We present a controlled perturbative approach to the low-temperature phase diagram of highly inhomogeneous Hubbard models in the limit of small coupling, $t^{\prime }$, between clusters. We apply this to the dimerized and checkerboard models. The dimerized model is found to behave like a doped semiconductor, with a Fermi-liquid groundstate with parameters (e.g., the effective mass) which are smooth functions of the Hubbard interaction, $U$. By contrast, the checkerboard model has a Fermi liquid phase at large $U>U_c=4.58$, a $d$-wave superconducting state with a full gap for $U_c>U>0$, and a narrow strip of an intermediate $d$-wave superconducting phase with gapless “nodal” quasiparticles for $\mid U-U_c\mid <\mathcal{O}\left(t^{\prime }\right)$.

• Received 20 December 2005

DOI:https://doi.org/10.1103/PhysRevB.73.214510