Majoranas with and without a 'character': hybridization, braiding and Majorana number

TL;DR

This paper investigates the conditions under which Majorana bound states can be assigned a 'character', linking it to topological class invariants, and explores implications for hybridization and braiding in topological superconductors.

Contribution

It clarifies the relationship between Majorana 'character' and topological class invariants, providing explicit constructions and analyzing hybridization and braiding effects.

Findings

Majorana 'character' relates to the topological class and invariant type.

Two MBS with the same character do not hybridize.

The character has no impact on braiding properties.

Abstract

In this paper we demonstrate under what conditions a pseudo-spin degree of freedom or character can be ascribed to the Majorana bound states (MBS) which can be created at the end of one dimensional non-interacting systems, corresponding to D, DIII and BDI in the usual classification scheme. We have found that such a character is directly related to the class of the topological superconductor and its description by a $\mathbb{Z}$, rather than a $\mathbb{Z}_2$, invariant which corresponds to the BDI class. We have also found that the DIII case with mirror symmetry, which supports multiple MBS, is in fact equivalent to the BDI class with an additional time-reversal symmetry. In all cases where a character can be given to the Majorana states we show how to construct the appropriate operator explicitly in various examples. We also examine the consequences of the Majorana character by considering possible hybridization of MBS brought into proximity and find that two MBS with the same character do not hybridize. Finally, we show that having this character or not has no consequence on the braiding properties of MBS.