Ground states of the generalized Falicov-Kimball model in one and two   dimensions
  A combination of small-cluster exact-diagonalization calculations and a well-controlled approximative method is used to study the ground-state phase diagram of the spin-one-half Falicov-Kimball model extended by the spin-dependent on-site interaction between localized ($f$) and itinerant ($d$) electrons. Both the magnetic and charge ordering are analysed as functions of the spin-dependent on-site interaction ($J$) and the total number of itinerant ($N_d$) and localized ($N_f$) electrons at selected $U$ (the spin-independent interaction between the $f$ and $d$ electrons). It is shown that the spin-dependent interaction (for $N_f=L$, where $L$ is the number of lattice sites) stabilizes the ferromagnetic (F) and ferrimagnetic (FI) state, while the stability region of the antiferromagnetic (AF) phase is gradually reduced. The precisely opposite effect on the stability of F, FI and AF phases has a reduction of $N_f$. Moreover, the strong coupling between the $f$ and $d$-electron subsystems is found for both $N_f=L$ as well as $N_f < L$. 
Pavol Farkasovsky and Hana Cencarikova