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Non-Abelian statistics in quantum Hall systems

Kareljan Schoutens, Eddy Ardonne

Published Jan 1, 2001• 0 citations
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Abstract

It has long been known that, under the laws of quantum mechanics, particles in two spatial dimensions may exhibit non-abelian braid statistics. In this note we briefly discuss particular states of matter, so-called paired and clustered quantum Hall states, which offer a concrete realization of this scenario: quasi-holes over these quantum Hall states satisfy non-abelian braid statistics. The associated degeneracies have their origin in combinatorial properties of (para-)fermions in conformal field theory. We also discuss the exclusion statistics of edge excitations in these quantum Hall systems.

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It has long been known that, under the laws of quantum mechanics, particles in two spatial dimensions may exhibit non-abelian braid statistics. In this note we briefly discuss particular states of matter, so-called paired and clustered quantum Hall states, which offer a concrete realization of this scenario: quasi-holes over these quantum Hall states satisfy non-abelian braid statistics. The associated degeneracies have their origin in combinatorial properties of (para-)fermions in conformal field theory. We also discuss the exclusion statistics of edge excitations in these quantum Hall systems.
The quasiparticles in quantum Hall liquids carry fractional charge and obey fractional quantum statistics. Of particular recent interest are those with non-Abelian statistics, since their braiding properties could, in principle, be used for robust coding of quantum information. There is already a good theoretical understanding of quasiholes in both Abelian and non-Abelian quantum Hall states. Here we develop conformal field theory methods that allow for an equally precise description of quasielectrons and explicitly construct two- and four-quasielectron excitations of the non-Abelian Moore-Read state.
The general formulation of quantum statistical mechanics hints at interesting generalizations of the usual Bose/Fermi framework in two spatial dimensions. Anyon statistics, which is essentially a continuous interpolation between Bose and Fermi statistics, is relevant to the Fractional Quantum Hall Effect in two-dimensional (i.e., thin layer) condensed matter systems. In addition, the possibility of non-abelian statistics, in which the multi-particle wavefunction transforms as a representation of a non-abelian group under the exchange of indistinguishable particles, has been explored. Spontaneously broken non-abelian gauge theories in (2 + 1) dimensions often have stable topological defects, called non-abelian vortices, that experience non-abelian statistics. In addition, it has been suggested that degenerate quasihole multiplets in Quantum Hall systems also transform as non-abelian representations of the braid group under particle exchange. In this thesis, I explore the braiding properties of systems of two-cycle flux vortices in a residual S₃ discrete gauge group. The individual vortices are uncharged, but multi-vortex states can have Cheshire charge. The uncharged sectors all have non-vanishing bosonic subspaces, as do the non-abelian charged trivial flux sectors. A kinetic Hamiltonian for vortices on a periodic lattice is constructed. There is a modification to the translational symmetry in the periodically identified direction for non-trivial Z₂ charged sectors. The ground state energies for various three and four vortex sectors is numerically determined. Typically, the ground state is bosonic, with a gap separating it from a non-abelian subspace.
We determine the exclusion statistics properties of the fundamental edge quasiparticles over a specific $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$ non-Abelian quantum Hall state known as the q=2q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2q=2 Pfaffian state. The fundamental excitations are the edge electrons of charge $\ensuremath{-}e$ and the edge quasiholes of charge +e/4+e/4+e/4. We explicitly determine thermodynamic distribution functions and establish a duality which generalizes the duality for fractional exclusion statistics as described by F. D. M. Haldane.