States with $v_1 = \lambda, v_2 = -\lambda$ and reciprocal equations in the six-vertex model(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

Abstract

The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of $n$ constants $v_{1},$ $\ldots,$ $v_{n}$ , the zeros of the function $Q(v)$ .A peculiar class of eigenvalues are those in which two of the constants $v_{1},$ $v_{2}$ are equal to $\lambda,$ $-\lambda$ , with $\Delta=-\cosh$ A and $\Delta$ related to the Boltzmann weights of the six-vertex model by the usual combination $\Delta=(a^{2}+b^{2}-c^{2})/2ab$ .The eigenvectors associated to these eigenvalues are Bethe states (although they seem not).We count the number of such states (eigenvectors) for $n=2,3,4,5$ when $N$ , the columns in a row of a square lattice, is arbitrary.The number obtained is independent of the value of $\Delta$ , but depends on $N$ .We give the explicit expres- sion of the eigenvalues in terms of $a,$ $b,$ $c$ (when possible) or in terms of the roots of a certain reciprocal polynomial, being very simple to reproduce numerically these special eigenvalues for arbitrary $N$ in the blocks $n$ considered.For real $a,$ $b,c$ such eigenvalues are real.