From charge to spin: analogies and differences in quantum transport coefficients

We review some recent results from the mathematical theory of transport of charge and spin in gapped crystalline quantum systems. The emphasis will be in transport coefficients like conductivities and conductances. As for the former, those are computed as appropriate expectations of current operators in a non-equilibrium almost-stationary state (NEASS), which arises from the perturbation of an equilibrium state by an external electric field. While for charge transport the usual double-commutator Kubo formula is recovered (also beyond linear response), we obtain formulas for appropriately-defined spin conductivities which are still explicit but more involved. Certain "Kubo-like" terms in these formulas are also shown to agree with corresponding contributions to the spin conductance. In addition to that, we employ similar techniques to show a new result, namely that even in systems with non-conserved spin there is no generation of spin torque, that is the spin torque operator has an expectation in the NEASS which vanishes faster than any power of the intensity of the perturbing field.