Signatures of the long-range phase transition in topological Josephson junctions

The extended Kitaev models spark a recent surge of interest due to its unprecedented topological phase with fractional invariants. In this extended model of topological superconductors, long-range pairing interactions can induce a pair of subgap massive edge modes (MEMs), posing novel challenges to the unambiguous detection of Majorana bound states (MBSs) sustained by the same model in the short-range limit. Here we investigate the effect of a power-law decayed long-range pairing on the dc and ac Josephson currents between two Majorana nanowires. By varying the magnetic field and long-range pairing, the nanowires can be driven into different phases hosting MEMs, MBSs, or trivial Andreev bound states (ABSs), which considerably modulate the Josephson current. For weakly linked Josephson junctions, we show that the MEMs emerging in the long-range phases can induce a zigzag profile in the dc current-phase relation featured by two sign reversals deviating from the point where the superconducting phase difference $\ensuremath{\phi}=\ensuremath{\pi}$. In contrast, the Josephson current indicates a sharp sign reversal at $\ensuremath{\phi}=\ensuremath{\pi}$ in the short-range phase that hosts MBSs, while a smooth sinelike dc Josephson current appears in the presence of ABSs. In the nonequilibrium junction biased by a voltage, we identify a $4\ensuremath{\pi}$ ac Josephson current in the short-range phases, indicating a MBSs-mediated tunneling process dominating the supercurrents. Differently, when the system enters MEMs-hosted long-range phases, the ordinary ac Josephson current with $2\ensuremath{\pi}$ periodicity can be restored. In addition, we find the ac Josephson current induced by MEMs is significantly suppressed under the strong long-range pairing interactions. These signatures can serve as probes to discriminate MBSs from MEMs and ABSs.