Symmetry classes and Topological properties of 1D Superconductors

We study the topological phases and the emergence of Majorana bound states (MBS) at the edge of 1D spinless p-wave superconductor (SC). We classify these systems according to the tenfold scheme on the basis of their topology, resulting from the presence or absence of discrete symmetries. The simplest system we study belongs to the BDI symmetry class, with Z topological invariant and integer number of edge modes. This is reflected by the presence of MBS doublets at the edges. In our analysis, we have considered two types of spinless p-wave pairing with the same and opposite signs of p-wave order parameter for the two spins. We study the type of edge states for these two cases for different values of the chemical potential (measured with respect to the SC gap). For μ>1/2, these MBS have an oscillating part along with a decay part. For 0<μ<1/2, they are purely decaying. Adding perturbations like s-wave pairing and Zeeman fields induces transitions from one topological class to another. We show that there are 3 symmetry classes for the first type of pairing and 6 for the second, with these perturbations. We also analyse the possible perturbations and their combinations, which can place the system in the above number of classes, for each of the two cases.