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

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.

21 pages, 7 figures; references added, typos corrected, minor changes according to referee's comments

References in corpus (10)

• Non-Abelian Anyons and Topological Quantum Computation
• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Non-Abelian quantum order in spin-orbit-coupled semiconductors: The search for topological Majorana particles in solid state systems
• Anyons and the quantum Hall effect - a pedagogical review
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Proximity effect at the superconductor - topological insulator interface
• Fractional topological insulators in three dimensions
• Anyonic interferometry without anyons: How a flux qubit can read out a topological qubit

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