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I was wondering why the plateaus of $\rho_{xy}$ in the integer quantum Hall effect are horizontal and do not scale linearly with the magnetic field $B$ since the Lorentz force should still be acting on the electrons.

I understand that by increasing the magnetic field B, new Landau levels become available which contribute to the discrete jumps of $\rho_{xy}$, but those electrons in the Landau levels that are already occupied should be affected by the Lorentz force (which is the underlying mechanism of the classical Hall effect) and hence I would expect them to contribute to a linear scaling of $\rho_{xy}$ as a function of B in between the discrete jumps. Instead of a linear relationship, we can see perfectly horizontal plateaus. Why is that?

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    $\begingroup$ As you seem to have understood, the lower Landau levels become, not only occupied, but filled. Hint: how can electrons in a filled Landau level respond to a small increase in $B$? $\endgroup$ Feb 16, 2022 at 18:46
  • $\begingroup$ @MariusLadegårdMeyer So maybe I don't quite get the hint, but are you implying that the electrons in the filled Landau levels are not influenced by the magnetic field at all? If so, then what about the electrons in the upmost Landau level - I would assume that those should be influenced by the magnetic field and are therefore prone to the Lorentz force. $\endgroup$
    – xabdax
    Feb 17, 2022 at 12:39
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    $\begingroup$ No, the hint is that filled Landau levels are often called "inert". In the presence of disorder (so in any real system), each Landau level lifts its degeneracy and broadens into a small band of energies. The ones near the middle are the edge states that are able to carry current through the sample, while the ones near the top and bottom energies of the band are localized states around the impurities. These cannot contribute to the current, so changing $B$ a little only affects these localized states, and the current does not change. Hence the resistance stays the same. $\endgroup$ Feb 17, 2022 at 13:43
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    $\begingroup$ A quick read that covers this is e.g. here, on page 12. $\endgroup$ Feb 17, 2022 at 13:44
  • $\begingroup$ @MariusLadegårdMeyer Yeah I think that makes sense, and thank you for the reference! $\endgroup$
    – xabdax
    Feb 17, 2022 at 16:36

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I guess the question is, given the Hall resistivity as \begin{equation*} R_{xy}=\frac{B}{ne} \end{equation*}

and with $\nu$ Landau levels filled (i.e, density of electrons fixed), why $R_{xy}$ does not go up with $B$.

The simple answer is that the degeneracy of each Landau level also goes up linearly with $B$, so the two contributions cancel:

\begin{align*} R_{xy}=\frac{B}{\nu\times eB/h}=\frac{h}{e^2\nu} \end{align*}

hence a plateau in $R_{xy}$.

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