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We investigate some foundational issues in the quantum theory of spintransport, in the general case when the unperturbed Hamiltonian operator $H_0$does not commute with the spin operator in view of Rashba interactions, as inthe typical models for the Quantum Spin Hall effect.  A gapped periodic one-particle Hamiltonian $H_0$ is perturbed by adding aconstant electric field of intensity $\varepsilon \ll 1$ in the $j$-thdirection, and the linear response in terms of a $S$-current in the $i$-thdirection is computed, where $S$ is a generalized spin operator. We derive ageneral formula for the spin conductivity that covers both the choice of theconventional and of the proper spin current operator. We investigate theindependence of the spin conductivity from the choice of the fundamental cell(Unit Cell Consistency), and we isolate a subclass of discrete periodic modelswhere the conventional and the proper $S$-conductivity agree, thus showing thatthe controversy about the choice of the spin current operator is immaterial asfar as models in this class are concerned. As a consequence of the generaltheory, we obtain that whenever the spin is (almost) conserved, the spinconductivity is (approximately) equal to the spin-Chern number. The methodrelies on the characterization of a non-equilibrium almost-stationary state(NEASS), which well approximates the physical state of the system (in the senseof space-adiabatic perturbation theory) and allows moreover to compute theresponse of the adiabatic $S$-current as the trace per unit volume of the$S$-current operator times the NEASS. This technique can be applied in ageneral framework, which includes both discrete and continuum models.

A new approach to transport coefficients in the quantum spin Hall effect