Charge and spin gaps of the ionic Hubbard model with density-dependent hopping

We calculate the charge gap $\Delta E_C$ and the spin gap $\Delta E_S$ of the ionic Hubbard chain including electron-hole symmetric density-dependent hopping. The vanishing of $\Delta E_C$ ($\Delta E_S$) signals a quantum critical point (QCP) in the charge (spin) sector. Between both critical points, the system is a fully gapped spontaneously dimerized insulator (SDI). We focus our study in this region. Including alternation in the hopping, it is possible to perform an adiabatic Thouless pump of one charge per cycle, but with a velocity limited by the size of the gaps.