Magnetic orderings and phase separations in a simple model of insulating systems

•A simple effective electronic model for (anti-)ferromagnetic insulators is studied.•Methods of analyses: variational MFA, MC simulations, rigorous solutions and others.•Phase diagrams for various lattice dimensions and electron densities are derived.•The interplay of different energy scales generates interesting critical behaviour.•This (magnetic) system exhibits phase separation phenomenon and tricritical points.

A simple effective model for a description of magnetically ordered narrow-band insulators is studied. The Hamiltonian considered consists of the effective on-site interaction (UU) and intersite magnetic exchange interactions (JzJz, JxyJxy) between nearest-neighbours. The phase diagrams and properties of this model for arbitrary chemical potential μμ and arbitrary electron density nn have been determined within several approaches: (i) the variational method (which treats the on-site interaction term exactly and the intersite interactions within the mean-field approximation) for any Jz,Jxy≠0Jz,Jxy≠0 (exact in the limit of infinite dimensions), (ii) the Monte Carlo simulations on a square lattice with periodic boundary conditions for Jxy=0Jxy=0, and (iii) other approximate methods (inter alia: random phase approximation and spin-wave approximation) as well as (iv) rigorous treatment to obtain results concerning the ground state phase diagrams (the two last also for Jz,Jxy≠0Jz,Jxy≠0). The investigations of the general case show that, depending on the values of interaction parameters and electron concentration nn, the system can exhibit not only homogeneous phases: (anti-)ferromagnetic (FαFα, α=z,xyα=z,xy) and nonordered (NO), but also phase separated states (PSαPSα: FαFα/NO). For a fixed nn one finds the following phase transitions (both continuous and discontinuous ones) and their sequences, which can occur with increasing temperature: Fα→NO, PSα→NO, PSα→Fα→NO, PSα→Fα→PSα→NO. The system analysed exhibits also tricritical behaviour.