Scaling and Crossover Functions for the Conductance in the Directed Network Model of Edge States

TL;DR

This paper demonstrates that the directed network model of edge states on a cylindrical surface is mathematically equivalent to a supersymmetric nonlinear sigma model and a random matrix approach, revealing universal conductance properties in the scaling limit.

Contribution

It establishes a mapping of the directed network model to a supersymmetric sigma model and a random matrix framework, providing a unified understanding of conductance behavior.

Findings

Universal crossover functions for conductance and variance are identified.

The conductance distribution matches that of disordered quasi-1D wires in the scaling limit.

The model explains and quantifies previous results by Chalker and Dohmen.

Abstract

We consider the directed network (DN) of edge states on the surface of a cylinder of length L and circumference C. By mapping it to a ferromagnetic superspin chain, and using a scaling analysis, we show its equivalence to a one-dimensional supersymmetric nonlinear sigma model in the scaling limit, for any value of the ratio L/C, except for short systems where L is less than of order C^{1/2}. For the sigma model, the universal crossover functions for the conductance and its variance have been determined previously. We also show that the DN model can be mapped directly onto the random matrix (Fokker-Planck) approach to disordered quasi-one-dimensional wires, which implies that the entire distribution of the conductance is the same as in the latter system, for any value of L/C in the same scaling limit. The results of Chalker and Dohmen are explained quantitatively.