  Magnetic adatom chains on surfaces constitute fascinating quantum spin
systems. Superconducting substrates suppress interactions with bulk electronic
excitations but couple the adatom spins to a chain of subgap Yu-Shiba-Rusinov
(YSR) quasiparticles. Using a scanning tunneling microscope, we investigate
such correlated spin-fermion systems by constructing Fe chains adatom by adatom
on superconducting NbSe$_2$. The adatoms couple entirely via the substrate,
retaining their quantum spin nature. In dimers, we observe that the deepest YSR
state undergoes a quantum phase transition due to Ruderman-Kittel-Kasuya-Yosida
interactions, a distinct signature of quantum spins. Chains exhibit coherent
hybridization and band formation of the YSR excitations, indicating
ferromagnetic coupling. Longer chains develop separate domains due to
coexisting charge-density-wave order of NbSe$_2$. Despite the
spin-orbit-coupled substrate, we find no signatures of Majoranas, possibly
because quantum spins reduce the parameter range for topological
superconductivity. We suggest that adatom chains are versatile systems for
investigating correlated-electron physics and its interplay with topological
superconductivity.
