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arXiv:1008.4173 · Physical Review Letters

Numerical Calculation of the Neutral Fermion Gap at the ν=5/2 Fractional Quantum Hall State

Parsa Bonderson, Adrian E. Feiguin, Chetan Nayak

Published May 5, 2011• 39 citations
arXivHTMLPDF

Abstract

We present the first numerical computation of the neutral fermion gap, ${\ensuremath{\Delta}}_{F}$, in the $\ensuremath{\nu}=5/2$ quantum Hall state, which is analogous to the energy gap for a Bogoliubov-de Gennes quasiparticle in a superconductor. We find ${\ensuremath{\Delta}}_{F}\ensuremath{\approx}0.027\frac{{e}^{2}}{\ensuremath{\epsilon}{\ensuremath{\ell}}_{0}}$, comparable to the charge gap. We also deduce an effective Fermi velocity ${v}_{F}$ for neutral fermions from the low-energy spectra for odd numbers of electrons, and thereby obtain a correlation length ${\ensuremath{\xi}}_{F}={v}_{F}/{\ensuremath{\Delta}}_{F}\ensuremath{\approx}1.3{\ensuremath{\ell}}_{0}$. We comment on implications for experiments, topological quantum information processing, and electronic mechanisms of superconductivity.

DOI
10.1103/physrevlett.106.186802

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Applying the transformation of fermion operators to the fermion quasiparticles introduced by Halperin, Lee, and Read, we estimate the scaling behavior of the ground-state energy and quasiparticle gaps as a function of filling fraction for a ``principal sequence'' of fractional-quantum-Hall-effect \ensuremath{\nu}=N/(2N\ifmmode\pm\else\textpm\fi{}1) states converging towards the gapless state at half filling. The exponent describing the shape of the cusp \ensuremath{\delta}E(\ensuremath{\nu})\ensuremath{\sim}\ensuremath{\Vert}\ensuremath{\delta}\ensuremath{\nu}${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\eta}}}$ is found to be greater than 1 and to depend nontrivially on the interaction potential. The dependence of quasiparticle gaps agrees with the results of recent measurements by R. R. Du et al.
Applying the transformation of fermion operators to the fermion quasiparticles introduced by Halperin, Lee, and Read, we estimate the scaling behavior of the ground-state energy and quasiparticle gaps as a function of filling fraction for a ''principal sequence'' of fractional-quantum-Hall-effect ν=N/(2N±1) states converging towards the gapless state at half filling. The exponent describing the shape of the cusp δE(ν)∼δνη is found to be greater than 1 and to depend nontrivially on the interaction potential. The dependence of quasiparticle gaps agrees with the results of recent measurements by R. R. Du et al.
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Single particle basis functions for composite fermions are obtained from which many-composite fermion states confined to the lowest electronic Landau level can be constructed in the standard manner, i.e. by building Slater determinants. This representation enables a Monte Carlo study of systems containing a large number of composite fermions, yielding new quantitative and qualitative information. The ground state energy and the gaps to charged and neutral excitations are computed for a number of fractional quantum Hall effect (FQHE) states, earlier off-limits to a quantitative investigation. The ground state energies are estimated to be accurate to ~0.1% and the gaps at the level of a few percent. It is also shown that at Landau level fillings smaller than or equal to 1/9 the FQHE is unstable to a spontaneous creation of excitons of composite fermions. In addition, this approach provides new conceptual insight into the structure of the composite fermion wave functions, resolving in the affirmative the question of whether it is possible to motivate the composite fermion theory entirely within the lowest Landau level, without appealing to higher Landau levels.
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