  Motivated by an exact mapping between anisotropic half integer spin quantum
Heisenberg models and asymmetric diffusions on the lattice, we consider an
anisotropic simple exclusion process with $N$ particles in a rectangle of
$\bbZ^2$. Every particle at row $h$ tries to jump to an arbitrary empty site at
row $h\pm 1$ with rate $q^{\pm 1}$, where $q\in (0,1)$ is a measure of the
drift driving the particles towards the bottom of the rectangle. We prove that
the spectral gap of the generator is uniformly positive in $N$ and in the size
of the rectangle. The proof is inspired by a recent interesting technique
envisaged by E. Carlen, M.C. Carvalho and M. Loss to analyze the Kac model for
the non linear Boltzmann equation. We then apply the result to prove precise
upper and lower bounds on the energy gap for the spin--S, ${\rm S}\in
\frac12\bbN$, XXZ chain and for the 111 interface of the spin--S XXZ Heisenberg
model, thus generalizing previous results valid only for spin $\frac12$.
