  We study proximity coupling between a superconductor and counter-propagating
gapless modes arising on the edges of Abelian fractional quantum Hall liquids
with filling fraction $\nu=1/m$ (with $m$ an odd integer). This setup can be
utilized to create non-Abelian parafermion zero-modes if the coupling to the
superconductor opens an energy gap in the counter-propagating modes. However,
when the coupling to the superconductor is weak an energy gap is opened only in
the presence of sufficiently strong attractive interactions between the edge
modes, which do not commonly occur in solid state experimental realizations. We
therefore investigate the possibility of obtaining a gapped phase by increasing
the strength of the proximity coupling to the superconductor. To this end, we
use an effective wire construction model for the quantum Hall liquid and employ
renormalization group methods to obtain the phase diagram of the system.
Surprisingly, at strong proximity coupling we find a gapped phase which is
stabilized for sufficiently strong repulsive interactions in the bulk of the
quantum Hall fluids. We furthermore identify a duality transformation that maps
between the weak coupling and strong coupling regimes, and use it to show that
the gapped phases in both regimes are continuously connected through an
intermediate proximity coupling regime.
