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 4pi-periodic Josephson effect. 
C. W. J. Beenakker, D. I. Pikulin, T. Hyart, H. Schomerus, and J. P.   Dahlhaus
cond-mat.supr-con cond-mat.mes-hall