Tomography of zero-energy end modes in topological superconducting wires

TL;DR

This paper provides a geometric characterization of Majorana zero modes in topological superconductor-semiconductor wires, revealing universal properties and explaining Josephson current behavior in various magnetic and spin-orbit configurations.

Contribution

It introduces a generalized Bloch coordinate framework to describe Majorana modes and analyzes their transformation under SU(2) rotations, offering a comprehensive understanding of their properties.

Findings

Identification of universal parameters when spin-orbit coupling and magnetic field are perpendicular.

Explanation of Josephson current behavior based on Majorana mode geometry.

Method to extract the full Majorana mode description from junction measurements.

Abstract

We characterize the Majorana zero modes in topological hybrid superconductor-semiconductor wires with spin-orbit coupling and magnetic field, in terms of generalized Bloch coordinates $\varphi, \theta, \delta$, and analyze their transformation under SU(2) rotations. We show that, when the spin-orbit coupling and the magnetic field are perpendicular, $\varphi$ and $\delta$ are universal in an appropriate coordinate system. We use these geometric properties to explain the behavior of the Josephson current in junctions of two wires with different orientations of the magnetic field and/or the spin-orbit coupling. We show how to extract from there, the angle $\theta$, hence providing a full description of the Majorana modes.