What are the topological phases of quantum Hall systems?

Fermionic modular categories and the 16-fold way classifies the topological phases of the fractional quantum Hall effect. The Laughlin states (Abelian anyons at filling factor $1/Q$, $Q$ odd) are discussed in example 2.6. The Moore-Read state (non-Abelian Ising anyons at half-integer filling factor) is in example 2.4.

We have a paper that contains lists of simple fermionic topological orders in 2+1D: https://arxiv.org/abs/1507.04673 . For fermionic topological without symmetry, there is no filling fraction. So our lists are based on the number of anyon types, together with their quantum dimensions and topological spins. Each entry in the table corresponds to a sequence of fermionic topological orders, differ by stacking of $p+ip$ invertible fermionic topological orders.

FQH states have additional U(1) symmetry and correspond to SET states, which do have filling fraction. Filling fraction do not determine the topological order. For a given filling fraction, there can be many different topological orders.

Every modular tensor category describes a bosonic topological order (up to $E_8$ invertible topological order). However, fermionic topological orders are not classified by modular tensor category. They are classified by a special type of braided fusion category, with modular extension. This is the main point of our work.