Josephson junction of minimally twisted bilayer graphene

We theoretically investigate the transport properties of a Josephson junction composed of superconductor/minimally twisted bilayer graphene/superconductor structures. In the presence of an out-of-plane electric field, the low-energy physics is best described by a network of domain-wall states. Depending on system parameters, they lead to the emergence of zigzag or pseudo-Landau-level modes with distinct transport characteristics. Specifically, we find zigzag modes feature linear dispersion of Andreev bound states, resulting in a $4\pi$-periodic Josephson current. In contrast, pseudo-Landau-level modes exhibit flat Andreev bound states and, consequently, a vanishing bulk Josephson current. Interestingly, edge states can give rise to $4\pi$-periodic Josephson response in the pseudo-Landau-level regime. We also discuss experimental signatures of such responses.