  Recently, a novel kind of hybrid atom-optomechanical system, consisting of
atoms in a lattice coupled to a membrane, has been experimentally realized
[Vochezer {\it et al.,} Phys. Rev. Lett. \textbf{120}, 073602 (2018)], which
promises a viable contender in the competitive field of simulating
non-equilibrium many-body physics. Here we are motivated to investigate a
spinor Bose gas coupled to a vibrational mode of a nano-membrane, focusing on
analyzing the role of the spinor degrees of freedom therein. Through an
adiabatic elimination of the degrees of freedom of the quantum oscillator, we
derive an effective Hamiltonian which reveals a competition between the force
localizing the atoms and the membrane displacement. We analyze the dynamical
stability of the steady state using Bogoliubov-de Gennes approach and derive
the stationary phase diagram in the parameter space. We investigate the
non-equilibrium quantum phase transition from a localized symmetric state of
the atom cloud to a shifted symmetry-broken state, where we present a detailed
analysis of the effects of the spin degree of freedom. Our work presents a
simple way to study the effects of the spinor degree of freedom on the
non-equilibrium nonlinear phenomena that is complementary to ongoing
experiments on the hybrid atom-optomechanical system.
