Low-Temperature Phases of Dissipative Superconductor--Fractional Quantum Hall Junctions

Fractional quantum Hall (FQH) edges with proximity-induced superconductivity can host parafermionic zero modes (PZMs), a promising platform for topological quantum computing. Recent experiments measured the edge conductance due to crossed Andreev reflection (CAR) between counterpropagating FQH edges separated by a narrow superconducting finger. The observed conductance was small and non-quantized, with two distinct behaviors: some FQH states exhibit temperature-independent conductance, while in others, it grows as the temperature decreases. We show that these observations can be explained by dissipation to sub-gap modes in the superconductor. We find that temperature-independent non-quantized conductance occurs only in the presence of an integer mode strongly coupled to the dissipative channel. The most notable example of states hosting such modes are ν=(p±1)/p states.

In contrast, in most observed fractions (e.g., Jain sequence), a non-quantized plateau as a function of temperature is impossible. These findings agree with the experimental results, which show a plateau for the ν=2/3 and integer states but not for ν=1/3 and ν=2/5 states. In addition, we find an intermediate phase in which PZMs are not present, but the conductance is similar to the gapped phase, implying that CAR rates are not a definitive indication of PZMs' presence.