Topology Classification using Chiral Symmetry and Spin Correlations in Graphene Nanoribbons.

We apply the topological classification theory using chiral symmetry to graphene nanoribbons (GNRs). This approach eliminates the requirement of time-reversal and spatial symmetry in previous Z2 topology theory, resulting in a Z classification with the conventional Z index in a new vector-formed expression called "chiral phase index" (CPI). Our approach is applicable to GNRs of arbitrary terminations and any quasi one-dimensional chiral structures, including magnetism. It naturally solves a recent experimental puzzle of junction states at a class of asymmetric GNR junctions. We moreover derive a simple analytic formula for the CPI of armchair GNRs. Since this approach enables access to electron spin behavior, based on the CPI, we design a novel GNR with periodic localized moments and strong spin-spin exchange coupling.