Clock model and parafermions in Rashba nanowires

We consider a semiconducting nanowire with Rashba spin-orbit interaction subjected to a magnetic field and in the presence of strong electron-electron interactions. When the ratio between Fermi and Rashba momenta is tuned to 1/2, two competing resonant multiparticle scattering processes are present simultaneously and the interplay between them brings the system into a gapless critical parafermion phase. This critical phase is described by a self-dual sine-Gordon model, which we are able to map explicitly onto the low-energy sector of the Z4 parafermion clock chain model. Finally, we show that by alternating regions in which only one of these two processes is present one can generate localized zero-energy parafermion bound states.