Advertisement

Orthogonal Polynomials, 6J-Symbols, and Statistical Weights of SOS Models

  • P. A. ValinevichEmail author
  • S. E. Derkachov
  • A. P. Isaev
  • A. V. Komisarchuk
Article

We describe a simple diagrammatic method that allows one to connect the Boltzmann weights of vertex models of statistical mechanics with those of SOS models. An analogy with the computation of 6j-symbols is pointed out. The construction of statistical weights heavily relies on the realization of the group SU(2) on the space of functions of one variable. A closed-form answer for some particular cases is obtained. It is shown that in the general case, the statistical weight of a SOS model, as well as the 6j-symbol, can be presented as the scalar product of two polynomials of a certain type. Bibliography: 16 titles.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press (1982).Google Scholar
  2. 2.
    V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Univ. Press (2010).Google Scholar
  3. 3.
    E. K. Sklyanin, L. A. Takhtadzhyan, and L. D. Faddeev, “Quantum inverse problem method. I,” Theor. Math. Phys., 40, No. 2, 688–706 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    G. E. Andrews, R. J. Baxter, and P. J. Forrester, “Eight-vertex SOS model and generalized Rogers–Ramanujan-type identities,” J. Stat. Phys., 35, 193–266 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, “Exactly solvable SOS models II,” Adv. Stud. Pure Math., 16, 17–122 (1988).CrossRefzbMATHGoogle Scholar
  6. 6.
    I. B. Frenkel and V. G. Turaev, “Trigonometric solutions of the Yang–Baxter equation, nets, and hypergeometric functions,” in: Functional Analysis on the Eve of the 21st Century, Vol. 1, Progress in Math., 131, Birkhäuser, Boston (1995), pp. 65–118.Google Scholar
  7. 7.
    I. B. Frenkel and V. G. Turaev, “Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions,” in: The Arnold–Gelfand Mathematical Seminars, Birkh¨auser, Boston (1997), pp. 171–204.Google Scholar
  8. 8.
    P. P. Kulish, N. Yu. Reshetikhin, and E. K. Sklyanin, “Yang–Baxter equation and representation theory: I,” Lett. Math. Phys., 5, 393–404 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Klimyk and K. Schmüdgen, Quantum Groups and Their Representations, Springer (1997).Google Scholar
  10. 10.
    I. M. Gelfand and M. I. Naimark, “Unitary representations of the classical groups,” Trudy Mat. Inst. Steklov, 36, 3–288 (1950).Google Scholar
  11. 11.
    D. P. Zhelobenko, Lections on Lie Group Theory [in Russian], Dubna (1965).Google Scholar
  12. 12.
    S. E. Derkachov, D. N. Karakhanyan, and R. Kirschner, “Yang–Baxter R-operators and parameter permutations,” Nucl. Phys. B, 785, 263–285 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    L. D. Faddeev, “How the algebraic Bethe ansatz works for integable models,” in: A. Connes, K. Gawedzki, and J. Zinn-Justin (eds.), Quantum Symmetries/Sym´etries Quantiques, Proceedings of the Les Houches summer school, LXIV, North Holland (1998), pp. 149–211.Google Scholar
  14. 14.
    V. Pasquier, “Etiology of IRF models,” Comm. Math. Phys., 118, 355–364 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. N. Kirillov and N. Yu. Reshetikhin, “Representations of the algebra U q(sℓ(2)), qorthogonal polynomials and invariants of links,” in: Infinite-Dimensional Lie Algebras and Groups, World Scientific (1989), pp. 285–339.Google Scholar
  16. 16.
    D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific (1988).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • P. A. Valinevich
    • 1
    Email author
  • S. E. Derkachov
    • 1
  • A. P. Isaev
    • 2
  • A. V. Komisarchuk
    • 1
  1. 1.St. Petersburg Department of Steklov Institute of MathematicsSt.PetersburgRussia
  2. 2.Bogoliubov Laboratory of Theoretical PhysicsJoint Institute of Nuclear ResearchDubnaRussia

Personalised recommendations