Majorana bound states in open quasi-1D and 2D systems with transverse Rashba coupling

TL;DR

This paper analyzes the conditions for Majorana states in open quasi-1D and 2D systems with Rashba coupling, revealing how geometry influences their formation and identifying quasi-Majorana states with local Majorana character.

Contribution

It provides an analytical calculation of the topological invariant for large systems and introduces the generalized Majorana polarization to study local Majorana features.

Findings

Majorana states depend on system geometry and size.

Quasi-Majorana states exhibit local Majorana character without global symmetry.

Analytical topological invariant matches numerical results for large systems.

Abstract

We study the formation of Majorana states in quasi-1D and 2D square lattices with open boundary conditions, with general anisotropic Rashba coupling, in the presence of an applied Zeeman field and in the proximity of a superconductor. For systems in which the length of the system is very large (quasi-1D) we calculate analytically the exact topological invariant, and we find a rich phase diagram which is strongly dependent on the width of the system. We compare our results with previous results based on a few-band approximation. We also investigate numerically open 2D systems of finite length in both directions. We use the recently introduced generalized Majorana polarization, which can locally evaluate the Majorana character of a given state. We find that the formation of Majoranas depends strongly on the geometry of the system and if the length and the width are comparable no Majorana states can form, however, one can show the formation of "quasi-Majorana" states that have a local Majorana character, but no global Majorana symmetry.