  The supersolid state of matter, exhibiting non-dissipative flow in solids,
has been elusive for thirty five years. The recent discovery of a non-classical
moment of inertia in solid 4He by Kim and Chan provided the first experimental
evidence, although the interpretation in terms of supersolidity of the ideal
crystal phase remains subject to debate. Using quantum Monte Carlo methods we
investigate the long-standing question of vacancy-induced superflow and find
that vacancies in a 4He crystal phase separate instead of forming a supersolid.
On the other hand, non-equilibrium vacancies relaxing on defects of
poly-crystalline samples could provide an explanation for the experimental
observations.
