 Solution

In a two-dimensional electron system under a perpendicular magnetic field, orbital energies form Landau levels with spacing $\hslash {\omega }_c$. Each resolved spin level has $eBA/h$ states in area $A$. At integer filling, the integer quantum Hall effect has Hall conductance ${\sigma }_{xy}=\nu e^2/h$ in magnitude and vanishing longitudinal conductance in the ideal plateau regime; inversion of the conductivity tensor gives Hall resistance $h/(\nu e^2)$.

A Hall-bar experiment cools the sample, passes a small current and records longitudinal and transverse voltages while sweeping the field. The Hall resistance develops plateaus while the longitudinal resistance develops minima. Disorder broadens the Landau levels and localizes many bulk states, allowing the chemical potential to move through a range of carrier density without changing the number of conducting edge channels. Extended states near a Landau-level center allow transitions between plateaus. Spin splitting determines which integer sequence is resolved. The robustness comes from the quantum state and its edge transport, rather than the Hall bar having a specially calibrated geometric width.