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

TL;DR

This paper demonstrates that in one-dimensional two-band insulators with inversion symmetry, the topological invariant depends on orbital embedding, revealing a new layer of topological classification influenced by intra-cell orbital positions.

Contribution

It shows that the $bZ_2$ topological invariant in 1D insulators depends on orbital embedding, contrasting with higher-dimensional cases where it does not.

Findings

The topological invariant $ heta$ depends on orbital embedding in 1D insulators.

The Shockley model exhibits a topological phase transition influenced by orbital positions.

SSH and CDW models have fixed topological phases regardless of embedding.

Abstract

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.