Local approximation to the spinless Falicov-Kimball model

The local approximation is applied to the spinless Falicov-Kimball model in one and two dimensions. The local approximation may be viewed as a zeroth-order expansion in 1/d that is exact in the weak-coupling limit. The local approximation rapidly becomes inaccurate as a function of interaction strength in one dimension, but is accurate in two dimensions. This result indicates that the 1/d expansion converges rapidly for the Falicov-Kimball model in d{≥}2 and that exact solutions of many-body problems in infinite dimensions are quantitatively relevant for physical dimensions.