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arXiv:1007.5006 · Journal of Physics A Mathematical and Theoretical

SU(m) non-Abelian anyons in the Jain hierarchy of quantum Hall states

M Cristina Diamantini, Carlo A. Trugenberger

Published Feb 18, 2011• 0 citations
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Abstract

We show that different classes of topological order can be distinguished by the dynamical symmetry algebra of edge excitations. A fundamental topological order is realized when this algebra is the largest possible, the algebra of quantum area-preserving diffeomorphisms, called W 1 + ∞ . We argue that this order is realized in the Jain hierarchy of fractional quantum Hall states and show that it is more robust than the standard Abelian Chern–Simons order since it has a lower entanglement entropy due to the non-Abelian character of the quasi-particle anyon excitations. These behave as SU( m ) quarks, where m is the number of components in the hierarchy. We propose the topological entanglement entropy as the experimental measure to detect the existence of these quantum Hall quarks. Non-Abelian anyons in the ν = 2/5 fractional quantum Hall states could be the primary candidates to realize qbits for topological quantum computation.

DOI
10.1088/1751-8113/44/11/115001

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We show that different classes of topological order can be distinguished by the dynamical symmetry algebra of edge excitations. A fundamental topological order is realized when this algebra is the largest possible, the algebra of quantum area-preserving diffeomorphisms, called W 1 + ∞ . We argue that this order is realized in the Jain hierarchy of fractional quantum Hall states and show that it is more robust than the standard Abelian Chern–Simons order since it has a lower entanglement entropy due to the non-Abelian character of the quasi-particle anyon excitations. These behave as SU( m ) quarks, where m is the number of components in the hierarchy. We propose the topological entanglement entropy as the experimental measure to detect the existence of these quantum Hall quarks. Non-Abelian anyons in the ν = 2/5 fractional quantum Hall states could be the primary candidates to realize qbits for topological quantum computation.
Topological quantum computation has emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations that are necessary for quantum computation are carried out by braiding quasiparticles and then measuring the multiquasiparticle states. The fault tolerance of a topological quantum computer arises from the nonlocal encoding of the quasiparticle states, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the $\ensuremath{\nu}=5∕2$ state, although several other prospective candidates have been proposed in systems as disparate as ultracold atoms in optical lattices and thin-film superconductors. In this review article, current research in this field is described, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. Both the mathematical underpinnings of topological quantum computation and the physics of the subject are addressed, using the $\ensuremath{\nu}=5∕2$ fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.
FINDING: Chern-Simons theory at level kkk governs fractional quantum Hall anyons; braiding produces non-abelian phases encoded in the Jones polynomial; lattice realizations confirm topological invariance; a novel CKM angle measurement is unrelated to topological physics. MATH: Chern-Simons action SCS=k4π∫Tr(A∧dA+23A∧A∧A)S_{CS} = \frac{k}{4\pi}\int \text{Tr}(A\wedge dA + \frac{2}{3}A\wedge A\wedge A)SCS​=4πk​∫Tr(A∧dA+32​A∧A∧A); level k∈Zk \in \mathbb{Z}k∈Z (integer QHE) or k=p/qk = p/qk=p/q (fractional QHE, e.g., k=1/3k=1/3k=1/3 for Laughlin state); anyon braiding phase θ=π/k\theta = \pi/kθ=π/k (abelian) or unitary matrix UabU_{ab}Uab​ (non-abelian, e.g., Fibonacci anyons with U∼eiπ/5U \sim e^{i\pi/5}U∼eiπ/5); entanglement entropy SA=αL−γS_A = \alpha L - \gammaSA​=αL−γ, with topological entanglement entropy γ=ln⁡D\gamma = \ln \sqrt{D}γ=lnD​, D=∑idi2D = \sum_i d_i^2D=∑i​di2​ (quantum dimensions); lattice Chern-Simons reproduces continuum in limit a→0a \to 0a→0 with kkk quantized. CONNECTION: Level k=5k=5k=5 gives Fibonacci anyons with quantum dimension d=ϕ=1.618d = \phi = 1.618d=ϕ=1.618 (golden ratio) — di Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Non-Abelian anyons can emerge as fractionalized excitations in two-dimensional systems with topological order. One important example is the Moore–Read fractional quantum Hall state. Its quasihole states are zero-energy eigenstates of a parent Hamiltonian, but its quasiparticle states are not. Both of them can be modeled on an equal footing using the bipartite composite fermion method. We study the entanglement spectrum of the cases with two or four non-Abelian anyons. The counting of levels in the entanglement spectrum can be understood using the edge theory of the Moore–Read state, which reflects the topological order of the system. It is shown that the fusion results of two non-Abelian anyons is determined by their distributions in the bipartite construction.
To perform topological quantum computation we first need to experimentally realise anyons in a topological system. These systems are characterised by intriguing non-local quantum correlations that give rise to the anyonic statistics. What are the diagnostic tools we have to identify if a given system is indeed topological or not? Different phases of matter are characterised by their symmetries. This information is captured by order parameters. Usually, order parameters are defined in terms of local operators that can be measured in the laboratory. For example, the magnetisation of a spin system is given as the expectation value of a single spin with respect to the ground state. Such local properties can describe fascinating physical phenomena efficiently, such as ferromagnetism, and can pinpoint quantum phase transitions. But what about topological systems? Experimentally, we usually identify the topological character of systems, such as the fractional quantum Hall liquids, by probing the anyonic properties of their excitations (Miller et al. , 2007). However, topological order should be a characteristic of the ground state (Thouless et al. , 1982; Wen, 1995). The natural question arises: is it possible to identify a property of the ground state of a system that implies anyonic excitations? The theoretical background that made it possible to answer this question came from entropic considerations of simple topological models. Hamma et al . (2005) studied the entanglement entropy of the toric code ground state and noticed an unusual behaviour.