Intertwined complex orbital order and ferrimagnetism in honeycomb molybdate Fe${}_2$Mo${}_3$O${}_8$

Motivated by the recent experimental interest in honeycomb molybdates A${}_2$Mo${}_3$O${}_8$, we return and develop the minimal models to capture the essential physics for Fe${}_2$Mo${}_3$O${}_8$. It is pointed out that, the magnetism in these materials arises from the mutual induction between the spin-orbit coupling of the tetrahedral sublattice with active orbitals and the superexchange with the spins from the octahedral sublattice. Essentially, the superexchange between the tetrahedral and octahedral sublattices directly drives the ferrimagnetism, and the residual spin-orbit coupling on the tetrahedral sublattice induces the complex orbital order. The hybrid magnon-phonon excitations are further explored together with their resulting Berry curvatures and thermal Hall effect. We discuss the experimental relevance with Fe${}_2$Mo${}_3$O${}_8$, and the framework can be extended to FeNiMo${}_3$O${}_8$, Ni${}_2$Mo${}_3$O${}_8$ and other isostructural magnets.