ABSTRACT - DRUM - University of Maryland

The Bogoliubov-de Gennes Hamiltonian H BdG takes the following form [20, 19]
⎛ ⎞
h ∆
H BdG = ⎜ ⎟
⎝ ⎠ , (6.7)
∆ † −h T
where h is the single-particle Hamiltonian [for spinless fermions it is simply h =
(
−
1
2m ∂2 r − µ ) δ(r − r ′ )] and ∆ is the gap operator. The BCS Hamiltonian can be
diagonalized by Bogoliubov transformation
∫
ˆγ † =
d 2 r [ u(r) ˆψ † (r) + v(r) ˆψ(r) ] . (6.8)
Here the wavefunctions u(r) and v(r) satisfy BdG equations:
⎛ ⎞ ⎛ ⎞
H BdG
⎜
u(r)
⎟
⎝ ⎠ = E u(r)
⎜ ⎟
⎝ ⎠ . (6.9)
v(r) v(r)
Throughout this work, we adopt the convention that operators which are hatted are
those acting on many-body Fock states while bold ones denote matrices in “lattice”
space.
The single-particle excitations ˆγ, known as Bogoliubov quasiparticles, are coherent
superpositions of particles and holes. The particle-hole symmetry implies
that the quasiparticle with eigenenergy E and that with eigenenergy −E are related
by ˆγ −E
= ˆγ † E
. Therefore, E = 0 state corresponds to a Majorana fermion
ˆγ 0 = ˆγ † 0 [76]. The existence of such zero-energy excitations also implies a non-trivial
degeneracy of ground states: when there are 2N such Majorana fermions, they
combine pair-wisely into N Dirac fermionic modes which can either be occupied or
unoccupied, leading to 2 N -fold degenerate ground states. The degeneracy is further
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