I almost solved the problem Equivalence of Bogoliubov-de Gennes Hamiltonian for nanowire. In the next steps I used the notation by arXiv:0707.1692:
Ψ†=((ψ†↑,ψ†↓),(ψ↓,−ψ↑))
and
Ψ=((ψ↑,ψ↓),(ψ†↓,−ψ†↑))T.
I'm trying to show that the Hamiltonian for a nanowire with proximity-induced superconductivity
ˆH=∫dx [∑σϵ{↑,↓}ψ†σ(ξp+αpσy+Bσz)ψσ+Δ(ψ†↓ψ†↑+ψ↑ψ↓)],
can be written as
ˆH=12∫dx Ψ†HΨ
with H=ξp1⊗τz+αpσy⊗τz+Bσz⊗1+Δ1⊗τx (here τi are the Pauli matrix for the particle-hole space and ⊗ means the Kronecker product).
Here I calculate as example the first und third term of Ψ†HΨ.
τzΨ=((ψ↑,ψ↓),−(ψ†↓,−ψ†↑))T=((ψ↑,ψ↓),(−ψ†↓,ψ†↑))T
⇒((ψ†↑,ψ†↓),(ψ↓,−ψ↑))ξp((ψ↑,ψ↓),(−ψ†↓,ψ†↑))T=(ψ†↑,ψ†↓)ξp(ψ↑,ψ↓)T+(ψ↓,−ψ↑)ξp(−ψ†↓,ψ†↑)T=ψ†↑ξpψ↑+ψ†↓ξpψ↓−ψ↓ξpψ†↓−ψ↑ξpψ†↑
Now I use the anticommutatorrelation {ψσ,ψ†σ′}=δσ,σ′⇔ψσψ†σ=1−ψ†σψσ
⇔2ψ†↑ξpψ↑+2ψ†↓ξpψ↓−2ξp
However, the term −2ξp here are wrong.
For the third term I obtain
ψ†↑Bσzψ↑+ψ†↓Bσzψ↓+ψ↓Bσzψ†↓+ψ↑Bσzψ†↑=ψ†↑Bσzψ↑+ψ†↓Bσzψ↓−ψ†↓Bσzψ↓−ψ†↑Bσzψ↑+2Bσz=2Bσz
Does anybody see my mistake?
Answer
I define H∼ψ†αξαβψβ, where I forget the sum/integrals and all these boring staff. I also define ξαβ≡ξ⋅σ=ξ0+ξxσx+ξyσy+ξzσz to have the most generic one-body Hamiltonian written in a compact form. The one body Hamiltonian then reads, in matrix notation H∼(ψ†↑ψ†↓)(ξ0+ξzξx−iξyξx+iξyξ0−ξz)(ψ↑ψ↓)
One now wants to add the particle-hole double space (Nambu space). One uses that (the anti-commutation relation) ψ†αξαβψβ=−ξαβψβψ†α+δαβξαβ=−ψβ(ξαβ)Tψ†α+δαβξαβ
Your ordering convention is found by an obvious change of basis from mine. Then you choose a representation for the tensor product and you're done. One more time, you can not avoid the final trace term, but most of the people forget to discuss it. It has almmost no role, except when you want to describe some effects related to the phase transition of superconductivity (for instance, to correctly write the free energy, you need it).
One more thing: the Hamiltonian you gave is a bit famous at the moment for hosting Majorana fermions. If you diagonalise the spin-part, you end up with a p-wave effective superconductivity at low energy.
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