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A semiconducting nanowire proximitized by an $s$-wave superconductor can betuned into a topological state by an applied magnetic field. This quantum phasetransition is marked by the emergence of Majorana zero modes at the ends of thewire. The fusion of Majorana modes at a junction between two nanowires resultsin a $4\pi$-periodic Josephson effect. We elucidate how the $4\pi$-periodicityarises across the topological phase transition in a highly-transparent shortnanowire junction. Owing to a high transmission coefficient, Majorana zeromodes coming from different wires are strongly coupled, with an energy scaleset by the proximity-induced, field-independent pairing potential. At the sametime, the topological spectral gap---defined by competition betweensuperconducting correlations and Zeeman splitting---becomes narrow in thevicinity of the transition point. The resulting hybridization of the fusedMajorana states with the spectral continuum strongly affects the electrondensity of states at the junction and its Josephson energy. We study themanifestations of this hybridization in the energy spectrum and phasedependence of the Josephson current. We pinpoint the experimentally observablesignatures of the topological phase transition, focusing on junctions with weakbackscattering.

Energy spectrum and current-phase relation of a nanowire Josephson junction close to the topological transition