ABSTRACT - DRUM - University of Maryland

pairing:
( )
p
H = ψ † 2
2m − µ + α(σ × p) · ẑ + V zσ z ψ + gψ † ↑ ψ† ↓ ψ ↓ψ ↑ . (1.30)
Here m is the mass of the atoms, α is the strength of spin-orbit coupling and V z
the Zeeman splitting. We have chosen the cold atomic gas to be confined in the xy
plane. Experimentally, Rashba spin-orbit coupling can be engineered by a variety
of ways [53, 54, 55, 56, 57].
The interaction can be treated by the standard BCS mean-field theory.
A
careful analysis of the pairing symmetry reveals that even in the presence of spinorbit
coupling, the short-range interaction we use in (1.30) only gives rise to s-
wave singlet pairing [48, 58].
So we can proceed with the following mean-field
Hamiltonian:
( )
p
H MF = ψ † 2
2m − µ + α(σ × p) · ẑ + V zσ ∗ z ψ + ∆ψ † ↑ ψ† ↓
+ h.c.. (1.31)
The value of the s-wave gap ∆ can be determined self-consistently from the gap
equation.
Intuitively, the Zeeman field opens a “magnetic” gap 2|V z | at the band crossing
point k = 0. When the Fermi level lies within the Zeeman gap, there is only one
Fermi surface, a “parent” state of chiral p-wave superconductor.
To determine
precisely the condition under which TSC exists, we need to calculate the parity of
the Chern number using the Pfaffian formula (1.21), and the result is
(−1) C = sgn(µ 2 + ∆ 2 − V 2
z ). (1.32)
Therefore if V 2
z
> µ 2 + ∆ 2 , the Chern number must be odd which ensures the
existence of unpaired Majorana zero modes in superconducting vortices. A more
23