Suppression of odd-frequency superconducting pairing by phase-disorder in a nanowire coupled to Majorana zero modes

Majorana zero modes (MZMs) exhibit pure odd-frequency superconducting pairing correlations. Recently, it was reported that a robust odd-frequency superconducting pairing can be induced in a spin-polarized wire due to an array of MZMs, where their coupling to the wire was considered real.

Considering the physically more realistic case of complex coupling strengths that generically appear for varying system parameters, we demonstrate that all superconducting correlations, including odd-frequency, suffer a considerable suppression due to phase-disorder averaging. In the first part we, evaluate the $T$-matrix within the second Born approximation, and demonstrate exponential suppression of the electron-hole pairing with phase-disorder strength. In the second part, we perform numerical calculations on a tight-binding lattice model and study the effects of phase-disorder averaging on the local density of states and superconducting correlations.

We attribute the suppression of superconducting correlations in the wire to the filling of the energy gap by in-gap Andreev Bound States appearing due to phase-matching conditions between spatially-separated MZMs. Our work helps clarify the conditions favorable for practical realization of the proposed odd-frequency superconducting system.