ABSTRACT - DRUM - University of Maryland

4.3 Conclusions
In this chapter, we address the problem of topological degeneracy lifting in
topological superconductors characterized by the presence of Majorana zero-energy
states bound to the vortex cores. We calculate analytically energy splitting of zeroenergy
modes due to the intervortex tunneling. We consider here canonical model of
topological superconductor, spinless p x + ip y superconductor, as well as the model
of Dirac fermions coupled to superconducting scalar field.
The latter is realized
at the topological insulator/s-wave superconductor interface. In the case of spinless
p x +ip y superconductor, we find that, in addition to the expected exponential decay,
the splitting energy for a pair of vortices oscillates with distance in weak-coupling
superconductor and these oscillations become over-damped as the magnitude of the
chemical potential is decreased. In the second model, the splitting energy oscillates
for finite chemical potential and vanishes at µ = 0. The vanishing of splitting energy
is a consequence of an additional symmetry, the chiral symmetry, emerging in the
model when chemical potential is exactly equal to zero.
We show that this fact
is not accidental but stems from the index theorem which relates the number of
zero modes of the Dirac operator to the topological index of the order parameter.
Finally, we discuss the implications of our results for many-anyon systems.
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