ABSTRACT - DRUM - University of Maryland

sponding quasiparticle operator can be identified with the Dirac fermion operator.
We explicitly show this for the case of spinless p x + ip y superconductor:
ĉ = √ 1
∫
(ˆγ 1 −iˆγ 2 ) =
2
d 2 r
[
]
ˆψ u∗ 1 − iu ∗ 2
√ + ˆψ † v∗ 1 − iv2
∗ √ . (4.1)
2 2
Therefore ĉ (ĉ † ) annihilates(creates) a quasiparticle on energy level E + . The original
two fold degeneracy between state with no occupation ĉ|0〉 = 0 and occupied |1〉 =
ĉ † |0〉 is lifted by energy splitting E + .
To calculate the energy of Ψ + , we employ the standard method based on
the wave function overlap [93]. Suppose the two vortices are placed symmetrically
with respect to y axis: R 1 = (R/2, 0) and R 2 = (−R/2, 0). BdG equations are
H BdG Ψ + = E + Ψ + , H BdG Ψ 1 = 0. We then multiply the first equation by Ψ ∗ 1 and
second by Ψ ∗ +, substract corresponding terms, and integrate over region Σ which is
the half plane x ∈ (0, ∞), y ∈ (−∞, ∞) arriving finally at the following expression
for E + :
E + =
∫
Σ d2 r Ψ † 1H BdG Ψ + − ∫ Σ d2 r Ψ † +H BdG Ψ 1
∫Σ d2 r Ψ † 1Ψ +
. (4.2)
This is the general expression for the energy splitting which is used to evaluate E +
in p x + ip y SC and TI/SC heterostructure.
4.1.1 Splitting in spinless p x + ip y superconductor
We now calculate splitting for two vortices in spinless p x + ip y
superconductor.
The denominator in Eq.(4.2) can be evaluated quite straightforwardly
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