On the equivalence between n-state spin and vertex models on the square lattice

TL;DR

This paper establishes a correspondence between n-state spin and vertex models on the square lattice, showing their partition functions coincide and exploring the integrability properties of the equivalent vertex models.

Contribution

It explicitly constructs the vertex model equivalent to an arbitrary n-state spin model and analyzes its algebraic structure, including the Yang-Baxter algebra and R-matrix.

Findings

Partition functions of spin and vertex models coincide for finite sizes.

Mapped Ising model in a magnetic field to an eight-vertex model with mixed arrow configurations.

Derived the R-matrix for the 27-vertex model related to a three-state spin model.

Abstract

In this paper we investigate a correspondence among spin and vertex models with the same number of local states on the square lattice with toroidal boundary conditions. We argue that the partition functions of an arbitrary $n$-state spin model and of a certain specific $n$-state vertex model coincide for finite lattice sizes. The equivalent vertex model has $n^3$ non-null Boltzmann weights and their relationship with the edge weights of the spin model is explicitly presented. In particular, the Ising model in a magnetic field is mapped to an eight-vertex model whose weights configurations combine both even and odd number of incoming and outcoming arrows at a vertex. We have studied the Yang-Baxter algebra for such mixed eight-vertex model when the weights are invariant under arrows reversing. We find that while the Lax operator lie on the same elliptic curve of the even eight-vertex model the respective $\mathrm{R}$-matrix can not be presented in terms of the difference of two rapidities. We also argue that the spin-vertex equivalence may be used to imbed an integrable spin model in the realm of the quantum inverse scattering framework. As an example, we show how to determine the $\mathrm{R}$-matrix of the 27-vertex model equivalent to a three-state spin model devised by Fateev and Zamolodchikov.