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Physical review. B, Condensed matter

Role of reversed spins in the correlated ground state for the fractional quantum Hall effect

Tapash Chakraborty, Fu‐Chun Zhang

Published Jun 15, 1984• 106 citations
Publisher

Abstract

The ground-state energy for the correlated liquid state of Laughlin with one-half of the electrons having spins antiparallel to the field is obtained at $\frac{2}{5}$ filling of the lowest Landau level. The energy is found to be lower compared with the potential energy of the spin-polarized state. The energy difference is small, however, compared with the difference of magnetic energies between the two states under the present experimental conditions. Some of the results obtained for the one-spin state are in good agreement with recent experimental observations and very accurate Monte Carlo results.

DOI
10.1103/physrevb.29.7032

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The ground-state energy for the correlated liquid state of Laughlin with one-half of the electrons having spins antiparallel to the field is obtained at 25\frac{2}{5}52​ filling of the lowest Landau level. The energy is found to be lower compared with the potential energy of the spin-polarized state. The energy difference is small, however, compared with the difference of magnetic energies between the two states under the present experimental conditions. Some of the results obtained for the one-spin state are in good agreement with recent experimental observations and very accurate Monte Carlo results.
We compare ground-state energies by variational Monte Carlo of the spin unpolarized Halperin 331 and the spin polarized Moore-Read (MR) Pfaffian fractional quantum Hall states at half filling of the lowest Landau level (LLL) and the second Landau level (SLL) as a function of small deviations around the Coulomb point via the finite-thickness effect and direct alterations to the the first two Haldane pseudopotentials. In the comparison we find that in the LLL, either the 331 state or the MR Pfaffian may be lower in energy depending on the deviations. In the SLL, however, the MR Pfaffian is consistently lower in energy except for large deviations. These results suggest that even under moderate deviations in the interaction potential (through various physical processes such as finite thickness, Landau-level mixing, etc.), the MR Pfaffian description is more energetically favorable than the Halperin 331 state in the half-filled SLL (i.e., $\ensuremath{\nu}=5/2$), consistent with recent experimental investigations.
Finite-size two-dimensional systems are investigated by numerical diagonalization of the Hamiltonian. Both fermion systems and boson systems are investigated. The energy of the liquidlike ground state of the fermion system shows a downward cusp at 13\frac{1}{3}31​ filling of the lowest Landau level. This behavior is consistent with the observed fractional quantum Hall effect. The short-range behavior of the pair distribution function of the ground state is analyzed and good agreement with Laughlin's trial wave function is obtained.
Two dimensional electrons in a strong magnetic field show the fractional quantum Hall effect at low temperatures. This effect is explained successfully by a discovery of a new liquid type ground state. However the infinitely strong magnetic field has been assumed in existing theories. Here the ground state around one third filling of the lowest Landau level is investigated at a finite magnetic field by numerical diagonalization of the Hamiltonian. When the cyclotron energy is not too small compared to a typical Coulomb energy, no qualitative change of the ground state is found: A natural generalization of the liquid state at the infinite magnetic field describes the ground state. Excitation energies of quasiparticles decrease as the magnetic field decreases.
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.
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.
Strongly correlated two-dimensional electronic systems subject to a perpendicular magnetic field at lowest Landau level (LLL) filling factors: 1/2, 1/4 and 1/6, are believed to be composite-fermion (CF) Fermi liquid phases. Even though a Bose Laughlin wave function cannot describe these filling factors, we investigate whether such a wave function provides a lower energy bound to the true CF Fermi liquid energies. By using Monte Carlo simulations in disk geometry we compute the Bose Laughlin energies and compare them to corresponding results for the spin-polarized LLL CF Fermi liquid state and avalable data from the literature. We find the unexpected result that, for filling factors ν = 1/4 and 1/6, the Bose Laughlin ground-state energy is practically identical to the true CF liquid energy while this is not the case at ν = 1/2 where the Bose Laughlin ground-state energy is sizeably lower than the energy of the CF Fermi liquid state.