• Open Access

Fractionalizing Majorana Fermions: Non-Abelian Statistics on the Edges of Abelian Quantum Hall States

Netanel H. Lindner, Erez Berg, Gil Refael, and Ady Stern
Phys. Rev. X 2, 041002 – Published 11 October 2012

Abstract

We study the non-Abelian statistics characterizing systems where counterpropagating 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 ν=1/m, while electrons of the opposite spin occupy a similar state with ν=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 2m. We calculate the unitary transformations that are associated with the braiding of these anyons, and we 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.

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  • Received 1 May 2012

DOI:https://doi.org/10.1103/PhysRevX.2.041002

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Netanel H. Lindner

  • Institute of Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
  • Department of Physics, California Institute of Technology, Pasadena, California 91125, USA

Erez Berg

  • Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

Gil Refael

  • Department of Physics, California Institute of Technology, Pasadena, California 91125, USA

Ady Stern

  • Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel

Popular Summary

Historically, quantum mechanics divided all elementary particles into two families, fermions and bosons, whose physical behavior are vastly different. For example, the fermionic nature of electrons (also known as “Pauli’s principle”) explains the periodic table of the elements. Bosonic physics explains superconductivity and lasers. However, the long-held belief that all particles must belong to one of these two categories has been proven wrong in the last three decades. We now know that for particles living in two spatial dimensions infinitely many categories exist, distinguished from one another by a property known as “quantum statistics.” Of these, the most interesting class, and the hardest to realize, goes under the name “non-Abelian particles.” The work presented here proposes a novel route to realizing non-Abelian particles on the one-dimensional edges of two-dimensional electronic systems.

A system made of non-Abelian particles has a multitude of ground states. In an ordinary quantum system, such a multitude is a consequence of symmetry (for example, the rotational symmetry of the hydrogen atom). For non-Abelian particles, however, the degeneracy is a topological property, and does not rely on any symmetry. It is, therefore, insensitive to any microscopic details. Indeed, topology plays a crucial, and interesting, role: The system can transform from one ground state to another through interchanges of non-Abelian particles, and this transformation depends only on the topology of the interchange. This topological robustness of non-Abelian particles makes them appealing candidates for the realization of a decoherence-free type of quantum computation.

Our work proposes that hybrid systems of fractional quantum Hall states and superconductors should host a novel type of particles with a rich non-Abelian nature. Since the non-Abelian particles we study live on a one-dimensional edge, we generalize the notion of the interchange of their positions, and analyze the resulting transformations within the ground-state manifold.

This new type of non-Abelian particles provides a strong impetus for understanding the experimental feasibility as well as the particle’s underlying mathematical structure. Perhaps most interesting, our study paves the way for seeking even more exotic non-Abelian particles, based on the paradigm of hybrid edges of two-dimensional fractionalized states of matter.

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Vol. 2, Iss. 4 — October - December 2012

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