Effective spin Hamiltonian and phase separation in the almost half-filled Hubbard model and the narrow-band $s$-$f$ model

An effective Hamiltonian of the spin degrees of freedom in the almost half-filled Hubbard model is constructed by means of functional integration in the static approximation. It is shown that in the narrow-band case formal perturbation theory in the ratio of the band width to the Coulomb repulsion coupling constant leads to terms which diverge as the temperature tends to zero. To remove these divergences, the so-called Cayley-tree approximation is used. In this approximation, in the leading orders in the small parameter an effective spin Hamiltonian of the model is derived. In a certain range of variation of the electron density, this Hamiltonian describes a separation of the system into a conducting ferromagnetic phase and a dielectric antiferromagnetic phase. It is shown that the interphase boundary has a thickness of the order of the lattice constant. It is also established that there is a connection between the narrow-band Hubbard model and the narrow-band $s$-$f$ model.