  We consider a hybrid atom-optomechanical system consisting of a mechanical
membrane inside an optical cavity and an atomic Bose-Einstein condensate
outside the cavity. The condensate is confined in an optical lattice potential
formed by a traveling laser beam reflected off one cavity mirror. We derive the
cavity-mediated effective atom-atom interaction potential, and find that it is
non-uniform, site-dependent, and does not decay as the interatomic distance
increases. We show that the presence of this effective interaction breaks the
Z$_2$ symmetry of the system and gives rise to new quantum phases and phase
transitions. When the long-range interaction dominates, the condensate breaks
the translation symmetry and turns into a novel self-organized lattice-like
state with increasing particle densities for sites farther away from the
cavity. We present the phase diagram of the system, and investigate the
stabilities of different phases by calculating their respective excitation
spectra. The system can serve as a platform to explore various self-organized
phenomena induced by the long-range interactions.
