Confused about Bogoliubov transformation

[JVanUW]

So I was told by my teacher today that I am doing the bogoliubov transformation wrong because I'm supposed to have 4 different vectors,

i.e

Instead of A_k and B_k

A_1k
A_2k

and
B_1k
B_2k

I'm wondering how this makes sense. I've seen papers with 2 different vectors and their complex counterparts but never 4 different vectors. Any help would be great.

 

[DrDu]

Maybe you can give us the expressions you obtained for A and B.