Random network models and quantum phase transitions in two dimensions

An overview of the random network model invented by Chalker and Coddington, and its generalizations, is provided. After a short introduction into the physics of the Integer Quantum Hall Effect, which historically has been the motivation for introducing the network model, the percolation model for electrons in spatial dimension 2 in a strong perpendicular magnetic field and a spatially correlated random potential is described. Based on this, the network model is established, using the concepts of percolating probability amplitude and tunneling. Its localization properties and its behavior at the critical point are discussed including a short survey on the statistics of energy levels and wave function amplitudes. Magneto-transport is reviewed with emphasis on some new results on conductance distributions. Generalizations are performed by establishing equivalent Hamiltonians. In particular, the significance of mappings to the Dirac model and the two-dimensional Ising model is discussed. A description of renormalization group treatments is given. The classification of two-dimensional random systems according to their symmetries is outlined. This provides access to the complete set of quantum phase transitions like the thermal Hall transition and the spin quantum Hall transition in two dimensions. The supersymmetric effective field theory for the critical properties of network models is formulated. The network model is extended to higher dimensions including remarks on the chiral metal phase at the surface of a multi-layer quantum Hall system.