Charge and spin orderings are studied on the simplest 1D and the 2D square lattice within the generalized Falicov-Kimball model with Hund coupling between localized and itinerant electrons. Using the restricted phase diagrams method (RPDM) a number of simple rules of formation of various sorts of ground state phases have been detected. In particular, relationships between density of current carriers (electrons or holes) and type of charge and magnetic arrangement has been determined. In 2D in the mixed valence regime only axial stripes (vertical or horizontal) have been found for intermediate values of the coupling constants. They are composed of ferromagnetic or antiferromagnetic chains interchanged with non-magnetic ones. For band fillings close to the half filling stripe phases oriented along one of the main diagonal direction are formed. The results suggest a possibility of tuning modulations of charge and magnetic superstructures with a change of doping.

The Falicov-Kimball model supplemented with the Ising-type Hund coupling describes charge and magnetic order induced by on-site, charge-, and spin-dependent interactions between itinerant electrons and localized ions. Motivated by a discovery of the rich structure of ground-state phase diagrams containing various charge and magnetic superstructures, we analyze the energy spectrum and determine numerically finite temperature properties of the model on the square 4 ϫ 4 cluster at half filling. For the density of magnetic ions equal to 1 and 1/2 and not too small coupling constants we show that many-electron collective excitations derived from changes in positions of the ions or inversions their spins have much lower energy than single-electron excitations. From an analysis of the excitation spectrum structure we derive effective interactions between the ions. When the coupling constants are large enough the interactions reduce to two-body forces between ions occupying the nearest-neighboring sites. Finite-temperature characteristics and temperatures of transformation from charge or magnetically ordered to disordered phases are determined in the wide range of the interaction couplings.

The Falicov-Kimball model enriched with Ising-type Hund coupling between spins of itinerant electrons and magnetic ions is studied exactly on finite 1D rings. At half filling (one electron per site) and for the density of magnetic ions equal to 1/2 it is shown that many-electron collective excitations coupled to spin reorientations of the magnetic ions have much lower energy than single electron excitations. This property is caused by energy gaps formed at the Fermi levels in one-electron energy spectra of all relevant magnetic configurations of the ions. Consequently, low temperature properties of the system are not driven by one-particle, elementary electron excitations but many-particle collective excitations, where both electrons and spins of magnetic ions are involved. In addition, it is shown that many-body excitation spectra derived both from spin fluctuations and from charge fluctuations have a regular structure.

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