Zeeman field induced non-trivial topology in a spin-orbit coupled superconductor

The hope to realize Majorana fermions at the vortex core of a two dimensional topological superconductor has led to a variety of proposals for devices which exhibit topological superconductivity. Many of these include superconductivity through the proximity effect and therefore require a layer of a conventional superconductor deposited on top of another system, which lends its topological properties\cite{Fu,Sau,Alicea}. The necessity of the superconducting layer poses some technical complications and, in particular, makes it harder to probe the Majorana state. In this work we propose to replace the proximity effect pairing by an innate tendency for pairing, mediated by interactions. We use a model system with spin orbit coupling and on-site repulsion and apply renormalization group to the interaction vertex. Without a Zeeman field this model exhibits pairing instabilities in different channels depending on the tuning of parameters. Once a Zeeman field is introduced the model favors topological superconductivity where the order parameter winds an odd number of times around the Fermi surface. This suggests that certain superconductors, with strong spin-orbit coupling, may go through a topological phase transition as a function of applied magnetic field.