Majorana fermions on a disordered triangular lattice

Vortices of several condensed matter systems are predicted to have zero-energy core excitations which are Majorana fermions (MFs). These exotic quasi-particles are neutral, massless and expected to have non-Abelian statistics. Furthermore, they make the ground state of the system highly degenerate. For a large density of vortices, an Abrikosov lattice is formed, and tunneling of MFs between vortices removes the energy degeneracy. In particular, the spectrum of MFs in a triangular lattice is gapped, and the Hamiltonian that describes such a system is antisymmetric under time reversal. We consider MFs on a disordered triangular lattice. We found that even for very weak disorder in the location of the vortices localized sub-gap modes appear. As the disorder becomes strong, a percolation phase transition takes place, and the gap is fully closed by extended states. The mechanism underlying these phenomena is domain walls between two time-reversed phases, which are created by flipping the sign of the tunneling matrix elements. The density of states in the disordered lattice seems to diverge at zero energy.