Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix
Tetsuo Deguchi
We discuss an algebraic method for constructing eigenvectors of the transfer matrix of the eight-vertex model at the discrete coupling parameters. We consider the algebraic Bethe ansatz of the elliptic quantum group E τ,η ( sl 2 ) for the case where the parameter η satisfies 2 N η = m 1 + m 2 τ for arbitrary integers N , m 1 and m 2 . When m 1 or m 2 is odd, the eigenvectors thus obtained have not been discussed previously. Furthermore, we construct a family of degenerate eigenvectors of the XYZ spin chain, some of which are shown to be related to the sl 2 loop algebra symmetry of the XXZ spin chain. We show that the dimension of some degenerate eigenspace of the XYZ spin chain on L sites is given by N 2 L/N , if L/N is an even integer. The construction of eigenvectors of the transfer matrices of some related interaction round a face models is also discussed.
arXiv:cond-mat/0109078•Journal of Physics A Mathematical and General•Jan 21, 2002•33 citations