Precursory Metal-Insulator transition in a small cluster of the `t-J' model: Exact analytic results

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, 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.