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.
We present exact analytical results for the differential conductance of a finite Kitaev chain in an N-S-N configuration, where the topological superconductor is contacted on both sides with normal leads. Our results are obtained with the Keldysh nonequilibrium Green's function technique, using the full spectrum of the Kitaev chain without resorting to minimal models. A closed formula for the linear conductance is given, and the analytical procedure to obtain the differential conductance for the transport mediated by higher excitations is described. The linear conductance attains the maximum value of e 2 /h only for the exact zero-energy states. Also, the differential conductance exhibits a complex pattern created by numerous crossings and anticrossings in the excitation spectrum. We reveal the crossings to be protected by inversion symmetry, while the anticrossings result from a pairing-induced hybridization of particlelike and holelike solutions with the same inversion character. Our comprehensive treatment of the Kitaev chain allows us also to identify the contributions of both local and nonlocal transmission processes to transport at arbitrary bias voltage. Local Andreev reflection processes dominate the transport within the bulk gap and diminish for higher excited states but reemerge when the bias voltage probes the avoided crossings. The nonlocal direct transmission is enhanced above the bulk gap but contributes also to the transport mediated by the topological states.
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