  Topological insulators in the presence of strong Coulomb interaction
constitute novel phases of matter. Transitions between these phases can be
driven by single-particle or many-body effects. On the basis of {\it ab-initio}
calculations, we identify a concrete material, {\it i.e.} Ca$_{2}$PtO$_{4}$,
that turns out to be a hole-doped weak topological insulator. Interestingly,
the Pt-$d$ orbitals in this material are relevant for the band inversion that
gives rise to the topological phase. Therefore, Coulomb interaction should be
of importance in Ca$_{2}$PtO$_{4}$. To study the influence of interactions on
the weak topological insulating phase, we look at a toy model corresponding to
a layer-stacked 3D version of the Bernevig-Hughes-Zhang model with local
interactions. For small to intermediate interaction strength, we discover novel
interaction-driven topological phase transitions between the weak topological
insulator and two Dirac semimetal phases. The latter correspond to gapless
topological phases. For strong interactions, the system eventually becomes a
Mott insulator.

Strongly Correlated Electrons;Materials Science