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, ν=1/m, m∈2Z+1, this leads to an electrical control of either Majorana (m=1) or parafermion (m≠1) zero modes. We also study edge reconstruction in these geometries.