  In this study, we carry out density functional theory calculations to
elucidate the polarization switching mechanism in charge-order-induced
ferroelectrics. Based on the investigations about (SrVO$_3$)$_1$(LaVO$_3$)$_1$
superlattice, we demonstrate that the charge ordering state couples strongly to
lattice modes, and charge-transfer induced polarization switching has to be
associated with changes of lattice distortions. Based on the type of lattice
mode strongly coupled to charge ordering states, we classify the charge
ordering materials in two type, namely polyhedral breathing and off-centering
displacement types. We further demonstrate that only in off-centering
displacement type charger ordering material, the polarization is switchable
under an external field. The implications of this theory to experimental
observations are also discussed and we successfully explain the different
electrical behaviors in LuFe$_2$O$_4$ and Fe$_3$O$_4$. This study aims to
provide guidance for searching and designing charge ordering ferroelectrics
with switchable polarization.
