Hubbard Model on Decorated Lattices

We introduce a family of lattices for which the Hubbard model and its natural extensions can be quasi-exactly solved, i.e. solved for the ground and low energy states. In particular, we show rigorously that the ground state of the Hubbard model with off-site Coulomb repulsions on a decorated Kagom\`{e} lattice is an ordered array of local currents. The low energy theory describing this chiral state is an $S=1/2$ XY model, where each spin degree of freedom represents the two possible chiralities of each local current.