  Certain lattices with specific geometries have one or more spectral bands
that are strictly flat, i.e. the electron energy is independent of the
momentum. This can occur robustly irrespective of the specific couplings
between the lattices sites due to the lattice symmetry, or it can result from
fine-tuned couplings between the lattice sites. While the theoretical picture
behind flat electronic bands is well-developed, experimental realization of
these lattices has proven challenging. Utilizing scanning tunnelling microscopy
(STM) and spectroscopy (STS), we manipulate individual vacancies in a chlorine
monolayer on Cu(100) to construct various atomically precise 1D lattices with
engineered flat bands. We realize experimentally both gapped and gapless flat
band systems with single or multiple flat bands. We also demonstrate
tuneability of the energy of the flat bands and how they can be switched "on"
and "off" by breaking and restoring the symmetry of the lattice geometry. The
experimental findings are corroborated by tight-binding calculations. Our
results constitute the first experimental realizations of engineered flat bands
in a 1D solid-state system and pave the way towards the construction of e.g.
topological flat band systems and experimental tests of flat-band-assisted
superconductivity in a fully controlled system.
