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

SIGMA 17 (2021), 025, 29 pages      arXiv:2010.10615
Contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan

Some Algebraic Aspects of the Inhomogeneous Six-Vertex Model

Vladimir V. Bazhanov a, Gleb A. Kotousov b, Sergii M. Koval a and Sergei L. Lukyanov cd
a) Department of Theoretical Physics, Research School of Physics, Australian National University, Canberra, ACT 2601, Australia
b) DESY, Theory Group, Notkestrasse 85, Hamburg, 22607, Germany
c) NHETC, Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849, USA
d) Kharkevich Institute for Information Transmission Problems, Moscow, 127994, Russia

Received October 30, 2020, in final form February 26, 2021; Published online March 16, 2021

The inhomogeneous six-vertex model is a 2$D$ multiparametric integrable statistical system. In the scaling limit it is expected to cover different classes of critical behaviour which, for the most part, have remained unexplored. For general values of the parameters and twisted boundary conditions the model possesses ${\rm U}(1)$ invariance. In this paper we discuss the restrictions imposed on the parameters for which additional global symmetries arise that are consistent with the integrable structure. These include the lattice counterparts of ${\mathcal C}$, ${\mathcal P}$ and ${\mathcal T}$ as well as translational invariance. The special properties of the lattice system that possesses an additional ${\mathcal Z}_r$ invariance are considered. We also describe the Hermitian structures, which are consistent with the integrable one. The analysis lays the groundwork for studying the scaling limit of the inhomogeneous six-vertex model.

Key words: solvable lattice models; Bethe ansatz; Yang-Baxter equation; discrete symmetries; Hermitian structures.

pdf (633 kb)   tex (44 kb)  


