  We prove that if $g\geq 2$ then the set of all Abelian differentials
$(M,\omega)$ for which the vertical flow is mildly mixing is dense in every
stratum of the moduli space $\mathcal{H}_g$. The proof is based on a sufficient
condition for special flows over irrational rotations and under piecewise
constant roof functions to be mildly mixing.
