Spin Conductance and Spin Conductivity in Topological Insulators: Analysis of Kubo-Like Terms

We investigate spin transport in 2-dimensional insulators, with the long-term goal of establishing whether any of the transport coefficients corresponds to the Fu-Kane-Mele index which characterizes 2d time-reversal-symmetric topological insulators. Inspired by the Kubo theory of charge transport, and by using a proper definition of the spin current operator (Shi et al. in Phys Rev Lett 96:076604, 2006), we define the Kubo-like spin conductance GKsz and spin conductivity sigmaKsz . We prove that for any gapped, periodic, near-sighted discrete Hamiltonian, the above quantities are mathematically well defined and the equality GKsz =sigmaKsz holds true. Moreover, we argue that the physically relevant condition to obtain the equality above is the vanishing of the mesoscopic average of the spin-torque response, which holds true under our hypotheses on the Hamiltonian operator. A central role in the proof is played by the trace per unit volume and by two generalizations of the trace, the principal value trace and its directional version.