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arXiv:solv-int/9903010 · Journal of Physics A Mathematical and General

Bethe ansatz for the three-layer Zamolodchikov model

Herman E. Boos, Vladimir V. Mangazeev

Published Jan 1, 1999• 6 citations
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Abstract

This paper is a continuation of our previous work (Boos H E and Mangazeev V V 1999 J. Phys. A: Math. Gen. 32 3041-54). We obtain two more functional relations for the eigenvalues of the transfer matrices for the sl (3) chiral Potts model at q 2 = -1. This model, up to a modification of boundary conditions, is equivalent to the three-layer three-dimensional Zamolodchikov model. From these relations we derive the Bethe ansatz equations.

DOI
10.1088/0305-4470/32/28/308

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This paper is a continuation of our previous work (Boos H E and Mangazeev V V 1999 J. Phys. A: Math. Gen. 32 3041-54). We obtain two more functional relations for the eigenvalues of the transfer matrices for the sl (3) chiral Potts model at q 2 = -1. This model, up to a modification of boundary conditions, is equivalent to the three-layer three-dimensional Zamolodchikov model. From these relations we derive the Bethe ansatz equations.
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.
Functional relations have previously been obtained for the eigenvalues of the transfer matrices of the chiral Potts model. Introducing skewed boundary conditions is equivalent to merely modifying the quantum number of the spin shift operator in the relations (which accounts for at least some of the previously noted "spurious solutions"). As a first step towards calculating the general interfacial tension, we consider the model in a zero-temperature limit. It is still non-trivial, there being near-vertical free interfaces separating domains of different spin value. These interfaces behave like the "lines of down arrows" in the ice model, so one may hope to follow Lieb and use the Bethe ansatz to evaluate the partition function. It turns out that this can indeed be done. There is no wetting of an interface by intermediate phases.