# Random network models with variable disorder of geometry

## TL;DR

This paper explores how varying the geometric disorder in random network models affects localization properties, revealing a line of critical points with a minimum at a specific disorder probability, aligning with quantum Hall effect observations.

## Contribution

It extends previous models by analyzing the impact of different disorder strengths on critical behavior, identifying a continuum of critical points with a minimum at p=1/3.

## Key findings

- Presence of a line of critical points with varying localization length indices.
- Minimum localization length index occurs at p=1/3.
- Results align with experimental quantum Hall transition data.

## Abstract

Recently it was shown (I.A.Gruzberg, A. Kl\"umper, W. Nuding and A. Sedrakyan, Phys.Rev.B 95, 125414 (2017)) that taking into account random positions of scattering nodes in the network model with $U(1)$ phase disorder yields a localization length exponent $2.37 \pm 0.011$ for plateau transitions in the integer quantum Hall effect. This is in striking agreement with the experimental value of $2.38 \pm 0.06$. Randomness of the network was modeled by replacing standard scattering nodes of a regular network by pure tunneling resp.reflection with probability $p$ where the particular value $p=1/3$ was chosen. Here we investigate the role played by the strength of the geometric disorder, i.e. the value of $p$. We consider random networks with arbitrary probability $0 <p<1/2$ for extreme cases and show the presence of a line of critical points with varying localization length indices having a minimum located at $p=1/3$.