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Equally spaced discrete harmonic oscillator levels, called Landau levels, are obtained for noninteracting electrons in 2D in presence of a magnetic field applied perpendicular to the plane. The Landau levels are highly degenerate and the energy eigenfunctions can also be obtained exactly.

In practice, however, the electrons are also moving in an underlying periodic lattice. When we put in a periodic potential, we usually get an electronic bandstructure.

What will happen to the energy spectrum of the theory, when both the effects are present i.e, for noninteracting electrons moving in 2D in presence of a magnetic field and a periodic potential? I was interested in this question because I want to know whether one needs to take into account the energy bandstructure due to a periodic potential to understand quantum Hall effect in addition to Landau levels.

If anyone knows a good reference that solves this problem, please let me know.

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This the famous Hofstadter problem. (Douglas Hofstadter is the author of Godel-Escher-Bach).

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    $\begingroup$ Can you suggest me a reference that solves the problem of 2D electron gas in presence of a magnetic field and a periodic potential. $\endgroup$ Aug 11, 2020 at 17:51
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    $\begingroup$ There's are refernces to the Hofstadter propbem in he wiki article. Just Goggle Hofstadter Butterfly to find many others. $\endgroup$
    – mike stone
    Aug 11, 2020 at 17:59
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Statistical Physics, Part 2 by Landau & Livshits contains a chapter on "The symmetry of electron states in a lattice in a magnetic field" (Translation might be unprecise, as I am using my Russian edition.) Basically, the presence of magnetic field breaks the translational symmetry and leads to some interesting quirks, but for weak magnetic fields on still can use Peierls substitution in the effective mass Hamiltonian.

This is an admittedly old resource, but it could serve as a good starting point for serious analysis.

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    $\begingroup$ I was interested in this question because I wonder if one needs to consider the energy bandstructure due to a periodic potential to understand quantum Hall effect in addition to Landau levels. $\endgroup$ Jan 27, 2021 at 15:53
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    $\begingroup$ The exact name of the chapter is almost as you translated, but without the leading "The", see the table of contents. $\endgroup$
    – Ruslan
    Jan 27, 2021 at 18:07

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