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

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 $ν_N = 1/3$ 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 $ν_N = 1/3$ and for $N \ge 3$ at $ν_N = 1/5$. This implies that the $1/3$ fractional quantum Hall effect cannot be observed for $N \ge 2$, in agreement with the experiment.