Quantum spin liquid and cluster Mott insulator phases in the Mo$_{3}$O$_{8}$ magnets

We unveil the microscopic origin of largely debated magnetism in the Mo$_{3}$O$_{8}$ cluster systems. Upon considering an extended Hubbard model at 1/6 filling on the anisotropic kagom\'e lattice formed by the Mo atoms, we argue that its ground state is determined by the competition between kinetic energy and intersite Coulomb interactions, which is controlled by the trimerisation of the kagom\'e lattice into the Mo$_{3}$O$_{13}$ clusters. Based on first-principles calculations, we show that the strong interaction limit is realised in LiZn$_{2}$Mo$_{3}$O$_{8}$ revealing a plaquette charge order with unpaired spins at the resonating hexagons, whose origin is solely related to the opposite signs of intracluster and intercluster hoppings, in contrast to all previous scenarios. On the other hand, both Li$_{2}$InMo$_{3}$O$_{8}$ and Li$_{2}$ScMo$_{3}$O$_{8}$ are demonstrated to fall into the weak interaction limit where the electrons are well localised at the Mo$_{3}$O$_{13}$ clusters. While the former is found to exhibit long-range antiferromagnetic order, the latter is more likely to reveal short-range order with quantum spin liquid-like excitations. Our results not only reproduce most of the experimentally observed features of these unique materials, but will also help to describe various properties in other quantum cluster magnets.