Retrieval of free-Majorana wavefunctions for the finite Kitaev chain using an appropriate momentum representation

TL;DR

This paper develops a method to retrieve Majorana wavefunctions in finite Kitaev chains using a discrete sine transform, enabling analysis of larger systems and confirming edge state properties.

Contribution

It introduces a new L-dimensional Nambu operator and a momentum representation approach for finite Kitaev chains, improving system size analysis and wavefunction characterization.

Findings

Successful retrieval of zero-energy Majorana modes as edge states

Excellent agreement between numerical results and perturbation theory

Wavefunctions exhibit exponential decay characteristic of edge states

Abstract

In quantum mechanics, the spaces of momentum and its conjugate, the position, are related via Fourier transforms and thus the properties are interwoven with their structure. In particular, for lattice systems possessing an underlying discrete position space, the momentum becomes finite. Moreover, if the lattice length is finite, $L<\infty$, the momentum space also becomes both finite and discrete breaking altogether the continuity of the dispersion relation. This aspect is relevant in new systems such as the topological materials. We address this point paving the path for new ways to the observation of Majorana quasi-particles. Furthermore, the Kitaev model, which is the simplest Hamiltonian supporting Majorana fermions, is therefore taken as a starting point for the theoretical description of the work. Our study focuses on finding the zero-energy modes in an artificial arrangement of a non-interacting superconducting finite nanowire by using a discrete-sine transform with the purpose of going from position to momentum space considering hard-wall boundary conditions in the process. Here, a new $L$-dimensional Nambu operator to diagonalize the system in the Bogoliubov-de Gennes formalism is proposed and, further, we arrive to a new space with a considerable reduced dimension allowing the treatment of larger system sizes. We also present a comparison between the numerical and third order perturbation theory for the weak coupling regime results presenting an excellent agreement. Finally, we analyze the wavefunctions of the retrieved zero-energy modes showing that they are indeed edge states and thus they have an exponential decay.