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Though the statement is made often, I've not seen any justification for neglecting electron-electron (Coulomb) interactions in the fully filled $\nu =1$ IQH state. I would highly appreciate if someone could provide an argument for the same.

Also, I don't have a background in solid-state physics, maybe I'm missing something obvious?

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You are not missing something obvious. In fact, for a long time, solid-state physicists wondered how interactions in solids could ever be neglected at all! This was especially true in light of the great experimental successes of band theory (which again neglects interactions).

The justification came when Landau formulated his theory of the Fermi liquid. Essentially what he said was that if one adiabatically turns on interactions, the "quasiparticles" (basically the screened electrons) near the Fermi energy act as if they are non-interacting except that they have a renormalized mass, and a few other similar qualitatively non-essential parameters. What this theory did at the end of the day, was justify the neglect of interactions in many cases.

In many quantum hall systems, this theory remains a good one and we can neglect the interactions. However, when one starts talking about the fractional quantum hall effect, one can no longer neglect interactions to describe experimental observations.

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