  Topological phases of matter give rise to exotic physics that can be
leveraged for next generation quantum computation and spintronic devices. Thus,
the search for topological phases and the quantum states that they exhibit have
become the subject of a massive research effort in condensed matter physics.
Topologically protected states have been produced in a variety of systems,
including artificial lattices, graphene nanoribbons (GNRs) and bismuth
bilayers. Despite these advances, the real-time manipulation of individual
topological states and their relative coupling, a necessary feature for the
realization of topological qubits, remains elusive. Guided by first-principles
calculations, we spatially manipulate robust, zero-dimensional topological
states by altering the topological invariants of quasi-one-dimensional
artificial graphene nanostructures. This is achieved by positioning carbon
monoxide molecules on a copper surface to confine its surface state electrons
into artificial atoms positioned to emulate the low-energy electronic structure
of graphene derivatives. Ultimately, we demonstrate control over the coupling
between adjacent topological states that are finely engineered and simulate
complex Hamiltonians. Our atomic synthesis gives access to an infinite range of
nanoribbon geometries, including those beyond the current reach of synthetic
chemistry, and thus provides an ideal platform for the design and study of
novel topological and quantum states of matter.
