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Interaction and dimensionality in the quantum Hall physics

While the composite fermion picture is so effective as to describe the excitation spectra including the spin wave for Laughlin's quantum liquid, ``how heavy and how strongly-interacting" remains a formidable question for the composite fermions, to which this article first addresses. The effective mass (purely interaction originated) defined from the excitation spectrum and obtained for various even- as well as odd-fractions exhibits a curious, step-like filling dependence basically determined by the number of flux quanta attached to each fermion, where the non-monotonic behaviour indicates a strong effect of gauge-field fluctuations. The excitation spectrum fits a Fermi liquid, but again a large effect of inter-composite fermion interaction appears as anomalous Landau's parameters. We have then moved on to see how the introduction of three-dimensionality (where the shape of the Fermi surface becomes relevant) affects the interacting electron system, and propose the magnetic-field induced SDW in three-dimensional systems. This should be a good candidate, in entirely realistic magnetic fields, for the integer QHE recently predicted by Koshino et al to occur in 3D on the fractal energy spectrum similar to Hofstadter's. The mechanism for the field-induced phase is an effect of interaction in Landau's quantisation on incompletely-nested (i.e., multiply-connected) Fermi surfaces, so the interplay of many-body physics and the magnetic quantisation on various Fermi surfaces may provide an interesting future avenue for 3D systems.