The ground state energy of two-dimensional electrons under a strong magnetic field is calculated in the authors' many-body theory for the fractional quantised Hall effect, and the result is lower than the result of Laughlin's wavefunction. The presence of the energy gap at fractional fillings provides a downward cusp in the correlation energy which makes those states stable to produce quantised Hall steps. The ground state has a broken symmetry and no pinning.
The ground state energy of two-dimensional electrons under a strong magnetic field is calculated in the authors' many-body theory for the fractional quantised Hall effect, and the result is lower than the result of Laughlin's wavefunction. The presence of the energy gap at fractional fillings provides a downward cusp in the correlation energy which makes those states stable to produce quantised Hall steps. The ground state has a broken symmetry and no pinning.
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 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.
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.
It is shown that the ground-state and the lowest excited-states of two-dimensional electron system (2DES), with ion jellium background, correspond to partial crystal-like (with the period Lx□=2mπℓ0) correlation order among N electrons of the main region (MR; Lx×Ly). Many-body variational ground-state wave function of 2DES is presented at the fractional and the integral filling factors ν=1/m; m=2ℓ+1 and ℓ=0,1,2,.... The ground-state manifests the broken symmetry liquid-crystal state with 2DES density that is periodic along the y− direction, with the period Lx□/m, and independent of x. At m=3,5, the ground-state has essentially lower energy per electron than the Laughlin, uniform liquid, ground-state; the same holds at m=1. At m≥3, the compound form of the many-body ground-state wave function leads to the compound structure of each electron within the main strip (MS; Lx□×Ly). Obtained compound exciton and compound spin-exciton states show finite excitation gaps. The excited compound electron (hole) is composed, within MS, from m strongly correlated quasielectrons (quasiholes) of the total charge e/m (−e/m) each. The activation gap is obtained: it is given by the excitation gap of relevant compound exciton, at m≥3, and by the gap of pertinent compound spin-exciton, at m=1. Quantized Hall conductance σH=e2/(2mπℏ) is obtained. The theory is in good agreement with experiments.
Ground-state of fractional and integral quantum Hall systems at ν≤1 and it excitations
Many-body variational ground-state wave function of two-dimensional electron system (2DES), localized in the main strip (MS)Lx□×Ly of the finite width Lx□=2πmℓ0 (and the periodic boundary condition (PBC) imposed along x−direction), is presented at the fractional and the integral filling factors ν=1/m for two different ion backgrounds: microscopical uniform ion background (UIB) and classical ion jellium background (IJB); ℓ0 is the magnetic length, m=2ℓ+1 and ℓ=0,1,2,.... It is shown that the ground-state and the lowest excited-state can correspond to partial crystal-like correlation order among N electrons of the main region (MR) Lx×Ly; then the study of 2DES of N electrons within MR is exactly reduced to the treatment of 2DES of N~=NLx□/Lx electrons localized within MS, with PBC along x. The ground-state manifests the broken symmetry liquid-crystal state with 2DES density that is periodic along the y− direction, with the period Lx□/m, and independent of x. For IJB, at m=3,5, the ground-state has essentially lower energy per electron than the Laughlin, uniform liquid, ground-state (the Laughlin model uses IJB); the same holds at m=1. Obtained compound exciton and compound spin-exciton states show finite excitation gaps. The excited compound electron (hole) is composed, within MS, from m strongly correlated quasielectrons (quasiholes) of the charge e/m (−e/m. Quantized Hall conductance σH=e2/(2mπℏ) is obtained. The theory is in good agreement with experiments.
A scaling procedure on the particle number is developed to construct the wave function of the uniform liquid ground state of the two-dimensional electron system in a strong magnetic field. When the filling factor is equal to the inverse of an odd number, this wave function is the correlated basis function introduced by Laughlin. In more general cases, when the filling factor is equal to a rational number, this wave function also can be regarded as a Laughlin wave function on the scale of a certain cluster. By this procedure, we can prove that an energy gap is always present between such a uniform liquid ground state and the excited states when the filling factor is equal to a rational number.
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.