Embedding integrable spin models in solvable vertex models on the square lattice

Exploring a mapping among $n$-state spin and vertex models on the square lattice we argue that a given integrable spin model with edge weights satisfying the rapidity difference property can be formulated in the framework of an equivalent solvable vertex model. The Lax operator and the $\mathrm{R}$-matrix associated to the vertex model are built in terms of the edge weights of the spin model and these operators are shown to satisfy the Yang-Baxter algebra. The unitarity of the $\mathrm{R}$-matrix follows from an assumption that the vertical edge weights of the spin model satisfy certain local identity known as inversion relation. We apply this embedding to the scalar $n$-state Potts model and we argue that the corresponding $\mathrm{R}$-matrix can be written in terms of the underlying Temperley-Lieb operators. We also consider our construction for the integrable Ashkin-Teller model and the respective $\mathrm{R}$-matrix is expressed in terms of sixteen distinct weights parametrized by theta functions. We comment on the possible extention of our results to spin models whose edge weights are not expressible in terms of the difference of spectral parameters.