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arXiv:cond-mat/9608139 · Physical review. B, Condensed matter

Exclusion statistics for fractional quantum Hall states on a sphere

SERGEI B. ISAKOV, Geoffrey Canright, M. D. Johnson

Published Mar 15, 1997• 16 citations
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Abstract

We discuss exclusion statistics parameters for quasiholes and quasielectrons excited above the fractional quantum Hall states near \ensuremath{\nu}=p/(2np+1). We derive the diagonal statistics parameters from the (``unprojected'') composite fermion (CF) picture. We propose values for the off-diagonal (mutual) statistics parameters as a simple modification of those obtained from the unprojected CF picture, by analyzing finite system numerical spectra in the spherical geometry.

DOI
10.1103/physrevb.55.6727

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For a finite number of electrons confined to a spherical surface, Jain's composite fermion (CF) picture gives a unique relationship between the value of the degeneracy, ${\mathit{N}}_{\mathit{L}}$=2S+1, of the lowest electron shell and the number of electrons for which a given incompressible quantum-liquid state of filling factor \ensuremath{\nu} occurs. For states close to the Laughlin \ensuremath{\nu}=1/3 state, two different mean-field CF descriptions can be applied. Some insight into quasielectron-quasihole symmetry and the nature of the lowest-energy sector of the energy spectrum can be obtained by comparing the different mean-field CF pictures. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matter•Dec 15, 1996•3 citations
We discuss exclusion statistics parameters for quasiholes and quasielectrons excited above the fractional quantum Hall states near \ensuremath{\nu}=p/(2np+1). We derive the diagonal statistics parameters from the (``unprojected'') composite fermion (CF) picture. We propose values for the off-diagonal (mutual) statistics parameters as a simple modification of those obtained from the unprojected CF picture, by analyzing finite system numerical spectra in the spherical geometry.
The energy spectra and wave functions of up to 14 interacting quasielectrons (QE's) in the Laughlin $\ensuremath{\nu}=\frac{1}{3}$ fractional quantum Hall (FQH) state are investigated using exact numerical diagonalization. It is shown that at sufficiently high density the QE's form pairs or larger clusters. This behavior, opposite to Laughlin correlations, invalidates the (sometimes invoked) reapplication of the composite fermion picture to the individual QE's. The series of finite-size incompressible ground states are identified at the QE filling factors ${\ensuremath{\nu}}_{\mathrm{QE}}=\frac{1}{2},$ 13,\frac{1}{3},31​, 23,\frac{2}{3},32​, corresponding to the electron fillings $\ensuremath{\nu}=\frac{3}{8},$ 411,\frac{4}{11},114​, 513.\frac{5}{13}.135​. The equivalent quasihole (QH) states occur at ${\ensuremath{\nu}}_{\mathrm{QH}}=\frac{1}{4},$ 15,\frac{1}{5},51​, 27,\frac{2}{7},72​, corresponding to $\ensuremath{\nu}=\frac{3}{10},$ 413,\frac{4}{13},134​, 517.\frac{5}{17}.175​. All these six novel FQH states were recently discovered experimentally. Detailed analysis indicates that QE or QH correlations in these states are different from those of well-known FQH electron states (e.g., Laughlin or Moore--Read states), leaving the origin of their incompressibility uncertain. Halperin's idea of Laughlin states of QP pairs is also explored, but is does not seem adequate.
Fractional quantum Hall (FQH) fluids host quasiparticle excitations that carry a fraction of the electronic charge. Moreover, in contrast to bosons and fermions that carry exchange statistics of 0 and $\ensuremath{\pi}$ respectively, these quasiparticles of FQH fluids, when braided around one another, can accumulate a Berry phase, which is a fractional multiple of $\ensuremath{\pi}$. Deploying the spherical geometry, we numerically demonstrate that composite fermion particle (CFP) excitations in the Jain FQH states carry Abelian fractional statistics. Previously, the exchange statistics of CFPs were studied in the disk geometry, where the statistics get obscured due to a shift in the phase arising from the addition of another CFP, making its determination cumbersome without prior knowledge of the shift. We show that on the sphere this technical issue can be circumvented and the statistics of CFPs can be obtained more transparently. The ideas we present can be extended to determine the statistics of quasiparticles arising in certain non-Abelian partonic FQH states.
Abstract Classical constraints on the reduced density matrix of quantum fluids in a single Landau level termed as local exclusion conditions (LECs) (Yang 2019 Phys. Rev. B 100 241302), have recently been shown to characterize the ground state of many fractional quantum Hall (FQH) phases. In this work, we extend the LEC construction to build the elementary excitations, namely quasiholes and quasielectrons, of these FQH phases. In particular, we elucidate the quasihole counting, categorize various types of quasielectrons, and construct their microscopic wave functions. Our extensive numerical calculations indicate that the undressed quasielectron excitations of the Laughlin state obtained from LECs are topologically equivalent to those obtained from the composite fermion theory. Intriguingly, the LEC construction unveils interesting connections between different FQH phases and offers a novel perspective on exotic states such as the Gaffnian and the Fibonacci state.
We study fractional quantum Hall states in the cylinder geometry with open boundaries. We introduce solvable Hamiltonians for which we are able to obtain exact results. We give a simple construction of the ground state, quasiholes, quasielectrons, and the magnetoroton branch of excited states for spin-polarized electrons at filling factor $\ensuremath{\nu}=1/3$ and spinless bosons at filling $\ensuremath{\nu}=1/2$. The wave functions are simple in the second-quantized language. These model Hamiltonians have all the features we expect from composite fermion theory.
We have studied here the charge and statistics of quasiparticle excitations in FQH states on the basis of the Berry phase approach incorporating the fact that even number of flux quanta can be gauged away when the Berry phase is removed to the dynamical phase. It is observed that the charge q and statistical parameter θ of a quasiparticle at filling factor ν = n/(2pn+1) are given by q = (n/2pn+1)e and θ = n/(2pn+1), with the fact that the charge of the quasihole is opposite to that of the quasielectron. Using Laughlin wave function for quasiparticles, numerical studies have been done following the work of Kjønsberg and Myrheim1 for FQH states at ν = 1/3 and it is pointed out that as in case of quasiholes, the statistics parameter can be well defined for quasielectrons having the value θ = 1/3.
For a finite number of electrons confined to a spherical surface, Jain's composite fermion (CF) picture gives a unique relationship between the value of the degeneracy, NL{\mathit{N}}_{\mathit{L}}NL​=2S+1, of the lowest electron shell and the number of electrons for which a given incompressible quantum-liquid state of filling factor \ensuremath{\nu} occurs. For states close to the Laughlin \ensuremath{\nu}=1/3 state, two different mean-field CF descriptions can be applied. Some insight into quasielectron-quasihole symmetry and the nature of the lowest-energy sector of the energy spectrum can be obtained by comparing the different mean-field CF pictures. \textcopyright{} 1996 The American Physical Society.