The mechanism of plateau formation in the fractional quantum Hall effect

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Abstract
Laughlin's fractionally charged quasi-holes and quasi-electrons are assumed to be pinned, and to be subject to a force j x a0 from the transport current. A force balance argument then explains the existence of Hall plateaus. The explanation of plateau formation in the integer quantum Hall effect (IQHE) is based on the existence of a mobility gap between extended states in different Landau levels and single-particle localisation associated with this gap. For the fractional quantum Hall effect (FQHE) it has been conjectured that, analogously, localisation of quasi-particles might account for the finite step width (Girvin 1987, Laughlin 1987). In the present Letter we propose a mechanism of plateau formation which supports the view that plateaus arise because of localisation of quasi-particles, i.e., Laughlin's fractionally charged quasi-holes and quasi-electrons. To facilitate our argumentation, which is of a hydrodynamic nature, we shall use the notions vortices and anti-vortices instead of quasi-holes and quasi-electrons. The vortices and the anti-vortices, which have opposite directions of rotation, are created in numbers proportional to the deviation of the magnetic flux from its mid-plateau value: vortices for positive deviation, and anti-vortices for negative deviation. The vortices (anti-vortices) are expected to remain pinned (Girvin 1987, Laughlin 1987) in the presence of a transport current that does not exceed the critical current where the FQHE breaks down. We assume that the vortices (anti-vortices) are subject to a force j x 'Do, where J is the current density, I'Dol = h/e, and where 'Do is parallel (antiparallel) to the magnetic field for vortices (anti-vortices). By inclusion of the reaction force from the pinning centres in a force balance equation for the electron system, we show that the Hall response, V,, is independent of the magnetic field in an interval around B, = l/m. Our picture is the following: suppose, for a two-dimensional system with Nelectrons at T = 0, we increase the magnetic field, B, relative to say B,=,. The perfect matching of electrons and magnetic flux quanta (crucial to the correlations of the v = 4 state) can then no longer be maintained everywhere. However, the electron liquid will try to maintain the favourable correlations of the v = 4 state, and as a consequence will yield