# Fractional Topological Superconductivity and Parafermion Corner States

## TL;DR

This paper proposes a system of coupled Rashba nanowires with strong interactions that can host fractional topological superconductivity and parafermion corner states, including Majorana states as a special case.

## Contribution

It introduces a novel setup of layered nanowires with phase difference and interactions that stabilizes fractional topological phases with parafermion edge and corner states.

## Key findings

- Strong electron interactions stabilize helical topological superconductivity.
- Zero-energy parafermion corner states are realized in the system.
- Majorana corner states appear in the non-interacting limit.

## Abstract

We consider a system of weakly coupled Rashba nanowires in the strong spin-orbit interaction (SOI) regime. The nanowires are arranged into two tunnel-coupled layers proximitized by a top and bottom superconductor such that the superconducting phase difference between them is $\pi$. We show that in such a system strong electron-electron interactions can stabilize a helical topological superconducting phase hosting Kramers partners of $\mathbb{Z}_{2m}$ parafermion edge modes, where $m$ is an odd integer determined by the position of the chemical potential. Furthermore, upon turning on a weak in-plane magnetic field, the system is driven into a second-order topological superconducting phase hosting zero-energy $\mathbb{Z}_{2m}$ parafermion bound states localized at two opposite corners of a rectangular sample. As a special case, zero-energy Majorana corner states emerge in the non-interacting limit $m=1$, where the chemical potential is tuned to the SOI energy of the single nanowires.