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International Journal of Modern Physics A

SOLUTIONS OF THE STAR-TRIANGLE EQUATIONS IN (Nα,Nβ) POTTS MODELS

Helen Au-Yang, Jacques H. H. Perk

Published Apr 1, 1992• 19 citations

Abstract

In this paper we present the solutions of the star-triangle equation for Boltzmann weights of the (Nα,Nβ) model. This model consists of two scalar Potts models (of Nα and Nβ states, respectively) coupled together by four-spin interaction terms and was introduced by Domany and Riedel to study the phase transitions in adsorbed thin films. If Nα,Nβ≠2, the star-triangle equations have nontrivial self-dual and non-self-dual solutions only for Nα=Nβ; whereas if Nα≠Nβ=2, we have other self-dual and non-self-dual solutions. After applying partial duality transformations several of our solutions can be identified with known solutions for Interaction-Round-a-Face (IRF) models. Our results also mean that our high-genus solutions found earlier are not the only solutions to the star-triangle equations in chiral Potts models with six or more states per site.

DOI: 10.1142/s0217751x92004166

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Star-Triangle Relation for a Three Dimensional Model

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•Jan 1, 1993•64 citations

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Double potts chain and exact results for some two-dimensional spin models

M. A. Yurishchev

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arXiv:cond-mat/0111401•Journal of Experimental and Theoretical Physics•Nov 1, 2001•1 citation

In this paper we present the solutions of the star-triangle equation for Boltzmann weights of the (Nα,Nβ) model. This model consists of two scalar Potts models (of Nα and Nβ states, respectively) coupled together by four-spin interaction terms and was introduced by Domany and Riedel to study the phase transitions in adsorbed thin films. If Nα,Nβ≠2, the star-triangle equations have nontrivial self-dual and non-self-dual solutions only for Nα=Nβ; whereas if Nα≠Nβ=2, we have other self-dual and non-self-dual solutions. After applying partial duality transformations several of our solutions can be identified with known solutions for Interaction-Round-a-Face (IRF) models. Our results also mean that our high-genus solutions found earlier are not the only solutions to the star-triangle equations in chiral Potts models with six or more states per site.

A new type of Yang-Baxter equation (YBE) for

R

-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the

Z_N

symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for

N>2

, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated

R

-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter

R

-matrix---realized as an intertwiner of cyclic representations of

U_q(\mathfrak{sl}_2)

at a root of unity---and Shastry's two-parameter

R

-operator obtained from the decorated YBE.

The solvable

sl(n)

-chiral Potts model can be interpreted as a three-dimensional lattice model with local interactions. To within a minor modification of the boundary conditions it is an Ising type model on the body centered cubic lattice with two- and three-spin interactions. The corresponding local Boltzmann weights obey a number of simple relations, including a restricted star-triangle relation, which is a modified version of the well-known star-triangle relation appearing in two-dimensional models. We show that these relations lead to remarkable symmetry properties of the Boltzmann weight function of an elementary cube of the lattice, related to spatial symmetry group of the cubic lattice. These symmetry properties allow one to prove the commutativity of the row-to-row transfer matrices, bypassing the tetrahedron relation. The partition function per site for the infinite lattice is calculated exactly.