Floquet Majorana fermions and parafermions in driven Rashba nanowires

We study a periodically driven nanowire with Rashba-like conduction and valence bands in the presence of a magnetic field. We identify topological regimes in which the noninteracting system hosts zero-energy bound states. We further investigate the effect of strong electron-electron interactions that give rise to parafermion zero energy modes hosted at the nanowire ends. The first setup we consider allows for topological phases by applying only static magnetic fields without the need of superconductivity. The second setup involves both superconductivity and time-dependent magnetic fields and supports topological phases without fine tuning of the chemical potential. Promising candidate materials are graphene nanoribbons due to their intrinsic particle-hole symmetry.

Figure 1

One-dimensional Rashba nanowire (orange cylinder) with the SOI vector $\mathbf{\alpha }$ pointing in the $z$ direction is aligned along the $x$ direction. The magnetic field $\mathbf{B}$ is chosen in the $x$ direction. A driving electric ac field $\mathcal{E}\left(t\right)$ of resonant frequency $\omega$, resulting in the coupling $t_F$ between bands, is applied in the transverse $z$ direction. In the topological regime ${\mathrm{\Delta }}_Z>t_F>0$, the system hosts zero energy bound states (blue curves) at each wire end.

Figure 2

The spectrum of the nanowire with band gap ${\mathrm{\Delta }}_g-2E_{so}$ separating valence $\left(\eta =\overline{1}\right)$ and conduction $\left(\eta =1\right)$ bands. The index $\sigma =1$ $\left(\sigma =\overline{1}\right)$ refers to the spin up (spin down) band shown in red (blue) color. The right- $\left(R_{\eta \sigma }\right)$ and left-mover $\left(L_{\eta \sigma }\right)$ fields are introduced close to the Fermi level. (a) The chemical potential $\mu$ is tuned to the SOI energy $E_{so}$ and the driving frequency $\hslash \omega ={\mathrm{\Delta }}_g$ results in resonant scattering between the two bands. If the Zeeman energy ${\mathrm{\Delta }}_Z$ exceeds the Floquet amplitude $t_F$, the system hosts zero energy modes. (b) To obtain parafermions, we tune $\mu$ to $E_{so}/9$ and readjust $\omega$ to $\hslash \omega ={\mathrm{\Delta }}_g-16E_{so}/9$. The leading term in the magnetic field $H_Z^{ee}$ (green arrows) involves two backscattering events and opens a partial gap in the spectrum only in the presence of strong electron-electron interactions. The driving term $H_d^{ee}$ (yellow arrows) commutes with $H_Z^{ee}$ and can be ordered simultaneously in the RG sense, leading to a fully gapped spectrum.

Figure 3

Floquet spectrum [see Eq. (4)] of Rashba nanowire driven by an electric field for ${\mathrm{\Delta }}_Z/E_{so}=0.6$ and $t_F/E_{so}=0.3$. The topological gap ${\mathrm{\Delta }}_0=2|{\mathrm{\Delta }}_Z-t_F|$ defined at $k=0$ closes for ${\mathrm{\Delta }}_Z=t_F$ signaling the topological phase transition. If $t_F>{\mathrm{\Delta }}_Z$, there is one zero energy bound state localized at each end of the nanowire.

Figure 4

The spectrum of the Rashba nanowire used in the second model. The index $\eta =1(\eta =\overline{1})$ is for upper and lower band, $\sigma =1(\sigma =\overline{1})$ for spin up (red) [down (blue)]. The chemical potentials ${\mu }_{1,\overline{1}}$ and the driving frequency $\omega$ of field $B\left(t\right)$ are chosen in such a way that the smallest Fermi wave vectors of two effective subbands coincide.

Figure 5

The spectrum of the Rashba nanowire modified by the proximity gap ${\mathrm{\Delta }}_{sc}$ and time-dependent magnetic field $B\left(t\right)$ in the strong electron-electron interaction regime. The index $\eta =1(\eta =\overline{1})$ is for upper (lower) band, $\sigma =1(\sigma =\overline{1})$ for spin up (red) [down (blue)]. The leading term in driving $H_F^{ee}$ (yellow arrows) involves two momentum-conserving backscattering terms. The superconductivity term $H_{sc}^{ee}$ (green arrows) commutes with $H_F^{ee}$ (orange arrows), therefore they can lead to simultaneous ordering of the corresponding bosonic fields, resulting in the fully gapped energy spectrum with zero-energy parafermion bound states localized at each wire end.