  An exact study of charge-spin separation, pairing fluctuations and pseudogaps
is carried out by combining the analytical eigenvalues of the four-site Hubbard
clusters with the grand canonical and canonical ensemble approaches in a
multidimensional parameter space of temperature (T), magnetic field (h),
on-site interaction (U) and chemical potential. Our results, near the average
number of electrons <N>=3, strongly suggest the existence of a critical
parameter U_{c}(T) for the localization of electrons and a particle-hole
binding (positive) gap at U>U_{c}(T), with a zero temperature quantum critical
point, U_{c}(0)=4.584. For U<U_{c}(T), particle-particle pair binding is found
with a (positive) pairing gap. The ground state degeneracy is lifted at
U>U_c(T) and the cluster becomes a Mott-Hubbard like insulator due to the
presence of energy gaps at all (allowed) integer numbers of electrons. In
contrast, for U< U_c(T), we find an electron pair binding instability at finite
temperature near <N>=3, which manifests a possible pairing mechanism, a
precursor to superconductivity in small clusters.
  In addition, the resulting phase diagram consisting of charge and spin
pseudogaps, antiferromagnetic correlations, hole pairing with competing
hole-rich (<N>=2), hole-poor (<N>=4) and magnetic (<N>=3) regions in the
ensemble of clusters near 1/8 filling closely resembles the phase diagrams and
inhomogeneous phase separation recently found in the family of doped high T_c
cuprates.
