Integrability of the odd eight-vertex model with symmetric weights

TL;DR

This paper explores the integrability of the odd eight-vertex model, showing its integrable conditions align with the even model under certain symmetries, and discusses the structure of its R-matrix.

Contribution

It demonstrates that the integrable manifold of the odd eight-vertex model matches that of the even model when weights are symmetric, and characterizes its R-matrix structure.

Findings

Integrability conditions are identical for odd and even models under arrow-inversion symmetry.

The R-matrix for the odd model differs from the Lax operator but satisfies Yang-Baxter equations.

Both models lead to a sheaf of R-matrices related to intertwiner relations.

Abstract

In this paper we investigate the integrability properties of a two-state vertex model on the square lattice whose microstates at a vertex has always an odd number of incoming or outcoming arrows. This model was named odd eight-vertex model by Wu and Kunz \cite{WK} to distinguish it from the well known eight-vertex model possessing an even number of arrows orientations at each vertex. When the energy weights are invariant under arrows inversion we show that the integrable manifold of the odd eight-vertex model coincides with that of the even eight-vertex model. The form of the $\mathrm{R}$-matrix for the odd eight-vertex model is however not the same as that of the respective Lax operator. Altogether we find that these eight-vertex models give rise to a generic sheaf of $\mathrm{R}$-matrices satisfying the Yang-Baxter equations resembling intertwiner relations associated to equidimensional representations.