Response of a strongly interacting spin-orbit coupling system to a Zeeman field

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.