Spinless Falicov-Kimball model (annealed binary alloy) from large to small dimensions

The phase diagram for the spinless Falicov-Kimball model on a hypercubic lattice is reexamined in the limits of large and small dimensions (for the half-filled ion case). This model is identical to the thermodynamical problem of an annealed binary alloy with diagonal disorder. The phase diagram for the infinite-dimensional case is remarkably similar to the conjectured phase diagram for the one-dimensional case. The system orders in short-period phases, orders in long-period (possibly incommensurate) phases, or segregates, depending on the interaction strength and the electron concentration. The analysis for this simple model provides hope that recently proposed solutions for other interacting fermion models in infinite dimensions will accurately represent the solutions in two and three dimensions.