Quadrupole moments and their interactions in the triangular lattice antiferromagnet FeI${}_2$

Motivated by the recent experiments on the triangular lattice antiferromagnet FeI${}_2$, we consider the presence of the quadrupole moments and their interactions between the spin-orbital-entangled local moments. In addition to the anisotropic pairwise interaction between the local moments, the interaction between the quadrupole moment arises from the interaction of the orbital occupation configurations. This is argued primarily from the time reversal symmetry and the different orbital configurations via their exchange paths. We discuss the implication of these interactions and expect this result to be complementary to the existing works in this system and other systems alike.