ABSTRACT - DRUM - University of Maryland

We summarize this section by providing an explicit expression for zero energy
eigenfunction:
[ (
Ψ 0 (r) = χ(r) exp i ϕ − π )
τ z − 1 ∫ r
]
dr ′ f(r ′ ) , (3.15)
2 v F 0
where χ(r) is given by Eq.(3.8) for ∆ 2 0 < 2mµvF 2 and Eq.(3.12) for ∆2 0 > 2mµvF 2 .
Using the zero energy solution obtained for one vortex one can be easily write
down wave function for multiple vortices spatially separated so that tunneling effects
can be ignored. Assume there are 2N vortices pinned at positions R i , i = 1, . . . , 2N.
The superconducting order parameter can be represented as
∆(r) =
2N∏
i=1
f(r − R i ) exp [ i ∑ i
ϕ i (r) ] , (3.16)
where ϕ i (r) = arg(r − R i ).
Near the k-th vortex core, the phase of the order
parameter is well approximated by ϕ k (r) + Ω k with Ω k
= ∑ i≠k ϕ i(R k ) which is
accurate in the limit of large inter-vortex separation. Then in the vicinity of k-th
vortex core, a zero energy bound state can be found [20]:
[
Ψ k (r) = e −iτz π 2 χ(rk ) exp − 1 ∫ rk
] [ (
dr ′ f(r ′ ) exp i ϕ k + Ω ) ]
k
τ z . (3.17)
v F 0
2
where r k = |r − R k |. Correspondingly, there are 2N Majorana fermion modes
localized in the vortex cores.
3.2 Bound states in the Dirac fermion model coupled with
s-wave superconducting scalar field.
We now discuss the zero energy bound states emerging in the model of Dirac
fermions interacting with the superconducting pairing potential.
This model is
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