Exact thermal studies of small (4-site and 8-site) Hubbard clusters with local electron repulsion yield intriguing insight into phase separation, charge-spin separation, pseudogaps, condensation, and, in particular, pairing fluctuations away from half-filling. These exact calculations, carried out in canonical and grand canonical ensembles, monitoring variations in temperature
T and magnetic field
h, show rich phase diagrams in a $T\ensuremath{-}\ensuremath{\mu}$ space consisting of pairing fluctuations and signatures of condensation. Corresponding electron pairing instabilities are seen when the on-site Coulomb interaction
U is smaller than a critical value
Uc(T) and they point to a possible electron pairing mechanism. The specific heat, magnetization, charge pairing, and spin pairing provide strong support for the existence of competing (paired and unpaired) phases near optimal doping in these clusters, and numerous similarities with experiments on high
Tc superconductors are pointed out. In addition, these ideas may be linked to superconducting carbon nanotubes where a purely electronic mechanism could be responsible for superconductivity.
We investigate the binding of holes and the emergence of competing spin-charge order in the simple and extended Hubbard model using exact diagonalization on the 3x4 cylindrical lattice. For the simple Hubbard model (V=0), we find weakly bound hole pairing mediated by magnetic correlations at intermediate repulsive U, without any evidence of phase separation. Introducing nearest-neighbor interaction V reveals a rich phase diagram: attractive V drives multi-hole clustering and phase separation with localized magnetic quenching, while repulsive V stabilizes charge-density-wave (CDW) order that coexists with bound hole pairs within a modulated magnetic background. At strong coupling (U=10), the competition sharpens, with attractive V overcoming on-site repulsion to form magnetically quenched clusters and repulsive V producing robust CDW order that constrains pairing. Real-space analysis of spin and charge correlations provides microscopic evidence of distinct binding mechanisms -- phase separation versus correlation-mediated pairing -- depending on the sign and strength of intersite interaction V . Our results establish a comprehensive picture of how nonlocal Coulomb interactions reshape the landscape of hole-binding and collective order in correlated electron systems.
With the growing interest of unconventional superconductors, it is believed that these materials have spatial inhomogeneities accompanied with the separate pairing of electrons and subsequent condensation. In doped high-Tc cuprate superconductors, iron pnictide and telluride nanomaterials, above properties can be traced from both repulsive and attractive interactions of electrons. The principle differences and similarities of electron coherent and incoherent pairing instabilities are analyzed using ensembles of small clusters under variation of chemical potential (or doping), interaction strength, temperature and magnetic field. The exact calculations of quantum critical points, the charge and spin collective excitations, and corresponding critical transition temperatures display a common mechanism of phase separation transitions, condensation and formation of spatial inhomogeneities in various cluster geometries. The dependence of the gap function on position of the apical oxygen atom near its sign change due to its vibration provides a simple microscopic explanation of supermodulation of the coherent pairing gap in Bi
2Sr
2CaCu
2O
8+δ seen recently at atomic scale in the scanning tunneling microscopy experiments. Some features of band structures obtained using the local spin density approximation for FeTe will be also discussed.