  The exchange interactions in insulators depend on the orbital state of
magnetic ions, obeying certain phenomenological principles, known as
Goodenough-Kanamori-Anderson rules. Particularly, the ferro order of alike
orbitals tends to stabilize antiferromagnetic interactions, while the antiferro
order of unlike orbitals favors ferromagnetic interactions. The Kugel-Khomskii
theory provides a universal view on such coupling between spin and orbital
degrees of freedom, based on the superexchange processes: namely, for a given
magnetic order, the occupied orbitals tend to arrange in a way to further
minimize the exchange energy. Then, if two magnetic sites are connected by the
spatial inversion, the antiferro orbital order should lead to the ferromagnetic
coupling and break the inversion symmetry. This constitutes the basic idea of
our work, which opens a new route for designing ferromagnetic ferroelectrics -
the rare but fundamentally and practically important multiferroic materials.
After illustrating the basic idea on toy-model examples, we propose that such
behavior can be indeed realized in the van der Waals ferromagnet VI$_3$,
employing for this analysis the realistic model derived from first-principles
calculations for magnetic $3d$ bands. We argue that the intraatomic Coulomb
interactions responsible for Hund's second rule, acting against the crystal
field, tend to restore the orbital degeneracy of the ionic $d^{2}$ state in
VI$_3$ and, thus, provide a necessary flexibility for activating the
Kugel-Khomskii mechanism of the orbital ordering. In the honeycomb lattice,
this orbital ordering breaks the inversion symmetry, stabilizing the
ferromagnetic-ferroelectric ground state. The symmetry breaking leads to the
canting of magnetization, which can be further controlled by the magnetic
field, producing a huge change of electric polarization.
