Magneto-Josephson effects in junctions with Majorana bound states

We investigate 1D quantum systems that support Majorana bound states at interfaces between topologically distinct regions. In particular, we show that there exists a duality between particle-hole and spin degrees of freedom in certain spin-orbit-coupled 1D platforms such as topological insulator edges. This duality results in a spin analogue of previously explored `fractional Josephson effects'---that is, the spin current flowing across a magnetic junction exhibits 4\pi-periodicity in the relative magnetic field angle across the junction. Furthermore, the interplay between the particle-hole and spin degrees of freedom results in unconventional magneto-Josephson effects, such that the Josephson current is a function of the magnetic field orientation with periodicity 4\pi.