“Classical” limit of Quantum Hall Effect

Question

Imagine a partially filled $\nu=1$ state of the integer quantum Hall effect (IQHE). One way to think about it is to imagine a gas of electrons where each particle is locked to the lowest quantum state of their circular motion (a Gaussian wave-packet in both $x$ and $y$ with the characteristic length being the magnetic length $l_0$). When this gas becomes degenerate, we get a fully filled $v=1$ Landau level and the bulk conductivity vanishes.

I wonder, how does a single-particle theory of a partially filled$\nu=1$ IQH state state look like in terms of such particles? What is the kinetic energy term? I seek a kind of field theory for this, valid on the scales of $k \ll l_0^{-1}$ and regularized in a consistent way on the scale of $l_0$. Perhaps a tight-binding-like lattice model?

I would bet it has been done (if possible at all) by some "high-flying" quantum field theorists out there , but I have not seen such concept being used in "down-to-Earth" experiment-oriented calculations of IQHE.

Answer 1

Actually, this sounds improbable, because the quantum hall effects are fundamentally about interactions. A fully filled Landau level makes it possible (after much handwaving, etc.) to argue that these may be neglected, but once you don't then the interactions dominate. In fact, it's exactly because of this that you get the Fraction Quantum Hall Effects.