Dissipative response of a topological Josephson junction at the critical point

A semiconducting nanowire proximitized by a superconductor is emerging as a building block of a topological qubit. Magnetic field applied to a proximitized nanowire drives it through the critical point into a topological state. We investigate signatures of this quantum phase transition in the dissipative response of a nanowire Josephson junction. The gap between the ground and excited states vanishes at the transition point. We find, that the low-frequency dissipation may remain weak despite the gap closing; this is markedly different from the conventional transition between the superconducting and normal states. In the absence of phase bias across the junction, the dissipative component of the admittance scales as σ(ω) ∼ ω². At a finite phase bias φ across the junction, σ(ω) ∼ φ² is frequency-independent in the interval ω ≤ Δ (here Δ is the value of the proximity-induced gap at zero field). We establish the complete scaling functions for the admittance in the vicinity of the transition. In the presence of a finite gap (i.e., upon detuning from the critical point) the scaling function for σ(ω) at a finite φ agrees with conventional Mattis-Bardeen formula. However, it is modified substantially and characterized by a much stronger frequency dependence if φ = 0.