Trapping and cooling fermionic atoms into the Mott and N\'eel states

We perform a theoretical study of a fermionic gas with two hyperfine states confined to an optical lattice. We derive a generic state diagram as a function of interaction strength, particle number, and confining potential. We discuss the central density, the double occupancy and their derivatives as probes for the Mott state, connecting our findings to the recent experiment of J\"ordens et al. Using entropic arguments we compare two different strategies to reach the antiferromagnetic state in the presence of a trapping potential.