Saturday, 30 July 2016

fourier transform - Edge states of Kitaev chain



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=i2jμc2j1c2j+(w+|Δ|)c2jc2j+1+(w+|Δ|)c2j1c2j+2


in terms of the Majorana operators c2j1 and c2j and constants μ,w,|Δ| 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=j(α+xj++αxj)c2j1


b


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.




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