Topological Properties of Gapped Graphene Nanoribbons with Spatial Symmetries.

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 [Formula: see text] topological classification exists for any type of termination. In these cases the Berry phase summed up over all occupied bands turns out to be π-quantized in the presence of the chiral symmetry. However, it does not always provide the correct corresponding [Formula: see text] 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 π-quantized values. The resulting [Formula: see text] 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.

Spin-resolved topology and partial axion angles in three-dimensional insulators.

Symmetry-protected topological crystalline insulators (TCIs) have primarily been characterized by their gapless boundary states. However, in time-reversal- ([Formula: see text]-) invariant (helical) 3D TCIs-termed higher-order TCIs (HOTIs)-the boundary signatures can manifest as a sample-dependent network of 1D hinge states. We here introduce nested spin-resolved Wilson loops and layer constructions as tools to characterize the intrinsic bulk topological properties of spinful 3D insulators. We discover that helical HOTIs realize one of three spin-resolved phases with distinct responses that are quantitatively robust to large deformations of the bulk spin-orbital texture: 3D quantum spin Hall insulators (QSHIs), "spin-Weyl" semimetals, and [Formula: see text]-doubled axion insulator (T-DAXI) states with nontrivial partial axion angles indicative of a 3D spin-magnetoelectric bulk response and half-quantized 2D TI surface states originating from a partial parity anomaly. Using ab-initio calculations, we demonstrate that ß-MoTe2 realizes a spin-Weyl state and that α-BiBr hosts both 3D QSHI and T-DAXI regimes.