The Absence of the Fractional Quantum Hall Effect at High Landau Levels

TL;DR

This paper investigates why the fractional quantum Hall effect at filling factors 1/3 and 1/5 is absent at high Landau levels, showing charge density waves are energetically favored over Laughlin liquids beyond certain Landau levels.

Contribution

The study demonstrates that charge density waves become more stable than Laughlin liquids at high Landau levels, explaining the absence of the fractional quantum Hall effect in these regimes.

Findings

Charge density wave period is approximately 3 times the cyclotron radius.

Charge density waves are energetically favored over Laughlin liquids for N ≥ 2 at 1/3 filling.

Fractional quantum Hall effect is absent for N ≥ 2, consistent with experimental observations.

Abstract

We compare the energies of the Laughlin liquid and a charge density wave in a weak magnetic field for the upper Landau level filling factors $\nu_N = 1/3$ and $1/5$. The charge density wave period has been optimized and was found to be $\simeq 3R_c$, where $R_c$ is the cyclotron radius. We conclude that the optimal charge density wave is more energetically preferable than the Laughlin liquid for the Landau level numbers $N \ge 2$ at $\nu_N = 1/3$ and for $N \ge 3$ at $\nu_N = 1/5$. This implies that the $1/3$ fractional quantum Hall effect cannot be observed for $N \ge 2$, in agreement with the experiment.