Many-terminal Majorana island: From topological to multichannel Kondo model

We study Kondo screening obtained by coupling Majorana bound states, located on a topological superconducting island, to interacting electronic reservoirs. At the charge degeneracy points of the island, we formulate an exact mapping onto the spin-$1/2$ multichannel Kondo effect. The coupling to Majorana fermions transforms the tunneling terms into effective fermionic bilinear contributions with a Luttinger parameter $K$ in the leads that is effectively doubled. For strong interactions $K=1/2$, the intermediate fixed point of the standard multichannel Kondo model is exactly recovered. It evolves with $K$ and connects to strong coupling in the noninteracting case $K=1$, with maximum conductance between the leads and robustness against channel asymmetries similarly to the topological Kondo effect. For a number of leads above four, there exists a window of Luttinger parameters in which a quantum phase transition can occur between the strong coupling topological Kondo state and the partially conducting multichannel Kondo state.

Figure 1

Sketch of the Majorana island. A superconducting box is connected through localized Majorana modes ${\gamma }_j$ to $M=5$ normal leads. The Majorana modes are typically realized as boundary bound states of topological nanowires deposited on the superconducting box, and therefore come in pairs. Only the Majorana modes coupled to electronic reservoirs are pictured here.

Figure 2

Phase diagram and flow for the single Majorana Kondo box far from charge degeneracy. The mobility is defined by $\mu =\frac{h{G}_{j,j}}{2e^2}\frac{M}{1-M}$. Solid line describes stable fixed points while pointed lines are unstable. The arrows depict the flow of the most relevant/less irrelevant operators of the corresponding fixed point.

Figure 3

Phase diagrams and flows for the single Majorana Kondo box at charge degeneracy. The mobility is defined by $\mu =\frac{h{G}_{j,j}}{2e^2}\frac{M}{1-M}$. The first graph describes the simple case of $M=3,4$ while the second is a schematic of what happens for $M\ge 5$.