We review some recent results from the mathematical theory of transport of charge and spin in gapped crystalline quantum systems. The emphasis will be on transport coefficients, such as 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 the 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.
We set ℏ = 1 henceforth. In addition, since f(ηt) vanishes for every t ≤ −1/η, one can actually set the initial datum ρɛ(t0) = Π0 at any t0 ≤ −1/η.
Note that the following map fKubo does not satisfy the properties required to be a switchingfunction, as it is not smooth but continuous and vanishes only as t → −∞.
The specification of the real part is needed due to the lack of periodicity for the proper spin current operator . In fact, in general, the cyclicity of τ does not hold for non-periodic operators [compare Proposition I.1(iv)], and consequently, one cannot deduce that the trace per unit volume defining the conductivity is real-valued, even if the operators involved are self-adjoint.
In this section, we follow the conventions of Ref. 17 so that whenever the configuration space is discrete, the coordinate xj ≡ nj of a point refers to the j-th coordinate of the corresponding element of Γ: in other words, points in the same unit cell have the same spatial coordinates. This affects the definition of the switching functions that we are about to give and of position operators Xj.