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Free Fermionic Elliptic Reflection Matrices and Quantum Group Invariance

Rodolfo Cuerno, A. González-Ruiz

Published Jan 1, 1993• 10 citations
OpenAlex

Abstract

Diagonal solutions for the reflection matrices associated to the elliptic R matrix of the eight vertex free fermion model are presented. They lead through the second derivative of the open chain transfer matrix to an XY hamiltonian in a magnetic field which is invariant under a quantum deformed Clifford–Hopf algebra.

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Journal of Physics A Mathematical and General•May 7, 1992•3 citations
The eigenvalues of the Corner Transfer Matrix Hamiltonian associated to the elliptic RRR matrix of the eight vertex free fermion model are computed in the anisotropic case for magnetic field smaller than the critical value. An argument based on generating functions is given, and the results are checked numerically. The spectrum consists of equally spaced levels.
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.