# Reference request on Bogoliubov de Gennes (BdG) formalism

I have tried to gain an understanding about the BdG formalism by just following the calculations I found here and there of people bringing their superconducting Hamiltonians into matrix forms but I am still quite unsure about the following things

1. Real space vs momentum space: while I feel like I could do the transformation in momentum space, I dont understand how it works in real space (for example here, the kiteav chain is dealt with in real space https://topocondmat.org/w1_topointro/1D.html)

2. The Bogoliubov transformation itself: Is it unitary or not? some write it is a member of $$SU(2)$$ some write it is only symplectic. I would love to understand the algebraic structure of these Hamiltonians, somewhere I saw people speaking about $$su(2)$$ subalgebras..

3. doubling degrees of freedom, the Hilbert space we are dealing with and the basis states
4. Nambu spinors? can you construct nambu spinors in real space

The list probably goes on, so I would really appreciate an coherent introduction to the topic, be it a textbook or lecture notes or whatnot.

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• You might find "Statistical Mechanics of Superconductivity" by T. Kita a very useful read (especially chapter 8). – Sunyam Jan 30 at 17:32