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arXiv:cond-mat/9408004 · Physical Review Letters

Composite-Fermion Picture for the Spin-Wave Excitation in the Fractional Quantum Hall System

Takashi Nakajima, Hideo Aoki

Published Dec 26, 1994• 40 citations
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Abstract

The spin-wave excitation mode from the spin-polarized ground state in the fractional quantum Hall system at the Landau level filling of $\ensuremath{\nu}=\frac{1}{(2m+1)}$ is shown, by comparing with the exact numerical result for finite systems, to be accurately described, for wavelengths exceeding the magnetic length, in terms of the composite-fermion picture for the spin-wave (magnon) theory formulated in the spherical geometry. This indicates that the composite-fermion approach extends to excited states and also provides the spin stiffness in terms of exchange interaction with transformed interparticle interactions.

DOI
10.1103/physrevlett.73.3568

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The spin-wave excitation mode from the spin-polarized ground state in the fractional quantum Hall system at the Landau level filling of \ensuremathν=1(2m+1)\ensuremath{\nu}=\frac{1}{(2m+1)}\ensuremathν=(2m+1)1​ is shown, by comparing with the exact numerical result for finite systems, to be accurately described, for wavelengths exceeding the magnetic length, in terms of the composite-fermion picture for the spin-wave (magnon) theory formulated in the spherical geometry. This indicates that the composite-fermion approach extends to excited states and also provides the spin stiffness in terms of exchange interaction with transformed interparticle interactions.
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.
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∕31∕31∕3 in the second Landau level (which corresponds to 7∕37∕37∕3 of experiment).
By developing an algorithm for evaluating the basis states for the composite fermions with negative effective magnetic field, we perform the composite-fermion-diagonalization study for the fully spin-polarized fractional quantum Hall states at the filling factors $\ensuremath{\nu}=3/10$, 4/13, and 5/17 in the range $2/7<\ensuremath{\nu}<1/3$. These observed states correspond to a partially filled second effective Landau level, for the composite fermions carrying four vortices, with filling factor $\overline{\ensuremath{\nu}}=1/2$, 1/3, and 2/3, respectively, analogous to the previously studied states of composite fermions with two attached vortices in the range $1/3<\ensuremath{\nu}<2/5$. We show that the character of these states in the range $2/7<\ensuremath{\nu}<1/3$ replicates the same for the states in the range $1/3<\ensuremath{\nu}<2/5$ having identical $\overline{\ensuremath{\nu}}$: chiral ppp-wave pairing with an anti-Pfaffian correlation of composite fermions carrying six quantized vortices produces an incompressible state at $\ensuremath{\nu}=3/10$; an unconventional interaction between composite fermions, resulting from the suppression of fermion pairs with relative angular momentum three and producing the fractional quantum Hall effect of composite fermions in the second effective Landau level with $\overline{\ensuremath{\nu}}=1/3$ and its particle-hole conjugate filling factor 2/3, reproduces incompressible states at 4/134/134/13 and 5/175/175/17 filling factors. We further estimate the thermodynamic limit of the ground-state energies and calculate the lowest energy gap for the neutral collective excitations of these states.
Collective modes capture the dynamical aspects of fractional quantum Hall (FQH) fluids. Depending on the active degrees of freedom, different types of collective modes can arise in a FQH state. In this work, we consider spinful FQH states in the lowest Landau level (LLL) along the Jain sequence of fillings ν=n/(2n±1)ν{=}n/(2n{\pm}1)ν=n/(2n±1) and compute the Coulomb dispersion of their spin-flip and spin-conserving collective modes in the spherical geometry. We use the LLL-projected density-wave and composite fermion (CF) exciton states as trial wave functions for these modes. To evaluate the dispersion of density-wave states, we derive the commutation algebra of spinful LLL-projected density operators on the sphere, which enables us to extract the gap of the density-wave excitations from the numerically computed density-density correlation function, i.e., the static structure factor, of the FQH ground state. We find that the CF excitons provide an accurate description of the collective modes at all wavelengths, while the density-wave states fail to do so. Specifically, the spin-flip density wave reliably captures the spin-flip collective mode only for the Laughlin and Halperin states, and that too only in the long-wavelength limit. Interestingly, for spin-singlet primary Jain states, the spin-conserving density mode is inaccurate even in the long-wavelength regime. We show that this discrepancy stems from the presence of an additional high-energy spin-conserving parton mode, similar to that found in fully polarized secondary Jain states at ν=n/(4n±1)ν{=}n/(4n{\pm}1)ν=n/(4n±1). We propose an ansatz for this parton mode and compute its Coulomb dispersion in the singlet state at ν=2/5ν{=}2/5ν=2/5. The predicted parton mode can be observed in circularly polarized inelastic light scattering experiments.
Using the Hamiltonian formulation for composite fermions developed recently, the temperature dependence of the spin polarization is computed for the translationally invariant fractional quantum Hall states at ν = 1/3 and ν = 2/5 in two steps. In the first step, the effect of particle-hole excitations on the spin polarization is computed in a composite-fermion Hartree-Fock approximation. The computed magnetization for ν = 1/3 lies above the experimental results for intermediate temperatures indicating the importance of long-wavelength spin fluctuations which are not correctly treated in the Hartree-Fock approximation. In the second step, spin fluctuations beyond the Hartree-Fock approximation are included for ν = 1/3 by mapping the problem onto the coarse-grained continuum quantum ferromagnet. The parameters of the description in terms of the effective continuum quantum ferromagnet are extracted from the preceding Hartree-Fock analysis. After the inclusion of spin fluctuations in a large- N approach, the results for the finite-temperature spin polarization are in quite good agreement with the experiments.
A system at filling factor 2/3 could be a candidate for a quantum Hall ferromagnet at integer filling factor of composite fermions. Using exact diagonalization with electrons on a torus we study the transition from the singlet ground state to the polarized ground state at this filling and look for signatures of quantum Hall ferromagnetism. Differences between the fractional and corresponding integer systems are emphasised. Most interestingly, we find around the transition a low excited half-polarized state which might become the ground state in the thermodynamical limit. We study its structure and compare it to the singlet and polarized ground states. A new interpretation of the singlet state is suggested and comparison of the filling factors 2/3 and 2/5 is presented. Adding magnetic inhomogeneities into the system we investigate the stability of all the three involved states and the tendency to build up domains like in conventional ferromagnets.