  We analyze the nucleation of supersolid order out of the superfluid ground
state of bosons on the triangular lattice. While the stability of supersolidity
against phase separation in this system is by now well established for
nearest-neighbor and long-range dipolar interactions, relevant for
two-dimensional arrays of ultra-cold polar molecules, here we address directly
the nature of the superfluid-to-supersolid transition. Based on symmetry
arguments and quantum Monte Carlo simulations, we conclude that this quantum
phase transition is driven first-order beyond the line of particle-hole
symmetry. Along this line, the transition is continuous and its scaling
behavior consistent with the three-dimensional (3D) XY universality class. We
relate this finding to a 3D Z6 clock model description of the enlarged symmetry
of the solid order parameter field. In the generic case however, the symmetry
reduces to that of a 3D Z3 clock model, which reflects the first-order nature
of the generic superfluid-to-supersolid quantum phase transition on the
triangular lattice.
