Non-Abelian parafermions in time-reversal invariant interacting helical systems

TL;DR

This paper demonstrates that time-reversal invariant interacting helical systems can host protected $ ext{Z}_4$ parafermions with unique braiding properties, arising from the interplay of spin-orbit coupling and electron interactions.

Contribution

It introduces a novel realization of $ ext{Z}_4$ parafermions in topological insulators, showing their braiding and potential for topological quantum computation.

Findings

Bound states at interfaces are $ ext{Z}_4$ parafermions.

Josephson current exhibits $8 ext{pi}$ periodicity.

Bound states are fourfold degenerate and protected by time-reversal symmetry.

Abstract

The interplay between bulk spin-orbit coupling and electron-electron interactions produces umklapp scattering in the helical edge states of a two-dimensional topological insulator. If the chemical potential is at the Dirac point, umklapp scattering can open a gap in the edge state spectrum even if the system is time-reversal invariant. We determine the zero-energy bound states at the interfaces between a section of a helical liquid which is gapped out by the superconducting proximity effect and a section gapped out by umklapp scattering. We show that these interfaces pin charges which are multiples of $e/2$, giving rise to a Josephson current with $8\pi$ periodicity. Moreover, the bound states, which are protected by time-reversal symmetry, are fourfold degenerate and can be described as $\mathbb{Z}_4$ parafermions. We determine their braiding statistics and show how braiding can be implemented in topological insulator systems.