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We investigate the spectrum above the kink ground states of the spin Jferromagnetic XXZ chain. Our main theorem is that there is a non-vanishing gapabove all ground states of this model for all values of J. Using a variety ofmethods, we obtain additional information about the magnitude of this gap,about its behavior for large \Delta, about its overall behavior as a functionof \Delta and its dependence on the ground state, about the scaling of the gapand the structure of the low-lying spectrum for large J, and about theexistence of isolated eigenvalues in the excitation spectrum. By combininginformation obtained by perturbation theory, numerical, and asymptotic analysiswe arrive at a number of interesting conjectures. The proof of the maintheorem, as well as some of the numerical results, rely on a comparison resultwith a Solid-on-Solid (SOS) approximation. This SOS model itself raisesinteresting questions in combinatorics, and we believe it will prove useful inthe study of interfaces in the XXZ model in higher dimensions.

The spectral gap for the ferromagnetic spin-J XXZ chain