Anomalous Josephson effect of s-wave pairing states in chiral double layers

We consider s-wave pairing in a double layer of two chiral metals due to Coulomb interaction, and study the Josephson effect near a domain wall, where the sign of the order parameter jumps. The domain wall creates two zero energy modes, whose superposition is associated with a current that flows in different direction in the two layers. Assuming a toroidal geometry, the effective Josephson currents winds around the domain walls, whose winding number is determined by the coefficients of the superimposed zero energy modes. This result can be understood as a competition of the conventional Josephson current perpendicular and the edge current parallel to a domain wall in a double layer of two chiral metals. As a realization we suggest the surface of a ring-shaped topological insulator. The duality between electron-electron and electron-hole double layers indicates that this effect should also be observable in excitonic double layers.