Journal of Statistical Physics

, Volume 84, Issue 1–2, pp 1–48 | Cite as

Interaction-round-a-face models with fixed boundary conditions: The ABF fusion hierarchy

  • Roger E. Behrend
  • Paul A. Pearce
  • David L. O'Brien


We use boundary weights and reflection equations to obtain families of commuting double-row transfer matrices for interaction-round-a-face models with fixed boundary conditions. In particular, we consider the fusion hierarchy of the Andrews-Baxter-Forrester (ABF) models, for which we obtain diagonal, elliptic solutions to the reflection equations, and find that the double-row transfer matrices satisfy functional equations with the same form as in the case of periodic boundary conditions.

Key Words

Andrews-Baxter-Forrester models exactly solvable lattice models fixed boundary conditions reflection equations Yang-Baxter equation 


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  1. 1.
    R. J. Baxter,Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982).Google Scholar
  2. 2.
    E. K. Sklyanin, Boundary conditions for integrable quantum systems,J. Phys. A 21:2375–2389 (1988).Google Scholar
  3. 3.
    J. L. Cardy, Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories,Nucl. Phys. B 275:200–218 (1986).Google Scholar
  4. 4.
    L. Mezincescu and R. I. Nepomechie, Integrable open spin chains with non-symmetricR-matrices,J. Phys. A 24:L17-L23 (1991).Google Scholar
  5. 5.
    L. Mezincescu and R. I. Nepomechie, Addendum: Integrability of open spin chains with quantum algebra symmetry,Int. J. Mod. Phys. A 7:5657–5659 (1992).Google Scholar
  6. 6.
    R. H. Yue and Y. X. Chen, IntegrableZ n ×Z n Belavin model with non-trivial boundary terms,J. Phys. A. 26:2989–2994 (1993).Google Scholar
  7. 7.
    S. Ghoshal and A. Zamolodchikov, BoundaryS matrix and boundary states in two-dimensional integrable quantum field theory,Int. J. Mod. Phys. A 9:3841–3885 (1994).Google Scholar
  8. 8.
    F. W. Wu, Ising model with four-spin interactions,Phys. Rev. B 4:2312–2314 (1971).Google Scholar
  9. 9.
    L. P. Kadanoff and F. J. Wegner, Some critical properties of the eight-vertex model,Phys. Rev. B 4:3989–3993 (1971).Google Scholar
  10. 10.
    R. J. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. II. Equivalence to a generalised ice-type lattice model,Ann. Phys. 76:25–47 (1973).Google Scholar
  11. 11.
    H. van Beijeren, Exactly solvable model for the roughening transition of a crystal surface,Phys. Rev. Lett. 38:993–996 (1977).Google Scholar
  12. 12.
    P. P. Kulish, Yang-Baxter equation and reflection equations in integrable models, Preprint [hep-th/9507070].Google Scholar
  13. 13.
    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).Google Scholar
  14. 14.
    L. Mezincescu and R. I. Nepomechie, Fusion procedure for open chains,J. Phys. A 25:2533–2543 (1992).Google Scholar
  15. 15.
    E. Date, M. Jimbo, T. Miwa, and M. Okado,Lett. Math. Phys. 12:209–215 (1986).Google Scholar
  16. 16.
    E. Date, M. Jimbo, T. Miwa, and M. Okado, Erratum and Addendum: Fusion of the eight vertex SOS model,Lett. Math. Phys. 14:97 (1987).Google Scholar
  17. 17.
    E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, Exactly solvable SOS models II: Proof of the star-triangle relation and combinatorial identities,Adv. Stud. Pure Math. 16:17–122 (1988).Google Scholar
  18. 18.
    Y. K. Zhou and P. A. Pearce, Fusion of A-D-E lattice models,Int. J. Mod. Phys. B 8:3531–3577 (1994).Google Scholar
  19. 19.
    V. V. Bazhanov and N. Y. Reshetikhin, Critical RSOS models and conformal field theory,Int. J. Mod. Phys. A 4:115–142 (1989).Google Scholar
  20. 20.
    A. Klümper and P. A. Pearce, Conformal weights of RSOS lattice models and their fusion hierarchies,Physica A 183304–350 (1992).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Roger E. Behrend
    • 1
  • Paul A. Pearce
    • 1
  • David L. O'Brien
    • 1
  1. 1.Department of MathematicsUniversity of MelbourneParkvilleAustralia

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