# Fractional Quantum Hall Effect

Can someone explain to me as simply as possible why the Laughlin states create energy gaps in the lowest landau level? I am trying to understand for a presentation why the Laughlin states correctly model the QHE effect when the filling factor is a odd fraction (1/3, 1/5, 1/7). As far as I understand it in order for there to be a quantized hall effect at v=1/3 the laughlin state must split the energy spectrum of the system such that there are discrete energy levels just like in the integer effect. I don't understand how this works. I know it is somehow connected with the fact the states are an incompressible liquid but I can't figure that part out either.

Do not confuse the fractional and integer quantum hall effects. At the level which you should be approaching it, they have very little in common. The IQHE is an independent electron effect --- you can understand it by thinking about one electron. The FQHE is fundamentally about the interactions between the electrons.

To study the Laughlin state requires heavy machinery. In fact, I believe (someone up to date with the literature should correct me) that there is no analytic way to prove that it is gapped. (Bearing in mind that gapped-ness is a property of the Hamiltonian + state, so this statement needs to be qualified by giving a Hamiltonian; usually, it is a simple n-particle Hamiltonian with only kinetic and mutual repulsion of an unspecified potential.)

EDIT: actually, it appears I lie. The m=3 state with Coulomb potential was solved quite a long time ago: http://prl.aps.org/abstract/PRL/v54/i6/p581_1

I imagine people have now extended this significantly. The calculation seems quite involved, however.

Yes the Laughlin wavefunction is a trial wavefunction for many interacting electrons, I understand that much. What I am trying to do is make the connection from the wavefunctions to how this relates to seeing the quantized plateaus in the hall resistance and why the resistance goes to zero as the special values of the filling factor 1/3, 1/5, 1/7.

In the IQHE it is because the energy levels of the electron is split into massively degenerate landau levels. Thus there is an energy gap between landau levels. Therefore if one landau level lets say the lowest is completely filled then one can imagine that resistivity which essentially is a measure of energy loss would go to zero because the only way to lose energy in such a system is to move a electron into the next landau level, but this is very difficult due to the energy gap between the two, thus in most systems at low temperature it can't be done. This is how I am understanding the integer effect.

In the FQHE described by the Laughlin function which as I can tell only describes the Lowest Landau Level. The power of the polynomial term (in the Laughlin wavefunction) is the reciporical of the filling factor. This then implies that for a power of 3 the LLL is 1/3 full. I guess my question is how does that 1/3 filled state create the similar effects seen in the IQHE? From what I have read there is supposed to be energy gaps within the LLL itself, but I can't see this from the wavefunction. Also the wavefunction is supposed to goto zero faster at the special values of the filling factor then for perturbations away from it but this doesn't make sense to me either.

Gokul43201
Staff Emeritus
Gold Member
I guess my question is how does that 1/3 filled state create the similar effects seen in the IQHE?
Have you come across the Composite Fermion/Boson picture? FQHE states can be pictured as IQHE states of composite particles (electrons attached to an even/odd number of flux quanta).

The standard reference is Perspectives in Quantum Hall Physics, by Pinczuk and Das Sarma.

A great introductory review (postscript file) by Steve Girvin can be downloaded here: www.bourbaphy.fr/girvin.ps[/URL]

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Have you come across the Composite Fermion/Boson picture? FQHE states can be pictured as IQHE states of composite particles (electrons attached to an even/odd number of flux quanta).

The problem there is that you can't be certain you don't have a continuum spectrum for unbinding the composite particle. It is post-hoc justified, but not a priori.

I stand by my original comment. To calculate the gap, look at the paper I linked to. It is not trivial.

Incidentally, the formation of plateaus in IQHE is *not* trivial. It is intrinsically tied up with disorder and Anderson localisation. This is often neglected in introductory texts, and I think it is a mistake; it makes people *think* they've understood something when in fact they have not at all.