Topological indices, defects, and Majorana fermions in chiral superconductors

We study theoretically the role of topological invariants to protect the Majorana fermions in a model of two-dimensional (2D) chiral superconductors which belong to class D of the topological periodic table. A rich phase diagram is revealed. Each phase is characterized by the topological invariants for 2D ($Z$) and 1D ($Z_2$), which lead to the Majorana fermion at the edge dislocation and the core of the vortex. Interference of the Majorana fermions originating from the different topological invariants is studied. The stability of the Majorana fermion with respect to the interlayer coupling, i.e., in 3D, is also examined.

Figure 1

The topological phase diagram of a model in Eq. (1) for chiral superconductors in 2D characterized by $\nu :{\nu }_x{\nu }_y$. The lines where the energy gap closes divide the ($t_x$,$t_y$) plane into nine domains. The system is a strong topological superconductor ($\nu =1$) in domains II, III, VII, and IX, with $\nu =0$ but some of $Z_2$ invariants being nonzero. In domain V, the system is the trivial strong coupling superconductor.

Figure 2

The edge dislocations (indicated by yellow points) in two- and three-dimensional systems. The 3D system is made by stacking 2D systems and adding a hopping integral along the $z$ direction with the edge dislocation isolated on a layer.

Figure 3

The energy levels and the probability distribution of zero-energy states at each site in the presence of edge dislocations: The left panel indicates the energy levels. The right panel indicates the probability distribution for the zero-energy states. The zero-energy Majorana bound states appear at each of the edge dislocations.

Figure 4

The interaction between zero-energy states localized at edge dislocations and zero-energy states localized at vortex cores: This figure shows the probability distribution of zero-energy states at each lattice site in each case. The interaction between zero-energy states at edge dislocations and zero-energy states at vortex cores eliminates zero-energy states when dislocations and vortex cores exist at the same positions.