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I know that Quantum Hall Effect and Fractional Quantum Hall Effect origin from Landau Level quantization.

In magnetic field, the energy of in-plane(plane perpendicular to magnetic field) degree of motion is quantized, which is $E=(n+1/2)\hbar\omega$, $n$ is integer.

In experiment, both QHE and FQHE are observed in low temperature and strong magnetic field, suggesting that landau level quantization is observable under the condition $k_B T<<\hbar\omega$, where $k_B$ is Boltzman constant.

I am not sure why this condition is important in experiment.

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    $\begingroup$ The first questions to ask yourself are always "What is the data?" and "What is the experimental signature of the thing we are looking for?". Once you have those answers (and I don't because I don't know these experiments) it is likely that the answer is clear. $\endgroup$ – dmckee Dec 16 '13 at 19:58
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In general, both IQHE and FQHE are rigid quantum states, whose rigidness is protected by the finite energy gap ($h\omega$ for IQHE) between the ground state(s) and the exited states. Finite temperature can support excitations to overcome the gap, which destroys the rigidness of the state. Under finite temperature, the quantization of the Hall conductivity is no longer exact. If the energy scale of temperature $k_BT$ is greater than the gap, the rigidness will be completely destroyed, and the quantization effect can not be observed anymore.

In particular for the IQHE you considered, the Landau level is still quantized under any temperature, but the fermion occupation is not. The fermion occupation follows the Fermi-Dirac distribution. Under finite temperature, the fermion surface is no longer sharp, so the fermions can not integer fill the Landau level, as a result, the quantized transport will be smeared out by the temperature.

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  • $\begingroup$ Thank you. I think u r right. So, I think QHE and FQHE should all refer to 'strongly degenerate fermi gas'. By the way, it is possible for bose system to have QHE and FQHE effect? Maybe yes? Since landau level quantization is independent of particles are fermions or bosons. $\endgroup$ – Blue Dec 17 '13 at 16:52
  • $\begingroup$ @Blue Yes, FQHE can also appear in boson system with strong interaction. But FQHE can not be understood by filling up the Landau levels. $\endgroup$ – Everett You Dec 18 '13 at 3:48
  • $\begingroup$ Thank you very much. Could you give me some reference about FQHE boson system? $\endgroup$ – Blue Dec 18 '13 at 5:11
  • $\begingroup$ @EverettYou can composite fermion understood by the charge attaching magnetic flux filled in the lowest Landau level? $\endgroup$ – Timothy Feb 7 '14 at 23:46
  • $\begingroup$ @Blue This paper (arxiv.org/abs/1203.3268) by X.-G. Wen and Z. Wang summarized both bosonic and fermionic FQH systems. $\endgroup$ – Everett You Feb 8 '14 at 22:17

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