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Modern Physics Letters B

SPIN-SINGLET LAUGHLIN-HALPERIN TYPE WAVEFUNCTIONS FOR THE QUANTUM HALL STATES

X. C. Xie, Fu‐Chun Zhang

Published Mar 10, 1991• 9 citations
Publisher

Abstract

We construct a class of spin-singlet wavefunctions for the filling factors ν=2/(2m±1), where m is an odd integer, and ν=1/2 quantum Hall states, based on Halperin’s generalization of the Laughlin wavefunction to systems without full spin polarization. The properties of the ν=2/(2m±1) states are consistent with the results obtained from the numerical calculations of small systems.

DOI
10.1142/s0217984991000551

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We construct a class of spin-singlet wavefunctions for the filling factors ν=2/(2m±1), where m is an odd integer, and ν=1/2 quantum Hall states, based on Halperin’s generalization of the Laughlin wavefunction to systems without full spin polarization. The properties of the ν=2/(2m±1) states are consistent with the results obtained from the numerical calculations of small systems.
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A simple one-dimensional model is proposed, in which N spinless repulsively interacting fermions occupy M > N degenerate states. It is argued that the energy spectrum and the wavefunctions of this system strongly resemble the spectrum and wavefunctions of 2D electrons in the lowest Landau level (the problem of the Fractional Quantum Hall Effect). In particular, Laughlin-type wavefunctions describe ground states at filling factors v = N/M = 1/ q , q odd.. Within this model the complimentary wavefunction for v =1−1/ q is found explicitly, and extremely simple ground state wavefunctions for arbitrary odd-denominator filling factors are proposed.
The most robust fractional quantum Hall states occur in the lowest Landau level at filling factors, 1/3 and 1/5. Such states are very well described by Laughlin's wave function. In this work, we have succeeded in calculating exactly the one-particle density function of the Laughlin states for some finite systems of particles in a disk geometry. The exact results we provide are not only important for the Laughlin states, but also for the general field of numerical calculations because they can serve as benchmarks to test the accuracy of various approaches, numerical schemes and computational methods used in studies of strongly correlated electronic systems.
We numerically study the spin polarization of the fractional quantum Hall state at filling factor $\ensuremath{\nu}=5/2$. By using both exact diagonalization and the density matrix renormalization group methods on the sphere, we are able to analyze more values of partial spin polarization (in addition to fully polarized and unpolarized) than any previous studies. We find that for the Coulomb interaction the exact finite-system ground state is fully polarized, for shifts corresponding to both the Moore-Read Pfaffian state and its particle-hole conjugate (anti-Pfaffian). This result is found to be robust against small variations in the interaction and change of shift. The low-energy excitation spectrum is consistent with spin-wave excitations of a fully magnetized ferromagnet.
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.