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(As suggested by Tobias, I shall indicate that I will write "IQHE" for "Integer quantum Hall effect" and "FQHE" for "Fractional quantum Hall effect" below.)

I am a mathematical physicist (with a more math-driven mind), working in the operator algebraic aspect. From my personal viewpoint, mathematicians have already achieved a fairly comprehensive understanding of the geometry of the Integer Quantum Hall Effect (IQHE for short below) after the work of Bellissard, van Elst and Schulz-Baldes. The geometry of IQHE is encoded by a noncommutative Brillouin zone, whose noncommutativity is essential and inevitable due to impurity and defects of the crystal. Then the physical observables (edge conductance) are expressed as an index pairing between a carefully constructed finitely summable Fredholm module, and a cyclic cocycle which is a semifinite trace. I hope I did not make a mistake here.

It is unclear to me whether there is already a nice and satisfying rigorous mathematical understanding of the geometry of the Fractional Quantum Hall Effect (FQHE). Topological orders seem to be essential here. But if I were understanding correctly, topological orders aim at classifying the geometry of the real space. This makes already a huge difference as IQHE cares only about the noncommutative geometry of the momentum space. Also crucial is that FQHE is sensitive to disorder whereas IQHE is not. So I have the following doubts in mind:

  1. Can IQHE be interpreted as a "degenerate" case of FQHE? Namely, let the interaction tend to $0$, does IQHE emerge naturally? In that case: why does disorder and weak external magnetic field lead to this degeneracy?

  2. What is the role of topological orders in FQHE? More precisely, is there an intuitive understanding why topological orders give rise to Laughlin states?

  3. Development in recent years claims that topological orders should be understood in terms of higher categories and tensor categories. How is FQHE understood within this new framework? (As a mathematician, and since I have quite some knowledge on higher category theory, I will be happy to look at a "mathematically clear" and "rigorous" account of the story).

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I will try to give a physicist's answer to your question. I hope it comes across helpfully given that I have not read the paper you mentioned. A few comments before getting into your three questions.

  1. I do not think of the IQHE as arising from a non-commutative Brillouin zone, although one can define it that way in the clean limit. It is a gapped state that is stable to both interactions and disorder (which breaks translation symmetry), so I do not understand why thinking of the Brillouin zone is helpful except at the clean limit. If it were true that the non-commutative Brillouin zone is the essential ingredient for IQHE, it would not survive disorder.

  2. Topological orders do not aim to classify the geometry of the real space. That word is used in many contexts, but I am not aware of such a definition / connection. Topological order very generally refers to stable gapped states that have anyonic excitations, which in the FQHE state are $e/m$ charged quasiparticles, where $m>1$ is odd for fermionic FQHE, even for bosonic FQHE.

  3. Neither FQHE nor IQHE are sensitive to disorder (in the sense that an arbitrarily small amount of disorder destroys the state). This can be a confusing statement since it is often mentioned that disorder is crucial for IQHE/FQHE. It is indeed crucial to see the plateaus in $\sigma_{xy}$, the Hall conductance -- this is because these states occur at very specific filling and disorder helps pin the additional particles when you shift the magnetic field slightly away from the exact fractional filling required to form the FQHE states. If you took a disorder-free, Galilean invariant system, you would not see the plateaus. But you would still have the IQHE/FQHE states if you tuned to the correct filling.

    In a related context, we need really high quality samples (with low disorder) to see FQHE compared to IQHE. This is because the fractionally filled lowest Landau level is a highly degenerate state (in the clean limit in the absence of disorder). Repulsive interactions take this highly degenerate state to the FQHE states. If the disorder is too strong, disorder by itself lifts the degeneracy and produces a boring state where the electrons are localized. In contrast, the IQHE is not that "sensitive" to disorder because the corresponding non-interacting state is gapped and non-degenerate, so some disorder cannot destroy the state.

Next, I will try to answer your questions.

  1. No. If you start from FQHE and take the interactions to zero, you end up with a $p/q$ filled Landau level which is a highly degenerate state. This is not the IQHE state. I am not sure what you mean by "why does disorder and weak external magnetic field lead to this degeneracy?".

  2. Topological orders do not "give rise" to Laughlin states. Laughlin state is an example of a topologically ordered state. Some other examples of topologically ordered states are the toric code, the double Semion model, and Levin-Wen string net models.

  3. Like I mentioned earlier, a general definition of topologically ordered states are gapped states that have anyonic excitations. Turns out in 2+1 dimensions, you can characterize the low-energy behavior of such states by specifying

    • The list of anyons
    • Its fusion properties, i.e. what is the new anyon that you get when you combine two anyons together.
    • It's braiding properties, i.e. what happens to the quantum state if you take two anyons and braid them around each other.

    If you are familiar with CFTs, this is analogous to specifying the CFT data, i.e. the primary fields, its scaling dimensions and OPEs and so on. This data is specified using higher category theory -- more specifically a unitary modular tensor category. The type of category you use gets more complicated if you go to higher spatial dimensions because you can have more exotic excitations.

Please comment if you have any questions, I'm sure there is a language barrier here and I am happy to clarify things.

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