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arXiv:1302.3025 · Journal of Physics A Mathematical and Theoretical

A new solution of the star–triangle relation

Andrew P. Kels

Published Jan 17, 2014• 26 citations
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Abstract

We obtain a new solution to the star–triangle relation for an Ising-type model with two kinds of spin variables at each lattice site, taking continuous real values and arbitrary integer values, respectively. The Boltzmann weights are manifestly real and positive. They are expressed through the Euler gamma function and depend on sums and differences of spins at the ends of an edge of the lattice.

DOI
10.1088/1751-8113/47/5/055203

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We obtain a new solution to the star–triangle relation for an Ising-type model with two kinds of spin variables at each lattice site, taking continuous real values and arbitrary integer values, respectively. The Boltzmann weights are manifestly real and positive. They are expressed through the Euler gamma function and depend on sums and differences of spins at the ends of an edge of the lattice.
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We obtain a new solution of the star-triangle relation with positive Boltzmann weights, which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known.This new master solution defines an exactly solvable two lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters.If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced on
The solvable sl(n)sl(n)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.
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.