ABSTRACT - DRUM - University of Maryland

Chapter 3
Majorana Bound States in Topological Defects
In this chapter, we review the analytic solutions of the Bogoliubov-de Gennes
equation for Majorana zero modes in a p x + ip y superconductor and at a topological
insulator/superconductor interface. From the explicit solutions we deduce the
generic Z 2 classification of Majorana zero-energy modes in superconducting vortices,
as well as the Z classification for Dirac-type Hamiltonian when an additional chiral
symmetry is present. Some of the results are used in the later chapters.
3.1 Bound states in p x + ip y superconductors
The BdG equation for p x + ip y superconductor has been derived in:
⎛ ⎞ ⎛ ⎞
u(r)
H BdG
⎜ ⎟
⎝ ⎠ = E u(r)
⎜ ⎟
⎝ ⎠ , (3.1)
v(r) v(r)
where the explicit form of the BdG Hamiltonian in real space is given by
⎛
⎞
− ∇2
2m − µ 1
{∆(r), ∂ x + i∂ y }
k
⎟
F
H BdG = ⎜
⎝
− 1
k F
{∆ ∗ (r), ∂ x − i∂ y }
∇ 2
2m + µ
with anti-commutator being defined as {a, b} = (ab + ba)/2.
⎟
⎠ (3.2)
The particle-hole symmetry of BdG Hamiltonian is represented by Ξ = τ x K
with K being complex conjugation operator and τ x being Pauli matrix in Nambu(particle-
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