Magnetism at low dimensions is a thriving field of research with exciting opportunities in technology. This Colloquium focuses on the properties of 1D magnetic systems on solid surfaces. From the emulation of 1D quantum phases to the potential realization of Majorana edge states, spin chains are unique systems to study. The advent of scanning tunneling microscope (STM) based techniques has permitted us to engineer spin chains in an atom-by-atom fashion via atom manipulation and to access their spin states on the ultimate atomic scale. Here the current state of research on spin correlations and dynamics of atomic spin chains as studied by the STM is presented. After a brief review of the main properties of spin chains on solid surfaces, spin chains are classified according to the coupling of their magnetic moments with the holding substrate. This classification scheme takes into account that the nature and lifetimes of the spin-chain excitations intrinsically depend on the holding substrate. Interest is shown of using insulating layers on metals, which generally results in an increase in the spin state's lifetimes such that their quantized nature gets evident and they are individually accessible. Next shown is the use of semiconductor substrates promising additional control through the tunable electron density via doping. When the coupling to the substrate is increased for spin chains on metals, the substrate conduction electron mediated interactions can lead to emergent exotic phases of the coupled spin chain-substrate conduction electron system. A particularly interesting example is furnished by superconductors. Magnetic impurities induce states in the superconducting gap. Because of the extended nature of the spin chain, the in-gap states develop into bands that can lead to the emergence of 1D topological superconductivity and consequently to the appearance of Majorana edge states. Finally, an outlook is given on the use of spin chains in spintronics, quantum communication, quantum computing, quantum simulations, and quantum sensors.

A magnetic atom inside a superconductor locally distorts superconductivity. It scatters Cooper pairs as a potential with broken time-reversal symmetry, leading to localized bound states with subgap excitation energies, named Shiba states. Most conventional approaches regarding Shiba states treat magnetic impurities as point scatterers with isotropic exchange interaction. Here, we show that the number and the shape of Shiba states are correlated to the spin-polarized atomic orbitals of the impurity, hybridized with the superconductor. Using scanning tunnelling spectroscopy, we spatially map the five Shiba excitations found on subsurface chromium atoms in Pb(111), resolving their particle and hole components. While particle components resemble d orbitals of embedded Cr atoms, hole components differ strongly from them. Density functional theory simulations correlate the orbital shapes to the magnetic ground state of the atom, and identify scattering channels and interactions, all valuable tools for designing atomic-scale superconducting devices.