Introduction to quantum Hall effects

The quantum Hall effect is a set of phenomena observed at low temperature in a two-dimensional electron gas subject to a strong perpendicular magnetic field. It manifests itself as a quantization of the nondiagonal elements (ρxy) of the resistivity tensor accompanied by simultaneous vanishing ρxx for ranges of magnetic field. For the integer quantum Hall effect ρxy =h/νe2, where h is the Planck constant, e - charge of an electron and ν is an integer, while for fractional Hall effect ν is a simple fraction. In spite of similar phenomenology deep and profound differences exist between these two effects. In the lecture the precision of the Hall quantization in the integer quantum Hall effect are discussed. Some recent theoretical and experimental discoveries connected with quantum Hall liquids are also mentioned. (author)