Superconductivity and fermionic dissipation in quantum Hall edges

Proximity-induced superconductivity in fractional quantum Hall edges is a prerequisite to proposed realizations of parafermion zero-modes. A recent experimental work [G\"{u}l et al., Phys. Rev. X 12, 021057 (2022)] provided evidence for such coupling, in the form of a crossed Andreev reflection signal, in which electrons enter a superconductor from one chiral mode and are reflected as holes to another, counter-propagating chiral mode. Remarkably, while the probability for crossed Andreev reflection was small, it was stronger for $\nu=1/3$ fractional quantum Hall edges than for integer ones. We theoretically explain these findings, including the relative strengths of the signals in the two cases and their qualitatively different temperature dependencies. An essential part of our model is the coupling of the edge modes to normal states in the cores of Abrikosov vortices induced by the magnetic field, which provide a fermionic bath. We find that the stronger crossed Andreev reflection in the fractional case originates from the suppression of electronic tunneling between the fermionic bath and the fractional quantum Hall edges. Our theory shows that the mere observation of crossed Andreev reflection signal does not necessarily imply the presence of localized parafermion zero-modes, and suggests ways to identify their presence from the behavior of this signal in the low temperature regime.