Network Models for Chiral Symmetry Classes of Anderson Localisation

We consider localisation problems belonging to the chiral symmetry classes, in which sublattice symmetry is responsible for singular behaviour at a band centre. As an account for the talk given in TH2002, we focus here on the chiral time-reversal invariant symmetry class. We formulate a model which has the relevant symmetries and which is a generalisation of the network model introduced previously in the context of the integer quantum Hall plateau transition. We show that this generalisation can be re-expressed as corresponding to the introduction of absorption and amplification into the original network model. This model represents a convenient starting point for analytic discussions and computational studies. It exhibits both localised and critical phases, and band-centre singularities in the critical phase approach more closely in small systems the expected asymptotic form than in other known realisations of the symmetry class. In addition, we demonstrate that by imposing appropriate constraints on disorder, a lattice version of the Dirac equation with a random vector potential can be obtained.