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arXiv:cond-mat/0504571 · Physical Review B

Fractional quantum Hall effect in the second Landau level: The importance of inter-composite-fermion interaction

Csaba Tőke, Michael R. Peterson, Gun Sang Jeon, Jainendra K. Jain

Published Sep 12, 2005• 31 citations
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Abstract

Exact diagonalization of a two-dimensional electron gas in a strong magnetic field in the disk geometry shows that there exists a filling-factor range in the second Landau level where the states significantly differ from those in the lowest Landau level. We show that the difference arises because the interaction between composite fermions is not negligible in higher Landau levels, as indicated by a substantial mixing between composite-fermion Landau-like levels, or $\ensuremath{\Lambda}$ levels. We find that the exact ground state is well reproduced by composite-fermion theory with $\ensuremath{\Lambda}$-level mixing incorporated at the lowest level of approximation. Using the same variational approach in the spherical geometry we estimate the excitation gap at filling $1∕3$ in the second Landau level (which corresponds to $7∕3$ of experiment).

DOI
10.1103/physrevb.72.125315

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Exact diagonalization of a two-dimensional electron gas in a strong magnetic field in the disk geometry shows that there exists a filling-factor range in the second Landau level where the states significantly differ from those in the lowest Landau level. We show that the difference arises because the interaction between composite fermions is not negligible in higher Landau levels, as indicated by a substantial mixing between composite-fermion Landau-like levels, or \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ levels. We find that the exact ground state is well reproduced by composite-fermion theory with \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-level mixing incorporated at the lowest level of approximation. Using the same variational approach in the spherical geometry we estimate the excitation gap at filling 1∕31∕31∕3 in the second Landau level (which corresponds to 7∕37∕37∕3 of experiment).
We compare quantum Hall systems at filling factors $\ensuremath{\nu}=2$ to $\ensuremath{\nu}=\frac{2}{3}$ and 25\frac{2}{5}52​, corresponding to the exact filling of two lowest electron or composite fermion (CF) Landau levels. The two fractional states are examples of CF liquids with spin dynamics. There is a close analogy between the ferromagnetic (spin polarization P=1P=1P=1) and paramagnetic (P=0)(P=0)(P=0) incompressible ground states that occur in all three systems in the limits of large and small Zeeman spin splitting. However, the excitation spectra are different. At $\ensuremath{\nu}=2$, we find spin domains at half-polarization (P=12)(P=\frac{1}{2})(P=21​), while antiferromagnetic order seems most favorable in the CF systems. The transition between P=0P=0P=0 and 1, as seen when, e.g., the magnetic field is tilted, is also studied by exact diagonalization in toroidal and spherical geometries. The essential role of an effective CF-CF interaction is discussed, and the experimentally observed incompressible half-polarized state is found in some models.
There is increasing experimental evidence for fractional quantum Hall effect at filling factor $\ensuremath{\nu}=2+3/8$. Modeling it as a system of composite fermions, we study the problem of interacting composite fermions by a number of methods. In our variational study, we consider the Fermi sea, the Pfaffian paired state, and bubble and stripe phases of composite fermions, and find that the Fermi sea state is favored for a wide range of transverse thickness. However, when we incorporate interactions between composite fermions through composite-fermion diagonalization on systems with up to 25 composite fermions, we find that a gap opens at the Fermi level, suggesting that inter-composite fermion interaction can induce fractional quantum Hall effect at $\ensuremath{\nu}=2+3/8$. The resulting state is seen to be distinct from the Pfaffian wave function.
The particle-hole (PH) symmetry of electrons is an exact symmetry of the electronic Hamiltonian confined to a specific Landau level, and its interplay with the formation of composite fermions has attracted much attention of late. We investigate an emergent symmetry in the fractional quantum Hall effect, namely, the PH symmetry of composite fermions, which relates states at composite fermion filling factors ${\ensuremath{\nu}}^{*}=n+\overline{\ensuremath{\nu}}$ and ${\ensuremath{\nu}}^{*}=n+1\ensuremath{-}\overline{\ensuremath{\nu}}$, where the integer nnn is the $\mathrm{\ensuremath{\Lambda}}$-level index and $0\ensuremath{\le}\overline{\ensuremath{\nu}}\ensuremath{\le}1$. Detailed calculations using the microscopic theory of composite fermions demonstrate the following for low-lying $\mathrm{\ensuremath{\Lambda}}$ levels (small nnn): (i) The two-body interaction between composite-fermion particles is very similar, apart from a constant additive term and an overall