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We investigate spin transport in 2-dimensional insulators, with the long-termgoal of establishing whether any of the transport coefficients corresponds tothe Fu-Kane-Mele index which characterizes 2d time-reversal-symmetrictopological insulators. Inspired by the Kubo theory of charge transport, and byusing a proper definition of the spin current operator, we define the Kubo-likespin conductance $G_K^{s_z}$ and spin conductivity $\sigma_K^{s_z}$. We provethat for any gapped, periodic, near-sighted discrete Hamiltonian, the abovequantities are mathematically well-defined and the equality $G_K^{s_z} =\sigma_K^{s_z}$ holds true. Moreover, we argue that the physically relevantcondition to obtain the equality above is the vanishing of the mesoscopicaverage of the spin-torque response, which holds true under our hypotheses onthe Hamiltonian operator. This vanishing condition might be relevant in view offurther extensions of the result, e.g. to ergodic random discrete Hamiltoniansor to Schr\"odinger operators on the continuum. A central role in the proof isplayed by the trace per unit volume and by two generalizations of the trace,the principal value trace and it directional version.

Spin Conductance and Spin Conductivity in Topological Insulators: Analysis of Kubo-like terms