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

Abstract

The topological invariants of band insulators are usually assumed to depend only on the connectivity between orbitals and not on their intracell 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 $\ensuremath{\vartheta}=0$ or $\ensuremath{\pi}$, related to the Zak phase, and show that $\ensuremath{\vartheta}$ 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 $\ensuremath{\vartheta}=0$ (resp. $\ensuremath{\pi}$). However, the Shockley model features a topological phase transition between $\ensuremath{\vartheta}=0$ and $\ensuremath{\pi}$. The key difference is whether the two orbitals per unit cell are at the same or different positions.