Title: Gap Dependency on Half Spaces in the Product Vacua and Boundary State
models

Abstract: We consider for a family of quantum spin $1/2$ systems called the
Product Vacua and Boundary State (PVBS) models defined on subsets of the
$d$ dimensional lattice $\mathbb{Z}^d$ with Hamiltonians composed of
sums of non-commuting local projections. For any set of parameters for
the PVBS model, we prove a simple geometric condition on the half-space
which determines the existence or non-existence of a spectral gap. As a
corollary, we prove the existence or non-existence of a spectral gap for
the model defined on the entire lattice $\mathbb{Z}^d$. This research
was supported in part by the National Science Foundation under Grant
DMS-1515850.