  We study two-component fermions in optical lattices with spatially
alternating on-site interactions using dynamical mean-field theory. Calculating
the quasi-particle weight, double occupancy, and order parameters for each
sublattice, we discuss the low-temperature properties of the system. When both
interactions are repulsive, the magnetically ordered state is realized at
half-filling. In the attractive case, the superfluid state is, in general,
realized with a particle number imbalance. On the other hand, when repulsive
and attractive interactions are comparable in the half-filled system, the
magnetically ordered and superfluid states are not realized, but the
metal-insulator transition occurs. This transition is characterized by the Mott
and pairing transitions discussed in the conventional repulsive and attractive
Hubbard models. In the doped system, commensurability emerges owing to the
repulsive interactions and the metal-insulator transition occurs down to
quarter-filling.
