In your title, you ask about **fractional QHE** but the Chern number description as far as I know holds for **integer QHE**. I don't know that much about FQHE, but let me say a bit about IQHE which I understand a bit better.

The famous paper of TKNN established that the Hall conductivity at integer filling is proportional to a topological invariant associated to the band structure of 2 dimensional Hamiltonians. This topological invariant, the Chern number, is an integer which tells us how the band structure "twists" over the Brillouin zone (this is the "torus" in your question) (more formally, the Chern number classifies the complex vector bundle associated to the band Hamiltonian). Remember for now that this is a property of the band structure, which comes out of a description where electron-electron interactions are neglected, i.e. we are not dealing with a "strongly-correlated system".

Side comment: considering a sort of $Z_2$ equivariant (roughly, time-reversal invariant) version of this topological invariant led to the current extremely hot topic of 3D topological insulators, as initiated by Fu and Kane (among others).

The fractional QHE does not admit a single-particle description -- that is, you can't understand the properties from band theory, as in IQHE -- it is a phase of electron behavior which results from **interactions**. Thus, I don't think the Chern number description carries over easily.

The FQHE admits a **Chern-Simons Landau-Ginzburg** description, which I've been reading a bit about in this 1992 review by Shou-Cheng Zhang. The Chern-Simons term in this field theory should not be confused with Chern numbers! (I'm not sure if that's what you're doing in the question, but I want to make that clear.) The notions are related mathematically, but I believe the physics here is distinct.

If you were just asking about IQHE, the insight of TKNN that the IQHE states are classified by a topological invariant probably rules out other independent descriptions. I might be confused as to your intent, but it seems unlikely that there could be a useful description of the IQHE states which doesn't use topology (topological invariant = stable to perturbations, which leads to the amazing plateaus, after all), and the topological situation is really pretty well understood at this point in time.

Please let me know if anything is unclear or if I've said something wrong. I too am just a learner in this field.

I might come back and add some stuff about FQHE if I ever get around to understanding it better.

This post imported from StackExchange Physics at 2014-04-04 16:31 (UCT), posted by SE-user j.c.