Magneto-roton Theory of Collective Excitations in the Fractional Quantum Hall Effect

We present a theory of the collective excitation spectrum in the fractional quantum Hall effect which is closely analogous to Feynman's theory of superfluid helium. The predicted spectrum has a large gap at k=0 and a deep magneto-roton minimum at finite wave vector, in excellent quantitative agreement with recent numerical calculations. We demonstrate that the magneto-roton minimum is a precursor to the gap collapse associated with the Wigner crystal instability occurring near \ensuremath{\nu}=(1/7). In addition to providing a simple physical picture of the collective excitation modes, this theory allows one to compute rather easily and accurately experimentally relevant quantities such as the susceptibility and the ac conductivity.