  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.
