We have formulated a microscopic many-body theory of the electron-hole interaction in the fractional-Hall-effect regime via an analysis based on the Bethe-Salpeter equation. Starting from the noninteracting vacuum suggested by Tao and Thouless we show that the generalized random-phase approximation used by these authors omits important excitonic effects and leads to a poor description of the system. When vertex corrections are included in the ladder approximation the system can undergo an excitonic instability for small values of the single-particle gap. Such an instability is a natural consequence of the fact that the ground state of the system is degenerate. Within our description fractionally charged excitations of the type first introduced by Laughlin can be viewed as domain walls separating different but otherwise equivalent and degenerate ground-state configurations.
We have formulated a microscopic many-body theory of the electron-hole interaction in the fractional-Hall-effect regime via an analysis based on the Bethe-Salpeter equation. Starting from the noninteracting vacuum suggested by Tao and Thouless we show that the generalized random-phase approximation used by these authors omits important excitonic effects and leads to a poor description of the system. When vertex corrections are included in the ladder approximation the system can undergo an excitonic instability for small values of the single-particle gap. Such an instability is a natural consequence of the fact that the ground state of the system is degenerate. Within our description fractionally charged excitations of the type first introduced by Laughlin can be viewed as domain walls separating different but otherwise equivalent and degenerate ground-state configurations.
It is shown that the wave function proposed by Tao and Thouless to explain the fractional quantization of the Hall current implies long-range correlations between the electrons that are very different from the properties expected from Laughlin's wave function. Calculations for a small number of electrons have shown that Laughlin's wave function is an excellent approximation to the true ground state, so it must be concluded that the Tao-Thouless wave function is not a good approximation. There is some discussion of the question of broken symmetry.
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.
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.
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.
We consider the Bethe Salpeter Equation (BSE) for a fractionally filled Landau level. A phenomenological discussion of the 1/3 Laughlin's state is performed by assuming an ansatz for the one-particle propagator. The BSE is solved in this approach and it predicts an instability under the formation of charge density oscillations for a wide range of the one-particle gap parameter values in contrast with previous single mode approximation results. However, the conclusion is compatible with the one obtained within a composite fermion description done by us before and with the saturation of the zero momentum oscillator strength sum rule by the cyclotronic resonance. Further studies should be done in order to understand the discrepancy.
We follow Fubini's suggestion to use vertex operators for describing electrons and holes in the two-dimensional set-up appropriate for the description of the fractional quantum Hall effect, i.e., on the gauge-fixed magnetic plane. Laughlin's wave function is thus reproduced as the correlator of primary conformal fields, represented as exponentials of a free scalar field. We generalize an Ansatz by Halperin and present a new wave function describing the ground-state and the excited states of a system of unpolarized electrons. We realize these wave functions as correlators of normal-ordered exponentials of two free fields. We also give an explicit representation for the creation operator of an excitation.