
  26. Kitanine N., Maillet J.M., Terras V., Form factors of the $XXZ$ Heisenberg spin-$\frac 12$ finite chain, Nuclear Phys. B 554 (1999), 647-678, math-ph/9807020.
  27. 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.
  28. Korepin V., Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982), 391-418.
  29. Korepin V., Zinn-Justin P., Thermodynamic limit of the six-vertex model with domain wall boundary conditions, J. Phys. A: Math. Gen. 33 (2000), 7053-7066, cond-mat/0004250.
  30. Lamers J., A pedagogical introduction to quantum integrability, with a view towards theoretical high-energy physics, PoS Proc. Sci. (2014), PoS(Modave2014), 001, 70 pages, arXiv:1501.06805.
  31. Lamers J., Integral formula for elliptic SOS models with domain walls and a reflecting end, Nuclear Phys. B 901 (2015), 556-583, arXiv:1510.00342.
  32. Lamers J., On elliptic quantum Integrability: vertex models, solid-on-solid models and spin chains, Ph.D. Thesis, Utrecht University, 2016, available at https://dspace.library.uu.nl/handle/1874/333998.
  33. Lieb E.H., Exact solution of the $F$ model of an antiferroelectric, Phys. Rev. Lett. 18 (1967), 1046-1048.
  34. Lieb E.H., Exact solution of the two-dimensional Slater KDP model of a ferroelectric, Phys. Rev. Lett. 19 (1967), 108-110.
  35. Lieb E.H., Residual entropy of square ice, Phys. Rev. 162 (1967), 162-172.
  36. Maillet J.M., Terras V., On the quantum inverse scattering problem, Nuclear Phys. B 575 (2000), 627-644, hep-th/9911030.
  37. Pakuliak S., Rubtsov V., Silantyev A., The SOS model partition function and the elliptic weight functions, J. Phys. A: Math. Theor. 41 (2008), 295204, 20 pages, arXiv:0802.0195.
  38. 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.
  39. Sklyanin E.K., Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen. 21 (1988), 2375-2389.
  40. 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.
  41. Sutherland B., Exact solution of a two-dimensional model for hydrogen-bonded crystals, Phys. Rev. Lett. 19 (1967), 103-104.
  42. Tarasov V., Varchenko A., Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque 246 (1997), vi+135 pages, q-alg/9703044.
  43. Tsuchiya O., Determinant formula for the six-vertex model with reflecting end, J. Math. Phys. 39 (1998), 5946-5951, solv-int/9804010.
  44. Wang Y.S., The scalar products and the norm of Bethe eigenstates for the boundary $XXX$ Heisenberg spin-1/2 finite chain, Nuclear Phys. B 622 (2002), 633-649.
  45. Zinn-Justin P., Six-vertex model with domain wall boundary conditions and one-matrix model, Phys. Rev. E 62 (2000), 3411-3418, math-ph/0005008.

Previous article  Next article   Contents of Volume 14 (2018)