An Ising-type formulation of the six-vertex model

TL;DR

This paper reformulates the six-vertex model as an Ising-type model with two-spin interactions, revealing new factorization properties and functional relations, and connecting integrability conditions to classical mathematical identities.

Contribution

It introduces an Ising-type reformulation of the six-vertex model, enabling a unified derivation of eigenvalue relations and Bethe ansatz solutions for higher spins.

Findings

Reformulation as Ising-type model simplifies analysis.

Derivation of all functional relations for eigenvalues.

Connection of integrability to classical Pfaff-Saalschuetz-Jackson identity.

Abstract

We show that the celebrated six-vertex model of statistical mechanics (along with its multistate generalizations) can be reformulated as an Ising-type model with only a two-spin interaction. Such a reformulation unravels remarkable factorization properties for row to row transfer matrices, allowing one to uniformly derive all functional relations for their eigenvalues and present the coordinate Bethe ansatz for the eigenvectors for all higher spin generalizations of the six-vertex model. The possibility of the Ising-type formulation of these models raises questions about the precedence of the traditional quantum group description of the vertex models. Indeed, the role of a primary integrability condition is now played by the star-triangle relation, which is not entirely natural in the standard quantum group setting, but implies the vertex-type Yang-Baxter equation and commutativity of transfer matrices as simple corollaries. As a mathematical identity the emerging star-triangle relation is equivalent to the Pfaff-Saalschuetz-Jackson summation formula, originally discovered by J.~F.~Pfaff in 1797. Plausibly, all vertex models associated with quantized affine Lie algebras and superalgebras can be reformulated as Ising-type models with only two-spin interactions.