Parafermions in the Fractional Quantum Hall Spin Transitions

Parafermion zero modes are promising for universal topological quantum computation because of their richer non-Abelian braiding properties. However, physical systems that are predicted to host these exotic excitations are rare and difficult to realize in experiments. In this work, we show that parafermion zero modes can emerge in the spin transitions in the fractional quantum Hall regime. Exact diagonalization of the Hamiltonian in a disk and torus geometries demonstrates formation of counter-propagating edge states with different spin polarizations at a boundary between polarized and unpolarized ν=2/3 phases. By analytical and numerical methods we find conditions for parafermion zero modes to emerge when these edge states are coupled to an s-wave superconductor. The phase diagram shows that the parafermionic phase, which is represented by the six-fold ground state degeneracy, is separated from gapped phases by a topological phase transition. Parafermion modes in fractional quantum Hall systems coupled to s-wave superconductors are experimentally feasible.