Parafermions in an Interacting Nanowire Bundle

We propose a scheme to induce ${\mathbb{Z}}_3$ parafermion modes, exotic zero-energy bound states that possess non-Abelian statistics. We consider a minimal setup consisting of a bundle of four tunnel coupled nanowires hosting spinless electrons that interact strongly with each other. The hallmark of our setup is that it relies only on simple one-dimensional wires, a uniform magnetic field, and strong interactions, but does not require the presence of superconductivity or exotic quantum Hall phases.

Figure 1

Bundle of four one-dimensional nanowires (green cylinders) hosting spinless electrons and aligned in $x$ direction. The upper (dark green) and lower (light green) wires are tunnel coupled to each other by the hopping amplitude $t_z$. Similarly, the two upper (lower) wires are tunnel coupled by $t_y$. A uniform magnetic field $\mathbf{B}=B\widehat{z}$ (blue arrow) penetrating the bundle in $z$ direction results in the opening of Peierls gap. For strong electron-electron interactions, zero-energy PFs emerge at the ends (yellow spheres).

Figure 2

The energy spectrum of the four wires. Each wire is labeled by two indices ($\tau ,\sigma$). The two upper (lower) wires with $\sigma =1$ ($\sigma =\overline{1}$) have positive (negative) mass. The hopping of strength $t_z$ between upper and lower wires (red dashed arrows) is momentum conserving. In contrast, the hopping of strength $t_y$ between left and right wires (blue solid arrows) results in scattering between two different momenta in the presence of a magnetic field $B$. In particular, at $B=B_1$ the momentum transfer is equal to $2k_F$, where $k_F$ is the Fermi momentum defined by the chemical potential $\mu$. Such resonant scattering between right and left mover fields leads to the opening of a Peierls gap at the Fermi level. In addition, if $t_y>t_z$, there are zero-energy bound states localized at the ends of the system.