Ordering the topological order in the fractional quantum Hall effect

TL;DR

This paper explores the topological order and quantum field theory of quantum Hall systems, emphasizing the role of global symmetries and anomalies in classifying minimal topological orders consistent with observed phenomena.

Contribution

It introduces the use of one-form global symmetry and anomalies as organizing principles to uniquely determine minimal topological orders in quantum Hall systems.

Findings

Almost all known topological orders are minimal theories.

Global symmetry constraints lead to a classification of topological orders.

The approach unifies different perspectives in the study of quantum Hall effects.

Abstract

We discuss the possible topological order/topological quantum field theory of different quantum Hall systems. Given the value of the Hall conductivity, we constrain the global symmetry of the low-energy theory and its anomaly. Specifically, the one-form global symmetry and its anomaly are presented as the organizing principle of these systems. This information is powerful enough to lead to a unique minimal topological order (or a small number of minimal topological orders). Almost all of the known experimentally discovered topological orders are these minimal theories. Since this work is interdisciplinary, we made a special effort to relate to researchers with different backgrounds by providing translations between different perspectives.