  A strongly spin-orbital coupled systems could be in a magnetic ordered phase
at zero field. However, a Zeeman field could drive it into different quantum or
topological phases. In this work, starting from general symmetry principle, we
construct various effective actions to study all these quantum phases and phase
transitions which take different forms depending on the condensation momenta
are commensurate or in-commensurate.
  We not only recover all these quantum phases and their excitations achieved
by the microscopic calculations, but also discover several novel classes of
quantum phase transitions with dynamic exponents $ z=1, z=2 $ and anisotropic
ones $ (z_x=3/2, z_y=3) $ respectively. We determine the relations between the
quantum spin and the order parameters of the effective actions which display
rich spin-orbital structures. We find a new type of dangerously irrelevant
operator we name type-II, in distinction from the known one we name type-I. We
explore a new phenomena called order parameter fractionization where one
complex order parameter split into two which is different than quantum spin
fractionization into a spinon and a $ Z_2 $ flux. Finite temperature
transitions are presented. The dynamic spin-spin correlation functions are
evaluated. Thermal Hall conductivities are discussed. The cases with the $
U(1)_{soc} $ symmetry explicitly broken are briefly outlined. In view of recent
experimental advances in generating 2d SOC for cold atoms in optical lattices,
these new many-body phenomena can be explored in the near future cold atom
experiments. Implications to various SOC materials such as MnSi,
Fe$_{0.5}$Co$_{0.5}$Si, especially 4d Kitaev materials $\alpha$-RuCl$_3$ in a
Zeeman field are outlined.
