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Physical review. B, Condensed matter

Origin of the ν=2/5 fractional quantum Hall effect

Edward H. Rezayi, Allan H. MacDonald

Published Oct 15, 1991• 36 citations
Publisher

Abstract

We present numerical evidence in support of the view that the physical origin of the fractional quantum Hall effect at \ensuremath{\nu}=n/(2n+1) is associated with the fact that this is the largest filling factor at which an n-component system of electrons can avoid having any pair of electrons occur in a state of relative angular momentum 1. Our calculations, for n=2, are based on a model in which the Hilbert space is truncated to the two lowest Landau levels and the Landau-level separation, \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, is a parameter of the model. For \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=0 and short-range repulsive interactions, a zero-energy incompressible ground state occurs at \ensuremath{\nu}=2/5. Our numerical results suggest that, in agreement with arguments advanced by J. K. Jain, this incompressibility survives to the strong-field limit \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=\ensuremath{\infty}.

DOI
10.1103/physrevb.44.8395

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We present numerical evidence in support of the view that the physical origin of the fractional quantum Hall effect at \ensuremath{\nu}=n/(2n+1) is associated with the fact that this is the largest filling factor at which an n-component system of electrons can avoid having any pair of electrons occur in a state of relative angular momentum 1. Our calculations, for n=2, are based on a model in which the Hilbert space is truncated to the two lowest Landau levels and the Landau-level separation, \ensuremath{\Elzxh}\ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​, is a parameter of the model. For \ensuremath{\Elzxh}\ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​=0 and short-range repulsive interactions, a zero-energy incompressible ground state occurs at \ensuremath{\nu}=2/5. Our numerical results suggest that, in agreement with arguments advanced by J. K. Jain, this incompressibility survives to the strong-field limit \ensuremath{\Elzxh}\ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​=\ensuremath{\infty}.
We provide numerical evidence that the ground state of a short-range-interaction model at \ensuremath{\nu}=1/2 is incompressible and spin singlet for a wide range of repulsive interactions. Furthermore it is accurately described by a trial wave function studied earlier. For the Coulomb interaction we find that this wave function provides a good description of the lowest-lying spin-singlet state, and propose that fractional quantum Hall effect would occur at \ensuremath{\nu}=1/2 if this state became the global ground state. Some conclusions of an earlier paper are invalidated by the present numerical study of a larger system; these will be indicated.
States of strongly interacting particles are of fundamental interest in physics and can produce exotic emergent phenomena and topological structures. We consider here two-dimensional electrons in a magnetic field, and, departing from the standard practice of restricting to the lowest LL, introduce a model short-range interaction that is infinitely strong compared to the cyclotron energy. We demonstrate that this model lends itself to an exact solution for the ground as well as excited states at arbitrary filling factors ν<1/2p and produces a fractional quantum Hall effect at fractions of the form ν=n/(2pn+1), where n and p are integers. The fractional quantum Hall states of our model share many topological properties with the corresponding Coulomb ground states in the lowest Landau level, such as the edge physics and the fractional charge of the excitations.
The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.
The authors discuss the existence of fractional quantised Hall states in other than the lowest (n=0) Landau level. They present the results of numerical studies of systems with up to nine particles in the n=1 Landau level. They find that a fractional quantised Hall state is likely for filling fractions of the n=1 Landau level, nu 1 , with nu 1 -1 =5. At nu 1 -1 =3 they find that the Laughlin trial wavefunctions is not a good candidate ground-state wavefunction. If there is a gap at nu 1 -1 it is predicted to be small. At nu 1 -1 = 7 / 2 they predict the existence of a fractional quantised Hall state despite the small or non-existent gap at nu 1 -1 =3. They also show that introducing truncated pseudopotential interactions is not always a physical procedure outside of the lowest Landau level.
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.
The nature of the state at low Landau-level filling factors has been a long-standing puzzle in the field of the fractional quantum Hall effect (FQHE). While theoretical calculations suggest that a crystal is favored at filling factors $\ensuremath{\nu}\ensuremath{\lesssim}\frac{1}{6}$, experiments show, at somewhat elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect at $\ensuremath{\nu}=\frac{1}{7}, \frac{2}{11}, \frac{2}{13}, \frac{3}{17}, \frac{3}{19}, \frac{1}{9}, \frac{2}{15}$, and 217\frac{2}{17}172​, which belong to the standard sequences $\ensuremath{\nu}=n/(6n\ifmmode\pm\else\textpm\fi{}1)$ and $n/(8n\ifmmode\pm\else\textpm\fi{}1)$. To address this paradox, we investigate the nature of some of the low-$\ensuremath{\nu}$ states, specifically $\ensuremath{\nu}=\frac{1}{7}, \frac{2}{13}$, and 19\frac{1}{9}91​, by variational Monte Carlo, density matrix renormalization group, and exact diagonalization methods. We conclude that in the thermodynamic limit, these are likely to be incompressible fractional quantum Hall liquids, albeit with strong short-range crystalline correlations. This suggests a natural explanation for the experimentally observed behavior and a rich phase diagram that admits, in the low-disorder limit, a multitude of crystal-FQHE liquid transitions as the filling factor is reduced.