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Journal of Mathematical Physics

The free energy of the three-dimensional Zamolodchikov model

M. L. Glasser

Published Nov 1, 1986• 0 citations
Publisher

Abstract

Details are given for the reduction of Baxter’s expression for the exact partition function per site of the Zamolodchikov model to explicit form.
DOI
10.1063/1.527302

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Starting with some observations due to A. B. Zamolodchikov, a short cut to the determination of the Baxter (eight-vertex) -model free energy is developed. The method involves factorizable SSS matrices in 1 + 1 dimensions, which are briefly reviewed. If the method generalizes, we may associate with each known SSS matrix a Baxter-like model with a known free energy.
It is shown that the Zamolodchikov model on the cubic lattice of NNN sites is equivalent to a spin model on a lattice of 2N2N2N sites with just two-spin and three-spin interactions. Together with the known transfer-matrix commutativity, this implies that the partition function ZZZ (for NNN large) factors into a product of single-variable functions. These functions, and hence ZZZ, are calculated by using symmetry relations. The model appears to be a critical free-fermion model.
In this talk, we present our recent results on the three-layer Zamolodchikov model. We discuss solutions to the Bethe ansatz equations following from functional relations. We consider two regimes I and II that differ by the signs of the spherical sides ( a 1 , a 2 , a 3 ) → (− a 1 , − a 2 , − a 3 ). Also, we accept the two-line hypothesis for regime I and the one-line hypothesis for regime II. In the thermodynamic limit, we derive integral equations for distribution densities and solve them exactly. Using this solution, we calculate the partition function for the three-layer Zamolodchikov model and check the compatibility of this result with functional relations. We also discuss the reasons for the discrepancy with Baxter’s result of 1986.
Abstract It is shown that the two-dimensional free fermion model is equivalent to a checkerboard Ising model, which is a special case of the general ‘Z-invariant’ Ising model. Expressions are given for the partition function and local correlations in terms of those of the regular square lattice Ising model. Corresponding results are given for the self-dual Potts model, and the application of the methods to the three-dimensional Zamolodchikov model is discussed. The paper ends with a discussion of the critical and disorder surfaces of the checkerboard Potts model.