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We explore the effect of inhomogeneity on electronic properties of thetwo-dimensional Hubbard model on a square lattice using dynamical mean-fieldtheory (DMFT). The inhomogeneity is introduced via modulated lattice hoppingsuch that in the extreme inhomogeneous limit the resulting geometry is a Lieblattice, which exhibits a flat-band dispersion. The crossover can be observedin the uniform sublattice magnetization which is zero in the homogeneous caseand increases with the inhomogeneity. Studying the spatially resolvedfrequency-dependent local self-energy, we find a crossover from Fermi-liquid tonon-Fermi-liquid behavior happening at a moderate value of the inhomogeneity.This emergence of a non-Fermi liquid is concomitant of a quasi-flat band. Forfinite doping the system with small inhomogeneity displays $d$-wavesuperconductivity coexisting with incommensurate spin-density order, inferredfrom the presence of oscillatory DMFT solutions. The $d$-wave superconductivitygets suppressed for moderate to large inhomogeneity for any finite doping whilethe incommensurate spin-density order still exists.

Magnetization, d-wave superconductivity and non-Fermi liquid behavior in a crossover from dispersive to flat bands