The Majorana spin in magnetic atomic chain systems

TL;DR

This paper demonstrates that spin-resolved measurements, particularly spin-polarized STM, can distinguish Majorana zero modes from trivial states in magnetic atomic chains embedded in superconductors, using theoretical and numerical analysis.

Contribution

It introduces a method to identify Majorana zero modes via spin density measurements, providing a practical diagnostic tool for topological superconductors.

Findings

Spin densities of Shiba states follow a sum rule related to the normal state.

Spin-polarized STM can differentiate Majorana modes from trivial states.

Numerical simulations confirm the theoretical predictions.

Abstract

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.