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
quare) lattices generated by 8- and 10-site Betts unit cells. The exact theory yields insights into the nature of quantum critical points, continuous transitions, dramatic phase separation instabilities and electron condensation in spatially inhomogeneous systems. The picture of coupled anti-parallel (singlet) spins and paired charged holes suggests full Bose condensation and coherent pairing in real space at zero temperature of electrons complied with the Bose-Einstein statistics. Separate pairing of charge and spin degrees at distinct condensation temperatures offers a new route to superconductivity different from the BCS scenario. The conditions for spin liquid behavior coexisting with unsaturated and saturated Nagaoka ferromagnetism due to spin-charge separation are established. The phase separation critical points and classical criticality found at zero and finite temperatures resemble a number of inhomogeneous, coherent and incoherent nanoscale phases seen near optimally doped high-$T_c$ cuprates, pnictides and CMR nanomaterials.
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
tive excitations and pseudogaps yield intriguing insights into level crossing degeneracies, phase separation and condensation. Criteria for spin-charge separation and reconciliation driven by interaction strength, next nearest coupling and temperature are found. Phase diagrams resemble a number of inhomogeneous, coherent and incoherent nanoscale phases seen recently in high T$_c$ cuprates, manganites and CMR nanomaterials.