scale factor, to that between composite-fermion holes in the same $\mathrm{\ensuremath{\Lambda}}$ level; and (ii) the three-body interaction for composite fermions is an order of magnitude smaller than the two-body interaction. Taken together, these results imply an approximate PH symmetry for composite fermions in low $\mathrm{\ensuremath{\Lambda}}$ levels, which is also supported by exact-diagonalization studies and available experiments. This symmetry, which relates states at electron filling factors $\ensuremath{\nu}=\frac{n+\overline{\ensuremath{\nu}}}{2(n+\overline{\ensuremath{\nu}})\ifmmode\pm\else\textpm\fi{}1}$ and $\ensuremath{\nu}=\frac{n+1\ensuremath{-}\overline{\ensuremath{\nu}}}{2(n+1\ensuremath{-}\overline{\ensuremath{\nu}})\ifmmode\pm\else\textpm\fi{}1}$, is not present in the original Hamiltonian and owes its existence entirely to the formation of composite fermions. With increasing $\mathrm{\ensuremath{\Lambda}}$-level index, the two-body and three-body pseudopotentials become comparable, but at the same time they both diminish in magnitude, indicating that the interaction between composite fermions becomes weak as we approach $\ensuremath{\nu}=1/2$.
We compute energy gaps for spin-polarized fractional quantum Hall states in the lowest Landau level (LL) at filling fractions $\ensuremath{\nu}=\frac{1}{3},$ 25,\frac{2}{5},52​, 37,\frac{3}{7},73​, and 49\frac{4}{9}94​ using exact diagonalization of systems with up to 16 particles and extrapolation to the infinite system-size limit. The gaps calculated for a pure Coulomb interaction and ignoring finite width effects, disorder and LL mixing agree well with the predictions of composite fermion theory provided the logarithmic corrections to the effective mass are included. This is in contrast with previous estimates, which, as we show, overestimated the gaps at $\ensuremath{\nu}=2/5$ and 3/7 by around 15%. We also study the reduction of the gaps as a result of the nonzero width of the two-dimensional (2D) layer. We show that these effects are accurately accounted for using either Gaussian or ``$z\ifmmode\times\else\texttimes\fi{}$ Gaussian'' (zG)(z\mathrm{G})(zG) trial wave functions, which we show are significantly better variational wave functions than the Fang-Howard wave function. The Gaussian and zGz\mathrm{G}zG wave functions give Haldane pseudopotential parameters which agree with those of self-consistent local density approximation calculations to better than $\ifmmode\pm\else\textpm\fi{}0.2%.$ For quantum well parameters typical of heterostructure samples, we find gap reductions of around 20%. The experimental gaps, after accounting heuristically for disorder, are still around 40% smaller than the computed gaps. However, for the case of tetracene layers in metal-insulator-semiconductor (MIS) devices we find that the measured activation gaps are close to those we compute. We discuss possible reasons why the difference between computed and measured activation gaps is larger in GaAs heterostructures than MIS devices. Finally, we present calculations using systems with up to 18 electrons of the gap at $\ensuremath{\nu}=\frac{5}{2}$ including width corrections.
Exact diagonalization studies have revealed that the energy spectrum of interacting electrons in the lowest Landau level splits, nonperturbatively, into bands, which is responsible for the fascinating phenomenology of this system. The theory of nearly free composite fermions has been shown to be valid for the lowest band, and thus to capture the low-temperature physics, but it overpredicts the number of states for the excited bands. We explain the state counting of higher bands in terms of composite fermions with an infinitely strong short-range interaction between an excited composite-fermion particle and the hole it leaves behind. This interaction, the form of which we derive from the microscopic composite-fermion theory, eliminates configurations containing certain tightly bound composite-fermion excitons. With this modification, the composite-fermion theory reproduces, for all well defined excited bands seen in exact diagonalization studies, an exact counting for $\ensuremath{\nu}>1/3$, and an almost exact counting for $\ensuremath{\nu}\ensuremath{\le}1/3$. The resulting insight clarifies that the corrections to the nearly free composite-fermion theory are not thermodynamically significant at sufficiently low temperatures, thus providing a microscopic explanation for why it has proved successful for the analysis of the various properties of the composite-fermion Fermi sea.
We study the fractional quantum Hall effect at filling fractions 7/3 and 5/2 in the presence of the spin-orbit interaction, using the exact diagonalization method and the density matrix renormalization group (DMRG) method in a spherical geometry. Trial wave functions at these fillings are the Laughlin state and the Moore-Reed-Pfaffian state. The ground state excitation energy gaps and pair-correlation functions at fractional filling factor 7/3 and 5/2 in the second Landau level are calculated. We find that the spin-orbit interaction stabilizes the fractional quantum Hall states.