Identifying the topological order of quantized half-filled Landau levels through their daughter states

Identifying the topological order of potentially non-Abelian half-filled quantum Hall states is notoriously difficult. While ``daughter states'' have been proposed as a signature, existing calculations were incomplete. This work provides a comprehensive framework based on hierarchical construction and flux attachment, predicting daughter states for all candidate parent states. Observing daughter states on both sides of half-filling enables the identification of the parent's topological order. Applied to experiments, this reveals a Pfaffian state in wide-well GaAs and a Pfaffian/anti-Pfaffian sequence in graphene, a contrast to PH-Pfaffian in narrow-well GaAs.