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Physical review. B, Condensed matter

Composite fermions on a Haldane sphere: Quasielectron-quasihole symmetry

Kyung-Soo Yi, Piotr Sitko, Anil Khurana, J. J. Quinn

Published Dec 15, 1996• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.54.16432

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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, 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.
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.
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.
Two- and three-body correlations of incompressible quantum liquids are studied numerically. Pairing of composite fermions (CFs) in the 1/3-filled second CF Landau level is found at ${\ensuremath{\nu}}_{e}=4∕11$. It is explained by reduced short-range repulsion due to ring-like single-particle charge distribution. Although Moore-Read state of CFs is unstable in the 1/2-filled second CF level, condensation of its quasiholes is a possible origin of incompressibility at ${\ensuremath{\nu}}_{e}=4∕11$. Electron pairing occurs at ${\ensuremath{\nu}}_{e}=7∕3$ and 13∕313∕313∕3, but with different pair-pair correlations. Signatures of triplets are found at higher fillings.
While an ordinary Fermi sea is perturbatively robust to interactions, the paradigmatic composite-fermion (CF) Fermi sea arises as a nonperturbative consequence of emergent gauge fields in a system where there was no Fermi sea to begin with. A mean-field picture suggests two Fermi seas, of composite fermions made from electrons or holes in the lowest Landau level, which occupy different areas away from half filling and thus appear to represent distinct states. Using the microscopic theory of composite fermions, which satisfies particle-hole symmetry in the lowest Landau level to an excellent approximation, we show that the Fermi wave vectors at filling factors ν and 1-ν are equal when expressed in units of the inverse magnetic length, and are generally consistent with the experimental findings of Kamburov et al. [Phys. Rev. Lett. 113, 196801 (2014)]. Our calculations suggest that the area of the CF Fermi sea may slightly violate the Luttinger area rule.
Strongly correlated two-dimensional electronic systems subject to a perpendicular magnetic field at lowest Landau level (LLL) filling factors: 1/2, 1/4 and 1/6, are believed to be composite-fermion (CF) Fermi liquid phases. Even though a Bose Laughlin wave function cannot describe these filling factors, we investigate whether such a wave function provides a lower energy bound to the true CF Fermi liquid energies. By using Monte Carlo simulations in disk geometry we compute the Bose Laughlin energies and compare them to corresponding results for the spin-polarized LLL CF Fermi liquid state and avalable data from the literature. We find the unexpected result that, for filling factors ν = 1/4 and 1/6, the Bose Laughlin ground-state energy is practically identical to the true CF liquid energy while this is not the case at ν = 1/2 where the Bose Laughlin ground-state energy is sizeably lower than the energy of the CF Fermi liquid state.
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.