Abstract
We present a comprehensive, analytical treatment of the finite Kitaev chain for arbitrary chemical potential and chain length. By means of an exact analytical diagonalization in the real space, we derive the momentum quantization conditions and present exact analytical formulas 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. Further, for a fixed value of the chemical potential, their properties are ruled by the ratio of the decay length to the chain length. A numerical analysis confirms the robustness of the topological states against disorder.
Highlights
The quest for topological quantum computation has drawn a lot of attention to Majorana zero energy modes (MZM), quasiparticles obeying non-Abelian statistics hosted by topological superconductors [1]
We present a comprehensive, analytical treatment of the finite Kitaev chain for arbitrary chemical potential and chain length
Due to its apparent simplicity, the Kitaev chain is often used as the archetypal example for topological superconductivity in one dimension
Summary
The quest for topological quantum computation has drawn a lot of attention to Majorana zero energy modes (MZM), quasiparticles obeying non-Abelian statistics hosted by topological superconductors [1]. As shown by Kitaev in the limit of an infinite chain, for a specific choice of parameters, the superconductor enters a topological phase where the chain can host a couple of unpaired zero energy Majorana modes at the end of the chain [2]. This model has become very popular due to its. Appendices A–G contain details of the factorisation of the characteristic polynomial in real space and the calculation of the associated eigenstates
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