The effects of the next-nearest-neighbour density-density interaction in the atomic limit of the extended Hubbard model

We have studied the extended Hubbard model in the atomic limit. The Hamiltonian analysed consists of the effective on-site interaction U and the intersite density-density interactions Wij (both nearest-neighbour and next-nearest-neighbour). The model can be considered as a simple effective model of charge ordered insulators. The phase diagrams and thermodynamic properties of this system have been determined within the variational approach, which treats the on-site interaction term exactly and the intersite interactions within the mean-field approximation. Our investigation of the general case taking into account for the first time the effects of longer-ranged density-density interaction (repulsive and attractive) as well as possible phase separations shows that, depending on the values of the interaction parameters and the electron concentration, the system can exhibit not only several homogeneous charge ordered (CO) phases, but also various phase separated states (CO-CO and CO-nonordered). One finds that the model considered exhibits very interesting multicritical behaviours and features, including bicritical, tricritical, and critical end points and isolated critical points.