  We study the stability of solid- and supersolid (SS) phases of a
three-dimensional spin- and a hardcore-Bose-Hubbard models on a body-centered
cubic lattice. To see the quantum effects on the stability of the SS phase, we
model the vacancies (interstitials) introduced in the solid, which are believed
responsible for the appearance of the SS phase, by spinwave bosons and adopt
the interaction between the condensed bosons as a criterion. A repulsive nature
of the low-energy effective interaction is the necessary condition for a
second-order solid-SS transition and, when this condition is met, normally the
SS phase is expected. In calculating the effective interaction, we use
expansions from the semiclassical- (i.e. large-S) and the Ising limit combined
with the ladder approximation. The impact of quantum fluctuations crucially
depends on the energy of the solid phase and that of the superfluid phase at
half filling. As an application to 4He, we study the parameter region in the
vicinity of the fitting parameter set given by Liu and Fisher. For this
parameters set, quantum fluctuations at the second order in 1/S destabilize the
solid phase, which is supposed to be stable within the mean field theory.
