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Acta Physica Polonica A

Fractional Quantum Hall States with Non-Abelian Statistics

Gunnar Möller, Arkadiusz Wójs, Nigel R. Cooper

Published Nov 1, 2009• 0 citations
Publisher

Abstract

Using exact numerical diagonalization we have studied correlated many-electron ground states in a partially filled second Landau level.We consider filling fractions ν = 1/2 and 2/5, for which incompressible quantum liquids with non-Abelian anion statistics have been proposed.Our calculations include finite layer width, Landau level mixing and arbitrary deformation of the interaction pseudopotential.Computed energies, gaps, and correlation functions support the non-Abelian ground states at both ν = 1/2 ("Pfaffian") and ν = 2/5 ("parafermion" state).

DOI
10.12693/aphyspola.116.847

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Using exact numerical diagonalization we have studied correlated many-electron ground states in a partially filled second Landau level.We consider filling fractions ν = 1/2 and 2/5, for which incompressible quantum liquids with non-Abelian anion statistics have been proposed.Our calculations include finite layer width, Landau level mixing and arbitrary deformation of the interaction pseudopotential.Computed energies, gaps, and correlation functions support the non-Abelian ground states at both ν = 1/2 ("Pfaffian") and ν = 2/5 ("parafermion" state).
We use exact numerical diagonalization in the search of fractional quantum Hall states with non-Abelian quasiparticle statistics. For the (most promising) states in a partially filled second Landau level, the search is narrowed to the range of filling factors 7/3 < v e < 8/3. In this range, the analysis of energy spectra and correlation functions, calculated including finite width and Landau level mixing, supports the prominent non-Abelian candidates at v e = 5/2 (paired Moore–Read "pfafian" state) and 12/5 (clustered Read–Rezayi "parafermion" state). Outside of this range, the noninteracting composite fermion model with four attached flux quanta is validated, yielding the family of quantum liquids with fractional, but Abelian statistics. The borderline v e = 7/3 state is shown to be adiabatically connected to the Laughlin liquid, but its short-range correlations are significantly different.
We examine the quantum phase diagram of the fractional quantum Hall effect (FQHE) in the lowest two Landau levels in half-filled bilayer structures as a function of tunneling strength and layer separation, i.e., we revisit the lowest Landau-level filling factor 1/2 bilayer problem and make predictions involving bilayers in the half-filled second Landau level (i.e., filling factor 5/2). Using numerical exact diagonalization we investigate the important question of whether this system supports a FQHE described by the non-Abelian Moore-Read Pfaffian state in the strong-tunneling regime. In the lowest Landau level, we find that although, in principle, increasing (decreasing) tunneling strength (layer separation) could lead to a transition from the Abelian two-component Halperin 331 to non-Abelian one-component Moore-Read Pfaffian state, the FQHE excitation gap is relatively small in the lowest Landau level Pfaffian regime, and we establish that all so far observed FQHE states in half-filled lowest Landau level bilayers are most likely described by the Abelian Halperin 331 state. In the second Landau level we make the prediction that bilayer structures would manifest two distinct branches of incompressible FQHE corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Moore-Read Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the possible quantum phase transition between them will be compelling evidence supporting the existence of the non-Abelian Moore-Read Pfaffian state in the second Landau level. We discuss our results in the context of existing experiments and theoretical works.
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The $\ensuremath{\nu}=5/2$ fractional quantum Hall effect is a system of intense experimental and theoretical interest as its ground state may host non-Abelian excitations, but the exact nature of the ground state is still undetermined. We present the results of an exact diagonalization study of an electron system in the disk configuration, including the effects of Landau level (LL) mixing and the finite thickness of the quantum well confining the electrons. The degeneracy between the two leading candidates for the ground state, the Pfaffian and anti-Pfaffian, is broken by interactions with a neutralizing background, in addition to the inclusion of two- and three-body interactions via LL mixing. As a result of the neutralizing background in the disk configuration, there is a phase transition from the anti-Pfaffian to the Pfaffian as LL mixing is turned on, in stark contrast to what is observed in spherical geometry. This behavior is in agreement with existing experiments, showing the appearance of the Pfaffian state at strong LL mixing before the system enters a compressible phase. The inclusion of LL mixing leads to an increased charge e/4e/4e/4 quasihole size. LL mixing interactions are also shown to overcome the effects of edge reconstruction. Due to finite thickness effects, these properties are enhanced dramatically. We also find that only the Pfaffian and anti-Pfaffian states continue to possess energy gaps at finite width, while gaps for compressible stripe states close, which is in agreement with available experimental data.
We study 'second generation' of fractional quantum Hall states which contain a partially filled composite fermion (CF) Landau level (LL). We explain the role of CF-CF interaction in the incompressibility of the underlying quantum electron liquid. In particular, using exact diagonalization on a Haldane sphere, we determine two- and three-body CF correlation functions for these liquids and show that the CFs form a paired state (instead of a Laughlin liquid) when filling v = 1/3 of their second LL. At v = 1/2 we show that the CFs group into even larger clusters rather than form a Moore—Read paired state. We also address the problem of spin polarization of interacting CFs and predict transition to the partially unpolarized ground state in sufficiently narrow quantum wells and/or weak magnetic fields.