Analytical and semianalytical tools to determine the topological character of Shiba chains

TL;DR

This paper introduces three analytical and semi-analytical methods to determine the topological nature of impurity Shiba chains, overcoming size-effects issues in previous numerical approaches, and validates them with a magnetic impurity chain example.

Contribution

The paper presents new analytical tools based on Green's functions and T-matrix formalism for topological analysis of Shiba chains, addressing limitations of existing numerical methods.

Findings

Methods accurately determine topological phases

Analytical solutions agree with numerical analysis

Tools applicable to various impurity chain configurations

Abstract

We introduce three new analytical and semi-analytical tools that allow one to determine the topological character of impurity Shiba chains. The analytical methods are based on calculating the effective Green's function of an infinite embedded chain using the T-matrix formalism and describing the chain as a {\it line impurity}. We thus provide a solution to the longstanding size-effects problem affecting the only general alternative method, the numerical tight-binding analysis. As an example we consider a chain of magnetic impurities deposited on an s-wave superconducting substrate with Rashba spin-orbit and we calculate its topological phase diagram as a function of the magnetic impurity strength and the chemical potential. We find a perfect agreement between all our new techniques and a numerical analysis.