  Shape-phase transitions in mixed parity boson systems are studied using mean
field theory. It is shown that without parity projection, shape transitions in
mixed parity systems are very similar to those in positive parity systems, both
requiring a finite interaction strength for the onset of deformation. The
shape-phase diagram in mixed parity systems, however, changes significantly
after parity projection, with the critical point for the onset of octupole
deformation moving to zero strength. Shape transitions in the Lipkin model are
shown to exhibit the same features, indicating that this is a direct
consequence of parity projection in finite, mixed parity systems independent of
the nature of interacting particles.
