Girvin–MacDonald–Platzman collective mode at general filling factors: magneto-roton minimum at half-filled Landau level

The single mode approximation has proved useful for the excitation spectrum at ν=1/3. We apply it to general fractions and find that it predicts n magneto-roton minima in the dispersion of the Girvin–MacDonald–Platzman collective mode for the fractional quantum Hall states at ν=n/(2n+1), and one magneto-roton minimum for both the composite Fermi sea and the paired composite fermion state. Experimental relevance of the results will be considered.