  We study the effect of hole hopping in a doped antiferromagnet described by
the `t-J' model, using exact analytic solutions for small clusters. In spite of
the small size, they reveal interesting details about the magnetic order, which
are not apparent in Mean Field treatments or in numerical calculations. The
4-site cluster with one hole yields the most interesting physics, displaying
different behaviors for the ground state: i) an antiferromagnetic phase for
t<<J, where the hole seems to be localized, not affecting the order of the
Heisenberg spins; ii)another regime for t~J that presents mixed ferro and
antiferro correlations and coexistence of metallic and insulating behaviors,
with the presence of charge and spin density waves, in what may be the analog
of the spiral phase obtained in Mean Field solutions; and finally iii)for t>>J,
we obtain strong ferromagnetic correlations (maximum spin) and no density
waves, with quantum fluctuations precluding the saturation of the magnetic
moment. This behavior shows traces of a metal-insulator transition as the hole
kinetic energy competes with the antiferromagnetic interactions.
