# Odd-frequency superconductivity induced by non-magnetic impurities

## TL;DR

This paper investigates the conditions under which odd-frequency superconductivity can be induced by non-magnetic impurities in a one-dimensional s-wave superconductor, revealing that not all impurities break translation symmetry to produce odd-frequency pairing.

## Contribution

It demonstrates that translation symmetry breaking alone does not induce odd-frequency pairing and distinguishes two different mechanisms by which specific impurities generate odd-frequency amplitudes.

## Key findings

- Chemical potential perturbations do not induce odd-frequency pairing.
- Quantum impurities induce odd-frequency pairing via quasiparticle tunneling.
- Gap perturbations lead to localized odd-frequency amplitudes.

## Abstract

A growing body of literature suggests that odd-frequency superconducting pair amplitudes can be generated in normal metal-superconductor junctions. The emergence of odd-frequency pairing in these systems is often attributed to the breaking of translation invariance. In this work, we study the pair symmetry of a one-dimensional $s$-wave superconductor in the presence of a single non-magnetic impurity and demonstrate that translation symmetry breaking is not sufficient for inducing odd-frequency pairing. We consider three kinds of impurities: a local perturbation of the chemical potential, an impurity possessing a quantum energy level, and a local perturbation of the superconducting gap. Surprisingly, we find local perturbations of the chemical potential do not induce any odd-frequency pairing, despite the fact that they break translation invariance. Moreover, although odd-frequency can be induced by both the quantum impurity and the perturbation of the gap, we find these odd-frequency amplitudes emerge from entirely different kinds of scattering processes. The quantum impurity generates odd-frequency pairs by allowing one of the quasiparticles belonging to an equal-time Cooper pair to tunnel onto the impurity state and then back to the superconductor, giving rise to odd-frequency amplitudes with a temporal broadening inversely proportional to the energy level of the impurity. In contrast to this, the perturbation of the gap leads to odd-frequency pairing by "gluing-together" normal state quasiparticles from different points in space and time, leading to odd-frequency amplitudes which are very localized in the time domain.