Topological Properties of Gapped Graphene Nanoribbons with Spatial Symmetries

Abstract: To date, almost all of the discussions on topological insulators (TIs) have focused on two- and three-dimensional systems. One-dimensional (1D) TIs manifested in real materials, in which localized spin states may exist at the end or near the junctions, have largely been unexplored. Previous studies have considered the system of gapped graphene nanoribbons (GNRs) possessing spatial symmetries (e.g. inversion) with only termination patterns commensurate with inversion- or mirror-symmetric unit cells. In this work, we prove that a symmetry-protected $\mathbb{Z}{2}$ topological classification exists for any type of termination. In these cases the Berry phase summed up over all occupied bands turns out to be $\pi$-quantized in the presence of the chiral symmetry. However, it does not always provide the correct corresponding $\mathbb{Z}{2}$ as one would have expected. We show that only the origin-independent part of the Berry phase gives the correct bulk-boundary correspondence by its $\pi$-quantized values. The resulting $\mathbb{Z}_{2}$ invariant depends on the choice of the 1D unit cell (defined by the nanoribbon termination) and is shown to be connected to the symmetry eigenvalues of the wave functions at the center and boundary of the Brillouin zone. Using the cove-edged GNRs as examples, we demonstrate the existence of localized states at the end of some GNR segments and at the junction between two GNRs based on a topological analysis. The current results are expected to shed light on the design of electronic devices based on GNRs as well as the understanding of the topological features in 1D systems.