ABSTRACT - DRUM - University of Maryland

where N 1 is the normalization constant determined by the following equation
∫
4π
rdr |u 0 (r)| 2 = 1. (3.9)
Evaluation of the integral yields:
N 2 1 =
8
3πk 2 1ξ 4 2F 1 ( 3 2 , 5 2 ; 3; −k2 1ξ 2 )
(3.10)
with its asymptotes given by
⎧
8 ⎪⎨ k
N1 2 3πk1 ∼
2 1 ξ ≪ 1
ξ4 . (3.11)
⎪⎩ k 1
k
2ξ 1 ξ ≫ 1
In the opposite limit ∆ 2 0 > 2mµvF 2 , the solution of Eq. (3.7) is given by first
order imaginary Bessel function:
√
χ(r) = N 2 I 1 (r ∆ 2 0/vF 2 − 2mµ). (3.12)
The function I n (r) diverges when r → ∞.
But the radial wave function u 0 (r)
remains bounded as long as µ > 0. This is consistent with the fact that µ = 0
separates topologically trivial phase (µ < 0) and non-Abelian phase (µ > 0) [19].
We now give expressions for N 2 . Explicitly, it is given by
4πN 2 2
∫ ∞
0
rdr I 2 1(k 2 r)e −2r/ξ = 1, (3.13)
where k 2 = √ ∆ 2 0/v 2 F − 2mµ. Since µ > 0, k 2ξ is always smaller than 1. We find N 2
is given by
N 2 2 =
8
3πk 2 2ξ 4 2F 1 ( 3 2 , 5 2 ; 3; k2 2ξ 2 )
(3.14)
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