Topological Majorana and Dirac Zero Modes in Superconducting Vortex Cores

We provide an argument based on flux insertion to show that certain superconductors with a nontrivial topological invariant have protected zero modes in their vortex cores. This argument has the flavor of a two-dimensional index theorem and applies to disordered systems as well. It also provides a new way of understanding the zero modes in the vortex cores of a spinless $p_x+i{p}_y$ superconductor. Applying this approach to superconductors with and without time-reversal and spin-rotational symmetry, we predict the necessary and sufficient conditions for protected zero modes to exist in their vortices.

Figure 1

A schematic plot of the energy $E$ versus $\varphi$, where $\varphi$ is the flux inserted through a plaquette of the insulator with Bogoliubov symmetry. Only states which lie close to the Fermi energy at $\varphi =0$ are shown. The dashed line is the single energy curve which traverses the gap as the flux is changed from 0 to $2\pi \hslash /e$.