© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:hep-th/9808083 · Journal of Physics A Mathematical and General

Classification of solutions to the reflection equation for two-component systems

Cong-xin Liu, Ju Guo-Xing, Shikun Wang, Ke Wu

Published Jan 1, 1999• 12 citations

Abstract

The symmetries, especially those related to the R -transformation, of the reflection equation (RE) for two-component systems are analysed. The classification of solutions to the RE for eight-, six- and seven-vertex-type R -matrices is given. All solutions can be obtained from those corresponding to the standard R -matrices by K -transformation. For the free-fermion models, the boundary matrices have property tr K + (0) = 0, and the free-fermion-type R -matrix with the same symmetry as that of a Baxter type corresponds to the same form of a K - -matrix for the Baxter type. We present the Hamiltonians for the open spin systems connected with our solutions. In particular, the boundary Hamiltonian of seven-vertex models is obtained with a generalization to the Sklyanin formalism.

DOI: 10.1088/0305-4470/32/19/304

Related papers

Classification of Exactly Solvable Two-Component Models

Kiyoshi Sogo, M. Uchinami, Yuko Akutsu +1 more

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.

Progress of Theoretical Physics•Aug 1, 1982•77 citations

Solution Transformations and Their Effects on the Systems with Open Boundary Conditions

JUGuo-Xing, ZHAOXian-Lin

The solution transformations and properties of the R-matrices for two-component systems uner these transformations are analyzed in details.Not all transformed R-matrices can be put into the Skalyanin's formalism.For those R-matrices with all required properties,the effects of solution transformations to the six- and eight-vertex systems with open boundary conditions are discussed.These effects can be one of the following types: The Hamiltonian is invariant or transposition-invariant or made in a similarity transformation,or its coupling coefficients are multiplied by an overall factor,or the spin of the system is rotated around the z axis or/and reflected with respect to some plane.In these cases,the transformed systems remain to be integrable.

理论物理通讯：英文版•Jan 1, 2002•0 citations

Solution Transformations and Their Effects on the Systems with Open Boundary Conditions

Ju Guo-Xing, Zhao Xian-Lin

The solution transformations and properties of the -matrices for two-component systems under these transformations are analyzed in details. Not all transformed -matrices can be put into the Skalyanin's formalism. For those -matrices with all required properties, the effects of solution transformations to the six- and eight-vertex systems with open boundary conditions are discussed. These effects can be one of the following types: The Hamiltonian is invariant or transposition-invariant or made in a similarity transformation, or its coupling coefficients are multiplied by an overall factor, or the spin of the system is rotated around the axis or/and reflected with respect to some plane. In these cases, the transformed systems remain to be integrable.

Communications in Theoretical Physics•Apr 15, 2002•0 citations

Reflection K-Matrices of the 19-Vertex Model and XXZ Spin-1 Chain with General Boundary Terms

Takeo Inami, Satoru Odake, Yao-Zhong Zhang

We derive and classify all solutions of the boundary Yang-Baxter equation (or the reflection equation) for the 19-vertex model associated with U_q(\\widehat{sl_2}). Integrable XXZ spin-1 chain hamiltonian with general boundary interactions is also obtained.

•Jan 1, 1996•27 citations

Fifteen-vertex models with non-symmetric $R$ matrices

R. S. Vieira

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 $R$ matrices containing several free-parameters. The corresponding reflection $K$ matrices, solutions of the boundary Yang-Baxter equation, were also found and classified. We found that there are three different families of regular $K$ matrices, regardless of what $R$ matrix we choose.

arXiv:1908.06932•Aug 19, 2019•8 citations

Classifying Nearest-Neighbor Interactions and Deformations of AdS

Marius de Leeuw, Chiara Paletta, Anton Pribytok +2 more

We classify all regular solutions of the Yang-Baxter equation of eight-vertex type. Regular solutions correspond to spin chains with nearest-neighbor interactions. We find a total of four independent solutions. Two are related to the usual six- and eight-vertex models that have R matrices of difference form. We find two new solutions of the Yang-Baxter equation, which are manifestly of nondifference form. These new solutions contain the S-matrices of the AdS_{2} and AdS_{3} integrable models as a special case. This can be used as a starting point to study and classify integrable deformations of these holographic integrable systems.

arXiv:2003.04332•Physical Review Letters•Jul 17, 2020•41 citations

The symmetries, especially those related to the R -transformation, of the reflection equation (RE) for two-component systems are analysed. The classification of solutions to the RE for eight-, six- and seven-vertex-type R -matrices is given. All solutions can be obtained from those corresponding to the standard R -matrices by K -transformation. For the free-fermion models, the boundary matrices have property tr K + (0) = 0, and the free-fermion-type R -matrix with the same symmetry as that of a Baxter type corresponds to the same form of a K - -matrix for the Baxter type. We present the Hamiltonians for the open spin systems connected with our solutions. In particular, the boundary Hamiltonian of seven-vertex models is obtained with a generalization to the Sklyanin formalism.

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.

The solution transformations and properties of the R-matrices for two-component systems uner these transformations are analyzed in details.Not all transformed R-matrices can be put into the Skalyanin's formalism.For those R-matrices with all required properties,the effects of solution transformations to the six- and eight-vertex systems with open boundary conditions are discussed.These effects can be one of the following types: The Hamiltonian is invariant or transposition-invariant or made in a similarity transformation,or its coupling coefficients are multiplied by an overall factor,or the spin of the system is rotated around the z axis or/and reflected with respect to some plane.In these cases,the transformed systems remain to be integrable.

The solution transformations and properties of the -matrices for two-component systems under these transformations are analyzed in details. Not all transformed -matrices can be put into the Skalyanin's formalism. For those -matrices with all required properties, the effects of solution transformations to the six- and eight-vertex systems with open boundary conditions are discussed. These effects can be one of the following types: The Hamiltonian is invariant or transposition-invariant or made in a similarity transformation, or its coupling coefficients are multiplied by an overall factor, or the spin of the system is rotated around the axis or/and reflected with respect to some plane. In these cases, the transformed systems remain to be integrable.

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

R

matrices containing several free-parameters. The corresponding reflection

K

matrices, solutions of the boundary Yang-Baxter equation, were also found and classified. We found that there are three different families of regular

K

matrices, regardless of what

R

matrix we choose.

We classify all regular solutions of the Yang-Baxter equation of eight-vertex type. Regular solutions correspond to spin chains with nearest-neighbor interactions. We find a total of four independent solutions. Two are related to the usual six- and eight-vertex models that have R matrices of difference form. We find two new solutions of the Yang-Baxter equation, which are manifestly of nondifference form. These new solutions contain the S-matrices of the AdS_{2} and AdS_{3} integrable models as a special case. This can be used as a starting point to study and classify integrable deformations of these holographic integrable systems.