Network models for localization problems belonging to the chiral symmetry classes

We consider localization problems belonging to the chiral symmetry classes, in which sublattice symmetry is responsible for singular behavior at a band center. We formulate models that have the relevant symmetries and that are generalizations of the network model introduced previously in the context of the integer quantum Hall plateau transition. We show that the generalizations required can be reexpressed as corresponding to the introduction of absorption and amplification into either the original network model, or the variants of it that represent disordered superconductors. 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, as well as new types of critical behavior. These models represent a convenient starting point for analytic discussions and computational studies, and we investigate in detail a two-dimensional example without time-reversal invariance. It exhibits both localized and critical phases, and band-center singularities in the critical phase approach the expected asymptotic form more closely in small systems than in other known realizations of the symmetry class.