Majorana fermions in vortex lattices

We consider Majorana fermions tunneling between vortices, within an array of such vortices in a 2D chiral p-wave superconductor. We calculate that the tunneling amplitude for Majorana fermions in a pair of vortices is proportional to the sine of half the difference between the global order parameter phases at the two vortices. Using this result we study tight-binding models of Majorana fermions in vortices arranged in a triangular or square lattice. In both cases we find that this phase-tunneling relationship leads to the creation of superlattices where the Majorana fermions form macroscopically degenerate `flat' bands at zero energy, in addition to other dispersive bands. This finding suggests that in vortex arrays tunneling processes do not change the energies of a finite fraction of Majorana fermions and hence brighten the prospects of topological quantum computing with a large number of Majorana states.