Modular Groups, Visibility Diagram and Quantum Hall Effect

TL;DR

This paper introduces a model based on the modular group action to classify quantum Hall states, visualized through visibility diagrams, and predicts resistivity plateau ratios confirmed by numerical simulations matching experimental data.

Contribution

It develops a novel classification model for quantum Hall states using visibility diagrams derived from modular group actions, linking mathematical structures to physical phenomena.

Findings

Visibility diagrams differentiate even and odd denominator fractions.

The model predicts ratios of resistivity plateau widths.

Numerical simulations align with experimental measurements.

Abstract

We consider the action of the modular group $\Gamma (2)$ on the set of positive rational fractions. From this, we derive a model for a classification of fractional (as well as integer) Hall states which can be visualized on two ``visibility" diagrams, the first one being associated with even denominator fractions whereas the second one is linked to odd denominator fractions. We use this model to predict, among some interesting physical quantities, the relative ratios of the width of the different transversal resistivity plateaus. A numerical simulation of the tranversal resistivity plot based on this last prediction fits well with the present experimental data.