Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states

TL;DR

This paper explores non-abelian anyons emerging from gapped edge modes of fractional quantum Hall states coupled to superconductors and ferromagnets, revealing richer braid group representations than Majorana fermions but not enabling universal quantum computation.

Contribution

It introduces a new class of non-abelian anyons from fractional quantum Hall edges and calculates their braiding properties, expanding the understanding of topological quantum states.

Findings

Each superconductor-ferromagnet interface hosts a non-abelian anyon with quantum dimension √2m.

Braiding these anyons yields richer non-abelian representations than Majorana fermions.

Topological manipulations with these anyons do not support universal quantum computation.

Abstract

We study the non-abelian statistics characterizing systems where counter-propagating gapless modes on the edges of fractional quantum Hall states are gapped by proximity-coupling to superconductors and ferromagnets. The most transparent example is that of a fractional quantum spin Hall state, in which electrons of one spin direction occupy a fractional quantum Hall state of $\nu= 1/m$, while electrons of the opposite spin occupy a similar state with $\nu = -1/m$. However, we also propose other examples of such systems, which are easier to realize experimentally. We find that each interface between a region on the edge coupled to a superconductor and a region coupled to a ferromagnet corresponds to a non-abelian anyon of quantum dimension $\sqrt{2m}$. We calculate the unitary transformations that are associated with braiding of these anyons, and show that they are able to realize a richer set of non-abelian representations of the braid group than the set realized by non-abelian anyons based on Majorana fermions. We carry out this calculation both explicitly and by applying general considerations. Finally, we show that topological manipulations with these anyons cannot realize universal quantum computation.