Inhomogeneous Hubbard Models: from Weak to Strong Coupling

We systematically study the ground-state phase diagram of two inhomogeneous Hubbard models -- the dimerized and checkerboard models -- in the limit in which the coupling between clusters, $t'$, is small. Using $t'$ as a small parameter, we can solve the problem for any strength of the Hubbard $U$, and so can trace the evolution of the ground state from the small $U$ (band structure) to the large $U$ (strongly correlated) limit. On both lattices, we have found that there are Fermi liquid phases with only modestly renormalizations of the effective mass as a function of $U$, and with residual repulsive interactions between the quasiparticles whose strength, likewise, varies smoothly as $U$ changes. In addition, there are also robust superconducting phases, especially on the checkerboard lattice, of various symmetries, despite the presence of only repulsive interactions in the microscopic model.