Existence of charge-density waves in two-dimensional ionic Hubbard model

We rigorously investigated the charge-charge correlation function of the ionic Hubbard model in two dimensions by reflection positivity. We prove the existence of charge-density waves for large staggered potential $\Delta$ (i.e., $\frac{\Delta}{2}+2V>U$) at low temperatures, where $U$ and $V$ are the on-site and nearest-neighbor Coulomb repulsions, respectively. The results are consistent with previous numerical simulation results. We argue that the absence of charge-density waves for $\Delta=0$ and $U$ are large enough (i.e., $U>\frac{\Delta}{2}$+2V).