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International Journal of Modern Physics B

INVERSION AND SYMMETRY RELATIONS FOR A THREE-DIMENSIONAL SOLVABLE MODEL

Vladimir V. Bazhanov

Published Sep 30, 1993• 6 citations
Publisher

Abstract

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. We discuss the spatial symmetry properties of the model and show that its partition function per site obeys the inversion relation. The connection with the free boson model is also discussed.

DOI
10.1142/s021797929300336x

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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. We discuss the spatial symmetry properties of the model and show that its partition function per site obeys the inversion relation. The connection with the free boson model is also discussed.
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.
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.
We discover an Ising-type duality in the general N -state chiral Potts model, which is the Kramers–Wannier duality of a planar Ising model when N = 2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small k ') to those of another chiral Potts model at a high temperature (of k ' −1 ). The τ (2) model and chiral Potts model on the dual lattice are established alongside of the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the sl 2 -loop-algebra symmetry of its associated spin- ( N −1)/2 XXZ chain through the identification of their eigenstates.