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Journal of the Physical Society of Japan

Boundary Boltzmann Weight for the Eight-Vertex SOS Model: Vertex-IRF Correspondence

Kazuhiro Hikami, Yasushi Komori

Published Jan 15, 1998• 2 citations
Publisher

Abstract

Integrable boundary Boltzmann weight for the eight-vertex solid-on-solid model is constructed. Based on the operator-valued solutions of both the Yang-Baxter equation and the reflection equation, we propose a simple method to give the Boltzmann weight using the vertex-IRF correspondence.
DOI
10.1143/jpsj.67.78

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Abstract Interaction-round the face (IRF) models are two-dimensional lattice models of statistical mechanics defined by an affine Lie algebra and admissibility conditions depending on a choice of representation of that affine Lie algebra. Integrable IRF models, i.e. the models the Boltzmann weights of which satisfy the quantum Yang–Baxter equation, are of particular interest. In this paper, we investigate trigonometric Boltzmann weights of integrable IRF models. By using an ansatz proposed by one of the authors in some previous works, the Boltzmann weights of the restricted IRF models based on the affine Lie algebras su ( 2 ) k and su ( 3 ) k are computed for fundamental and adjoint representations for some fixed levels k. New solutions for the Boltzmann weights are obtained. We also study the vertex-IRF correspondence in the context of an unrestricted IRF model based on su ( 3 ) k (for general k) and discuss how it can be used to find Boltzmann weights in terms of the quantum R ^ matrix when the adjoint representation defines the admissibility conditions.
A bstract This paper represents a continuation of our previous work, where the Boltzmann weights (BWs) for several Interaction-Round-the Face (IRF) lattice models were computed using their relation to rational conformal field theories. Here, we focus on deriving solutions for the Boltzmann weights of the Interaction-Round the Face lattice model, specifically, the unrestricted face model, based on the su(3)k \mathfrak{su}{(3)}_k su(3)k​ su 3 k affine Lie algebra. The admissibility conditions are defined by the adjoint representation. We find the BWs by determining the quantum R matrix of the Uq(sl(3)) {U}_q\left(\mathfrak{sl}(3)\right) Uq​(sl(3)) U q sl 3 quantum algebra in the adjoint representation and then applying the so-called Vertex-IRF correspondence. The Vertex-IRF correspondence defines the BWs of IRF models in terms of R matrix elements.
In this paper we study, in the manner of Baxter [R. J. Baxter Exactly Solved Models in Statistical Mechanics (Academic, New York, 1982)], the possible integrable manifolds of the asymmetric eight-vertex model. As expected, they occur when the Boltzmann weights are either symmetric or satisfy the free-fermion condition; but our analysis clarifies the reason why both manifolds need to share a universal invariant. We also show that the free-fermion condition implies three distinct classes of integrable models.