Onset of the 4π-periodic Josephson effect and dissipative response at the Majorana phase transition in a semiconducting nanowire junction

We study the appearance of the 4π-periodicity of the Josephson energy in a single-channel semiconducting nanowire junction much shorter than the coherence length. We compute analytically the Josephson energy of the junction as a function of magnetic field, taking into account the contributions of both discrete and continuous spectrum. We show that, in the limit of perfect transmission, the spectrum of the junction is gapless for a finite range of phase difference and a finite range of magnetic fields on the topological side of the transition. We also study the finite-frequency admittance of the junction, paying particular attention to the critical regime. We show that the expected critical behavior near the phase transition only appears in the current-current correlation function at sub-leading order in the ratio of the induced superconducting gap to the spin-orbit energy, a fact that can be understood within the critical theory of a helical, neutral Majorana mode.