Superconducting analogue of the parafermion fractional quantum Hall states

Read and Rezayi $Z_k$ parafermion wavefunctions 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 talk, we discuss the $\nu=2/k$ FQH-superconductor heterostructure and through analytical and numerical calculations we argue that it can yield the superconducting analogue of the $Z_k$ parafermion FQH state. The resulting topological state has a gapless chiral edge state with $Z_k$ parafermion conformal field theory description. For instance, we find that a $\nu= 2/3$ FQH in proximity to a superconductor produces a $Z_3$ parafermion superconducting state. This state can host Fibonacci anyons capable of performing universal quantum computation through braiding operations. We finally discuss our experimental proposal for realizing parafermion superconductors. Reference: arXiv:1307.8069