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Journal of the Physical Society of Japan

The Gordon-Generalization Hierarchy of Exactly Solvable IRF Models

Atsuo Kuniba, Yasuhiro Akutsu, Miki Wadati

Published Oct 15, 1986• 8 citations
Publisher

Abstract

For any k -state model among the Gordon-Generalization (GG) hierarchy (J. Phys. Soc. Jpn. 55 (1986) 1092 and 2166), a systematic procedure to obtain the parametrized Boltzmann weights is presented. An exact expression for the free energy of the general k -state model in the GG hierarchy is explicitly given. The procedure is successfully applied to determine the parametrization for the 5-state model in the GG hierarchy. It is concluded that each k -state model in the GG hierarchy is exactly solvable.

DOI
10.1143/jpsj.55.3338

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For any k -state model among the Gordon-Generalization (GG) hierarchy (J. Phys. Soc. Jpn. 55 (1986) 1092 and 2166), a systematic procedure to obtain the parametrized Boltzmann weights is presented. An exact expression for the free energy of the general k -state model in the GG hierarchy is explicitly given. The procedure is successfully applied to determine the parametrization for the 5-state model in the GG hierarchy. It is concluded that each k -state model in the GG hierarchy is exactly solvable.
It is shown that a k -state IRF model (σ=0, 1, ···, k -1), with a condition k -2≦σ i +σ j ≦ k on adjacent spins σ i and σ j , is exactly solvable for all k ( k ≧3). This proves the existence of a new hierarchy of solvable IRF models. It is also shown that the k -state IRF model is equivalent to a 3 k -state solid on solid (SOS) model. Considering all the known results, it is predicted that for an arbitrary set of integers L and f ( L ≦0, f ≦1) there exists a solvable IRF model with the hard core condition L ≧σ i +σ j ≦ L + f .
Multi-state generalization, which is applicable to arbitrary IRF model, is presented. The generalization is made in such a way that each spin is replaced by an S N -symmetric multiplet. If the original IRF model is exactly solvable satisfying the star-triangle equation, so is the generalized one. A simple symmetry-breaking procedure is also presented. In the special case N =2, another kind of generalization is proposed and an exactly solvable 4-state model with S 2 -symmetry between the highest spins is newly constructed.
A bstract This paper represents a continuation of our previous work, where the Boltzmann weights (BWs) for several Interaction-Round-the Face (IRF) lattice models were computed using their relation to rational conformal field theories. Here, we focus on deriving solutions for the Boltzmann weights of the Interaction-Round the Face lattice model, specifically, the unrestricted face model, based on the su(3)k \mathfrak{su}{(3)}_k su(3)k​ su 3 k affine Lie algebra. The admissibility conditions are defined by the adjoint representation. We find the BWs by determining the quantum R matrix of the Uq(sl(3)) {U}_q\left(\mathfrak{sl}(3)\right) Uq​(sl(3)) U q sl 3 quantum algebra in the adjoint representation and then applying the so-called Vertex-IRF correspondence. The Vertex-IRF correspondence defines the BWs of IRF models in terms of R matrix elements.