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(Fractional) quantum Hall systems are $2+1$-dimensional models which are said to possess topological order. One (maybe even complete) set of invariants of topological phases in $2+1$ dimensions is given that anyon statistics, or in mathematical terms, the (2-extended) axiomatic TQFT of the phase. Both are labelled by modular fusion categories.

Question: Which are the modular fusion categories associated to quantum Hall systems? Is there a table somewhere in the literature that associates to each value of the "filling fraction" (and whatever other parameters there are in quantum Hall models) a modular fusion category?

Also: Quantum Hall systems involve fermionic degrees of freedom. Are they proper fermionic phases? Or are the fermions somehow restricted to local islands of even parity, such that the resulting phases are still bosonic? In the former case, the anyon statistics are not described by modular fusion categories but their fermionic analogue, something like "modular super-fusion categories".

I guess this is written down at a lot of places, I'm just having a hard time finding it without having to skim through lots of condensed matter literature that I don't understand.

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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.

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  • $\begingroup$ Thanks! From your reference I extract that the answer to the "also" question is yes, quantum Hall systems are proper fermionic phases, with their anyon statistics described by super-modular fusion categories? $\endgroup$
    – Andi Bauer
    Commented Jun 28, 2019 at 11:37
  • $\begingroup$ Are the Laughlin and Moore-Read theories the only ones describing quantum Hall systems? Or are there other theories for other filling factors, corresponding to other topological phases? $\endgroup$
    – Andi Bauer
    Commented Jun 28, 2019 at 11:40
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    $\begingroup$ the Laughlin and Moore-Read phases are the two topological phases that are most likely to correspond to physical reality in a single layer system, meaning that they have the lowest energy; double-layer systems allow for other phases, including the Read-Rezayi state (Fibonacci anyons at filling factor 2/3). Table 1 of the Zoo of quantum-topological phases of matter gives an overview (the states labelled 2F are quantum Hall states). $\endgroup$
    – Carlo Beenakker
    Commented Jun 28, 2019 at 12:30
  • $\begingroup$ Concerning the need to account for the fermionic nature of the quasiparticles: in a simplified description (known in physics as "bosonisation"), one can ignore that. One would then still recover the correct braiding algebra and fusion rules, but the fermionic spin introduces additional Abelian phase factors that are missed in the bosonic treatment. $\endgroup$
    – Carlo Beenakker
    Commented Jun 28, 2019 at 15:24
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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.

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