Zero-energy states in the Kitaev finite and semi-infinite model

Abstract

The Kitaev chain models a p -wave superconductor and hosts two Majorana bound states at the ends of the chain in the topological phase, for example if μ = 0 , Δ = t , where μ , Δ and t are chemical potential, superconducting pairing potential, and the next-nearest neighbor hopping amplitude, respectively. We consider finite and semi-infinite chains with close parameters μ = 0 and Δ = t + ɛ where ɛ is small, near the point Δ − t = 0 . Using the Dyson equation and the Green’s function for the infinite Kitaev chain, we analytically study the conditions for the appearance of zero-energy states, as well as their wave functions. We proove that in the finite chain such states exist only if Δ > t and the number of sites is odd. Zero-energy states disappear in the finite chain in the presence of an impurity potential, which indicates their instability. But in the semi-infinite chain, for Δ > t there is a single Majorana state and it is robust against an impurity.Thus the bulk-boundary correspondence may be violated for the Kitaev chain near the singular point. • We study the Kitaev chain with the pair potential close to the hopping amplitude. • Zero-energy states exist only for odd sites number and a large enough pair potential. • For a large pair potential there is Majorana state in the semi-infinite Kitaev chain. • The bulk-boundary correspondence is not always satisfied for the Kitaev chain.