Emergent orbitals in the cluster Mott insulator on a breathing Kagome lattice

Motivated by the recent developments on cluster Mott insulating materials such as the cluster magnet LiZn$_2$Mo$_3$O$_8$, we consider the strong plaquette charge ordered regime of the extended Hubbard model on a breathing Kagome lattice and reveal the properties of the cluster Mottness. The plaquette charge order arises from the inter-site charge interaction and the collective motion of three localized electrons on the hexagon plaquettes. This model leads naturally to a reduction of the local moments by 2/3 as observed in LiZn$_2$Mo$_3$O$_8$. Furthermore, at low temperatures each hexagon plaquette contains an extra orbital-like degree of freedom in addition to the remaining spin 1/2. We explore the consequence of this emergent orbital degree of freedom. We point out the interaction between the local moments is naturally described by a Kugel-Khomskii spin-orbital model. We develop a parton approach and suggest a spin liquid ground state with spinon Fermi surfaces for this model. We further predict an emergent orbital order when the system is under a strong magnetic field. Various experimental consequences for LiZn$_2$Mo$_3$O$_8$ are discussed, including an argument that the charge ordering much be short ranged if the charge per Mo is slightly off stoichiometry.