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

TL;DR

This work develops a general spin-vertex correspondence that embeds integrable $n$-state spin models into solvable $n$-state vertex models on the square lattice by constructing a vertex Lax operator and an $R$-matrix from spin-edge weights. It shows that star-triangle relations among spin weights imply the Yang-Baxter algebra for the $R$-matrix and that unitarity follows from inversion relations, enabling commuting transfer matrices. The authors apply the framework to the scalar Potts model, obtaining an $R$-matrix expressed via Temperley-Lieb generators, and to the Ashkin-Teller model, obtaining a 16-weight $R$-matrix expressed with theta-function parametrizations, illustrating rich algebraic structures beyond difference form. They also discuss a generalized vertex construction with non-difference-form $R$-matrices and corresponding Hamiltonians with extra interactions, signaling new integrable deformations of quantum spin chains. Overall, the approach unifies spin- and vertex-model integrability on the square lattice and opens avenues for extending to models with non-difference spectral parameter dependence.

Abstract

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.

Figures (4)

• Figure 1: Schematic representation of the horizontal $W_{h}(i,j|x)$ and the vertical $W_{v}(j,k|x)$ local Boltzmann weights of the spin models where $(i,j,k)$ indicate spin states.
• Figure 2: Schematic representation of the diagonal-to-diagonal transfer matrix of spin models.
• Figure 3: The local Boltzmann weights of vertex models with spin variables $i,j,k,l$ taken values on the set $\{1,\dots,n\}$.
• Figure 4: Schematic representation of the row-to-row transfer matrix of vertex models.