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) + ip(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.