Spontaneous fractional Josephson current in parafermion junctions

Abstract

We study a parafermion Josephson junction comprising a pair of counterpropagating edge modes of two quantum Hall systems, proximitized by an $s$-wave superconductor. We show that the difference between the lengths (which can be controlled by external gates) of the two counterpropagating chiral edges at the Josephson junction, can act as a source of spontaneous phase bias. For the Laughlin filling fractions, $\ensuremath{\nu}=1/m, m\ensuremath{\in}2\mathbb{Z}+1$, this leads to an electrical control of either Majorana ($m=1$) or parafermion ($m\ensuremath{\ne}1$) zero modes.