Zero modes of two-dimensional chiral $p$-wave superconductors

We discuss fermionic zero modes in the two-dimensional chiral $p$-wave superconductors. We show quite generally that without fine tuning, in a macroscopic sample there is only one or zero of such Majorana-fermion modes depending only on whether the total vorticity of the order parameter is odd or even, respectively. As a special case of this, we find explicitly the one zero mode localized on a single odd-vorticity vortex and show that, in contrast, zero modes are absent for an even-vorticity vortex. One zero mode per odd vortex persists, within an exponential accuracy, for a collection of well-separated vortices, shifting to finite $\pm E$ energies as two odd vortices approach. These results should be useful for the demonstration of the non-Abelian statistics that such zero-mode vortices are expected to exhibit and for their possible application in quantum computation.