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arXiv:hep-th/9210063 · Modern Physics Letters A

YANG-BAXTER EQUATION FOR BROKEN $\mathbb{Z}_N^{\otimes n-1}$ MODELS

YAS-HIRO QUANO, AKIRA FUJII

Published Jun 7, 1993• 16 citations
arXivHTMLPDF

Abstract

We construct a factorized representation of the [Formula: see text]-Sklyanin algebra from the vertex-face correspondence. Using this representation, we obtain a new solvable model which gives an [Formula: see text]-generalization of the broken ℤN model. We further prove the Yang-Baxter equation for this model.
DOI
10.1142/s021773239300132x

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YANG-BAXTER EQUATION FOR BROKEN ZN⊗n−1\mathbb{Z}_N^{\otimes n-1}ZN⊗n−1​ MODELS
We study three properties of the ℤn⊗ℤn-symmetric lattice model; i.e. the initial condition, the unitarity and the crossing symmetry. The scalar factors appearing in the unitarity and the crossing symmetry are explicitly obtained. The [Formula: see text]-Sklyanin algebra is introduced in the natural framework of the inverse problem for this model. We build both finite- and infinite-dimensional representations of the [Formula: see text]-Sklyanin algebra, and construct an [Formula: see text] generalization of the broken ℤN model. Furthermore, the Yang-Baxter equation for this new model is proved.
In this paper we review the theory of the Yang–Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R-matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). In the simplest case of the spin s=1/2 this R-matrix naturally reduces to the R-matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the Q-operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin s. We also give all functional equations satisfied by the transfer-matrices and Q-operators.
In the case of two-component models, all possible solutions for factorization equations of S-matrix are obtained. These solutions give a classification of solvable vertex models. It is found that Yang-Baxter relation is more general than the factorization equation. The six vertex model by Yang, Yang and Sutherland is derived not from the factorization equation but from Yang-Baxter relation. Thus it is concluded that Yang-Baxter relation lies at the basis of all the known solvable two component models. Furthermore, spin Hamiltonians corresponding to our vertex models are presented in explicit forms.
An analog of the Yang-Baxter Algebra (YBA) is defined in face language. The operators tαα′,ββ′(θ) introduced here enjoy all essential properties of the vertex language YBA. Using this face YBA an algebraic Bethe Ansatz (BA) is constructed for SOS models (unrestricted IRF models). The face dual of the six-vertex model and the critical ABF model are worked out explicitly. Eigenvectors and eigenvalues of the transfer matrix are found and the corresponding BA equations derived and compared with the six vertex BAE.