  Superradiant phase transition represents an important quantum phenomenon that
shows the collective excitations based on the coupling between atoms and cavity
modes. The spin-orbit coupling is another quantum effect which induced from the
interaction of the atom internal degrees of freedom and momentum of
center-of-mass. In this work, we consider the cavity assisted dynamical
spin-orbit coupling which comes from the combination of these two effects. It
can induce a series of interesting quantum phenomena, such as the flat spectrum
and the singularity of the excitation energy spectrum around the critical point
of quantum phase transition. We further discuss the influence of atom decay and
nonlinear coupling to the phase diagram. The atom decay suppresses the
singularity of the phase diagram and the nonlinear coupling can break the
symmetric properties of the phase transition. Our work provide the theoretical
methods to research the rich quantum phenomena in this dynamic many-body
systems.
