  We introduce and analyze a class of quantum spin models defined on
d-dimensional lattices Lambda subset of Z^d, which we call `Product Vacua with
Boundary States' (PVBS). We characterize their ground state spaces on arbitrary
finite volumes and study the thermodynamic limit. Using the martingale method,
we prove that the models have a gapped excitation spectrum on Z^d except for
critical values of the parameters. For special values of the parameters we show
that the excitation spectrum is gapless. We demonstrate the sensitivity of the
spectrum to the existence and orientation of boundaries. This sensitivity can
be explained by the presence or absence of edge excitations. In particular, we
study a PVBS models on a slanted half-plane and show that it has gapless edge
states but a gapped excitation spectrum in the bulk.
