Spontaneous fractional Josephson current from parafermions

We study a parafermion Josephson junction (JJ) comprising a pair of counter-propagating edge modes of two quantum Hall (QH) 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 counter-propagating chiral edges at the Josephson junction, can act as a source of spontaneous phase bias. For the Laughlin filling fractions, $\nu =1\mathrm{/}m,m\in 2\mathbb{Z}+1$, this leads to an electrical control of either Majorana $(m=1)$ or parafermion $(m\mathrm{\ne }1)$ zero modes.