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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.

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Sorry for being so vague, it's more a search of a question than than asking a well-defined answer. – Slaviks Aug 20 '11 at 17:54
up vote 4 down vote accepted

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.

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Sorry, this is wrong. Integer quantum Hall effect does not require interactions, it is a Fermi gas problem in a magnetic field + confining potential. For parabolic confinement it is an exactly solvable model (Fock-Darwin). – Slaviks Aug 21 '11 at 9:18
It seems I was too quick to judge and missed your point. So is the essence of you answer that a "rarified" ($\nu <1$) 2D gas is interaction-dominated? – Slaviks Aug 21 '11 at 9:22
Yes. The point is that inside each Landau level the kinetic energy (which is all you have without interactions) is exactly the same for all electrons, in all states. If you partially fill that level, you end up with a huge number of states (binomially large) which are degenerate in energy. These will be strongly split by anything you then introduce. – genneth Aug 21 '11 at 15:42
... including, e.g. disorder. Thank for elucidating the QHE picture for me! – Slaviks Aug 21 '11 at 15:48
Actually, disorder is not so important --- it modifies each electrons separately and does not force correlated motion. – genneth Aug 21 '11 at 21:59

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