  We study the effect of different open boundary conditions on the insulating
ground states of the one-dimensional extended Bose-Hubbard model at and near
unit filling. To this end, we employ the density matrix renormalization group
method with system sizes up to 250 sites. To characterize the system, various
order parameters and entanglement entropies are calculated. When opposite edge
potentials are added to the two ends of the chain, the inversion symmetry is
explicitly broken, and the regular bulk phases appear. On the other hand,
simple open boundary conditions often exhibit non-degenerate ground states with
a domain wall in the middle of the chain, which induces a sign-flip of an order
parameter. Such a domain wall can lead to an algebraic behavior of the
off-diagonals of the single particle density matrix. We show that this
algebraic behavior adds only a finite contribution to the entanglement entropy,
which does not diverge as the system size increases. Therefore, it is not an
indication of a superfluid phase. We confirm this picture by analytical
calculations based on an effective Hamiltonian for a domain wall.
