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Springer eBooks

Ensemble Density Functional Theory for Inhomogeneous Fractional Quantum Hall Systems

Olle Heinonen, M. I. Lubin, M. D. Johnson

Published Jan 1, 1998• 0 citations
Publisher
DOI
10.1007/978-1-4899-0316-7_22

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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.

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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.
Abstract The fractional quantum Hall effect remains a captivating area in condensed matter physics, characterized by strongly correlated topological order, which manifests as fractionalized excitations and anyonic statistics. Numerical simulations, such as exact diagonalization, density matrix renormalization groups, matrix product states, and Monte Carlo methods are essential for examining the properties of strongly correlated systems. Recently, density functional theory has been employed in this field within the framework of composite fermion theory. This paper systematically evaluates how density functional theory approaches have addressed fundamental challenges in fractional quantum Hall systems, including ground state and low-energy excitations. Special attention is given to the insights provided by density functional theory regarding composite fermion behavior, edge effects, and the nature of fractional charge and magnetoroton excitations. The discussion critically examines both the advantages and limitations of these approaches, while highlighting the productive interplay between numerical simulations and theoretical models. Future directions are explored, particularly the promising potential of time-dependent density functional theory for modeling non-equilibrium dynamics in quantum Hall systems.
We derive the density-density response function in the fractional quantum Hall effect using the Jastrow theory. The approximation where the two-particle correlation function is kept fixed gives as a result the single-mode approximation and the excitation spectrum derived by Girvin, MacDonald, and Platzman [Phys. Rev. Lett. 54, 581 (1985); Phys. Rev. B 33, 2481 (1986)]. The continuity equation satisfied by the system is also obtained. It is speculated that the quasiexcitons could be described by allowing also the two-particle correlation factor to change.