How Do Landau Levels Function in Solid State Band Theory?

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SUMMARY

Landau levels represent the quantized energy levels of electrons in a magnetic field, crucial for understanding solid state physics. In band theory, these levels arise from the Hamiltonian of electrons interacting with a vector potential, allowing for a transformation into a harmonic oscillator model to derive eigenvalues. The de Haas-van Alphen effect is linked to the extremal orbit of the Fermi surface, where metallic systems can be approximated as a free electron gas, leading to the classification of Fermi liquids with long-lived quasiparticles. These quasiparticles exhibit properties akin to free electrons, despite interactions that renormalize certain quantities.

PREREQUISITES
  • Understanding of solid state physics concepts
  • Familiarity with Hamiltonian mechanics
  • Knowledge of the de Haas-van Alphen effect
  • Basic principles of quantum mechanics and band theory
NEXT STEPS
  • Study the derivation of Landau levels from the Hamiltonian in solid state physics
  • Explore the implications of the de Haas-van Alphen effect in various metallic systems
  • Investigate the properties of Fermi liquids and their quasiparticles
  • Learn about the quantum Hall effect (QHE) and its relation to cyclotron orbits
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Physicists, solid state researchers, and students interested in advanced topics in band theory and quantum mechanics.

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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?
 
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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.
 

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