Collective excitations of fractional Hall states and Wigner crystallization in higher Landau levels

An expression has been derived for the collective-excitation dispersion for fractional Hall states which occur in higher orbital Landau levels in terms of the electron pair-correlation function in these states. Explicit results for the n=1 Landau level have been obtained at fractional filling factors {ν}=(1/3) and {ν}=(1/5) based on Laughlin's trial wave functions for the ground state. The results at {ν}=(1/3) are qualitatively different from those in the lowest Landau level and are consistent with a weak quantum Hall effect at this fraction for n=1. The results for {ν}=(1/5) are similar to those in the n=0 Landau level but the collective excitations have a higher energy. We associate this increase with a decrease in the fractional filling factor at which Wigner crystallization occurs. A moment sum rule is derived for pair-correlation functions in higher Landau levels.