Topological Superconductivity in open boundary condition graphene

Recently, van-Hove doped graphene was achieved for the first time, thus bringing us closer to realize topological superconductivity in this material. The behavior of the order parameter at the boundaries is especially interesting, as there majorana fermions can exist in a zeeman field. Thus, it is of importance to understand the behavior of the topological superconductor at the boundary, which was already studied on a mean field level with attractive interactions as input. Here, we go beyond this by using the recently developed realspace truncated unity functional renomalization group (RS-TUfRG) and the simpler random phase approximation (RPA) to obtain a more realistic effective interaction in one dimensional graphene slabs. On the basis of this effective interactions we perform a self consistent Bogoliubov-de-Gennes approach to obtain the pairing gap. We then discuss the interplay between the boundaries and the pairing interaction and the possible influences which arise due to the open boundaries on the level of the BdG decoupling of the effective interaction. We also highlight the differences of the effective interaction generated by the two methods and discuss the importance of the inclusion of the selfenergy.