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 ξ). We calculate the maximum (critical) current I(c) that can flow without dissipation along a single edge, going beyond the short-junction restriction L << ξ of earlier work, and find a dependence on the fermion parity of the ground state when L becomes larger than ξ. 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 π-periodic Josephson effect.