ABSTRACT - DRUM - University of Maryland

with λ = µξ/v. It has the following asymptotes:
⎧
1 ⎪⎨ λ ≪ 1
N3 2 πξ
∼
2 ⎪⎩ λ
λ ≫ 1.
2ξ 2
(3.29)
The case of µ = 0 is special due to the presence of an additional symmetry of BdG
Hamiltonian - chiral symmetry. For l > 0 the solution of Eq.(3.25) becomes
⎛ ⎞ ⎛ ⎞
χ ↑
⎜ ⎟
⎝ ⎠ ∝ r m ⎜ ⎟
∫ r
⎝ ⎠ e−λ 0 dr′ f(r ′) , (3.30)
χ ↓ 0
and for l < 0,
⎛ ⎞ ⎛ ⎞
χ ↑
⎜ ⎟
⎝ ⎠ ∝ 0
⎜ ⎟
⎝
χ ↓ r −(m+1)
⎠ e−λ ∫ r
0 dr′ f(r ′) . (3.31)
Again the normalizability requires λ = 1.
Because the chiral symmetry also relates eigenstates with positive energies to
those with negative energies which follows from γ 5 H BdG γ 5 = −H BdG , one can always
require the zero-energy eigenstates to be eigenstates of γ 5 . The wave function in
Eq.(3.30) is an eigenstate of γ 5 with eigenvalue 1 while wave function (3.31) has
eigenvalue −1. We define eigenstates of γ 5 with eigenvalue ±1 as ± chirality.
To summarize we have obtained the Majorana zero-energy bound state attached
to the vortex with odd vorticity:
⎛
⎞
e −iπ/4 χ ↑ (r)
e iπ/4 χ ↓ (r)e iϕ
Ψ 0 (r) = e i(l−1)ϕ/2 e −iπ/4 χ ↓ (r)e −ilϕ
⎜
⎟
⎝
⎠
−e iπ/4 χ ↑ (r)e −i(l−1)ϕ
(3.32)
57