  In this paper, we establish that Majorana zero modes emerging from a
topological band structure of a chain of magnetic atoms embedded in a
superconductor can be distinguished from trivial localized zero energy states
that may accidentally form in this system using spin resolved measurements. To
demonstrate this key Majorana diagnostics, we study the spin composition of
magnetic impurity induced in-gap Shiba states in a superconductor using a
quantum impurity model (at the mean-field level). By examining the spin and
spectral densities in the context of the Bogoliubov-de Gennes (BdG)
particle-hole symmetry, we derive a sum rule that relates the spin densities of
localized Shiba states with those in the normal state without
superconductivity. Extending our investigations to ferromagnetic chain of
magnetic impurities, we identify key features of the spin properties of the
extended Shiba state bands, as well as those associated with a localized
Majorana end mode when the effect of spin-orbit interaction is included. We
then formulate a phenomenological theory for the measurement of the local spin
densities with spin-polarized scanning tunneling microscopy (STM) techniques.
By combining the calculated spin densities and the measurement theory, we show
that spin-polarized STM measurements can reveal a sharp contrast in spin
polarization between an accidentally-zero-energy trivial Shiba state and a
Majorana zero mode in a topological superconducting phase in atomic chains. We
further confirm our results with numerical simulations that address generic
parameter settings.
