$6\pi$ Josephson effect in Majorana box devices

We study Majorana devices featuring a competition between superconductivity and multi-channel Kondo physics. Our proposal extends previous work on single-channel Kondo systems to a topologically nontrivial setting of non-Fermi liquid type, where topological superconductor wires (with gap $\Delta$) represent leads tunnel-coupled to a Coulomb-blockaded Majorana box. On the box, a spin degree of freedom with Kondo temperature $T_K$ is nonlocally defined in terms of Majorana states. For $\Delta\gg T_K$, the destruction of Kondo screening by superconductivity implies a $4\pi$-periodic Josephson current-phase relation. Using a strong-coupling analysis in the opposite regime $\Delta \ll T_K$, we find a $6\pi$-periodic Josephson relation for three leads, with critical current $I_c\approx e\Delta^2/ \hbar T_K$, corresponding to the transfer of fractionalized charges $e^*=2e/3$.