Calculations of in-gap states of ferromagnetic spin chains on s -wave wide-band superconductors

Magnetic impurities create in-gap states on superconductors. Recent experiments explore the topological properties of one-dimensional arrays of magnetic impurities on superconductors, because in certain regimes $p$-wave pairing can be locally induced leading to new topological phases. A byproduct of the new accessible phases is the appearance of zero-energy edge states that have non-Abelian exchange properties and can be used for topological quantum computation. Despite the large amount of theory devoted to these systems, most treatments use approximations that render their applicability limited when comparing with usual experiments of 1D impurity arrays on wide-band superconductors. These approximations either involve tight-binding-like approximations where the impurity energy scales match the minute energy scale of the superconducting gap and are many times unrealistic, or they assume strongly-bound in-gap states. Here, we use a theory for $s$-wave superconductors based on a wide-band normal metal, with any possible energy scale for the magnetic impurities and develop an efficient way of computing the well-known topological invariants of infinite chains. We perform concrete calculations on ferromagnetic spin chains using BCS Green's function for the superconductor and including Rashba coupling to compare with recent experimental results. The infinite-chain properties can be analytically obtained, giving us a way to compare with finite-chain calculations. We show that it is possible to converge to the infinite limit by doing finite-size numerical calculation, paving the way for numerical calculations not based on analytical Green's functions. As an application, we show that energy oscillations around zero with increasing number of atoms in the spin chain does not reflect the topological origin of the low-energy state.