  Information about double parton distributions (DPDs) can be obtained by
calculating four-point functions on the lattice. We continue our study on the
first DPD Mellin moment of the unpolarized proton by considering interference
effects w.r.t. the quark flavor. In our simulation we employ an $n_f = 2 + 1$
ensemble with inverse coupling $\beta = 3.4$, and pseudoscalar masses of $m_\pi
= 355~\mathrm{MeV}$ and $m_K = 441~\mathrm{MeV}$. The results are converted to
the $\overline{\mathrm{MS}}$-scheme at the scale $\mu = 2~\mathrm{GeV}$. We
analyze the dependence of the considered Mellin moments on the quark
polarization and compare our results with quark model predictions.
