  Knowledge of the topology of the electronic ground state of materials has led
to deep insights to novel phenomena such as the integer quantum Hall effect and
fermion-number fractionalization, as well as other properties of matter.
Joining two insulators of different topological classes produces fascinating
boundary states in the band gap. Another exciting recent development is the
bottom-up synthesis (from molecular precursors) of graphene nanoribbons (GNRs)
with atomic precision control of their edge and width. Here we connect these
two fields, and show for the first time that semiconducting GNRs of different
width, edge, and end termination belong to different topological classes. The
topology of GNRs is protected by spatial symmetries and dictated by the
terminating unit cell. We have derived explicit formula for their topological
invariants, and show that localized junction states developed between two GNRs
of distinct topology may be tuned by lateral junction geometry. The topology of
a GNR can be further modified by dopants, such as a periodic array of boron
atoms. In a superlattice consisted of segments of doped and pristine GNRs, the
junction states are stable spin centers, forming a Heisenberg antiferromagnetic
spin 1/2 chain with tunable exchange interaction. The discoveries here are not
only of scientific interest for studies of quasi one-dimensional systems, but
also open a new path for design principles of future GNR-based devices through
their topological characters.
