# Majorana states in prismatic core-shell nanowires

## TL;DR

This paper explores how prismatic core-shell nanowires with polygonal cross sections can host Majorana states, showing that corner localization influences the stability and nature of these states under various geometric and physical conditions.

## Contribution

It systematically analyzes the low-energy spectrum and topological phases of prismatic nanowires with different geometries, highlighting the role of corner localization in Majorana state stability.

## Key findings

- Corner localization depends on shell geometry and affects Majorana stability.
- Prismatic geometries can host pseudo Majorana states that evolve into true Majoranas.
- Magnetic field and chemical potential influence the topological phase diagram.

## Abstract

We consider core-shell nanowires with conductive shell and insulating core, and with polygonal cross section. We investigate the implications of this geometry on Majorana states expected in the presence of proximity-induced superconductivity and an external magnetic field. A typical prismatic nanowire has a hexagonal profile, but square and triangular shapes can also be obtained. The low-energy states are localized at the corners of the cross section, i.e. along the prism edges, and are separated by a gap from higher energy states localized on the sides. The corner localization depends on the details of the shell geometry, i.e. thickness, diameter, and sharpness of the corners. We study systematically the low-energy spectrum of prismatic shells using numerical methods and derive the topological phase diagram as a function of magnetic field and chemical potential for triangular, square, and hexagonal geometries. A strong corner localization enhances the stability of Majorana modes to various perturbations, including the orbital effect of the magnetic field, whereas a weaker localization favorizes orbital effects and reduces the critical magnetic field. The prismatic geometry allows the Majorana zero-energy modes to be accompanied by low-energy states, which we call pseudo Majorana, and which converge to real Majoranas in the limit of small shell thickness. We include the Rashba spin-orbit coupling in a phenomenological manner, assuming a radial electric field across the shell.