Topology Classification using Chiral Symmetry: Chiral Zak Phase and Spin Correlations*

Topology classification theory has been broadly applied to explain and gain deeper insight of many physical phenomena. On the other hand, the power of this theory for one-dimensional (1D) systems has not been as widely used. Quasi 1D graphene nanoribbons (GNRs) can now be synthesized with atomic precision via bottom-up molecular precursor techniques and are discovered to possess nontrivial topological properties, as predicted by Cao, Zhao and Louie under a Z₂ classification using time reversal and spatial symmetries. However, the use of these symmetries constrains the classification to nonmagnetic materials with symmetric unit cell. Here we develop another approach to use only chiral symmetry (based on chiral Zak phase) to classify 1D materials, which gives an Z classification. Using bulk-edge correspondence, we connect the chiral Zak phase with an edge-index that predicts the characters of topology-induced states at junctions of two 1D materials belonging to different classes. Moreover, using the GNRs as an example, we investigate the spin-spin correlations between topologically protected junction states via first-principles calculations.