  Topological modes in one- and two-dimensional systems have been proposed for
numerous applications utilizing their exotic electronic responses. The
zero-energy, topologically protected end modes can be realized in the
Su-Schrieffer-Heeger (SSH) model, which has been experimentally implemented in
atomic-scale solid-state structures and in ultra-cold atomic gases. While the
edge modes in the SSH model are at exactly the mid-gap energy, other
paradigmatic 1D models such as trimer and coupled dimer chains have non-zero
energy boundary states. However, these chains have not been realized in an
atomically tuneable system that would allow explicit control of the edge modes.
Here, we demonstrate atomically controlled trimer and coupled dimer chains
realized using chlorine vacancies in the c$(2\times2)$ adsorption layer on
Cu(100). This system allows wide tuneability of the domain wall modes that we
experimentally demonstrate using low-temperature scanning tunneling microscopy
(STM). In the future, these modes may be used to realize well-defined
fractional charge states or find applications in exotic quantum devices with
atomically well-defined geometries.
