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A stochastic limit approach to the fractional quantum Hall effect

Luigi Accardi, Fabio Bagarello

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

We study three diffent approximations of the usual many-body model for the fractional quantum Hall effect (FQHE) consisting in a two-dimensional electron gas (2DEG) in a uniform electric and magnetic field.For each of them we consider the stochastic limit and we find the quantum Langevin equation and the generators of the master equation.This allows us to calculate the explicit form of the conductivity and the resistivity tensors.The relevance of the levels beyond the lowest Landau's one is explained, and a fine tuning relation between the electric and the magnetic fields is obtained from the stochastic resonance principle.It is shown that this relation could be at the origin of the quantization of the resistivity tensor.

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International Journal of Modern Physics B•Feb 28, 1994•5 citations
We study three diffent approximations of the usual many-body model for the fractional quantum Hall effect (FQHE) consisting in a two-dimensional electron gas (2DEG) in a uniform electric and magnetic field.For each of them we consider the stochastic limit and we find the quantum Langevin equation and the generators of the master equation.This allows us to calculate the explicit form of the conductivity and the resistivity tensors.The relevance of the levels beyond the lowest Landau's one is explained, and a fine tuning relation between the electric and the magnetic fields is obtained from the stochastic resonance principle.It is shown that this relation could be at the origin of the quantization of the resistivity tensor.
After more than three decades the fractional quantum Hall effect still poses challenges to contemporary physics. While many features are phenomenologically understood, a detailed comprehension of its microscopic origin still lacks. Here, we address this problem by showing that the Hamiltonian describing interacting electrons in a strong magnetic field, restricted to the lowest Landau level, can be mapped onto a one-dimensional lattice gas with repulsive interactions. The statistical mechanics of such models leads to interpret the sequence of Hall plateaux as a fractal phase diagram. We derive an expression for the ground states, forming Wigner crystals in the reciprocal lattice of the degenerate Landau states, and predict the form of the fractional plateaux landscape.
We extend the quantum Hall matrix model to include couplings to external electric and magnetic fields. The associated current suffers from matrix ordering ambiguities even in the classical model. We calculate the linear response at low momenta---this is unambigously defined. In particular, we obtain the correct fractional quantum Hall conductivity, and the expected density modulations in response to a weak and slowly varying magnetic field. In the quantum theory the ordering ambiguities are more severe, but we are able to construct a good density operator, satisfying the pertinent lowest Landau level commutator algebra, for the special case of $2\ifmmode\times\else\texttimes\fi{}2$ matrices. Our results support the conjecture that the quantum Hall matrix models do describe important aspects of the dynamics of electrons in the lowest Landau level.
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 fractional quantum Hall effect (FQHE) occurs at certain magnetic field strengths B*(n) in a two-dimensional electron gas of density n at strong magnetic fields perpendicular to the plane of the electron gas. At these magnetic fields strengths, the system is incompressible, i.e., there is a finite cost in energy for creating charge density fluctuations in the bulk, while the boundary of the electron gas has gapless modes of density waves. The bulk energy gap arises because of the strong electron-electron interactions. While there are very good models for infinite homogeneous systems and for the gapless excitations of the boundary of the electron gas, computational methods to accurately model finite, inhomogeneous systems with more then about ten electrons have not been available until very recently. We will here review an ensemble density functional approach to studying the ground state of large inhomogeneous spin polarized FQHE systems.
Exact diagonalization of a two-dimensional electron gas in a strong magnetic field in the disk geometry shows that there exists a filling-factor range in the second Landau level where the states significantly differ from those in the lowest Landau level. We show that the difference arises because the interaction between composite fermions is not negligible in higher Landau levels, as indicated by a substantial mixing between composite-fermion Landau-like levels, or $\ensuremath{\Lambda}$ levels. We find that the exact ground state is well reproduced by composite-fermion theory with $\ensuremath{\Lambda}$-level mixing incorporated at the lowest level of approximation. Using the same variational approach in the spherical geometry we estimate the excitation gap at filling 1∕31∕31∕3 in the second Landau level (which corresponds to 7∕37∕37∕3 of experiment).
The formalism introduced in a previous paper is used for discussing the Coulomb interaction of many electrons moving in two space-dimensions in the presence of a strong magnetic field. The matrix element of the Coulomb interaction is evaluated in the new basis, whose states are invariant under discrete translations (up to a gauge transformation). This paper is devoted to the case of low filling factor, thus we limit ourselves to the lowest Landau level and to spins all oriented along the magnetic field. For the case of filling factor νf = 1/u we give an Ansatz on the state of many electrons which provides a good approximated solution of the Hartree–Fock equation. For general filling factor νf = u′/u a trial state is given which converges very rapidly to a solution of the self-consistent equation. We generalize the Hartree–Fock equation by considering some correlation: all quantum states are allowed for the u′ electrons with the same translation quantum numbers. Numerical results are given for the mean energy and the energy bands, for some values of the filling factor (νf = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5). Our results agree numerically with the Charge Density Wave approach. The boundary conditions are shown to be very important: only large systems (degeneracy of Landau level over 200) are not affected by the boundaries. Therefore results obtained on small scale systems are somewhat unreliable. The relevance of the results for the Fractional Quantum Hall Effect is briefly discussed.