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Communications in Theoretical Physics

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

Ju Guo-Xing, Zhao Xian-Lin

Published Apr 15, 2002• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/37/4/397

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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.
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 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.
The general solutions for the factorization equations of the reflection matrices Kpm(theta)K^{\\pm}(\\theta)Kpm(theta) for the eight vertex and six vertex models (XYZ, XXZ and XXX chains) are found. The associated integrable magnetic Hamiltonians are explicitly derived, finding families dependig on several continuous as well as discrete parameters.