Theory of strongly correlated electron systems on the basis of a diagrammatic technique for Hubbard operators.

Abstract

The Hubbard model with infinite intrasite Coulomb interaction is studied in terms of the diagrammatic technique for Hubbard operators. Exact diagrammatic representations for single-particle electron Green's functions ${\mathit{scrG}}_{\mathrm{\ensuremath{\uparrow}}}$,${\mathit{scrG}}_{\mathrm{\ensuremath{\downarrow}}}$, as well as for the spin Green's function scrD are constructed. All three Green's functions are expressed through a unique system of four-point and three-point vertex parts, for which graphical expressions are given in terms of diagrammatic series. The general graphical representations are given for both the paramagnetic and the ferromagnetic phases. Evaluation of the four-point vertex parts for a three-dimensional system are performed within the generalized random-phase approximation (GRPA), which consists in summation of all loop-type diagrams. The GRPA expressions for the vertex parts are used to calculate the Green's functions. It is shown that the electron Green's functions ${\mathit{scrG}}_{\mathrm{\ensuremath{\uparrow}}}$ and ${\mathit{scrG}}_{\mathrm{\ensuremath{\downarrow}}}$ in the paramagnetic phase have a propagator character, whereas the spin Green's function scrD is more complex exhibiting existence of incoherent states.The propagator-type and incoherent-type contributions correspond to two terms in the expression for the dynamical susceptibility. One of them has the typical Pauli character for itinerant magnetism, and the other is of Curie form (\ensuremath{\propto}1/T), manifesting the appearance of localized magnetic moments. It has been shown that there is a critical electron concentration ${\mathit{n}}_{\mathit{c}}$, for which a change of the regime from the itinerant-type magnetism to the magnetic behavior typical for localized magnetic moments takes place. Simultaneously, at the same critical concentration the paramagnetic phase becomes unstable against the ferromagnetic ordering. In the simplest approximation for the electron Green's-function lines which enter the loop diagrams, it turns out that ${\mathit{n}}_{\mathit{c}}$=0.66 for the case of constant density of states of the unperturbed band. The exact diagrammatic expressions for all three Green's functions are also given in the ferromagnetic phase. It is shown that two of them, ${\mathit{scrG}}_{\mathrm{\ensuremath{\downarrow}}}$ and scrD, are of the propagator type, whereas ${\mathit{scrG}}_{\mathrm{\ensuremath{\uparrow}}}$ (for the spin aligned against the spontaneous magnetization) contains incoherent contributions. The presented theory can be further developed in many different directions.