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.