  It is well-known that the liquid properties in a strongly confined system can
be very different from their ordinary behaviors in an extended system, due to
the competition between the thermal energy and the interaction energy. Here we
show that, in a low-dimensional self-assembled dipolar crystal, the parabolic
optical confinement potential can also strongly affect the quantum many-body
properties in the low temperature regime. For example, by changing the
confinement aspect ratio, the bulk of the system can undergo a quantum phase
transition between a liquid state and a solid state via a nonmonotonic pattern
formation of the domain wall. Furthermore, the entropy of a trapped dipolar
crystal can be much larger than the liquid state in the weak dipole limit,
indicating an intrinsic polarization cooling mechanism via increasing the
external field. These highly correlated confinement effects are very important
to the experimental preparation of a self-assembled dipolar crystal using
ultracold polar molecules.
