  We propose an effective model for the study of the interplay between
correlation and topology by decorating the Kane-Mele model with a set of
localized interacting orbitals hybridized to just one sublattice, breaking the
inversion symmetry. We show that in the time-reversal symmetric case, the
interplay between interactions and hybridization extends the stability of the
topological phase and depending on the driving mechanism very different
behaviors are observed after the topological phase transition (TPT). We discuss
the fate of the TPT in presence of weak ferromagnetic order, by introducing a
weak local magnetic field at the localized orbitals, which splits the two band
inversion points. One of the platforms to apply this model to are ferrovalley
compounds, which are characterized by two independent band inversion points.
Understanding this family of materials is crucial for the development of the
valleytronics. An alternative to spintronics, which uses valley polarization as
opposed to spin degrees of freedom as the building block, promises great
opportunities for the development of information storage.
