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Journal of Physics A Mathematical and General

A remark on solutions of reflection equation for the critical ZN-symmetric vertex model

Yuji Yamada

Published Dec 15, 2003• 2 citations
Publisher

Abstract

We consider solutions to the reflection equation for the critical Z N -symmetric vertex model, which is the trigonometric limit of the elliptic Z N -symmetric R -matrix of Belavin. These critical R -matrices have two parameters u and η. The transfer matrices constructed from this R -matrix R ( u , η) under the cyclic boundary condition are commutative among different u when η is in common, . We prove that an arbitrary solution to the reflection equation is independent of the parameter η.

DOI
10.1088/0305-4470/37/2/019

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We consider solutions to the reflection equation for the critical Z N -symmetric vertex model, which is the trigonometric limit of the elliptic Z N -symmetric R -matrix of Belavin. These critical R -matrices have two parameters u and η. The transfer matrices constructed from this R -matrix R ( u , η) under the cyclic boundary condition are commutative among different u when η is in common, . We prove that an arbitrary solution to the reflection equation is independent of the parameter η.
In this work, we employ the algebraic-differential method recently developed by the author to solve the Yang-Baxter equation for arbitrary fifteen-vertex models satisfying the ice-rule. We show that there are four different families of such regular RRR matrices containing several free-parameters. The corresponding reflection KKK matrices, solutions of the boundary Yang-Baxter equation, were also found and classified. We found that there are three different families of regular KKK matrices, regardless of what RRR matrix we choose.
Belavin's n -symmetric elliptic model with boundary reflection is considered on the basis of the boundary CTM bootstrap. We find non-diagonal K -matrices for n >2 that satisfy the reflection equation (boundary Yang-Baxter equation), and also find non-diagonal Boltzmann weights for the A (1) n -1 -face model even for n ⩾2. We derive difference equations of the quantum Knizhnik-Zamolodchikov type for correlation functions of the boundary model. The boundary spontaneous polarization is obtained by solving the simplest difference equations in the case of the free boundary condition. The resulting quantity is the square of the spontaneous polarization for the bulk n -symmetric model, up to a phase factor.
Integrable An−1(1)A^{(1)}_{n-1}An−1(1)​ IRF Model with Reflecting Boundary Conditions