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Physical review. B, Condensed matter

Microscopic wave functions for the fractional quantized Hall states at ν=25 and 27

R. Morf, N. d’Ambrumenil, Bertrand I. Halperin

Published Aug 15, 1986• 101 citations
Publisher

Abstract

New microscopic trial wave functions are proposed for fractional quantized Hall states at $\ensuremath{\nu}=\frac{2}{(2p+1)}$. Results for $\ensuremath{\nu}=\frac{2}{5} \mathrm{and} \frac{2}{7}$ based on Monte Carlo and exact numerical computations are presented. For $N\ensuremath{\le}10$ electrons on a sphere, energies differ from the exact ground-state energy by less than 0.3%. For even $p$, such as $\ensuremath{\nu}=\frac{2}{5}$, the trial wave function may be regarded as a microscopic realization of the condensed quasiparticle state postulated in hierarchical schemes.

DOI
10.1103/physrevb.34.3037

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New microscopic trial wave functions are proposed for fractional quantized Hall states at \ensuremathν=2(2p+1)\ensuremath{\nu}=\frac{2}{(2p+1)}\ensuremathν=(2p+1)2​. Results for \ensuremathν=25and27\ensuremath{\nu}=\frac{2}{5} \mathrm{and} \frac{2}{7}\ensuremathν=52​and72​ based on Monte Carlo and exact numerical computations are presented. For N\ensuremath≤10N\ensuremath{\le}10N\ensuremath≤10 electrons on a sphere, energies differ from the exact ground-state energy by less than 0.3%. For even ppp, such as \ensuremathν=25\ensuremath{\nu}=\frac{2}{5}\ensuremathν=52​, the trial wave function may be regarded as a microscopic realization of the condensed quasiparticle state postulated in hierarchical schemes.
With the use of Haldane's spherical model, the fractional quantum Hall effect is studied by means of large-scale configuration-interaction calculations. The exact wave function and the ground-state energy are computed numerically up to ten electrons for filling \ensuremath{\nu}=(1/3), and up to seven electrons for filling \ensuremath{\nu}=(1/5). Quasiparticle, quasihole, and exciton energies are computed. The mutual relation between wave functions on the sphere and wave functions on the disk is fully analyzed. A wave function originally proposed by Halperin for the disk geometry is tested on the sphere; the numerical results up to six electrons for \ensuremath{\nu}=(2/5) and \ensuremath{\nu}=(2/9) are favorable.
Trial wave functions which, in the thermodynamic limit, describe states relevant to the fractional quantum Hall effect are explicitly constructed for up to eight electrons. Estimates are given for the ground-state energies and pair-correlation functions and for the low-lying excitation energies for states associated with the filling factors \ensuremath{\nu}=(1/3), (2/7), and (2/5). The results help explain why the filling factors \ensuremath{\nu}=${\ensuremath{\nu}}_{m}$=m/(2m+1), m=1,2,..., and \ensuremath{\nu}=1-${\ensuremath{\nu}}_{m}$ show strong quantum Hall effects.
New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry ±1/4\pm1/4±1/4 of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta NϕN_ϕNϕ​ is shown to be related to the number of electrons NNN by Nϕ=2N−(5−q)N_ϕ = 2N-(5-q)Nϕ​=2N−(5−q) with qqq being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at Nϕ=2N−3N_ϕ = 2N-3Nϕ​=2N−3. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.
It has long been known that Laughlin's wave functions, describing the fractional quantum Hall effect at filling fractions ν=1/(2k+1), can be obtained as correlation functions in conformal field theory. We show how to generalize this approach to construct explicit trial wave functions for all states in the quantum Hall hierarchy corresponding to quasiparticle (as opposed to quasihole) condensates, including the recently observed state at ν= 4/11. At the filling fractions ν= n/(2np+1) this construction exactly reproduces Jain's composite fermion wave functions. An explicit connection is made to Wen's topological classification of FQH states.
We propose trial wave functions for quasiparticle and exciton excitations of the Moore-Read Pfaffian fractional quantum Hall states, both for bosons and for fermions, and study these numerically. Our construction of trial wave functions employs a picture of the bosonic Moore-Read state as a symmetrized double layer composite fermion state. We obtain the number of independent angular momentum multiplets of quasiparticle and exciton trial states for systems of up to 20 electrons. We find that the counting for quasielectrons at large angular momentum on the sphere matches that expected from the conformal field theory (CFT) that describes the Moore-Read state's boundary theory. In particular, the counting for quasielectrons is the same as for quasiholes, in accordance with the idea that the CFT describing both sides of the Fractional Quantum Hall plateau should be the same. We also show that our trial wave functions have good overlaps with exact wave functions obtained using various interactions, including second Landau level Coulomb interactions and the three-body delta interaction for which the Pfaffian states and their quasiholes are exact ground states. We discuss how these results relate to recent work by Sreejith et al. on a similar set of trial wave functions for excitations over the Pfaffian state as well as to earlier work by Hansson et al., which has produced trial wave functions for quasiparticles based on conformal field theory methods and by Bernevig and Haldane, which produced trial wave functions based on clustering properties and ``squeezing.''