Time-reversal invariant parafermions in interacting Rashba nanowires

We propose a scheme to generate pairs of time-reversal invariant parafermions. Our setup consists of two quantum wires with Rashba spin-orbit interactions coupled to an $s$-wave superconductor, in the presence of electron-electron interactions. The zero-energy bound states localized at the wire ends arise from the interplay between two types of proximity-induced superconductivity: the usual intrawire superconductivity and the interwire superconductivity due to crossed Andreev reflections. If the latter dominates, which is the case for strong electron-electron interactions, the system supports Kramers pair of parafermions. Moreover, the scheme can be extended to a two-dimensional sea of time-reversal invariant parafermions.

Figure 1

Sketch of two Rashba QWs (yellow strips) coupled to an $s$-wave superconductor (blue strip). The SOI field points in positive (negative) direction, say, along the $z$ azis for the upper (lower) QW, $\tau =1$ ($\tau =\overline{1}$). The intrawire proximity-induced superconductivity of strength ${\Delta }_{\tau }$ corresponds to a Cooper pair (pair of green dots) tunneling as a whole into the $\tau$ wire. The interwire proximity-induced superconductivity of strength ${\Delta }_c$ corresponds to crossed Andreev reflection into both QWs, which dominates for strong electron-electron interaction assumed here.

Figure 2

The spectrum of two QWs with positive (negative) Rashba SOI for $\tau =1$ ($\tau =\overline{1}$). The solid (dashed) lines correspond to electrons (holes). The chemical potential $\mu$ is tuned to the crossing point between spin up (blue) and spin down (red). The superconductivity couples states with opposite momenta and opposite spins belonging to the same $\tau$ wire (${\Delta }_{\tau }$) and belonging to different wires (${\Delta }_c$). The spectrum is gapless at $k=0$ for ${\Delta }_c^2={\Delta }_1{\Delta }_{\overline{1}}$, marking the topological phase transition that separates the topological phase with two localized midgap bound states at each wire end from the trivial phase without them.

Figure 3

The spectrum of two QWs with positive Rashba SOI in both QWs. The solid (dashed) lines correspond to electrons (holes). The chemical potential ${\mu }_{\tau }$ is tuned to the crossing point between spin up (blue) and spin down (red). The superconductivity couples states with opposite momenta and opposite spins belonging to the same $\tau$ wire (${\Delta }_{\tau }$) and belonging to different wires (${\Delta }_c$). The spectrum is gapless at $k=0$ for ${\Delta }_c^2={\Delta }_1{\Delta }_{\overline{1}}$, marking the topological phase transition that separates the topological phase with two localized midgap bound states at each wire end from the trivial phase without them.

Figure 4

The momentum-conserving scattering events corresponding to (a) ${\mathcal{H}}_c^{ee}$ and (b) ${\mathcal{H}}_{s,\tau }^{ee}$ for the chemical potential ${\mu }_{1/3,\tau }=E_{\text{so},\tau }/9$ with associated Fermi wave vectors $\pm k_{\text{so}}(1\pm 1/3)$. See the caption of Fig. 2 for notations.

Figure 5

Two-dimensional system of parafermions consisting of an array of coupled QWs with proximity-induced interwire and intrawire superconductivity, see Fig. 1 in the main part. The transition between the interwire-pairing-dominant phase (${\mu }_{1/3}$) and the intrawire-pairing-dominant phase (${\mu }_o$) is controlled by electrical gates (green slabs). Parafermions are formed initially at the boundaries between these two phases. The tunneling $t$ between two neighboring QWs not separated by a superconductor results in deconfinement [36] and in a sea of time-reversal invariant parafermions.