  We analyze a class of quantum spin models defined on half-spaces in the
$d$-dimensional hypercubic lattice bounded by a hyperplane with inward unit
normal vector $m\in\mathbb{R}^d$. The family of models was previously
introduced as the single species Product Vacua with Boundary States (PVBS)
model, which is a spin-$1/2$ model with a XXZ-type nearest neighbor
interactions depending on parameters $\lambda_j\in (0,\infty)$, one for each
coordinate direction. For any given values of the parameters, we prove an upper
bound for the spectral gap above the unique ground state of these models, which
vanishes for exactly one direction of the normal vector $m$. For all other
choices of $m$ we derive a positive lower bound of the spectral gap, except for
the case $\lambda_1 =\cdots =\lambda_d=1$, which is known to have gapless
excitations in the bulk.
