  Electronic states in the gap of a superconductor inherit intriguing many-body
properties from the superconductor. Here, we create these in-gap states by
manipulating Cr atomic chains on the $\beta$-Bi$_2$Pd superconductor. We find
that the topological properties of the in-gap states can greatly vary depending
on the crafted spin chain. These systems make an ideal platform for non-trivial
topological phases because of the large atom-superconductor interactions and
the existence of a large Rashba coupling at the Bi-terminated surface. We study
two spin chains, one with atoms two-lattice-parameter apart and one with
square-root-of-two lattice parameters. Of these, only the second one is in a
topologically non-trivial phase, in correspondence with the spin interactions
for this geometry.
