Strong-coupling topological Josephson effect in quantum wires

TL;DR

This paper analyzes the Josephson effect in quantum wires with Majorana modes, revealing a transition at strong coupling where energy levels merge, leading to a $4\,\pi$ periodic current characteristic of topological superconductivity.

Contribution

It provides an exact solution for the Josephson effect in quantum wires with Majorana modes, including the behavior at strong coupling and the resulting degeneracies, beyond perturbative approaches.

Findings

Energy gap closes at critical coupling $g_c=\sqrt{2}w$

Exact energy levels show merging of three Bogoliubov states at critical point

Josephson current exhibits $4\pi$ periodicity characteristic of topological effects

Abstract

We investigate the Josephson effect for a setup with two lattice quantum wires featuring Majorana zero energy boundary modes at the tunnel junction. In the weak-coupling, the exact solution reproduces the perturbative result for the energy containing a contribution $\sim \pm\cos(\phi/2)$ relative to the tunneling of paired Majorana fermions. As the tunnel amplitude $g$ grows relative to the hopping amplitude $w$, the gap between the energy levels gradually diminishes until it closes completely at the critical value $g_c=\sqrt{2}w$. At this point the Josephson energies have the principal values $E_{m\sigma}=2\sigma\sqrt{2}w\cos[\phi/6+2\pi (m-1)/3]$, where $m=-1,0,1$ and $\sigma=\pm 1$, a result not following from perturbation theory. It represents a transparent regime where three Bogoliubov states merge, leading to additional degeneracies of the topologically nontrivial ground state with odd number of Majorana fermions at the end of each wire. We also obtain the exact tunnel currents for a fixed parity of the eigenstates. The Josephson current shows the characteristic $4\pi$ periodicity expected for a topological Josephson effect. We discuss the additional features of the current associated with a closure of the energy gap between the energy levels.