# Chiral current-phase relation of topological Josephson junctions: A signature of the $4\pi$-periodic Josephson effect

## TL;DR

This paper introduces a chiral current-phase relation as a signature of the $4\pi$-periodic Josephson effect in topological junctions, providing a new way to detect Majorana modes without fixing fermionic parity.

## Contribution

It proposes a novel chiral CPR model for the $4\\pi$-periodic Josephson effect, applicable to spin-polarized topological junctions, and details methods to realize it experimentally.

## Key findings

- Chiral CPR described as $J(\\phi) \\propto C \\ |\\\sin(\\rac{\\\phi}{2})|$.
- Chirality $C$ equals the Chern number of the spin band.
- Three scenarios for achieving chiral CPR: Zeeman splitting, combined Zeeman and bias currents, external magnetic field.

## Abstract

The $4\pi$-periodic Josephson effect is an indicator of Majorana zero modes and a ground-state degeneracy which are central to topological quantum computation. However, the observability of a $4\pi$-periodic Josephson current-phase relation (CPR) is hindered by the necessity to fix the fermionic parity. As an alternative to a $4\pi$-periodic CPR, this paper proposes a chiral CPR for the $4\pi$-periodic Josephson effect. This is a CPR of the form $J(\phi) \propto C \, |\sin(\phi/2)|$, describing a unidirectional supercurrent with the chirality $C= \pm 1$. Its non-analytic dependence on the Josephson phase difference $\phi$ translates into the $4\pi$-periodic CPR $J(\phi) \propto \sin(\phi/2)$. The proposal requires a spin-polarized topological Josephson junction which is modeled here as a short link between spin-split superconducting channels at the edge of a two-dimensional topological insulator. In this case, $C$ coincides with the Chern number of the occupied spin band of the topological insulator. The paper details three scenarios of achieving a chiral CPR: By only Zeeman-like splitting, by Zeeman splitting combined with bias currents, and by an external out-of-plane magnetic field.