  Topological electronic phases exist in a variety of naturally occurring
materials but can also be created artificially. We used a cryogenic scanning
tunneling microscope to create dimerized chains of identical quantum dots on a
semiconductor surface and to demonstrate that these chains give rise to
one-dimensional topological phases. The dots were assembled from charged
adatoms, creating a confining potential with single-atom precision acting on
electrons in surface states of the semiconductor. Quantum coupling between the
dots leads to electronic states localized at the ends of the chains, as well as
at deliberately created internal domain walls, in agreement with the
predictions of the Su-Schrieffer-Heeger model. Scanning tunneling spectroscopy
also reveals deviations from this well-established model manifested in an
asymmetric level spectrum and energy shifts of the boundary states. The
deviations arise because the dots are charged and hence lead to an onsite
potential that varies along the chain. We show that this variation can be
mitigated by electrostatic gating using auxiliary charged adatoms, enabling
fine-tuning of the boundary states and control of their quantum superposition.
The experimental data, which are complemented by theoretical modeling of the
potential and the resulting eigenstates, reveal the important role of
electrostatics in these engineered quantum structures.
