Abstract
Read-Rezayi parafermion wave functions describe 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 paper, we study the FQH-superconductor heterostructure and find the superconducting analogue of the parafermion FQH state. Our main tool is the mapping of the FQH into coupled one-dimensional chains, each with a pair of counterpropagating modes. We show that by inducing intrachain pairing and charge preserving backscattering with identical couplings, the one-dimensional chains flow into gapless parafermions when . By studying the effect of interchain coupling, we show that every parafermion mode becomes massive except for the two outermost ones. Thus, we achieve a fractional topological superconductor whose chiral edge state is described by a parafermion conformal field theory. For instance, we find that a FQH in proximity to a superconductor produces a parafermion superconducting state. This state is topologically indistinguishable from the non-Abelian part of the Read-Rezayi state. Both of these systems can host Fibonacci anyons capable of performing universal quantum computation through braiding operations.
- Received 27 August 2013
DOI:https://doi.org/10.1103/PhysRevX.4.031009
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Published by the American Physical Society
Popular Summary
Electronic interactions can lead to exotic phases with emergent excitations that carry fractional charge and statistics. These fractionalized excitations are called non-Abelian anyons (non-Abelions) if their ground-state degeneracy depends exponentially on their numbers. The non-Abelions are protected against disorder, thermal fluctuation, and interactions; these observations have inspired the use of non-Abelions as building blocks of topological quantum computers. The Fibonacci anyon is capable of performing the entire set of quantum gates needed for a quantum computer through braiding operations. The fractional quantum Hall system has been the focal point of the search for non-Abelions, but the search has thus far been unsuccessful. A fundamental question that arises is whether we can find alternative systems with non-Abelions such as Fibonacci anyons as their local excitations. We provide an affirmative answer to this critical question and explicitly show that the heterostructure of an -wave superconductor with a fractional quantum Hall state at a filling fraction yields the parafermion state that hosts Fibonacci anyons.
We first show how a fractional quantum Hall state can be viewed as an array of quantum chains coupled through backscattering. Next, we use conformal field theory and study the effect of superconducting pairing on individual chains. We show that when the intrachain electron pairing and backscattering have the same strength, individual chains are critical. We also consider the interchain couplings and demonstrate that they induce a bulk gap; a chiral gapless mode with a parafermion conformal field theory description survives. Thus, the resulting fractional topological superconductor, defined as a fractional quantum Hall system with a filling fraction of that has superconducting pairing, is described by parafermion conformal field theory.
Since for this system has the same edge states as the non-Abelian part of the fractional quantum Hall state, we conclude that these two seemingly different states are described by a related underlying topological field theory. Both systems can host Fibonacci anyons capable of performing quantum computations.
