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Acta Physica Polonica A

Ising Ferromagnetism of Composite Fermions

Karel Výborný, Ondřej Čertı́k, Daniela Pfannkuche, D. Wodziński, Arkadiusz Wójs, J. J. Quinn

Published Sep 1, 2006• 3 citations
Publisher

Abstract

The quantum Hall ferromagnets at the half-filling of a pair of degenerate electron or composite fermion Landau levels are studied by exact numerical diagonalization.The results obtained using open and closed geometries (rectangular -with periodic boundary conditions and spherical) are compared.The ferro-and paramagnetic ground states are identified in finite-size energy spectra, and the pair-correlation functions are used in search of the domain structure at half-polarization.

DOI
10.12693/aphyspola.110.409

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Journal of Physics Condensed Matter•May 10, 1993•1 citation
The quantum Hall ferromagnets at the half-filling of a pair of degenerate electron or composite fermion Landau levels are studied by exact numerical diagonalization.The results obtained using open and closed geometries (rectangular -with periodic boundary conditions and spherical) are compared.The ferro-and paramagnetic ground states are identified in finite-size energy spectra, and the pair-correlation functions are used in search of the domain structure at half-polarization.
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.
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.
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).
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.
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.
Finite-size exact diagonalization of the Hamiltonian for four, eight and 16 electrons in a half-filled Landau level under periodic boundary conditions in a rectangular geometry was carried out. The magnitude of the peak value of the structure factor is found to increase with the system size to the power 0.36, consistent with a picture of a quasi-solid discussed by Chui. The energy difference of the lowest-energy state of the two different particle-hole symmetry manifolds decreases as the system size is increased, suggestive of broken particle-hole symmetry.