Parity Anomaly and Spin Transmutation in Quantum Spin Hall Josephson Junctions

We study the Josephson effect in a quantum spin Hall system coupled to a localized magnetic impurity. As a consequence of the fermion parity anomaly, the spin of the combined system of impurity and spin-Hall edge alternates between half-integer and integer values when the superconducting phase difference across the junction advances by 2π. This leads to characteristic differences in the splittings of the spin multiplets by exchange coupling and single-ion anisotropy at phase differences, for which time-reversal symmetry is preserved. We discuss the resulting 8π-periodic (or Z_{4}) fractional Josephson effect in the context of recent experiments.