ABSTRACT - DRUM - University of Maryland

Of particular importance is the sign of splitting as noted in Ref. [92].
It
determines which state is energetically favored when tunneling interaction is present.
If E + > 0, |0〉 is favored whereas E + < 0 favors |1〉. We note here that the definition
of states |0〉 and |1〉 relies on how we define the Dirac fermion operator ĉ and ĉ † .
Due to the presence of a constant term together with trigonometric function, the
sign of splitting can change. To figure out when the sign oscillates, we require the
amplitude of the trigonometric part is greater than the constant part which gives
√
1 + 4 λ > 2(1 + λ2 ) 1/4
.
2 λ
Solving this inequality yields λ = kξ > 8.
Therefore in this parameter regime
the sign of splitting changes with distance R. Otherwise the splitting still shows
oscillatory behavior but the sign is fixed to be positive.
In weak-coupling superconductors where ∆ 0 ≪ ε F or equivalently k F ξ ≫ 1,
the expression for the energy splitting (4.4) can be considerably simplified. In this
case, µ ≈ ε F and k ≈ k F . Keeping only terms that are leading order in (k F ξ) −1 , we
find
√
2
E + ≈
π ∆ cos(k F R + π)
(
4
0 √ exp − R )
, (4.6)
kF R
ξ
which is the expression reported in Ref. [[91]]. A similar expression for splitting of a
pair of Majorana bound states on superconductor/2D topological insulator/magnet
interface is found in Ref. [95].
Next we consider a different limit ∆ 2 0 > 2mµv 2 F
in which the wave function
of Majorana bound state for a single vortex doesn’t show any spatial oscillations.
Thus, we expect that tunneling splitting will show just an exponential decay without
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