$\mathrm{U}(2)$ Chern-Simons-Ginzburg-Landau Theory of Fractional Quantum Hall Hierarchies

TL;DR

This paper develops a unified effective field theory framework for Abelian and non-Abelian fractional quantum Hall hierarchies, capturing known states and topological orders.

Contribution

It introduces a $ ext{U}(2)$ Chern-Simons-Ginzburg-Landau approach that reproduces all known fractional quantum Hall hierarchy states and their topological properties.

Findings

Reproduces all filling fractions from wavefunction and categorical data.

Uniquely determines the topological orders of hierarchy states.

Identifies a particle-hole symmetry relating different hierarchy sequences.

Abstract

We construct effective $\mathrm{U}(2)$ Chern-Simons-Ginzburg-Landau theories for Abelian and non-Abelian fractional quantum Hall hierarchies for those which had previously been described only through categorical data or trial wavefunctions. Our framework captures both Abelian hierarchy states built on half-filled Pfaffian-type parents and non-Abelian hierarchies emerging from Abelian states. It reproduces all filling fractions obtained from wavefunction and categorical constructions and, moreover, uniquely determines the corresponding topological orders. We also identify an intriguing particle-hole symmetry relating two hierarchy sequences, one built on a trivial insulator and the other on the $\nu=1$ integer quantum Hall state, which respectively generate the Read-Rezayi sequences and their particle-hole conjugates under the same hierarchy construction.