Superconducting Analogue of the Parafermion Fractional Quantum Hall States

Read-Rezayi Z(k) parafermion wave functions describe nu = 2 + (k/kM + 2) fractional quantum Hall (FQH) states. These states support non-Abelian excitations from which protected quantum gates can be designed. However, there is no experimental evidence for these non-Abelian anyons to date. In this paper, we study the nu = 2/k FQH-superconductor heterostructure and find the superconducting analogue of the Z(k) parafermion FQH state. Our main tool is the mapping of the FQH into coupled one-dimensional chains, each with a pair of counterpropagating modes. We show that by inducing intrachain pairing and charge preserving backscattering with identical couplings, the one-dimensional chains flow into gapless Z(k) parafermions when k < 4. By studying the effect of interchain coupling, we show that every parafermion mode becomes massive except for the two outermost ones. Thus, we achieve a fractional topological superconductor whose chiral edge state is described by a Z(k) parafermion conformal field theory. For instance, we find that a nu = 2/3 FQH in proximity to a superconductor produces a Z(3) parafermion superconducting state. This state is topologically indistinguishable from the non-Abelian part of the nu = 12/5 Read-Rezayi state. Both of these systems can host Fibonacci anyons capable of performing universal quantum computation through braiding operations.