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Journal of the Physical Society of Japan

Inhomogeneous Eight-Vertex SOS Model and Solvable IRF Hierarchies

Yasuhiro Akutsu, Atsuo Kuniba, Miki Wadati

Published Sep 15, 1986• 12 citations
Publisher

Abstract

It is shown that an exactly solvable inhomogeneous eight-vertex SOS (8VSOS) model ( Z -invariant 8VSOS model) reproduces most of the konwn hierarchies of solvable IRF spin models. It is also shown that the Z -invariance gives a simple explanation for the coincidence found by Baxter and Andrews between the expressions of the local spin densities for the two different hierarchies of the IRF models, the Gordon-Generalization hierarchy and the Andrews-Baxter-Forrester hierarchy.

DOI
10.1143/jpsj.55.2907

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Physical Review Letters•Apr 13, 1987•37 citations
It is shown that an exactly solvable inhomogeneous eight-vertex SOS (8VSOS) model ( Z -invariant 8VSOS model) reproduces most of the konwn hierarchies of solvable IRF spin models. It is also shown that the Z -invariance gives a simple explanation for the coincidence found by Baxter and Andrews between the expressions of the local spin densities for the two different hierarchies of the IRF models, the Gordon-Generalization hierarchy and the Andrews-Baxter-Forrester hierarchy.
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 .
It is shown that ( r -1)-state restricted eight-vertex SOS model (8VSOS) by Andrews, Baxter and Forrester is equivalent to q -state Potts model with q =4·cos 2 (π/ r ) ( r =4, 5, 6···) at criticality. The equivalence follows from the Temperley-Lieb algebra shared by the models which is nothing but C * -algebra A q , n defined by Jones in von Neumann algebra theory. This implies the correspondence of indices for subfactors of type II 1 factors and discrete series of Virasoro algebra which labels the universality classes of the 8VSOS models.
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.
A new functional equation satisfied by the commuting row-to-row transfer matrices is derived for the eight-vertex model. Functional equations of the same form are satisfied by hard hexagons, magnetic hard squares, the self-dual Potts models, the Andrews-Baxter-Forrester models, and others. These new functional equations are called inversion identities because they generalize the inversion relation for local transfer matrices. It is conjectured that all solvable models satisfying Yang-Baxter equations possess such inversion identities and that, in general, these functional equations can be solved for the transfer-matrix eigenvalues.