# The sixteenfold way and the quantum Hall effect at half-integer filling factors

## TL;DR

This paper reviews the topological orders and experimental signatures of half-integer fractional quantum Hall states across various materials, emphasizing the role of the sixteenfold way classification in understanding their properties.

## Contribution

It provides a comprehensive analysis of all possible composite-fermion states at half-integer filling, unifying their topological orders using Kitaev's sixteenfold way framework.

## Key findings

- Identifies experimental signatures for different topological orders.
- Introduces a unified description of topological orders in these states.
- Analyzes implications for interferometry and thermal transport experiments.

## Abstract

Fractional quantum Hall states at half-integer filling factors have been observed in many systems beyond the $5/2$ and $7/2$ plateaus in GaAs quantum wells. This includes bilayer states in GaAs, several half-integer plateaus in ZnO-based heterostructures, and quantum Hall liquids in graphene. In all cases, Cooper pairing of composite fermions is believed to explain the plateaus. The nature of Cooper pairing and the topological order on those plateaus are hotly debated. Different orders are believed to be present in different systems. This makes it important to understand experimental signatures of all proposed orders. We review the expected experimental signatures for all possible composite-fermion states at half-integer filling. We address Mach-Zehnder interferometry, thermal transport, tunneling experiments, and Fabry-P\'{e}rot interferometry. For this end, we introduce a uniform description of the topological orders of Kitaev's sixteenfold way in terms of their wave-functions, effective Hamiltonians, and edge theories.