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arXiv:2306.13972 · Physical review. B./Physical review. B

Laughlin's quasielectron as a nonlocal composite fermion

Alberto Nardin, Leonardo Mazza

Published Nov 8, 2023• 1 citation
arXivHTMLPDF

Abstract

We discuss the link between the quasielectron wave functions proposed by Laughlin and by Jain and show both analytically and numerically that Laughlin's quasielectron is a nonlocal composite-fermion state. Composite-fermion states are typically discussed in terms of the composite-fermion Landau levels (also known as lambda levels). In standard composite-fermion quasielectron wave functions the excited lambda levels have subextensive occupation numbers. However, once the Laughlin's quasielectron is reformulated as a composite fermion, an overall logarithmic occupation of the first lambda level is made apparent, which includes orbitals that are localized at the boundary of the droplet. Even though the wave function proposed by Laughlin features a localized quasielectron with well-defined fractional charge, it exhibits some nontrivial boundary properties which motivate our interpretation of Laughlin's quasielectron as a nonlocal object. This has an important physical consequence: Laughlin's quasielectron fractionalizes an incorrect spin, deeply related to the anyonic braiding statistics. We conclude that Laughlin's quasielectron is not a good candidate for a quasielectron wave function.

DOI
10.1103/physrevb.108.l201106

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We discuss the link between the quasielectron wave functions proposed by Laughlin and by Jain and show both analytically and numerically that Laughlin's quasielectron is a nonlocal composite-fermion state. Composite-fermion states are typically discussed in terms of the composite-fermion Landau levels (also known as lambda levels). In standard composite-fermion quasielectron wave functions the excited lambda levels have subextensive occupation numbers. However, once the Laughlin's quasielectron is reformulated as a composite fermion, an overall logarithmic occupation of the first lambda level is made apparent, which includes orbitals that are localized at the boundary of the droplet. Even though the wave function proposed by Laughlin features a localized quasielectron with well-defined fractional charge, it exhibits some nontrivial boundary properties which motivate our interpretation of Laughlin's quasielectron as a nonlocal object. This has an important physical consequence: Laughlin's quasielectron fractionalizes an incorrect spin, deeply related to the anyonic braiding statistics. We conclude that Laughlin's quasielectron is not a good candidate for a quasielectron wave function.
Two-and three-body correlation functions (number of pairs or triplets vs. relative angular momentum) of electrons or Laughlin quasielectrons (i.e., composite fermions in their first excited Landau level) are studied numerically in several fractional quantum Hall liquids.It is shown directly that the νe = 4/11 liquid (corresponding to a ν = 1/3 filling of composite fermions in their first excited Landau level) is a paired state of quasielectrons, hence interpreted as a condensate of "second-generation" quasiholes of Moore-Read ν = 1/2 state of composite fermions.
Variational Monte Carlo calculations of the quasielectron and quasihole excitation energies in the fractional quantum Hall effect have been carried out at filling fractions $\ensuremath{\nu}=1/3,$ 1/5, and 1/7. For the quasielectron both the trial wave function originally proposed by Laughlin and the composite-fermion wave function proposed by Jain have been used. We find that for long-range Coulomb interactions the results obtained using these two wave functions are essentially the same, though the energy gap obtained using the composite-fermion quasielectron is slightly smaller, and closer to extrapolated exact-diagonalization results.
Matrix product state techniques provide a very efficient way to numerically evaluate certain classes of quantum Hall wave functions that can be written as correlators in two-dimensional conformal field theories. Important examples are the Laughlin and Moore-Read ground states and their quasihole excitations. In this paper, we extend the matrix product state techniques to evaluate quasielectron wave functions, a more complex task because the corresponding conformal field theory operator is not local. We use our method to obtain density profiles for states with multiple quasielectrons and quasiholes, and to calculate the (mutual) statistical phases of the excitations with high precision. The wave functions we study are subject to a known difficulty: the position of a quasielectron depends on the presence of other quasiparticles, even when their separation is large compared to the magnetic length. Quasielectron wave functions constructed using the composite fermion picture, which are topologically equivalent to the quasielectrons we study, have the same problem. This flaw is serious in that it gives wrong results for the statistical phases obtained by braiding distant quasiparticles. We analyze this problem in detail and show that it originates from an incomplete screening of the topological charges, which invalidates the plasma analogy. We demonstrate that this can be remedied in the case when the separation between the quasiparticles is large, which allows us to obtain the correct statistical phases. Finally, we propose that a modification of the Laughlin state, that allows for local quasielectron operators, should have good topological properties for arbitrary configurations of excitations.
Abstract Matrix product state techniques provide a very efficient way to numerically evaluate certain classes of quantum Hall wave functions that can be written as correlators in two-dimensional conformal field theories. Important examples are the Laughlin and Moore-Read ground states and their quasihole excitations. In this paper, we extend the matrix product state techniques to evaluate quasielectron wave functions, a more complex task because the corresponding conformal field theory operator is not local. We use our method to obtain density profiles for states with multiple quasielectrons and quasiholes, and to calculate the (mutual) statistical phases of the excitations with high precision. The wave functions we study are subject to a known difficulty: the position of a quasielectron depends on the presence of other quasiparticles, even when their separation is large compared to the magnetic length. Quasielectron wave functions constructed using the composite fermion picture, which are topologically equivalent to the quasielectrons we study, have the same problem. This flaw is serious in that it gives wrong results for the statistical phases obtained by braiding distant quasiparticles. We analyze this problem in detail and show that it originates from an incomplete screening of the topological charges, which invalidates the plasma analogy. We demonstrate that this can be remedied in the case when the separation between the quasiparticles is large, which allows us to obtain the correct statistical phases. Finally, we propose that a modification of the Laughlin state, that allows for local quasielectron operators, should have good topological properties for arbitrary configurations of excitations.
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.
We present model wave functions for quasielectron (as opposed to quasihole) excitations of the unitary Z_{k} parafermion sequence (Laughlin, Moore-Read, or Read-Rezayi) of fractional quantum Hall states. We uniquely define these states through two generalized clustering conditions: they vanish when either a cluster of k+2 electrons is put together or when two clusters of k+1 electrons are formed at different positions. For Abelian fractional quantum Hall states (k=1), our construction reproduces the Jain quasielectron wave function and elucidates the difference between the Jain and Laughlin quasielectrons.