ABSTRACT - DRUM - University of Maryland

here and leaves its proof to Appendix A:
(−1) C = ∏ K
Pf[H K Ξ]. (1.21)
Here the product is taken over all symmetric points K satisfying K ≡ −K in the
first Brillouin zone.
1.4 Majorana Zero Modes and TQC
As we have reviewed in the previous section, superconductors have particlehole
symmetry as a result of the redundant representation. The energy spectrum is
symmetric with respect to zero and γ −E = γ † E
. Thus the zero-energy states are very
special because γ E=0 = γ † E=0
. Such a quasiparticle is self-conjugate, being identical
to its “antiparticle”. Given two self-conjugate quasiparticles at E = 0 denoted by
γ 1 and γ 2 , it is straightforward to check that they still obey fermionic commutation
relation: {γ i , γ j } = δ ij . It immediately follows that γ 2 = 1 . Such self-conjugate
2
fermionic quasiparticles are called Majorana fermions(MF). Loosely speaking it can
be regarded as half of an ordinary fermion, since an ordinary fermion can always
be represented as two degenerate MFs: given a fermion annihilation operator c
satisfying {c, c † } = 1, we can form two MFs γ 1 = c+c† √
2
, γ 2 = c−c† √
2i
and c = γ 1 + iγ 2 .
One may then wonder what is special about MF here in a topological superconductor
since they are just the usual fermionic operators in disguise. It is important
to clarify that what we are interested are unpaired (non-degenerate), localized Majorana
fermionic excitations, which can not be obtained by naively rewriting a usual
fermionic operator as two MF operators. The particle-hole symmetry ensures that
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