Extended Falicov-Kimball model at weak onsite and intersite Coulomb interactions

We analyze in detail a behavior of the order parameter in the half-filled extended Falicov-Kimball model for small Coulomb interactions (both onsite $U$ and intersite $V$). The parameter is defined as the difference of localized electron concentrations in both sublattices of the Bethe lattice (in the limit of large coordination number). Using two methods, namely, the dynamic mean field theory and the Hartree-Fock approximation, we found the ranges of $U$ and $V$ for which the anomalous temperature dependence of the order parameter, characterized by the sharp reduction near $T \approx T_{\textrm{C}}/2$, occurs ($T_{\textrm{C}}$ is the temperature of the continuous order-disorder transition). In order to quantitatively describe this anomaly, we defined a function that measures the departure of the order parameter dependence from the standard mean-field $S=1/2$ Ising-like curve. We determined the $U$-dependent critical value $V_{\textrm{C}}$ of $V$ above which the anomaly disappears. Indicators of the anomalous behavior of the parameter dependence can be also observed in the temperature dependence of the specific heat.