Orbital embedding and topology of one-dimensional two-band insulators

The topological invariants of band insulators are usually assumed to depend only on the connectivity between orbitals and not on their intra-cell position (orbital embedding), which is a separate piece of information in the tight-binding description. For example, in two dimensions, the orbital embedding is known to change the Berry curvature but not the Chern number. Here, we consider one-dimensional inversion-symmetric insulators classified by a $\mathbb{Z}_2$ topological invariant $\vartheta=0$ or $\pi$, related to the Zak phase, and show that $\vartheta$ crucially depends on orbital embedding. We study three two-band models with bond, site or mixed inversion: the Su-Schrieffer-Heeger model (SSH), the charge density wave model (CDW) and the Shockley model. The SSH (resp. CDW) model is found to have a unique phase with $\vartheta=0$ (resp. $\pi$). However, the Shockley model features a topological phase transition between $\vartheta=0$ and $\pi$. The key difference is whether the two orbitals per unit cell are at the same or different positions.