Strong-coupling topological Josephson effect in quantum wires

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 $\sim \pm \cos (\varphi /2)$ 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 $E_{m\sigma }=2\sigma \sqrt{2}w\cos [\varphi /6+2\pi (m-1)/3]$, 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.

20 pages, 6 figures; v3: Extensively revised version; added new figures and corrected the calculation of the current by taking the fixed parity ensemble into account; calculation of the Josephson energy levels remains unchanged

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Cited by in corpus (9)

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• Fermion-parity anomaly of the critical supercurrent in the quantum spin-Hall effect
• Time-Reversal-Invariant $Z_4$ Fractional Josephson Effect
• Fractional Josephson Effect with and without Majorana Zero Modes
• Time-reversal anomaly and Josephson effect in time-reversal invariant topological superconductors
• Fractional Josephson effect in a quadruple quantum dot
• Josephson junction on one edge of a two dimensional topological insulator affected by magnetic impurity
• Non-equilibrium $8\pi$ Josephson Effect in Atomic Kitaev Wires
• Superconductor/normal-metal/superconductor junction of topological superconductors revisited: Fractional Josephson current, fermion parity, and oscillating wavefunctions