We consider the anisotropic three dimensional XXZ Heisenberg ferromagnet in a cylinder with axis along the 111 direction and boundary conditions that induce ground states describing an interface orthogonal to the cylinder axis. Let L be the linear size of the basis of the cylinder. Because of the breaking of the continuous symmetry around the ^ z axis, the Goldstone theorem implies that the spectral gap above such ground states must tend to zero as L !1. In [3] it was proved that, by perturbing in a sub{cylinder with basis of linear size R L the interface ground state, it is possible to construct excited states whose energy gap shrinks as R . Here we prove that, uniformly in the height of the cylinder and in the location of the interface, the energy gap above the interface ground state is bounded from below by const.L . We prove the result by rst mapping the problem into an asymmetric simple exclusion process on Z then by adapting to the latter the recursive analysis to estimate from below the spectral gap of the associated Markov generator developed in [7]. Along the way we improve some bounds on the equivalence of ensembles already discussed in [3] and we establish an upper bound on the density of states close to the bottom of the spectrum. 2000 MSC: 82B10, 82B20, 60K35 Key words and phrases: XXZ model, quantum interface, asymmetric exclusion process, equivalence of ensembles, spectral gap. Date: June 25, 2001. 1.

We investigate the spectrum above the kink ground states of the spin J ferromagnetic XXZ chain with Ising anisotropy ∆. Our main theorem is that there is a non-vanishing gap above all ground states of this model for all values of J. Using a variety of methods, we obtain additional information about the magnitude of this gap, about its behavior for large ∆, about its overall behavior as a function of ∆ and its dependence on the ground state, about the scaling of the gap and the structure of the low-lying spectrum for large J, and about the existence of isolated eigenvalues in the excitation spectrum. By combining information obtained by perturbation theory, numerical, and asymptotic analysis we arrive at a number of interesting conjectures. The proof of the main theorem, as well as some of the numerical results, rely on a comparison result with a Solid-on-Solid (SOS) approximation. This SOS model itself raises interesting questions in combinatorics, and we believe it will prove useful in the study of interfaces in the XXZ model in higher dimensions.