The group $\Gamma (2)$ and the fractional quantum hall effect

TL;DR

This paper studies the action of the modular group Gamma(2) on the quantum Hall effect, classifying fractional quantum Hall states and discussing phase transitions between them.

Contribution

It provides a comprehensive classification of fractional quantum Hall numbers using the Gamma(2) group action, extending previous frameworks like Jain's.

Findings

Classified fractional quantum Hall states via Gamma(2) action.

Identified possible phase transitions between quantum Hall states.

Connected mathematical group actions to physical quantum Hall phenomena.

Abstract

We analyze the action of the inhomogeneous modular group $\Gamma (2)$ on the three cusps of its principal fundamental domain in the Poincare half plane. From this, we obtain an exhaustive classification of the fractional quantum Hall numbers. This classification is somehow similar to the one given by Jain. We also present some resulting remarks concerning direct phase transitions between the different quantum Hall states.