Majorana zero modes in tubular nanowires

Abstract

Majorana zero modes have recently been proposed as a solution to obtain fault-tolerant quantum computation. They are quasiparticle excitations in quantum systems at the nanoscale with the potential to function as qubits, the basic building blocks of quantum computers. Their topological properties hold promise to counter the decoherence problem, where quantum coherence is lost due to coupling with the environment, which is one of the main challenges for upscaling the number of qubits in a quantum computer. In this work, Majorana zero modes in core-shell nanowire systems are investigated. Core-shell nanowires are tubular conductors that commonly have polygonal cross-section geometry due to crystallographic qualities. Semiconductors with proximity-induced superconductivity, large g-factors and Rashba spin-orbit coupling, can be tuned by an external magnetic field to host Majorana zero modes. Corner-localization of the lowest energy states in polygonal core-shell nanowires allows for hosting multiple Majorana zero modes within a single nanowire system. Three-dimensional nanowires are modeled using the Bogoliubov-de Gennes Hamiltonian, which is solved numerically by diagonalization. Effects of variable core and shell geometry are analyzed and triangular wires with hexagonal cores are found to have particularly large energy separations between the first and second groups of energy states, which is favorable for experimental hosting of Majorana zero modes. Braiding in such a system is discussed and for the realization of real space braiding, a dual core wire is suggested to include the necessary degrees of freedom for the simplest non-commutative braiding operation. The relation between Andreev reflection and propagation of the superconductivity property is studied by modeling partially proximitized wire shells, with radial, angular and longitudinal interfaces, corresponding to common experimental platforms. Varying compatibility is found with Andreev reflection. Flux-periodic oscillations in the energy spectra of proximitized shells are explored along with the effects of geometry, Zeeman and spin-orbit interaction. Instances are shown where the periodicity of the lowest energy state is found to separately display normal, superconducting and fractional flux quanta. To gain insight into the structure of topological invariants, the first Chern number in particular, Stokes' theorem for bivector-valued fields is analyzed and visualized using geometric calculus. The findings show how the multiple components necessary for the emergence of Majorana zero modes each have complex subtleties and interdependencies, and in what way they should be investigated to come closer to the technological realization of Majorana zero modes.