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

In this paper we demonstrate under what conditions a pseudo-spin degree of freedom or character can be ascribed to Majorana bound states (MBS). These exotic states can be created at the boundaries of non-interacting systems, corresponding to D, DIII and BDI in the usual classification scheme, and we focus on one dimension. We have found that such a character is directly related to the class of the topological superconductor and its description by a Z, rather than a Zs, 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 local operator explicitly with 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.