Testing the formation of Majorana states using the Majorana Polarization

TL;DR

This paper introduces a local order parameter called Majorana polarization to identify Majorana states in superconducting systems, demonstrating its effectiveness in one- and two-dimensional models with spin-orbit coupling and magnetic fields.

Contribution

It defines the Majorana polarization vector and criterion, providing a new method to characterize Majorana states locally in finite-size superconducting systems.

Findings

Majorana polarization effectively identifies Majorana states.

The criterion works for 1D wires and 2D finite systems.

Local order parameter is essential for finite-size systems.

Abstract

We study the formation of Majorana states in superconductors using the Majorana polarization, which can locally evaluate the Majorana character of a given state. We introduce the definition of the Majorana polarization vector and the corresponding criterion to identify a Majorana state, and we apply it to some simple cases such as a one-dimensional wire with spin-orbit coupling, subject to a Zeeman magnetic field, and proximitized by a superconductor, as well as to an NS junction made with such a wire. We also apply this criterion to two-dimensional finite-size strips and squares subject to the same physical conditions. Our analysis demonstrates the necessity of using the Majorana polarization local order parameter to characterize the Majorana states, particularly in finite-size systems.