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Search for parafermions and Fibonacci anyons, which are excitations obeyingnon-Abelian statistics, is driven both by the quest for deeper understanding ofnature and prospects for universal topological quantum computation. However,physical systems that can host these exotic excitations are rare and hard torealize in experiments. Here we study the domain walls and the edge statesformed in spin transitions in the fractional quantum Hall effect. Effectivetheory approach and exact diagonalization in a disk and torus geometries provesthe existence of the counter-propagating edge modes with opposite spinpolarizations at the boundary between the two neighboring regions of thetwo-dimensional electron liquid in spin-polarized and spin-unpolarized phases.By analytical and numerical analysis, we argue that these systems can hostparafermions when coupled to an s-wave superconductor and are experimentallyfeasible. We investigate settings based on $\nu=\frac{2}{3}$, $\nu=\frac{4}{3}$and $\nu=\frac{5}{3}$ spin transitions and analyze spin-flipping interactionsthat hybridize counter-propagating modes. Finally, we discuss spin-orbitinteractions of composite fermions.

Parafermions, induced edge states and domain walls in the fractional quantum Hall effect spin transitions