ABSTRACT - DRUM - University of Maryland

states near the Fermi circle. We first diagonalize the single-particle Hamiltonian:
E ± (p) = ±v F |p| − µ. (1.28)
And the eigenvectors are given by |±, p〉 = 1 √
2
(1, ±e iθp ) T . Projecting onto the +
band, we find the effective Hamiltonian is given by
H eff = ∑ p
f † p(v F |p| − µ)f p − 1 2 ∆e−iθp f † pf † −p + h.c.. (1.29)
Here f p = 1 √
2
(ψ ↑p +e −iθp ψ ↓p ). So in this basis, the chirality of the Dirac Hamiltonian
results in the p x + ip y pairing symmetry. However, we would like to remark that the
surface states do not break time-reversal symmetry while p x + ip y superconductors
break time-reversal symmetry.
We can further solve the corresponding BdG equation with superconducting
vortices and find a single Majorana zero-energy bound state in a hc
2e
explicit solution is displayed in Chapter 3.
vortex. The
We now turn to the second proposal, where effective p-wave pairing is realized
in cold fermionic atoms [48]. The idea is that for spin-1/2 fermions, a single Fermi
surface can be created by simply applying a Zeeman field to polarize the fermions
and tune the Fermi energy in the Zeeman gap. Notice that the Zeeman splitting
explicitly breaks the time-reversal symmetry for spin-1/2 fermions. S-wave interactions
between the fermions lead to the formation of a BCS superfluid with s-wave
singlet pairing. So Rashba spin-orbit coupling is needed to allow pairing on the same
Fermi surface. Therefore we are led to the following theoretical model describing
spin-orbit coupled fermions subject to a Zeeman field and s-wave superconducting
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