The Spectral Gap of the Ferromagnetic XXZ-Chain

We prove that the spectral gap of the spin- ferromagnetic XXZ-chain with HamiltonianH=–_x S^{(1)}_xS^{(1)}_{x+1}+S^{(2)}_xS^{(2)}_{x+1}+\Delta S^{(3)}_xS^{(3)}_{x+1}, is given by -1 for all 1. This is the gap in the spectrum of the infinite chainin any of its ground states, the translation invariant ones as well asthe kink ground states, which contain an interface between an up and a down region.In particular, this shows that the lowest magnon energy is not affected by the presence of a domain wall. This surprising fact is a consequence of the SU _q (2)quantum group symmetry of the model.