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

TL;DR

This paper shows that phase disorder in Majorana zero mode-coupled nanowires suppresses odd-frequency superconductivity, highlighting the need for precise parameter control in realizing such exotic phases.

Contribution

It demonstrates analytically and numerically that phase disorder significantly diminishes odd-frequency correlations in Majorana-coupled nanowires, setting constraints for experimental realization.

Findings

Phase disorder suppresses odd-frequency superconducting correlations.

Superconducting peaks are smeared by bound states due to phase mismatch.

Full parameter control is essential for realizing bulk odd-frequency superconductivity.

Abstract

Odd-frequency superconductivity is an exotic phase of matter in which Cooper pairing between electrons is entirely dynamical in nature. Majorana zero modes exhibit pure odd-frequency superconducting correlations due to their specific properties. Thus, by tunnel-coupling an array of Majorana zero modes to a spin-polarized wire, it is in principle possible to engineer a bulk one-dimensional odd-frequency spinless $s$-wave superconductor. We here point out that each tunnel coupling element, being dependent on a large number of material-specific parameters, is generically complex with sample variability in both its magnitude and phase. Using this, we demonstrate that, upon averaging over phase-disorder, the induced superconducting, including odd-frequency, correlations in the spin-polarized wire are significantly suppressed. We perform both a rigorous analytical evaluation of the disorder-averaged $T$-matrix in the wire, as well as numerical calculations based on a tight-binding model, and find that the anomalous, i.e. superconducting, part of the $T$-matrix is highly suppressed with phase disorder. We also demonstrate that this suppression is concurrent with the filling of the single-particle excitation gap by smearing the near-zero frequency peaks, due to formation of bound states that satisfy phase-matching conditions between spatially separated Majorana zero modes. Our results convey important constraints on the parameter control needed in practical realizations of Majorana zero mode structures and suggest that the achievement of a bulk 1D odd-$\omega$ superconductivity from MZMs demand full control of the system parameters.