$8\pi$-periodic Josephson effects in a quantum dot / quantum spin-Hall josephson junction system

Josephson junctions made of conventional $s$-wave superconductors display $2\pi$ periodicity. On the other hand, $4\pi$-periodic fractional Josephson effect is known to be a characteristic signature of topological superconductors and Majorana fermions [1]. Zhang and Kane have shown that Josephson junctions made of topological superconductors are $8\pi$-periodic if interaction is used to avoid dissipation [2]. Here we present a general argument for how time-reversal symmetry and $Z_2$ non-trivial topology constrains the Josephson periodicity to be $8\pi$. We then illustrate this through a microscopic model of a quantum dot in a quantum spin-hall Josephson junction.