  In this letter, we investigate the ground state properties of an
optomechanical system consisting of a coupled cavity and mechanical modes. An
exact solution is given when the ratio $\eta$ between the cavity and mechanical
frequencies tends to infinity. This solution reveals a coherent photon
occupation in the ground state by breaking continuous or discrete symmetries,
exhibiting an equilibrium quantum phase transition (QPT). In the $U(1)$-broken
phase, an unstable Goldstone mode can be excited. In the model featuring $Z_2$
symmetry, we discover the mutually (in the finite $\eta$) or unidirectionally
(in $\eta \rightarrow \infty$) dependent relation between the squeezed vacuum
of the cavity and mechanical modes. In particular, when the cavity is driven by
a squeezed field along the required squeezing parameter, it enables modifying
the region of $Z_2$-broken phase and significantly reducing the coupling
strength to reach QPTs. Furthermore, by coupling atoms to the cavity mode, the
hybrid system can undergo a QPT at a hybrid critical point, which is
cooperatively determined by the optomechanical and light-atom systems. These
results suggest that this optomechanical system complements other phase
transition models for exploring novel critical phenomena.
