$6\pi$ 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 $\mathrm{\Delta }$) 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 $\mathrm{\Delta }\gg T_K$, 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 $\mathrm{\Delta }\ll T_K$, we find a $6\pi$-periodic Josephson relation for three leads, with critical current $I_c\approx e{\mathrm{\Delta }}^2\mathrm{/}\mathrm{\hslash }{T}_K$, corresponding to the transfer of fractionalized charges $e^{\ast }=2e\mathrm{/}3$.