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arXiv:0710.1088 · Physical Review Letters

Effect of Landau Level Mixing on Braiding Statistics

Steven H. Simon

Published Mar 21, 2008• 15 citations
arXivHTMLPDF

Abstract

We examine the effect of Landau level mixing on the braiding statistics of quasiparticles of Abelian and non-Abelian quantum Hall states. While path dependent geometric phases can perturb the Abelian part of the statistics, we find that the non-Abelian properties remain unchanged to an accuracy that is exponentially small in the distance between quasiparticles.

DOI
10.1103/physrevlett.100.116803

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We examine the effect of Landau level mixing on the braiding statistics of quasiparticles of Abelian and non-Abelian quantum Hall states. While path dependent geometric phases can perturb the Abelian part of the statistics, we find that the non-Abelian properties remain unchanged to an accuracy that is exponentially small in the distance between quasiparticles.
Pairing of composite fermions provides a possible mechanism for fractional quantum Hall effect at even denominator fractions and is believed to serve as a platform for realizing quasiparticles with non-Abelian braiding statistics. We present results from fixed-phase diffusion Monte Carlo calculations which predict that substantial Landau level mixing can induce a pairing of composite fermions at filling factors ν=1/2 and ν=1/4 in the l=-3 relative angular momentum channel, thereby destabilizing the composite-fermion Fermi seas to produce non-Abelian fractional quantum Hall states.
Some models describing unconventional fractional quantum Hall states predict quasiparticles that obey non-Abelian quantum statistics. The most prominent example is the Moore-Read model for the $\ensuremath{\nu}=5∕2$ state, in which the ground state is a superconductor of composite fermions, the charged excitations are vortices in that superconductor, and the non-Abelian statistics is closely linked to the degeneracy of the ground state in the presence of vortices. In this paper we develop a physical picture of the non-Abelian statistics of these vortices. Considering first the positions of the vortices as fixed, we define a set of single-particle states at and near the core of each vortex, and employ general properties of the corresponding Bogolubov--de Gennes equations to write the ground states in the Fock space defined by these single-particle states. We find all ground states to be entangled superpositions of all possible occupations of the single-particle states near the vortex cores, in which the probability for all occupations is equal, and the relative phases vary from one ground state to another. Then, we examine the evolution of the ground states as the positions of the vortices are braided. We find that as vortices move, they accumulate a geometric phase that depends on the occupations of the single-particle states near the cores of other vortices. Thus, braiding of vortices changes the relative phase between different components of a superposition, in which the occupations of these states differ, and hence transform the system from one ground state to another. These transformations, that emanate from the quantum entanglement of the occupations of single-particle states and from the dependence of the geometric phase on these occupations, are the source of the non-Abelian statistics. Finally, by exploring a ``self-similar'' form of the many-body wave functions of the various ground states, we show the equivalence of our picture, in which vortex braiding leads to a change in the relative phase of components in a superposition, and pictures derived previously, in which vortex braiding seemingly affects the occupations of states in the cores of the vortices.
The effect of Landau level (LL) mixing on the $\ensuremath{\nu}=5∕2$ fractional quantum Hall state is studied directly in a minimal, yet still numerically tractable, approximation. Specifically, the inter-LL excitations with one unit of cyclotron energy are included in the exact diagonalization of a small yet representative number of electrons on a sphere. Significant reduction of the excitation gap found already in this simple model indicates that LL mixing must play an important role in more realistic systems, with a full spectrum of allowed inter-LL excitations. This suggests a possible route to the explanation of the troubling discrepancy between earlier calculations and experiments. On the other hand, even within our limited approach the LL mixing is found to considerably reduce overlaps with the Moore-Read wave function, raising the question of the actual realization of non-Abelian quasiparticles (QPs) at $\ensuremath{\nu}=5∕2$. In view of the discussed limitations of exact numerics, conclusive resolution of this question will probably require an experiment directly probing the QP statistics.
We report on exact diagonalization studies for fully spin polarized 5/2 fractional quantum Hall effect, incorporating Landau-level mixing through the Bishara-Nayak effective interaction. We find that there is an experimentally accessible region in the phase diagram where the Pfaffian model accurately describes not only the ground state but also the neutral and charged excitations. These results are consistent with the observed persistence of the 5/2 Hall effect down to very low magnetic fields; they are also relevant to the experimental attempts to detect non-Abelian braid statistics.
Particles obeying non-Abelian braid statistics have been predicted to emerge in the fractional quantum Hall effect. In particular, a model Hamiltonian with short-range three-body interaction (${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}}_{3}^{\text{Pf}}$) between electrons confined to the lowest Landau level provides exact solutions for quasiholes, and thereby allows a proof of principle for the existence of quasiholes obeying non-Abelian braid statistics. We construct, in terms of two- and three-body Haldane pseudopotentials, a model Hamiltonian that can be solved exactly for both quasiholes and quasiparticles, and provide evidence of non-Abelian statistics for the latter as well. The structure of the quasiparticle states of this model is in agreement with that predicted by the bipartite composite-fermion model of quasiparticles. We further demonstrate, for systems for which exact diagonalization is possible, adiabatic continuity for the ground state, the ordinary neutral excitation, and the topological exciton as we deform our model Hamiltonian continuously into the lowest Landau-level ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}}_{3}^{\text{Pf}}$ Hamiltonian.