Network Models of Quantum Percolation and Their Field-Theory Representations

Abstract

We obtain the field-theory representations of several network models that are relevant to 2D transport in high magnetic fields. Among them, the simplest one, which is relevant to the plateau transition in the quantum Hall effect, is equivalent to a particular representation of an antiferromagnetic SU(2N) ($N\rightarrow 0$) spin chain. Since the later can be mapped onto a $\theta\ne 0$, $U(2N)/U(N)\times U(N)$ sigma model, and since recent numerical analyses of the corresponding network give a delocalization transition with $\nu\approx 2.3$, we conclude that the same exponent is applicable to the sigma model.