Topology concepts have significantly deepened of our understanding in recent years of the electronic properties of one-dimensional (1D) nano structures such as the graphene nanoribbons. Controlling topological electronic properties of GNRs has been demonstrated in both theoretical studies and experimental realization. Most previous works rely on classification theory requiring both time reversal and spatial symmetry of a unit cell in the 1D bulk material that is commensurate to its boundary. To access boundary structures that lead to unit cell with no spatial symmetry and to generalize the theory, we propose here another classification scheme, using chiral symmetry, to arrive at a Z classification that is not only applicable to GNRs with arbitrary terminations, but also to any general 1D chiral structures. This theory, combining with Liebs theorem, moreover enables access to the electrons spin degree of freedom, allowing for investigation of spin physics.
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