Abstract: The "flux-energy" diagram for the Hamiltonian of the Bloch electron in a uniform magnetic field is known as the "Hofstadter butterfly" (or "Azbel-Hofstadter butterfly"); it is among the best knownfractal structures of condensed matter physics. Originally this diagram was constructed numerically by Hofstadter for tight-binding Hamiltonian; now these diagrams are known for a few periodic potentials.
In the case of three-dimensional Bloch elecrons, the electron motion along the field is not quantized by the field, and the most gaps of the transversal motion are smeared; therefore, the fractal structure of the "flux-energy" diagram disappears . Recently it was shown that the fractal structure appears in the "angle-energy" diagram for the tight-binding Hamiltonian (the strenghth of the field is fixed but the direction is varied); the result is published in
[1] M.Koshino, H.Aoki, K.Kuroki, S.Kagoshima, T.Osada: Hofstadter butterfly and integer quantum Hall effect in three dimensions, Phys. Rev. Lett. 86 (2001), 1062-1065.
[2] M..Koshino, H.Aoki, T.Osada, K.Kuroki, S.Kagoshima: Phase diagramm for the Hofstadter butterfly and integer quantum Hall effect in three dimensions, Phys. Rev. B65 (2002), 045310.
In the talk, the results concerning the fractal structure of the "angle-energy" diagram for the Hamiltonian of an elecron in the presence of a uniform magnetic field and a periodic array of point scatterers will be reported. The results are obtained in collaboration with V.Demidov.