Chiral supercurrent through a quantum Hall weak link and its current-phase relation

We calculate properties of the supercurrent carried by edge states of a quantum Hall weak link. This ``chiral" supercurrent is qualitatively distinct from the usual supercurrents since it cannot be mediated by a single edge alone. Moreover, this supercurrent was previously shown to obey an unusual current-phase relation with the period $2 \phi_0=h/e$, which is twice the usual period of Josephson junctions. We show that ``chiral" nature of this supercurrent is robust to interactions to infinite order in perturbation theory. We compare our results with experimental findings of Amet et al [Science, 352(6288)] and find that, quantitative agreement in magnitude of the supercurrent can be attained by making reasonable but critical assumptions about the superconductor quantum Hall interface. We show that in the regime probed by this experiment, flux dependence of the chiral supercurrent is exponentially suppressed. This explains the paradoxical $ \phi_0=h/2e$ observed periodicity of the supercurrent since in this regime, flux dependent part of the edge state (chiral) supercurrent becomes smaller than the estimated value for residual (non-chiral) supercurrent carried by the bulk.