Relation between symmetry breaking and the anomalous Josephson effect

We report a numerical study and symmetry analysis of the anomalous Josephson current in junctions coupled with a two-dimensional electron gas (2DEG). Taking the 2DEG with the coexistence of spin-orbit coupling (SOI) and Zeeman field as an example, we determine the symmetry criterion for the appearing of an anomalous supercurrent at zero phase difference. When the Zeeman field is unsuitably oriented with SOI, the system possesses some symmetries which make the anomalous supercurrent zero and these symmetries are identified here. The symmetry criterion for the anomalous Josephson effect found here is general and suitable for any other junctions contacted with two conventional s-wave superconductors, for example, the superconductor-(S-) ferromagnet (F) hybrid system and the S/F/S junction on the surface of topological insulator.