Landau levels and integer quantum Hall effect
A perpendicular magnetic field organizes two-dimensional electron motion into quantized cyclotron levels.
$$E_n=\hbar\omega_c\left(n+\frac12\right)$$
What this means
A perpendicular magnetic field organizes two-dimensional electron motion into quantized cyclotron levels.
Deep dive: mathematical and physical mechanism
For a parabolic band, $\omega_c=|eB|/m$ and each Landau level has large degeneracy. When the Fermi energy lies in a mobility gap, the Hall conductance is quantized. Edge channels arise at boundaries where confining potentials bend Landau levels.
How it appears in quantum phenomena
Two-dimensional electron gases at low temperatures and strong fields.
Assumptions, caveats and limitations
Landau levels alone do not explain plateau widths; disorder and localization matter.
Connect the ideas
For context, see the Topological Band Theory topic guide and the reading references.