Exact eigenvectors and eigenvalues of the finite Kitaev chain and its topological properties

TL;DR

This paper provides an exact analytical solution for the eigenvalues and eigenvectors of the finite Kitaev chain, elucidating its topological boundary states and their robustness, for arbitrary chemical potential.

Contribution

It derives exact formulas for the energy spectrum and eigenstates of the finite Kitaev chain, including boundary and bulk states, and analyzes their topological properties.

Findings

Boundary states can have zero, small, or finite energy.

Boundary states are topological and robust against disorder.

Analytical formulas apply for arbitrary chemical potential.

Abstract

We present a comprehensive, analytical treatment of the finite Kitaev chain for arbitrary chemical potential. We derive the momentum quantization conditions and present exact analytical formulae for the resulting energy spectrum and eigenstate wave functions, encompassing boundary and bulk states. In accordance with an analysis based on the winding number topological invariant, and as expected from the bulk-edge correspondence, the boundary states are topological in nature. They can have zero, exponentially small or even finite energy. A numerical analysis confirms their robustness against disorder.