Interaction contributions to the spin Hall effect in the Kane-Mele-Hubbard model

Different from the charge Hall effect in which Coulomb interaction is necessary for establishing steady states, the spin Hall effect (SHE) has usually been understood as a non-interacting effect, where the steady-state is reached by spin-orbit-coupling-induced intrinsic or extrinsic spin relaxation. However, in nonmagnetic systems proximate to a magnetic instability, interaction may play a significant role in the SHE. Here we study the interaction contributions to the SHE by using the nonequilibrium Green function approach in a concrete model: the Kane-Mele-Hubbard model. The model involves nearest-neighbor spin-independent hopping and 2nd nearest-neighbor spin-dependent hopping of s electrons on a honeycomb lattice. The on-site Hubbard interaction is treated by an unrestricted Hartree-Fock decoupling self-consistently. By solving the non-equilibrium Green function of a finite system coupled to two leads, we calculate the spin accumulation on the lateral edges which is a direct experimental observable. We compare the cases with and without the Hubbard interaction, and study the connection between the SHE and local spin susceptibility at the edges.