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Exact calculations of collective excitations and charge/spin (pseudo)gaps in an ensemble of bipartite and nonbipartite clusters yield level crossing degeneracies, spin-charge separation, condensation and recombination of electron charge and spin, driven by interaction strength, inter-site couplings and temperature. Near crossing degeneracies, the electron configurations of the lowest energies control the physics of electronic pairing, phase separation and magnetic transitions. Rigorous conditions are found for the smooth and dramatic phase transitions with competing stable and unstable inhomogeneities. Condensation of electron charge and spin degrees at various temperatures offers a new mechanism of pairing and a possible route to superconductivity in inhomogeneous systems, different from the BCS scenario. Small bipartite and frustrated clusters exhibit charge and spin inhomogeneities in many respects typical for nano and heterostructured materials. The calculated phase diagrams in various geometries may be linked to atomic scale experiments in high T$_c$ cuprates, manganites and other concentrated transition metal oxides.
Electron pairing and ferromagnetism in various cluster geometries are studied with emphasis on tetrahedron and square pyramid under variation of interaction strength, electron doping and temperature. These exact calculations of charge and spin collec
Spontaneous phase separation instabilities with the formation of various types of charge and spin pairing (pseudo)gaps in $U>0$ Hubbard model including the {it next nearest neighbor coupling} are calculated with the emphasis on the two-dimensional (s
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
There is growing evidence that the unconventional spatial inhomogeneities in the doped high-Tc superconductors are accompanied by the pairing of electrons, subsequent quantum phase transitions (QPTs), and condensation in coherent states. We show that
The magnetic-field, temperature, and angular dependence of the interlayer magnetoresistance of two different quasi-two-dimensional (2D) organic superconductors is reported. For $kappa$-(BEDT-TTF)$_2$I$_3$ we find a well-resolved peak in the angle-dep