I heard someone said that Landau level is degenerate due to translation symmetry, but I don't know how to see that. Can anyone explain a little bit?
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Schematically, in the presence of a strong magnetic field, electrons in a 2D system will orbit in circles. This occurs classically, and pretty much also Quantum Mechanically (besides some choice of Gauge etc.). If there is translational symmetry, these circles can be located anywhere on the sample spatially without an energy cost (except possibly near the edges). This freedom of choice for the circle position is what gives you the degeneracy.
In such a case you have approximately $N \sim \frac{\text{Area of System}}{\text{Area of Circle}}$ degenerate states in the system, where $\text{Area of Circle}$ is a tiny microscopic number usually on the scale of Nano meters.
KF Gauss
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