  The boundary of a manifold can alter the phase of a theory in the bulk. We
explore the possibility of a boundary-induced phase transition for the chiral
symmetry of QCD. In particular, we investigate the consequences of imposing
homogeneous Dirichlet boundary conditions on the quark fields. Such boundary
conditions are sometimes employed in lattice gauge theory computations, for
example, when including external electromagnetic fields, or when computing
quark propagators with a reduced temporal extent. Homogeneous Dirichlet
boundary conditions force the chiral condensate to vanish at the boundary, and
thereby obstruct the spontaneous breaking of chiral symmetry in the bulk. We
show the restoration of chiral symmetry due to a boundary is a non-perturbative
phenomenon depending upon the mechanism of spontaneous symmetry breaking, and
utilize the sigma model to exemplify the issues. Within this model, we find
that chiral symmetry is completely restored if the length of the compact
direction is less than 2.0 fm. For lengths greater than about 4 fm, an
approximately uniform chiral condensate forms centered about the midpoint of
the compact direction. While the volume-averaged condensate approaches the
infinite volume value as the compact direction becomes very long, the
finite-size corrections are shown to be power law rather than exponential.
