How Do Landau Levels Function in Solid State Band Theory?

[taishizhiqiu]

Landau level is the energy levels of free electron gas in a magnetic field. However, this term is also frequently used in solid state physics. I have the following questions:
1. what does this term exactly mean in band theory? After all, electrons are not free here.
2. why is de Haas-van Alphen period related to extremal orbit of the Fermi surface?

 

[radium]

A lot of metallic systems can be modeled as a free electron gas. If they have a certain type of interactions they can be classified as Fermi liquid meaning there are long lived quasiparticles near the surface. The volume enclosed by the FS is proportional to the electron density. The charge, spin and momentum of the electrons remain the same but other quantities are renormalized. So basically the quasiparticles behave like free electrons.

Landau levels are obtained from the Hamiltonian of electrons coupled to a vector potential. It turns out you can manipulate the expression to make it look like a harmonic oscillator to obtain the eigenvalues.

A semiclassical picture in 2d (like the QHE) has the electrons in cyclotron orbits with the radius eB/m. The centers drift in an E field.