## Superconducting analogue of the parafermion fractional quantum Hall states    [PDF]

Abolhassan Vaezi
We address the long standing problem of finding the superconducting analogue of the $Z_k$ parafermion fractional quantum Hall (FQH) states. By studying the FQH-superconductor heterojunction, we show that the low energy description of the system is identical to the non-Abelian part of the Read-Rezayi $Z_{k}$ parafermion state where $k=2/\nu$ and $\nu$ is the filling fraction of the parent FQH state. Our main tool is the 2D bosonization of the FQH states that maps the system into coupled 1D chains with two counter-propagating modes. We show that by inducing intra-chain pairing and charge preserving backscattering with identical couplings these 1D chains will be described by $Z_{k}$ parafermion conformal field theory (CFT) when $k\leq 4$. We then study the effect of inter-chain couplings and show that every parafermion mode becomes massive except for the two outermost ones. Consequently, we achieve a fractional topological superconductor whose chiral edge state is described by a $Z_k$ parafermion CFT. The most notable case is a $\nu=2/3$ FQH state in proximity of a superconductor that undergoes a topological phase transition into a $Z_3$ parafermion superconducting state. This state is topologically indistinguishable from the non-Abelian part of the $\nu=13/5$ Read-Rezay state which is believed to host Fibonacci anyons with $d_{\rm F}=\frac{1+\sqrt{5}}{2}$ quantum dimension capable of performing universal quantum computation. We finally discuss our experimental proposal to realize the fractional Chern insulator at the surface of a three dimensional topological insulator. The superconducting pairing can be induced into this state through the proximity effect. Our proposal is based on the experimental observation of the nearly flat surface state of the Iron-doped ${\rm Bi_2Se_3}$ topological insulator as well as the superconducting gap at its surface state via the proximity effect.
View original: http://arxiv.org/abs/1307.8069