ABSTRACT - DRUM - University of Maryland

Here τ x is the Pauli matrix acting on the particle-hole space. Notice that this
is an anti-unitary symmetry for the BdG Hamiltonian matrix since the complex
conjugation is involved. It implies that the solutions of BdG equation always come
in pairs: for each solution Ψ E = (u E , v E ) T with energy eigenvalue E, there is a
corresponding solution Ψ −E = τ x Ψ ∗ E
with energy −E. In terms of the Bogoliubov
quasiparticles, we readily have γ −E = γ † E
. This again confirms that the particle-hole
symmetry reflects the doubling of the degrees of freedom: creating a hole excitation
by γ † −E is equivalent to annihilating a particle excitation γ E. The Hamiltonian is
now diagonalized using γ operators:
Here E 0 = − 1 2
∑
E n>0 E n is a constant.
H = ∑ E n>0
E n γ † nγ n + E 0 . (1.9)
To reveal the topological nature of the p x +ip y superconductor, we first review
the general framework of the topological classification of superconductors. Here by
superconductor we mean fermionic systems described by BCS mean-field Hamiltonians.
We do not assume any symmetries present in the system. Without loss of
generality we consider lattice models of fermions with periodic boundary conditions,
since any continuum model can be approached as a limiting case of a lattice model.
A generic BCS Hamiltonian can be expressed in the momentum space as:
⎛ ⎞
H = ∑ k
Ψ † k H kΨ k , Ψ k = ⎜
⎝
ψ k
ψ † −k
⎟
⎠ . (1.10)
Here k is the lattice momentum taking value in the first Brillouin zone. ψ k can
have internal degrees of freedom, such as spin and orbital indices. H k is the BdG
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