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arXiv:cond-mat/9803169 · Physical Review Letters

Exclusion Statistics for Non-Abelian Quantum Hall States

Kareljan Schoutens

Published Aug 31, 1998• 22 citations
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Abstract

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 $q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ Pfaffian state. The fundamental excitations are the edge electrons of charge $\ensuremath{-}e$ and the edge quasiholes of charge $+e/4$. 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.

DOI
10.1103/physrevlett.81.1929

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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 quasihole states of several paired states, the Pfaffian, Haldane-Rezayi, and 331 states, which under certain conditions may describe electrons at filling factor \ensuremath{\nu}=1/2 or 5/2, are studied analytically and numerically in the spherical geometry, for the Hamiltonians for which the ground state are known exactly. We also find all the ground states (without quasiparticles) for these systems in the toroidal geometry. In each case, a complete set of linearly independent functions that are energy eigenstates of zero energy is found explicitly. For fixed positions of the quasiholes, the number of linearly independent states is ${2}^{\mathit{n}\mathrm{\ensuremath{-}}1}$ for the Pfaffian, and ${2}^{2\mathit{n}\mathrm{\ensuremath{-}}3}$ for the Haldane-Rezayi state; these degeneracies are needed if these systems are to possess non-Abelian statistics, and they agree with predictions based on conformal field theory. The dimensions of the spaces of states for each number of quasiholes agree with numerical results for moderate system sizes. The effects of tunneling and of the Zeeman term are discussed for the 331 and Haldane-Rezayi states, as well as the relation to Laughlin states of electron pairs. A model introduced by Ho, which was supposed to connect the 331 and Pfaffian states, is found to have the same degeneracies of zero-energy states as the 331 state, except at its Pfaffian point where it is much more highly degenerate than either the 331 or the Pfaffian. We introduce a modification of the model which has the degeneracies of the 331 state everywhere including the Pfaffian point; at the latter point, tunneling reduces the degeneracies to those of the Pfaffian state. An experimental difference is pointed out between the Laughlin states of electron pairs and the other paired states, in the current-voltage response when electrons tunnel into the edge. An appendix contains results for the permanent state, in which the zero modes can be occupied by composite bosons, rather than by composite fermions as in the other cases; the system is found to have an incipient instability toward a spin-polarized state. \textcopyright{} 1996 The American Physical Society.