St\v{r}eda formula for charge and spin currents

We consider a 2-dimensional Bloch--Landau--Pauli Hamiltonian for a spinful electron in a constant magnetic field subject to a periodic background potential. Assuming that the $z$-component of the spin operator is conserved, we compute the linear response of the associated spin density of states to a small change in the magnetic field, and identify it with the spin Hall conductivity. This response is in the form of a spin Chern marker, which is in general quantized to a half-integer, and to an integer under the further assumption of time-reversal symmetry. Our result is thus a generalization to the context of the quantum spin Hall effect to the well-known formula by St\v{r}eda, which is formulated instead for charge transport.