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Journal of the Physical Society of Japan

Exactly Solvable IRF Models. III. A New Hierarchy of Solvable Models

Yasuhiro Akutsu, Atsuo Kuniba, Miki Wadati

Published Jun 15, 1986• 20 citations
Publisher

Abstract

An infinite series of exactly solvable multi-state IRF models is constructed. Each model in the new hierarchy is the special S 2 -generalization, proposed in the previous paper II, of the corresponding model in the `eight-vertex SOS' hierarchy. The Boltzmann weights and their parametrizations are given with a detailed proof that they satisfy the star-triangle equation.

DOI
10.1143/jpsj.55.1880

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An infinite series of exactly solvable multi-state IRF models is constructed. Each model in the new hierarchy is the special S 2 -generalization, proposed in the previous paper II, of the corresponding model in the `eight-vertex SOS' hierarchy. The Boltzmann weights and their parametrizations are given with a detailed proof that they satisfy the star-triangle equation.
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.
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 .
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.
In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local “spin variables” can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent “dual” formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra Uq(slˆ(2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model.