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 Laughlinʹs quantum liquid, “how heavy and how strongly-interacting” remains a formidable question for the composite fermions, 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 nonmonotonic 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 Landauʹs 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 spin density wave in three-dimensional (3D) systems. This should be a good candidate, in entirely realistic magnetic fields, for the integer quantum Hall effect recently predicted by Koshino et al., to occur in 3D on the fractal energy spectrum similar to Hofstadterʹs. The mechanism for the field-induced phase is an effect of interaction in Landauʹs 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.