Fractional spin Josephson effect and electrically controlled magnetization in quantum spin Hall edges

We explore a spin Josephson effect in a system of two ferromagnets coupled by a tunnel junction formed of two-dimensional (2D) time-reversal invariant topological insulators. In analogy with the more commonly studied instance of the Josephson effect for charge in superconductors, we investigate properties of the phase-coherent spin current resulting from the misalignment of the in-plane magnetization angles of the two ferromagnets. We show that the topological insulating barrier offers the exciting prospect of hosting a fractional spin Josephson effect mediated by bound states at the ferromagnet-topological insulator interface. We provide multiple perspectives to understand the $4\ensuremath{\pi}$ periodic nature of this effect. We discuss several measurable consequences, such as the generation of a transverse voltage signal which allows for purely electrical measurements, an inverse of this effect where an applied voltage gives rise to a transverse spin current, and a fractional ac spin Josephson effect.