Pietro Caputo, Fabio Martinelli
Asymmetric diffusion and the energy gap above the 111 ground state of 
the quantum XXZ model
(586K, Postscript)

ABSTRACT.  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.