Criticality in the Crossed Andreev Reflection of a Quantum Hall Edge

We develop a theory of the nonlocal transport of two counterpropagating ν=1 quantum Hall edges coupled via a narrow disordered superconductor. Contrary to the predictions developed for the clean case, the edge states proximitized in this way do not turn into a topological superconductor. Instead, they are naturally tuned to the critical point between trivial and topological phases. This occurs due to the competition between tunneling processes with and without particle-hole conversion. The critical conductance is a random, sample-specific quantity with a zero average and unusual bias dependence. The negative values of conductance are relatively stable against variations of the carrier density, which may make the critical state appear as a topological one. Our results offer an interpretation of recent experiments [G.-H. Lee et al., Nat. Phys. 13, 693 (2017)NPAHAX1745-247310.1038/nphys4084; O. Gül et al., Phys. Rev. X 12, 021057 (2022)PRXHAE2160-330810.1103/PhysRevX.12.021057].