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.
Electron pairing and formation of various types of magnetic correlations for ensembles of small clusters of different geometries are studied with emphasis on tetrahedrons and square pyramids under variation of interaction strength, electron doping and temperature. These exact calculations of charge and spin collective excitations and pseudogaps yield intriguing insights into level crossing degeneracies, phase separation and condensation. Obtained coherent and incoherent pairings provide a route for possible superconductivity different from the conventional BCS theory. Criteria for spin-charge separation, reconciliation and recombination driven by interaction strength, next-nearest coupling and temperature are found. Resulting phase diagrams resemble a number of inhomogeneous, coherent and incoherent nanoscale phases seen recently in high-
Tc cuprates, manganites and colossal magnetoresistive (CMR) nanomaterials.
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 $(\ensuremath{\mu})$. Our results, near the average number of electrons $⟨N⟩\ensuremath{\approx}3$, strongly suggest the existence of a critical parameter
Uc(T) for the localization of electrons and a particle-hole binding (positive) gap ${\ensuremath{\Delta}}^{e\text{\ensuremath{-}}h}(T)>0$ at
U>Uc(T), with a zero temperature critical value,
Uc(0)=4.584. For
U<Uc(T), the particle-particle pair binding is found with a (positive) pairing gap ${\ensuremath{\Delta}}^{P}(T)>0$. The ground state degeneracy is lifted at
U>Uc(T) and the cluster becomes a Mott-Hubbard like insulator due to the presence of energy gaps at all (allowed) integer numbers $(1\ensuremath{\le}N\ensuremath{\le}8)$ of electrons. In contrast, for $U\ensuremath{\le}{U}_{c}(T)$, we find an electron pair binding instability at finite temperature near $⟨N⟩\ensuremath{\approx}3$, which manifests a possible pairing mechanism, a precursor to superconductivity in small clusters. Rigorous criteria for the existence of many-body Mott-Hubbard like particle-hole and particle-particle pairings, spin pairing, (spin) pseudogap, and (spin) antiferromagnetic critical crossover temperatures, at which the corresponding pseudogaps disappear, are also formulated. In particular, 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-
Tc cuprates.
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.
Electron coherent and incoherent pairings and other types of correlations are studied exactly for the ensemble of small clusters with different geometries under variation of interaction strength, electron doping and temperature. Charge and spin collective excitations yield intriguing insights into level crossing degeneracies, phase separation and condensation. Criteria for spin-charge separation and recombination driven by interaction strength and temperature are found. Phase diagrams resemble a number of inhomogeneous, nanoscale phases and pseudogap regions seen recently in high TC cuprates, manganites, multiferroics and CMR nanomaterials.
In order to understand various local phenomena related to electronic phase separation discovered recently in strongly correlated materials, quantum cluster methods are applied to a theoretical model, i.e., Hubbard model, which has successfully predicted basic properties of strongly correlated systems. Our calculations are performed at two levels: exact calculations based on the isolated small clusters via exact diagonalization and approximate calculations applied to large lattices using the variational cluster approximation (VCA).\nWe first study phase separation instabilities with the formation of a pairing gap in the two-dimensional Hubbard model for 8-site Betts clusters. The exact diagonalization method is applied to extract exact ground and excited states of the Hubbard model. The results show that the electronic states with one hole off half filling are unstable and they energetically prefer creating the spatial phase separation. The effect of next nearest hopping is also discussed in this part.\nWe studied the variation of the pairing when the out-of-plane correlation is present by adding an interacting apical site above the two-dimensional cluster plane. The calculations indicate that the out-of-plane correlation can be detrimental to the in-plane pairing effect. When it is not too strong to destroy electron pairs on the plane, the modulation of the apical site can drive the pairing gap to show a sinusoidal variation which is consistent with recent experimental discoveries.\nVariational cluster approximation is introduced to explore local properties of larger lattices. The local nematic state is studied in the square lattice using this method, which indicates that it is possible that the electronic states can locally break the C4 symmetry (90o$ rotational symmetry) to form a pattern where the electronic properties, such as charge and spin correlations, are different in x and y directions.\nThe spatial phase separation in the square and honeycomb lattices is also investigated by the variational cluster approximation. The study shows that Coulomb interaction and lattice geometry are both crucial for electronic phase separation. We propose that electronic phase separation is the result of an instability of the Fermi surface at the boundary of the first Brillouin zone of antiferromagnetic states.