= Solution
On an integer plateau the sheet transverse resistivity has magnitude
$$
|\rho_{xy}|=\frac{h}{\nu e^2},
$$
where the integer $\nu$ counts occupied resolved <quantum Hall edge states>. Increasing $B$ at fixed sheet density reduces the <Landau filling factor> $\nu_{\mathrm{fill}}=n_sh/(eB)$. The observed trace is therefore a staircase toward larger $|\rho_{xy}|$, with transitions between plateaux rather than the classical straight line. In the plateau regions $\rho_{xx}\simeq0$; dissipative peaks occur at transitions.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2023/paper-2-hall-response.png]
{title=Classical Hall line and illustrative integer quantum Hall plateaux}
{description=Original schematic responses, with the classical slope fixed by the later calculation. Plateau widths and dissipative peak shapes are illustrative rather than fitted to an experimental trace.}
{source=paper-2-hall-response.py}
It is not correct to demand an exactly integer value of $n_sh/(eB)$ everywhere along a finite-width plateau. Localization accommodates changes in total population without changing the number of extended current-carrying channels. If spin remains unresolved, the usual spin-degenerate sequence contains even integers rather than every integer.
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