  The exact numerical diagonalization and thermodynamics in an ensemble of
small Hubbard clusters in the ground state and finite temperatures reveal
intriguing insights into the nascent charge and spin pairings, Bose
condensation and ferromagnetism in nanoclusters. The phase diagram off half
filling strongly suggests the existence of subsequent transitions from electron
pairing into unsaturated and saturated ferromagnetic Mott-Hubbard like
insulators, driven by electron repulsion. Rigorous criteria for the existence
of quantum critical points in the ground state and corresponding crossovers at
finite temperatures are formulated. The phase diagram for 2x4-site clusters
illustrates how these features are scaled with cluster size. The phase
separation and electron pairing, monitored by a magnetic field and electron
doping, surprisingly resemble phase diagrams in the family of doped high Tc
cuprates.
