Splitting of Majorana-Fermion Modes due to Intervortex Tunneling in a $p_x+i{p}_y$ Superconductor

We consider a two-dimensional ($p_x+i{p}_y$) superconductor in the presence of multiple vortices, which support zero-energy Majorana-fermion states in their cores. Intervortex tunnelings of the Majorana fermions lift the topological state degeneracy. Using the Bogoliubov–de Gennes equation, we calculate splitting of the zero-energy modes due to these tunneling events. We also discuss superconducting fluctuations and, in particular, their effect on the energy splitting.

Figure 1

Schematic plot of a two-vortex configuration. The shaded region corresponds to the order-parameter profile $f(\mathit{r})$ with vortices located at ${\mathit{R}}_1$ and ${\mathit{R}}_2$. The solid (blue) line represents the real part of the wave function $u_1(\mathit{r})+i{u}_2(\mathit{r})$.