  We demonstrate using direct numerical diagonalization and extrapolation
methods that boundary conditions have a profound effect on the bulk properties
of a simple $Z(N)$ model for $N \ge 3$ for which the model hamiltonian is
non-hermitian. For $N=2$ the model reduces to the well known quantum Ising
model in a transverse field. For open boundary conditions the $Z(N)$ model is
known to be solved exactly in terms of free parafermions. Once the ends of the
open chain are connected by considering the model on a ring, the bulk
properties, including the ground-state energy per site, are seen to differ
dramatically with increasing $N$. Other properties, such as the leading
finite-size corrections to the ground-state energy, the mass gap exponent and
the specific heat exponent, are also seen to be dependent on the boundary
conditions. We speculate that this anomalous bulk behaviour is a topological
effect.
