Spin Hall conductivity in insulators with nonconserved spin
We study the linear response of a spin current to a small electric field in a two-dimensional crystalline insulator with nonconserved spin. We adopt the spin current operator proposed in J. Shi et al. [Phys. Rev. Lett. 96, 076604 (2006)PRLTAO0031-900710.1103/PhysRevLett.96.076604], which satisfies a continuity equation and fits the Onsager relations. We use the time-independent perturbation theory to present a formula for the spin Hall conductivity, which consists of a "Chern-type"term, reminiscent of the Kubo formula obtained for the quantum Hall systems, and a correction term that accounts for the nonconservation of spin. We illustrate our findings on the Bernevig-Hughes-Zhang model and the Kane-Mele model for time-reversal-symmetric topological insulators and show that the correction term scales quadratically with the amplitude of the spin-conservation-breaking terms. In both models, the spin Hall conductivity deviates from the quantized value when spin is not conserved.