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arXiv:0810.1766 · Journal of Statistical Mechanics Theory and Experiment

Reflection matrices for theUq[spo(2n|2m)] vertex model

A Lima-Santos

Published Apr 6, 2009• 5 citations
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Abstract

We propose a classification of the solutions of the graded reflection equations for the U q [ spo (2 n |2 m )] vertex model. We find twelve distinct classes of reflection matrices such that four of them are diagonal. In the non-diagonal matrices the number of free parameters depends on the number of bosonic (2 n ) and fermionic (2 m ) degrees of freedom while in the diagonal ones we find solutions with at most one free parameter.

DOI
10.1088/1742-5468/2009/04/p04005

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We propose a classification of the solutions of the graded reflection equations for the U q [ spo (2 n |2 m )] vertex model. We find twelve distinct classes of reflection matrices such that four of them are diagonal. In the non-diagonal matrices the number of free parameters depends on the number of bosonic (2 n ) and fermionic (2 m ) degrees of freedom while in the diagonal ones we find solutions with at most one free parameter.
We consider the insertion of integrable boundaries for a class of supersymmetric quantum models. The generic conditions for constructing purely bosonic, purely fermionic or mixed type solutions of the graded reflection equation are extracted. Focusing on models associated with g l ( m | n ) or 𝒰 q ( g l ( m | n )) symmetry, we first consider purely bosonic reflection matrices with special structures, for general values of m , n . These solutions provide the bosonic parts to construct fuller reflection matrices, containing fermionic degrees of freedom as well.
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.