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 H0 = ǫ 0 k + A(k)σx − B(k)σy, where A(k) = η i ki + η A ijkikj + η A ijlkikjkl + · · · , B(k) = η i ki + η ijkikj + η ijlkikjkl + · · · , and η’s are the specific system-dependent coefficients. Specifically, we find that in the intrinsic case the magnitude of torque spin-Hall conductivity σz xy(0) is always twice larger than the conventional spin-Hall conductivity σz xy(0), and the two conductivities have the opposite signs, i.e., σ τz xy(0) = −2σz xy(0). This universal relation, therefore, suggests that in the intrinsic case, the total spin Hall conductivity σ xy(0) in the 2D systems is equal to conventional spin Hall conductivity in magnitude but has the opposite sign, namely, σ xy(0) = σ τz xy(0) + σ sz xy(0) = −σz 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 Iz = `