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

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.

Figure 1

Alternating superconducting and Mott insulating sections ($N=2$) with periodic boundary conditions. The phase fields ${\theta }_i$ (${\varphi }_i$) are pinned in the $i\text{th}$ blue superconducting (green Mott insulating) region. Bound states ${\chi }_i$ (stars) emerge at the interfaces.

Figure 2

A 2D topological insulator (green) with movable edge states and a set of gates for locally switching the proximity effect on (dark blue) and off (light blue) allow the implementation of the braiding protocol (17) for the bound states ${\chi }_i$. In step (a), the bound states ${\chi }_1$ and ${\chi }_2$ are coupled. Deforming the edge states also makes it possible to couple ${\chi }_2$ and ${\chi }_3$ or ${\chi }_2$ and ${\chi }_4$, as shown in (b) and (c), respectively.