An Index Theorem for the Majorana Zero Modes in Chiral P-Wave Superconductors*

We show that the Majorana fermion zero modes in the cores of odd winding number vortices of a 2D $p_x+ip_y$-paired superconductor is due to an index theorem. This theorem is analogous to that proven by Jackiw and Rebbi for the existence of localized Dirac fermion zero modes on the mass domain walls of a 1D Dirac theory. The important difference is that, in our case, the theorem is proven for a two component fermion field theory where the first and second components are related by parity reversal and hermitian conjugation. The vortices with Majorana zero modes can be used, in principle, to build a topological quantum computer.