Fractional topological superconductivity and parafermion corner states

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 noninteracting limit $m=1$, where the chemical potential is tuned to the SOI energy of the single nanowires.

Figure 1

The setup consists of two layers of coupled Rashba nanowires where the index $\tau =1$ ($\tau =\overline{1}$) denotes the upper (lower) layer. The strength of the Rashba SOI associated with propagation along the $x$ direction is given by $\alpha$ ($-\alpha$) for the upper (lower) layer. Both layers are brought into proximity to an $s$-wave bulk superconductor such that there is a phase difference of $\pi$ between them. In addition, the two layers are strongly coupled by interlayer tunneling of strength $\mathrm{\Gamma }$. Neighboring nanowires of the same layer are weakly coupled via a spin-conserving hopping term of strength $t_z$, via a spin-flip hopping term of strength $\beta$ ($-\beta$) associated with Rashba SOI along the $z$ direction, and via a crossed-Andreev superconducting term of strength ${\mathrm{\Delta }}_c$ ($-{\mathrm{\Delta }}_c$), where the last two terms are again of opposite sign for the two layers. Finally, a weak in-plane magnetic field of strength $B$ is applied. We show that in specific regions of parameter space, this system hosts two zero-energy corner states (here denoted by ${\gamma }_1$ and ${\gamma }_2$) at two opposite corners of a rectangular sample. In the noninteracting case, these states are Majorana zero modes, whereas strong electron-electron interactions lead to ${\mathbb{Z}}_{2m}$ parafermion corner states, where $m$ is an odd integer depending on the position of the chemical potential.

Figure 2

Probability density of low-energy states of $H$ [see Eqs. (1, 2, 3, 4)] obtained numerically. (a) For ${\mathrm{\Delta }}_Z=0$, the system is a helical TSC with Kramers partners of gapless Majorana modes propagating along the edges. (b) In the presence of a small in-plane magnetic field, ${\mathrm{\Delta }}_Z>0$, we find Majorana bound states localized at two opposite corners of the system. The inset shows the spectrum confirming that these two states (red dots) are indeed at zero energy. The numerical parameters are $N=100, \mu =0, k_{so}L=85, \mathrm{\Gamma }/E_{so}\approx 0.6, \mathrm{\Delta }/E_{so}\approx 0.55, t_z/E_{so}\approx 0.01, \beta /E_{so}\approx 0.28, {\mathrm{\Delta }}_c/E_{so}\approx 0.11$, and, in (b), ${\mathrm{\Delta }}_Z/E_{so}\approx 0.07$ and $\varphi =-\pi /16$. We note that the found topological phases are stable against disorder and do not rely on spatial symmetries.