ABSTRACT - DRUM - University of Maryland

satisfied for m = 0. The singlevalueness of the wavefunction then requires l to be an
odd integer. We thus see that Majorana bound state only exist in vortices with odd
vorticity, which justifies the Z 2 classification of Majorana zero modes in vortices.
The radial part of the BdG equations in m = 0 channel then reads
⎛
⎞ ⎛ ⎞
[
]
− 1
2m
⎜
(∂2 r + 1∂ r r − n2
1
) − µ
r 2 k F
f(r)(∂ r + 1 ) + f ′ (r)
2r 2
u 0 (r)
⎟ ⎜ ⎟
⎝ [
]
⎠ ⎝ ⎠ = 0. (3.5)
− 1
k F
f(r) (∂ r + 1 ) + f ′ (r) 1
2r 2 2m (∂2 r + 1∂ r r − n2 ) + µ v
r 2 0 (r)
Given that the radial part of the BdG equation (3.5) is real, one can choose u 0 (r)
and v 0 (r) to be real. Then the condition ΞΨ 0 = Ψ 0 reduces to v 0 = λu 0 with
λ = ±1. Using this constraint, the differential equation for u 0 becomes:
{(∂ 2 r + 1 r ∂ r − n2
r 2 )
− 2mµ − 2λ [ (
f ∂ r + 1 v F 2r
)
+ f ′
One can seek the solution of the above equation in the form
2
]}
u 0 = 0.
[ ∫ r
]
u(r) = χ(r) exp λ dr ′ f(r ′ ) , (3.6)
0
which leads to
χ ′′ + χ′
r + (
2mµ − f 2
v 2 F
− n2
r 2 )
χ = 0. (3.7)
Here the profile f(r) = ∆ 0 tanh(r/ξ) vanishes at the origin and reaches ∆ 0 away
from vortex core region. For our purpose, it’s sufficient to consider the behavior of
solution outside the core region where f(r) is equal to its asymptotic bulk value ∆ 0 .
It is obvious now that λ = −1 yields the only normalizable solution.
When ∆ 2 0 < 2mµv 2 F
which is the case for weak-coupling BCS superconductors,
Eq.(3.7) becomes first order Bessel equation. Thus, the solution is given by Bessel
function of the first kind J n (x):
√
χ(r) = N 1 J 1 (r 2mµ − ∆ 2 0/vF 2 ), (3.8)
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