Shape of the magnetoroton atν=1/3andν=7/3in real samples

We revisit the theory of the collective neutral excitation mode in the fractional quantum Hall effect at Landau level filling fractions $\ensuremath{\nu}=1/3$ and $\ensuremath{\nu}=7/3$. We include the effect of finite thickness of the two-dimensional electron gas and use extensive exact diagonalizations in the torus geometry. In the lowest Landau level the collective gapped mode, i.e., the magnetoroton always merges in the continuum in the long-wavelength limit. In the second Landau level the mode is well defined only for wave vectors smaller than a critical value and disappears in the continuum beyond this point. Its curvature near zero momentum is opposite to that of the LLL. It is well separated from the continuum even at zero momentum, and the gap of the continuum of higher-lying states is twice the collective mode gap at $k=0$. The shape of the dispersion relation survives a perturbative treatment of Landau level mixing.