Class C models

The dimensionality and the symmetries of the Hamiltonian determine the universality classes of the system for scaling behaviour. In one of these classes, known as class C, properties of suitably chosen quantum lattice models can be expressed in terms of observables for a classical model defined on the same lattice. Models in class C arise from the Bogoliubov de-Gennes Hamiltonian for quasiparticles in a gapless, disordered spin-singlet superconductor with broken time-reversal symmetry for orbital motion but negligible Zeeman splitting. Here the special energy is the chemical potential and pairs of levels are related by particle-hole symmetry, which has profound consequences for the influence of disorder on quasiparticle eigenstates. The quantum to classical mapping provides a framework within which quasiparticle properties can be studied in great detail.

We have studied the disorder-induced localisation transition in a three-dimensional class C network model. We can calculate the conductance and density of states as averages in a classical system of dense, interacting random walks. Using this mapping, we presented a more precise numerical study of critical behaviour at an Anderson transition than has been possible previously in any context. We study the spin quantum Hall transitions in a two-dimensional network model consisting in several coupled layers.