Quantized polarization in a generalized Rice-Mele model at arbitrary filling

We discuss the charge polarization in a generalized Rice-Mele model at arbitrary particle filling per site as a model of charge-ordered systems in one dimension. The model possesses neither the conventional bond-centered inversion symmetry nor the one site translation symmetry alone but has combinations of these symmetries. We show that the charge polarization in the ground state is quantized by the combined symmetry and is characterized solely by the filling. Especially, the polarization can be $1/2$ (mod 1) in the zero filling limit. Under the open boundary condition, there exist excess charges accumulated near edges of the system irrespective of existence or absence of edge modes. Correspondingly, we decompose the polarization into a bulk contribution and an edge contribution and numerically demonstrate that the polarization is dominated by the former (latter) when the energy gap is small (large). We also discuss a simple generalization of our model and examine absence/existence of a gapless edge mode protected by the inversion symmetry by introducing intra unit cell and inter unit cell contributions of the charge polarization.