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$\mathbb{Z}_4$ parafermions can be realized in a strongly interacting quantumspin Hall Josephson junction or in a spin Hall Josephson junction stronglycoupled to an impurity spin. In this paper we study a system that has bothfeatures, but with weak (repulsive) interactions and a weakly coupled spin. Weshow that for a strongly anisotropic exchange interaction, at low temperaturesthe system enters a strong coupling limit in which it hosts two $\mathbb{Z}_4$parafermions, characterizing a fourfold degeneracy of the ground state. Weconstruct the parafermion operators explicitly, and show that they facilitatefractional $e/2$ charge tunneling across the junction. The dependence of theeffective low-energy spectrum on the superconducting phase difference revealsan $8\pi$ periodicity of the supercurrent.

$\mathbb{Z}_4$ Parafermions in an interacting quantum spin Hall Josephson junction coupled to an impurity spin