Pairing Symmetry and Classification of Majorana Zero Modes in a Superconducting Vortex

We present a complete topological classification of Majorana Zero Modes (MZMs) bound to the core of a superconducting vortex in 2D in the presence of additional crystalline symmetries. For each relevant magnetic point group G, we first find all the fermion symmetry groups G_f that are characterized by inequivalent central extensions of G with respect to fermion parity Z^f₂. For each G_f we employ Kitaev’s K-theory methodology, together with the Teo-Kane approach for topological defects, to classify the MZM multiplets using the bulk topological index of an effective one-dimensional chain with on-site symmetries. We analyze systems with both weak and strong spin orbit coupling. We also study the effect of interactions on the non-interacting Bogoliubov-deGennes classification. We observe that crystalline symmetries can protect MZM multiplets at the vortex core which can otherwise be gapped out. We discuss the close correspondence between pairing symmetry, G_f and MZM classification which, in certain symmetry classes, can make MZMs a potential probe of pairing symmetry.