Classification of Solutions to Reflection Equation of Two-Component Systems

TL;DR

This paper classifies solutions to the reflection equation for two-component systems with various vertex models, revealing their derivation from standard R-matrices and exploring boundary conditions in related spin systems.

Contribution

It provides a comprehensive classification of reflection equation solutions for multiple vertex models, including new boundary Hamiltonians and generalizations of Sklyanin's formalism.

Findings

Solutions derived from standard R-matrices via K-transformations.

Boundary matrices for free-Fermion models have trace-zero property.

Hamiltonians for open spin systems with classified boundary conditions.

Abstract

The symmetries, especially those related to the $R$-transformation, of the reflection equation(RE) for two-component systems are analyzed. 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 Baxter type corresponds to the same form of $K_-$-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 was obtained with a generalization to the Sklyanin's formalism.