  Antiferromagnetic, Mott insulator, d-wave and gossamer superfluid phases are
calculated for 2D square lattices from the extended Hubbard (t-J-U) model using
the Gutzwiller projection method and renormalized mean field theory. Phase
separation between antiferromagnetic and d-wave superfluid phases is found near
half filling when the on-site repulsion exceeds $U\ga7.3t$, and coincides with
a first order transition in the double occupancy. Phase separation is thus
predicted for 2D optical lattices with ultracold Fermi atoms whereas it is
inhibited in cuprates by Coulomb frustration which instead may lead to stripes.
In a confined optical lattice the resulting density distribution is
discontinuous an with extended Mott plateau which enhances the
antiferromagnetic phase but suppresses the superfluid phase. Observation of
Mott insulator, antiferromagnetic, stripe and superfluid phases in density and
momentum distributions and correlations is discussed.
