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Journal of the Physical Society of Japan

Exactly Solvable IRF Models. II.SN-Generalizations

Yasuhiro Akutsu, Atsuo Kuniba, Miki Wadati

Published May 15, 1986• 27 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.55.1466

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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 .

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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.

Journal of the Physical Society of Japan•Oct 15, 1986•8 citations

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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•May 1, 1994•10 citations
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 pt.I see ibid, vol.17, p.1251 (1984). The authors continue their study of the star-triangle relation for two-component spin models, with the help of the inversion relation. The conclusion is that the most general interaction-round-a-face models (16 non-zero parameters) that satisfy the star-triangle relations are mainly Baxter (1972) and Fan and Wu (1970) models. In this way they recover Krichever's result (1981) for the classification of the star-triangle relations for vertex models. They sketch a generalisation of this approach to models with excluded configurations, to staggered models, and to three-dimensional models (tetrahedron relations).
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.
We have introduced a type of universal transformation between spin variables si{\mathit{s}}_{\mathit{i}}si​ and ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$ to map the general spin-1 model onto the spin-1/2 Ising model and have employed the exact results of the latter to find the exactly solvable cases of the spin-1 model. Our transformations are not one-to-one correspondences between si{\mathit{s}}_{\mathit{i}}si​ and ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$ and they are applicable to all lattices, depending on the lattice structure only through the coordination number. On square, triangle, and honeycomb lattices the exact critical points are shown in certain subspaces of parameter space (K, J, L, H, and \ensuremath{\Delta}).