Positions of the magnetoroton minima in the fractional quantum Hall effect

TL;DR

This paper uses microscopic composite fermion theory to accurately predict the positions of magnetoroton minima in fractional quantum Hall states, confirming previous theories and showing insensitivity to microscopic interactions.

Contribution

It provides detailed calculations of magnetoroton minima positions for various filling factors, validating and extending existing theoretical predictions.

Findings

Magnetoroton minima positions agree with Chern-Simons and Golkar et al. theories.

Minima positions are insensitive to microscopic interaction details.

Charge and neutral gaps are obtained for specific Landau levels.

Abstract

The multitude of excitations of the fractional quantum Hall state are very accurately understood, microscopically, as excitations of composite fermions across their Landau-like $\Lambda$ levels. In particular, the dispersion of the composite fermion exciton, which is the lowest energy spin conserving neutral excitation, displays filling-factor-specific minima called "magnetoroton" minima. Simon and Halperin employed the Chern-Simons field theory of composite fermions [Phys. Rev. B {\bf 48}, 17368 (1993)] to predict the magnetoroton minima positions. Recently, Golkar \emph{et al.} [Phys. Rev. Lett. {\bf 117}, 216403 (2016)] have modeled the neutral excitations as deformations of the composite fermion Fermi sea, which results in a prediction for the positions of the magnetoroton minima. Using methods of the microscopic composite fermion theory we calculate the positions of the roton minima for filling factors up to 5/11 along the sequence $s/(2s+1)$ and find them to be in reasonably good agreement with both the Chern-Simons field theory of composite fermions and Golkar \emph{et al.}'s theory. We also find that the positions of the roton minima are insensitive to the microscopic interaction in agreement with Golkar \emph{et al.}'s theory. As a byproduct of our calculations, we obtain the charge and neutral gaps for the fully spin polarized states along the sequence $s/(2s\pm 1)$ in the lowest Landau level and the $n=1$ Landau level of graphene.