Imagine a gas of free electrons. When we apply a magnetic field, the electrons begin to orbit. Their energy states become quantized in the direction perpendicular to the field. These quantized levels are called Landau levels. Each of the Landau levels has a large degeneracy, propotional to the value of the magnetic field. For a sufficiently large magnetic field, the degeneracy becomes so high that all the free electrons fall into the lowest few Landau levels. If we can measure the transitions between Landau levels in a given material, we may in this way understand the band structure.

In this colloquium, I will illustrate how to apply magneto-optical spectroscopy to understand the low energy behavior of systems with small carrier densities, small effective masses, and small or vanishing band gaps, whose properties are in some conditions expected to be topologically nontrivial.