Ensemble density functional theory for inhomogeneous fractional quantum hall systems

Abstract

The fractional quantum Hall effect (FQHE) occurs at a 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 than about 10 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. © 1996 John Wiley & Sons, Inc.

Ensemble density-functional approach to charge-spin textures in inhomogeneous quantum Hall systems

“…IN ordinary DFT the occupancies f i,α are zero or one. In our ensemble DFT calculations, we obtain fractional occupation numbers using a method of running averages described in our earlier work 14,15 .…”

“…(14), and the single-particle density matrix found self-consistently using Eq. (15). We now make the simplifying assumption that the polar angle θ(r) of the spin density is a function of the radial coordinate alone, θ(r) = θ(r), and that the azimuthal angle ϕ(r) is of the form…”

“…We have previously developed an ensemble density functional approach for spin-polarized systems 14,15 , and subsequently generalized that approach to include the electron spin, but with the spin quantization axis constrained to be parallel to the external magnetic field 16 . We present here a further extension which is able to deal with a rotating spin quantization axis.…”

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

“…(18)] can be used. However, a procedure to compute the fractional occupancies f α has not existed 32 , and one major advance in our work is that we have found a simple way to generate the occupancies, at least for the FQHE 14,15 . Applying ensemble DFT to the FQHE, we have found that fractionallyoccupied KS orbitals are indeed degenerate at the Fermi energy, consistent with our demonstration above that the FQHE is in general not pure-state v-representable.…”

“…Motivated by these ideas, we have generalized our ensemble DFT approach to include spin degrees of freedom within the LSDA [Eq. (15)]. We have applied this generalization to parabolic maximum density droplets to study the edge reconstruction as a function of confinement strength (magnetic field) and Zeeman coupling.…”