Network Models of Quantum Percolation and Their Field-Theory Representations

TL;DR

This paper develops field-theory representations for network models relevant to 2D quantum transport in high magnetic fields, linking them to sigma models and analyzing critical exponents for delocalization transitions.

Contribution

It introduces field-theory representations for key network models, connecting quantum Hall transition models to sigma models and providing insights into critical exponents.

Findings

The network model for quantum Hall transition maps to a specific sigma model.

Numerical analysis suggests a delocalization transition with critical exponent approximately 2.3.

The sigma model's critical behavior aligns with network model results.

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) ($N\to 0$) 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.