Quantum Hall Effect with Composites of Magnetic Flux Tubes and Charged Particles

Composites formed from charged particles and magnetic flux tubes, proposed by Wilczek, are one model for anyons—particles obeying fractional statistics. Here we propose a scheme for realizing charged flux tubes, in which a charged object with an intrinsic magnetic dipole moment is placed between two semi-infinite blocks of a high-permeability (${\mu }_r$) material, and the images of the magnetic moment create an effective flux tube. We show that the scheme can lead to a realization of Wilczek’s anyons, when a two-dimensional electron system, which exhibits the integer quantum Hall effect, is sandwiched between two blocks of the high-${\mu }_r$ material with a temporally fast response (in the cyclotron and Larmor frequency range). The signature of Wilczek’s anyons is a slight shift of the resistivity at the plateau of the IQHE. Thus, the quest for high-${\mu }_r$ materials at high frequencies, which is underway in the field of metamaterials, and the quest for anyons, are here found to be on the same avenue.

Figure 1

The scheme which gives rise to Wilczek’s flux-tube-charge composites. (a) A charged object with an intrinsic magnetic dipole (blue circle with a red arrow) induces an array of image magnetic dipole moments within high-${\mu }_r$ blocks (shaded gray), which can be interpreted as a flux-tube-charge composite (central illustration). (b) A 2DEG in a uniform magnetic field ${\mathbf{B}}_0$ (in the IQHE state) is sandwiched between two blocks of high-${\mu }_r$ material. Dipole magnetic moments of the electrons are aligned with ${\mathbf{B}}_0$ and behave as Wilczek’s flux-tube-charge composites via the mechanism depicted in (a).

Figure 2

Parameter $-\mathrm{\Delta }=-e{\pi }^{-1}{\hslash }^{-1}\oint \mathbf{A}·\mathbf{d}\mathbf{l}$ as a function $r/d$, for three values of ${\mu }_r$; $d=10\mathrm{nm}$ (see text for details).