M. D. Johnson

Addition spectra of quantum dots in strong magnetic fields.

We consider the magnetic field dependence of the chemical potential for parabolically confined quantum dots in a strong magnetic field. Approximate expressions based on the notion that the size of a dot is determined by a competition between confinement and interaction energies are shown to be consistent with exact diagonalization studies for small quantum dots. Fine structure is present in the magnetic field dependence which cannot be explained without a full many-body description and is associated with ground-state level crossings as a function of confinement strength or Zeeman interaction strength. Some of this fine structure is associated with precursors of the bulk incompressible states responsible for the fractional quantum Hall effect

Quantum dots in strong magnetic fields: stability criteria for the maximum density droplet

In this article we discuss the ground state of a parabolically confined quantum dot in the limit of very strong magnetic fields where the electron system is completely spin-polarised and all electrons are in the lowest Landau level. Without electron-electron interactions the ground state is a single Slater determinant corresponding to a droplet centred on the minimum of the confinement potential and occupying the minimum area allowed by the Pauli exclusion principle. Electron-electron interactions favour droplets of larger area. We derive exact criteria for the stability of the maximum density droplet against edge excitations and against the introduction of holes in the interior of the droplet. The possibility of obtaining exact results in the strong magnetic field case is related to important simplifications associated with broken time-reversal symmetry in a strong magnetic field.

Financial Market Dynamics

A necessary precondition for modeling financial markets is a complete understanding of their statistics, including dynamics. Distributions derived from nonextensive Tsallis statistics are closely connected with dynamics described by a nonlinear Fokker–Planck equation. The combination shows promise in describing stochastic processes with power-law distributions and superdiffusive dynamics. We investigate intra-day price changes in the S&P500 stock index within this framework. We find that the power-law tails of the distributions, and the indexs anomalously diffusing dynamics, are very accurately described by this approach. Our results show good agreement between market data and Fokker–Planck dynamics. This approach may be applicable in any anomalously diffusing system in which the correlations in time can be accounted for by an Ito–Langevin process with a simple time-dependent diffusion coefficient.

Magnetic oscillations of a fractional Hall dot.

We show that a quantum dot in the fractional Hall regime exhibits mesoscopic oscillations in its thermodynamic properties with a period which is a multiple of the period for free electrons. Our calculations are performed for parabolic quantum dots with hard-core electron-electron interactions and are exact in the strong field limit for k B T smaller than the fractional Hall gap. Explicit expressions are given for the temperature dependence of the amplitude of the oscillations

Exclusion statistics for fractional quantum Hall states on a sphere

We discuss exclusion statistics parameters for quasiholes and quasielectrons excited above the fractional quantum Hall states near � = p/(2np + 1). We derive the diagonal statistics parameters from the (“unprojected”) composite fermion (CF) picture. We propose values for the off-diagonal (mutual) statistics parameters as a simple modification of those obtained from the unprojected CF picture, by analyzing finite system numerical spectra in the spherical geometry.

ENSEMBLE DENSITY FUNCTIONAL THEORY OF THE FRACTIONAL QUANTUM HALL EFFECT

We develop an ensemble density functional theory for the fractional quantum Hall effect using a local density approximation. Model calculations for edge reconstructions of a spin-polarized quantum dot give results in good agreement with semiclassical and Hartree-Fock calculations, and with small system numerical diagonalizations. This establishes the usefulness of density functional theory to study the fractional quantum Hall effect, which opens up the possibility of studying inhomegeneous systems with many more electrons than has heretofore been possible.

Spin-ensemble density-functional theory for inhomogeneous quantum Hall systems

We review an ensemble density functional approach to spin-polarized inhomogeneous quantum Hall systems. Recent work on generalizations to include spin degrees of freedom is summarized at the end of the manuscript.

Mesoscopic transport beyond linear response.

We present an approach to steady-state mesoscopic transport based on the masimum entropy principle formulation of nonequilibrium statistical mechanics. Our approach is not limited to the linear response regime. This approach yields the quantization observed in the integer quantum Hall effect at large currents, which until now has been unexplained. We also predict new behaviors of nonlocal resistances at large currents in the presence of dirty contacts.

Observability of counterpropagating modes at fractional quantum Hall edges

When the bulk filling factor is

Nonlinear steady-state mesoscopic transport: Formalism.

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