Spontaneous fractional Josephson current in parafermion junctions

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, ν=1/m, m∈2 Z +1, this leads to an electrical control of either Majorana (m=1) or parafermion (m≠1) zero modes.