We present study on a multi-state vertex model solved by Sogo et al. at one of first order phase transition points. The origin of the level crossings is analyzed in terms of vertex weights and the Bethe ansatz equation. Singularity at this point is discussed in terms of quantum enveloping algebra.
In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi $ and energy $\Lambda $, which satisfy $V\psi =\Lambda \psi $, where V is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights a=b=1 and c>0. We also show that the same vector $\psi $ satisfies $H\psi =E\psi $, where H is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value E computed explicitly. Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors [5] that amounts to proving that the random-cluster model on Z2 with cluster weight q>4 exhibits a first-order phase transition.