ABSTRACT - DRUM - University of Maryland

the zero mode (see Chapter 3 for details). Besides the zero modes, there are other
eigenstates with energy eigenvalues well below the bulk gap, the so-called midgap
states [31, 32, 33, 34]. A semiclassical argument, treating the vortex core as a hole
of size ξ ∼ v F
∆0
, gives an estimate of the energy of midgap states to be of the order
1
mξ 2 ≈ ∆2 0
E F
where E F is the Fermi energy. As long as k F ξ ≫ 1, this energy scale is
much smaller than the bulk gap.
2
1
2 1
Figure 1.1: Braiding of Majorana fermions bound to vortices.
We now demonstrate the very peculiar non-Abelian braiding statistics of MFs
in superconducting vortices, first derived by Ivanov [35]. It is crucial to keep track of
the branch cut where the superconducting phase jumps by 2π to uniquely define the
superconducting phase everywhere (except at the vortex cores). Pictorially we can
attach a “string” to each vortex, which goes all the way to infinity (or the system
boundary) to represent the branch cut. As the vortices are transported adiabatically,
the branch cuts are also “dragged” along with the vortices and we have to make sure
that the vortices do not cross the branch cuts, as depicted in Fig. 1.1. We denote the
local phases seen by vortices 1 and 2 by χ 1 and χ 2 respectively. Before the exchange,
χ 1 = π + 0 + , χ 2 = 0 + and after the exchange χ ′ 1 = 2π − 0 + , χ ′ 2 = π + 0 + . The gauge
transformation then implies γ 1 picks up a phase e i(χ 1−χ ′ 1 )/2 = −1. Consequently,
γ 1 is replaced by γ 2 but γ 2 is replaced by −γ 1 .
We therefore conclude that the
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