Fermion-Parity Anomaly of the Critical Supercurrent in the Quantum Spin-Hall Effect

The helical edge state of a quantum spin-Hall insulator can carry a supercurrent in equilibrium between two superconducting electrodes (separation L, coherence length xi). We calculate the maximum (critical) current I-c that can flow without dissipation along a single edge, going beyond the short-junction restriction L << xi of earlier work, and find a dependence on the fermion parity of the ground state when L becomes larger than xi. Fermion-parity conservation doubles the critical current in the low-temperature, long-junction limit, while for a short junction I-c is the same with or without parity constraints. This provides a phase-insensitive, dc signature of the 4 pi-periodic Josephson effect. DOI: 10.1103/PhysRevLett.110.017003