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 $H_0=\epsilon_k^0+A(k)\sigma_x-B(k)\sigma_y$, where $A(k)=\eta^A_ik_i+\eta^A_{ij}k_ik_j+\eta^A_{ijl}k_ik_jk_l+...$, $B(k)=\eta^B_ik_i+\eta^B_{ij}k_ik_j+\eta^B_{ijl}k_ik_jk_l+...$, and $\eta$'s are the specific system-dependent coefficients. Specifically, we find that in the intrinsic case the magnitude of torque spin-Hall conductivity $\sigma^{\tau_z}_{xy}(0)$ is always twice larger than the conventional spin-Hall conductivity $\sigma^{s_z}_{xy}(0)$, and the two conductivities have the opposite signs, i.e., $\sigma^{\tau_z}_{xy}(0)=-2\sigma^{s_z}_{xy}(0)$. 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 $<{1/2}\{I_z,v_x\}>$ vanishes, and this leads to a hyper selection rule for the conventional spin-Hall current.