Quantum Hall Effect

The history and the experimental conditions leading to the discovery of the quantum Hall effect are discussed with a view to compare and contrast with the classical version of the effect. Landau levels are obtained for electrons confined in two dimensions (2D) in the presence of a strong transverse magnetic field. Their characteristics, such as, huge degeneracy, conductance properties, incompressibility etc. are discussed. The role of conduction via the edge modes in quantum Hall samples, and that it earned them the nomenclature of topological insulators, is emphasized. The Hall resistivity is computed using the Kubo formula, and the quantization of the hall plateaus is shown to be directly related to a topological invariant called the Chern number. A comparison of the above scenario observed in a 2D electron gas is performed by computing the Landau levels in graphene which yields feasibility of realizing the quantum Hall effect at the room temperature. Subsequently, the above discussion of the integer quantum Hall effect is supplemented by introducing the fractional quantum Hall effect, where the quantization of the hall plateaus is observed at fractional values which underscores the role of electronic interactions. We have stated the properties of the variational wavefunction due to Laughlin, and its success in explaining the odd-denominator fractions observed in experiments. Next, the idea of composite fermions due to Jain is shown to yield a much simpler and significantly intuitive picture of an enormously complicated many-particle problem. Eventually, to explain a lot of other fractions observed in the experiments, a discussion of the hierarchy scenario is invoked.