I am reading paper about Kitaev chain of electrons, which can exhibit famous Majorana fermions at ends of wire. The Hamiltonian (his Eq. (6)) reads
$H = frac{i}{2} sum_j – mu c_{2j-1}c_{2j} +(w+|Delta|)c_{2j}c_{2j+1} +(-w+|Delta|)c_{2j-1}c_{2j+2}$
in terms of the Majorana operators $c_{2j-1}$ and $c_{2j}$ and constants $mu, w, |Delta|$ can be thought of as parameters controlling Fermi level, hopping and gap.
Kitaev shows there are solutions of Hamiltonian $H$ with zero-energy. He gives the operators associated to zero-energy solutions in ansatz form (his Eq. (14)):
$b’ = sum_j left( alpha_{+}’ x_{+}^{j} + alpha_{-}’ x_{-}^{j}right) c_{2j-1}$
$b” = sum_j left( alpha_{+}” x_{+}^{-j} + alpha_{-}” x_{-}^{-j}right) c_{2j}$
where all $alpha$ are constants and $x_{pm}$ are unknowns to be found.
Q1) How to find $x_{pm}$?
Q2) How to show $x_{pm} = frac{-mu pm sqrt{mu^2 – 4 w^2 + 4 |Delta|^2}}{2 (w+|Delta|)}$?
I attempted to show it by computing $[b', H]=[b'', H]=0$, but it did not give correct answer.
Any help appreciated.
