Bethe ansatz and boundary energy of the open spin-1/2 XXZ chain

We review recent results on the Bethe ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters α_?, α₊, β_?, β₊ are nonzero. A generalization of the BaxterT-Q equation that involves more than one independentQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.