Majorana fermions in finite-size strips with in-plane magnetic fields

TL;DR

This paper investigates the emergence of Majorana bound states in finite-size quasi-one-dimensional systems with Rashba spin-orbit coupling under in-plane magnetic fields, using numerical and analytical methods to map their topological phases.

Contribution

It introduces a comprehensive analysis combining numerical diagonalization and Hamiltonian singular point methods to determine topological phases and Majorana states in disordered systems with in-plane magnetic fields.

Findings

Topological phase diagrams are consistent across methods.

Majorana states with even pairs are not topologically protected.

The $ ext{Z}_2$ invariant correctly predicts Majorana parity.

Abstract

We study the Majorana bound states arising in quasi-one-dimensional systems with Rashba spin-orbit coupling in the presence of an in-plane Zeeman magnetic field. Using two different methods, first, the numerical diagonalization of the tight-binding Hamiltonian, and second, finding the singular points of the Hamiltonian (see Refs. [1-4]), we obtain the topological phase diagram for these systems as a function of the chemical potential and the magnetic field, and we demonstrate the consistency of these two methods. By introducing disorder into these systems we confirm that the states with even number of Majorana pairs are not topologically protected. Finally, we show that a formal calculation of the $\mathbb{Z}_2$ topological invariants recovers correctly the parity of the number of Majorana bound states pairs, and it is thus fully consistent with the phase diagrams of the disordered systems.