The Fractional Quantum Hall Effect: Phonons, Rotons and Fractionally Charged Vortices

Abstract

Despite the fact that one is dealing with fermions and not bosons, there are close analogies between the fractional quantum Hall effect (FQHE) and superfluidity in helium films. A theory has been developed [1] for the collective excitation spectrum in the FQHE which is closely analogous to Feynman's theory of superfluid 4He. The physical content of the theory is the notion that the continuum of single-particle excitations is quenched by the magnetic field, leaving a single collective mode (per Landau level) which absorbs nearly all of the available oscillator strength. Thus, despite the Fermi statistics, the system behaves much like a boson superfluid. At long wavelengths the collective-mode is a phonon (densitywave) but unlike the case of 4He, the phonon has a large gap. At the characteristic wave vector associated with the interparticle spacing, the collective-mode energy suffers a deep "magneto-roton" minimum analogous to the roton minimum in helium. The single mode approximation is quantitatively quite accurate and allows one to rather easily and accurately compute experimentally relevant quantities such as the AC conductivity and the static susceptibility in addition to the mode energy itself.Pursuing the superfluidity analogy further shows that the analogs of quantized vortices are Laughlin's fractionally charged quasi-particles. The existence of a finite rather than divergent vortex energy (as occurs in helium) is associated with the absence of a gapless Goldstone mode via a remarkable analog of the Anderson-Higgs mechanism which leads, not to flux quantization, but rather to (fractional) charge quantization. Some first steps towards a Landau-Ginsburg theory incorporating these ideas will be presented.