An exact study of charge-spin separation, pairing fluctuations and   pseudogaps in four-site Hubbard clusters
  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. 
Armen N. Kocharian, Gayanath W. Fernando, Kalum Palandage and James W.   Davenport