Network models for localisation problems belonging to the chiral symmetry classes

TL;DR

This paper develops network models for chiral symmetry class localization problems, revealing new critical behaviors and providing a basis for analytical and computational studies of disordered systems.

Contribution

It introduces generalized network models with absorption and amplification, linking them to lattice Dirac equations and exploring critical phenomena in two-dimensional systems.

Findings

Models exhibit localized and critical phases.

Band-centre singularities approach expected asymptotic forms.

New types of critical behavior identified.

Abstract

We consider localisation problems belonging to the chiral symmetry classes, in which sublattice symmetry is responsible for singular behaviour at a band centre. We formulate models which have the relevant symmetries and which are generalisations of the network model introduced previously in the context of the integer quantum Hall plateau transition. We show that the generalisations required can be re-expressed 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 behaviour. 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 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.