Topological Kondo effect

We review recent theoretical progress in the understanding of the topological Kondo effect in Coulomb-blockaded Majorana devices and generalizations thereof. The central ingredient in Majorana devices is the so-called Majorana box which encodes a spin-1/2 degree of freedom in the Majorana subspace that can be addressed by electron cotunneling processes. In particular, after explaining the basic physics of the topological Kondo effect in a Majorana box connected to a set of normal-conducting leads, we discuss the Josephson current-phase relation in a superconducting multi-terminal setup where the central junction is again formed by a Majorana box but the leads are phase-biased superconductors. For large Kondo temperature, one finds that the competition between two-channel Kondo physics and the gap opening in the leads results in a 6π periodicity of the current-phase relation. This periodicity is due to the fractionalized charge excitations characterizing the non-Fermi liquid two-channel Kondo fixed point. We also explore generalizations of this Majorana-based topological Kondo setup to platforms hosting parafermionic excitations.