Torque and conventional spin-Hall currents in two-dimensional spin-orbit coupled systems: Universal relation and hyper-selection rule

We investigate torque and also conventionally defined spin-Hall currents in two-dimensional (2D) spin-orbit coupled systems of spin-1/2 particles within the linear response Kubo formalism. We obtain some interesting relations between the conventional and torque spin-Hall conductivities for the generic effective Hamiltonian $H_0={ϵ}_k^0+A(k){\sigma }_x-B(k){\sigma }_y$, where $A(k)={\eta }_i^A{k}_i+{\eta }_{ij}^A{k}_i{k}_j+{\eta }_{ijl}^A{k}_i{k}_j{k}_l+\mathrm{.}\mathrm{.}\mathrm{.}$, $B(k)={\eta }_i^B{k}_i+{\eta }_{ij}^B{k}_i{k}_j+{\eta }_{ijl}^B{k}_i{k}_j{k}_l+\mathrm{.}\mathrm{.}\mathrm{.}$, and $\eta$’s are the specific system-dependent coefficients. Specifically, we find that in the intrinsic case the magnitude of torque spin-Hall conductivity ${\sigma }_{xy}^{{\tau }_z}(0)$ is always twice larger than the conventional spin-Hall conductivity ${\sigma }_{xy}^{s_z}(0)$, and the two conductivities have the opposite signs, i.e., ${\sigma }_{xy}^{{\tau }_z}(0)=-2{\sigma }_{xy}^{s_z}(0)$. This universal relation also holds in the presence of an uniform in-plane magnetic field. We also find that if the energy dispersion is rotationally invariant, there exists a hyper-angular momentum $I_z = (k\times \partial\theta/\partial k)_z s_z + L_z$ which is conserved. Furthermore, the hyper-angular momentum current $<1\mathrm{/}2\{I_z,v_x\}>$ vanishes, and this leads to a hyper selection rule for the conventional spin-Hall current.