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Asymmetric diffusion and the energy gap above the 111 ground state of the quantum XXZ model

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 $\hat z$ axis, the Goldstone theorem implies that the spectral gap above such ground states must tend to zero as $L\to \infty$. In \cite{BCNS} it was proved that, by perturbing in a sub--cylinder with basis of linear size $R\ll L$ the interface ground state, it is possible to construct excited states whose energy gap shrinks as $R^{-2}$. 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 $\text{const.}L^{-2}$. We prove the result by first mapping the problem into an asymmetric simple exclusion process on $\Z^3$ and then by adapting to the latter the recursive analysis to estimate from below the spectral gap of the associated Markov generator developed in \cite{CancMart}. Along the way we improve some bounds on the equivalence of ensembles already discussed in \cite{BCNS} and we establish an upper bound on the density of states close to the bottom of the spectrum.