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arXiv:cond-mat/0108363 · Birkhäuser Boston eBooks

A Rapidity-Independent Parameter in the Star-Triangle Relation

R. J. Baxter

Published Jan 1, 2002• 16 citations
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Abstract

The normalization factor in the star-triangle relation can be evaluated in a simple form by taking determinants. If we combine this with the rotation symmetries, then we can show that a certain simple quantity I has to be independent of the rapidities. In this sense it is an invariant. We evaluate it for several particular models and find it is one for self-dual models, and is related to the modulus k (or k’) for the Ising, Kashiwara—Miwa and chiral Potts models.

DOI
10.1007/978-1-4612-0087-1_3

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Journal of Physics A Mathematical and General•Apr 21, 1984•5 citations
The normalization factor in the star-triangle relation can be evaluated in a simple form by taking determinants. If we combine this with the rotation symmetries, then we can show that a certain simple quantity I has to be independent of the rapidities. In this sense it is an invariant. We evaluate it for several particular models and find it is one for self-dual models, and is related to the modulus k (or k’) for the Ising, Kashiwara—Miwa and chiral Potts models.
Symmetry relations such as the star-triangle or the inverse relation are very useful in determining the partition function of two-dimensional exactly soluble models. A common construction of the three-dimensional equivalents of these symmetry relations is presented. They are used to derive, in a geometric way applying simultaneously to different kinds of spin models, the consequent global properties, i.e. the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function. The usefulness of the inverse relation is illustrated by an application to the three-dimensional Ising model.
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).