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
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)
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..
- doubling degrees of freedom, the Hilbert space we are dealing with and the basis states
- 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.