Orbital embedding and topology of one-dimensional two-band insulators
Jean-Noël Fuchs, Frédéric Piéchon
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.