Star-Triangle Relation for a Three Dimensional Model

TL;DR

This paper introduces a modified star-triangle relation for a three-dimensional lattice model, revealing symmetry properties that enable exact calculation of the partition function and proof of transfer matrix commutativity.

Contribution

It presents a new star-triangle relation for a 3D model, leading to symmetry insights and exact solutions, bypassing the tetrahedron relation.

Findings

Derived a modified star-triangle relation for 3D lattice models

Proved symmetry properties of Boltzmann weights related to cubic lattice symmetry

Calculated the exact partition function per site for the infinite lattice

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. 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.