Hartree theory for the negative-Uextended Hubbard model: Ground state

We have investigated the ground-state properties of the negative-$U$ extended Hubbard model in the weak-coupling regime with the Hartree theory formulated in terms of the Bogoliubov variational approach. While the pure phases of charge order (CO), singlet superconducting (SS), and nonordered can be solved analytically with a model density of states, the mixed phase of CO and SS requires numerical computation. The ground-state phase diagram is derived, and the order parameters and the chemical potential of the ground state are analyzed. The characteristic difference between the behaviors of the SS and the CO order parameters at very weak coupling suggests a complicated competition between various interactions. Present results are compared with those derived earlier for the strong-coupling regime.