The transfer matrix of the general integrable open XXZ quantum spin chain obeys certain functional relations at roots of unity. By exploiting these functional relations, we determine the Bethe ansatz solution for the transfer matrix eigenvalues for the special cases where all but one of the boundary parameters are zero, and the bulk anisotropy parameter is iπ/3, iπ/5, ....

We propose a Bethe-Ansatz-type solution of the open spin-1/2 integrable XXZ quantum spin chain with general integrable boundary terms and bulk anisotropy values iπ/(p+1), where p is a positive integer. All six boundary parameters are arbitrary, and need not satisfy any constraint. The solution is in terms of generalized T −Q equations, having more than one Q function. We find numerical evidence that this solution gives the complete set of 2 N transfer matrix eigenvalues, where N is the number of spins. 4)

We propose a generalization of the Baxter T − Q relation which involves more than one independent Q(u). We argue that the eigenvalues of the transfer matrix of the open XXZ quantum spin chain are given by such generalized T − Q relations, for the case that at most two of the boundary parameters {α − , α + , β − , β + } are nonzero, and the bulk anisotropy parameter has values η =

Bethe Ansatz solutions of the open spin-1 2 integrable XXZ quantum spin chain at roots of unity with nondiagonal boundary terms containing two free boundary parameters have recently been proposed. We use these solutions to compute the boundary energy (surface energy) in the thermodynamic limit.

We consider the open spin-s XXZ quantum spin chain with nondiagonal boundary terms. By exploiting certain functional relations at roots of unity, we propose the Bethe ansatz solution for the transfer matrix eigenvalues for cases where atmost two of the boundary parameters are set to be arbitrary and the bulk anisotropy parameter has values η = iπ 3 , iπ 5 , . . .. We present numerical evidence to demonstrate completeness of the Bethe ansatz solutions derived for s = 1/2 and s = 1. 1