Electric polarization of one-dimensional inversion-symmetric two-band insulators

The bulk electric polarization $P$ of one-dimensional crystalline insulators is defined modulo a polarization quantum $P_q$. The latter is a measurable quantity that depends on the number $n_s$ of sites per unit cell. For two-band models, $n_s=1$ or $2$ and $P_q=g/n_s$ ($g=1$ or $2$ being the spin degeneracy). For inversion-symmetric crystals either $P=0$ or $P_q/2$ mod $P_q$. Depending on the position of the two inversion centers with respect to the ions, three situations arise: bond, site or mixed inversion. As representative two-band examples of these three cases, we study the Su-Schrieffer-Heeger (SSH), charge density wave (CDW) and Shockley models. SSH has a unique phase with $P=0$ mod $g/2$, CDW has a unique phase with $P=g/4$ mod $g/2$, and Shockley has two distinct phases with $P=0$ or $g/2$ mod $g$. In all three cases, as long as inversion symmetry is present, chiral symmetry is found to be irrelevant for $P$. As a generalization of SSH and CDW, we analytically compute $P$ for the RM model and illustrate the role of the unusual $P_q=g/2$ on edge and soliton fractional charges and on adiabatic pumping.