  We consider bosonic atoms loaded into optical lattices with cavity-mediated
infinite-range interactions. Competing short- and global-range interactions
cultivates a rich phase diagram. With a systematic field-theoretical
perspective, we present an $\emph{analytical}$ construction of global
ground-state phase diagram. We find that the infinite-range interaction
enhances the fluctuation of the number density. In the strong coupling regime,
we find four branches of elementary excitations with two being "partilce-like"
and two being "hole-like", and that the excitation gap becomes soft at the
phase boundary between compressible phases and incompressible phases. We derive
an effective theory describing compressible superfluid and supersolid states.
To complement this perturbative study, we construct a self-consistent
mean-field theory and find numerical results consistent with our theoretical
analysis. We map out the phase diagram and find that a charge density wave may
undergo a structure phase transition to a different charge density wave before
it finally enters into the supersolid phase driven by increasing the hopping
amplitude.
