Charge-ordered states and the phase diagram of the extended Hubbard model on the Bethe lattice

We study the extended Hubbard model (EHM) with both onsite Hubbard interaction and the intersite density–density interaction between nearest-neighbors using the standard Hartree mean-field approximation (MFA) on the Bethe lattice. We found that, at the ground state, the system can be in a charge-ordered band-insulating (COI), a charge-order metallic (COM) or a non-charge-ordered (NO) state. Moreover, the finite-temperature phase diagrams are presented. Several observables like a charge-order parameter, a spectral function, and particularly at finite temperatures, a thermally-excited charge carrier concentration (to visualize the degree of metallicity) are analyzed. The results show that increasing onsite repulsion suppresses charge order and change the properties of the system from insulating to metallic. Worth noting, that a number of phenomena can be found within the MFA, where their analysis is much simpler than in more advanced approaches. The method used for the EHM on the Bethe lattice also allows for a series of analytical derivations and simplifications to see general geometry-independent features and analytical results, avoiding the numerical inaccuracies and other issues that appear with a purely numerical solution.