  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.

Mesoscale and Nanoscale Physics;Quantum Physics