  1. Alcaraz F.C., Barber M.N., Batchelor M.T., Conformal invariance, the $XXZ$ chain and the operator content of two-dimensional critical systems, Ann. Physics 182 (1988), 280-343.
  2. Baxter R.J., Private communication, 2020.
  3. Baxter R.J., Generalized ferroelectric model on a square lattice, Stud. Appl. Math. 50 (1971), 51-69.
  4. Baxter R.J., Partition function of the eight-vertex lattice model, Ann. Physics 70 (1972), 193-228.
  5. 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.
  6. Baxter R.J., Solvable eight-vertex model on an arbitrary planar lattice, Philos. Trans. Roy. Soc. London Ser. A 289 (1978), 315-346.
  7. Bazhanov V., Bobenko A., Reshetikhin N., Quantum discrete sine-Gordon model at roots of $1$: integrable quantum system on the integrable classical background, Comm. Math. Phys. 175 (1996), 377-400.
  8. Bazhanov V.V., Kotousov G.A., Koval S.M., Lukyanov S.L., On the scaling behaviour of the alternating spin chain, J. High Energy Phys. 2019 (2019), no. 8, 087, 30 pages, arXiv:1903.05033.
  9. Bazhanov V.V., Kotousov G.A., Koval S.M., Lukyanov S.L., Scaling limit of the ${\mathcal Z}_2$ invariant inhomogeneous six-vertex model, Nuclear Phys. B 965 (2021), 115337, 156 pages, arXiv:2010.10613.
  10. Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B., Integrable structure of conformal field theory. II. ${\rm Q}$-operator and DDV equation, Comm. Math. Phys. 190 (1997), 247-278, arXiv:hep-th/9604044.
  11. Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B., Integrable structure of conformal field theory. III. The Yang-Baxter relation, Comm. Math. Phys. 200 (1999), 297-324, arXiv:hep-th/9805008.
  12. Bazhanov V.V., Mangazeev V.V., Sergeev S.M., Faddeev-Volkov solution of the Yang-Baxter equation and discrete conformal symmetry, Nuclear Phys. B 784 (2007), 234-258, arXiv:hep-th/0703041.
  13. Bethe H., Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys. 71 (1931), 205-226.
  14. Bobenko A., Kutz N., Pinkall U., The discrete quantum pendulum, Phys. Lett. A 177 (1993), 399-404.
  15. Bobenko A.I., Springborn B.A., Variational principles for circle patterns and Koebe's theorem, Trans. Amer. Math. Soc. 356 (2004), 659-689, arXiv:math.GT/0203250.
  16. Candu C., Ikhlef Y., Nonlinear integral equations for the ${\rm SL}(2,{\mathbb R})/{\rm U}(1)$ black hole sigma model, J. Phys. A: Math. Theor. 46 (2013), 415401, 31 pages, arXiv:1306.2646.
  17. Destri C., de Vega H.J., Light-cone lattice approach to fermionic theories in $2$D. The massive Thirring model, Nuclear Phys. B 290 (1987), 363-391.
  18. Faddeev L., Volkov A.Yu., Hirota equation as an example of an integrable symplectic map, Lett. Math. Phys. 32 (1994), 125-135, arXiv:hep-th/9405087.
  19. Faddeev L.D., Reshetikhin N.Yu., Integrability of the principal chiral field model in $1+1$ dimension, Ann. Physics 167 (1986), 227-256.
  20. Frahm H., Martins M.J., Phase diagram of an integrable alternating ${\rm U}_q[\mathfrak{sl}(2|1)]$ superspin chain, Nuclear Phys. B 862 (2012), 504-552, arXiv:1202.4676.
  21. Frahm H., Seel A., The staggered six-vertex model: conformal invariance and corrections to scaling, Nuclear Phys. B 879 (2014), 382-406, arXiv:1311.6911.
  22. Gaudin M., McCoy B.M., Wu T.T., Normalization sum for the Bethe's hypothesis wave functions of the Heisenberg-Ising chain, Phys. Rev. D 23 (1981), 417-419.
  23. Ikhlef Y., Jacobsen J., Saleur H., A staggered six-vertex model with non-compact continuum limit, Nuclear Phys. B 789 (2008), 483-524, arXiv:cond-mat/0612037.
  24. Ikhlef Y., Jacobsen J., Saleur H., The $\mathbb Z_2$ staggered vertex model and its applications, J. Phys. A: Math. Theor. 43 (2010), 225201, 37 pages, arXiv:0911.3003.
  25. Ikhlef Y., Jacobsen J., Saleur H., An integrable spin chain for the ${\rm SL}(2,{\mathbb R})/{\rm U}(1)$ black hole sigma model, Phys. Rev. Lett. 108 (2012), 081601, 6 pages, arXiv:1109.1119.
  26. Jacobsen J.L., Saleur H., The antiferromagnetic transition for the square-lattice Potts model, Nuclear Phys. B 743 (2006), 207-248, arXiv:cond-mat/0512058.
  27. Japaridze G.I., Nersesyan A.A., Wiegmann P.B., Regularized integrable version of the one-dimensional quantum sine-Gordon model, Phys. Scripta 27 (1983), 5-7.
  28. Kadanoff L.P., Brown C.A., Correlation functions on the critical lines of the Baxter and Ashkin-Teller models, Ann. Physics 121 (1979), 318-342.
  29. Khoroshkin S.M., Tolstoy V.N., Universal $R$-matrix for quantized (super)algebras, Comm. Math. Phys. 141 (1991), 599-617.
  30. Korepin V.E., Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982), 391-418.
  31. Lieb E.H., Residual entropy of square ice, Phys. Rev. 162 (1967), 162-172.
  32. Lieb E.H., Wu F.Y., Two-dimensional ferroelectric models, in Phase Transitions and Critical Phenomena, Editors C. Domb, M.S. Green, Academic Press, London, 1972, 331-490.
  33. Luther A., Peschel I., Calculation of critical exponents in two-dimensions from quantum field theory in one-dimension, Phys. Rev. B 12 (1975), 3908-3917.
  34. Pauling L., The structure and entropy of ice and of other crystals with some randomness of atomic arrangement, J. Amer. Chem. Soc. 57 (1935), 2680-2684.
  35. Polyakov A., Wiegmann P.B., Theory of nonabelian Goldstone bosons in two dimensions, Phys. Lett. B 131 (1983), 121-126.
  36. Reshetikhin N.Yu., Lectures on the integrability of the six-vertex model, in Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing, Oxford University Press, Oxford, 2010, 197-266, arXiv:1010.5031.
  37. Sklyanin E.K., Takhtadzhyan L.A., Faddeev L.D., Quantum inverse problem method. I, Theoret. and Math. Phys. 40 (1979), 688-706.
  38. Sutherland B., Exact solution of a two-dimensional model for hydrogen-bonded crystals, Phys. Rev. Lett. 19 (1967), 103-104.
  39. Sutherland B., Yang C.N., Yang C.P., Exact solution of a model of two-dimensional ferroelectrics in an arbitrary external electric field, Phys. Rev. Lett. 19 (1967), 588-591.
  40. Takhtadzhan 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.
  41. Yang C.P., Exact solution of a model of two-dimensional ferroelectrics in an arbitrary external electric field, Phys. Rev. Lett. 19 (1967), 586-588.

Previous article  Next article  Contents of Volume 17 (2021)