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Exotic tiling patterns of quasicrystals have motivated extensive studies ofquantum phenomena such as critical states and phasons. Nevertheless, thesystematic understanding of the Landau levels of quasicrystals in the presenceof the magnetic field has not been established yet. One of the main obstaclesis the complication of the quasiperiodic tilings without periodic lengthscales, thus it has been thought that the system cannot possess any universalfeatures of the Landau levels. In this paper, contrary to these assertions, wedevelop a generic theory of the Landau levels for quasicrystals. Focusing onthe two dimensional quasicrystals with rotational symmetries, we highlight thatquasiperiodic tilings induce anomalous Landau levels where electrons arelocalized near the rotational symmetry centers. Interestingly, the localizationlength of these Landau levels has a universal dependence on n for quasicrystalswith n-fold rotational symmetry. Furthermore, macroscopically degenerate zeroenergy Landau levels are present due to the chiral symmetry of the rhombictilings. In this case, each Landau level forms an independent island whereelectrons are trapped at given fields, but with field control, the interferencebetween the islands gives rise to an abrupt change in the local density ofstates. Our work provide a general scheme to understand the electronlocalization behavior of the Landau levels in quasicrystals.

Length scale formation in the Landau levels of quasicrystals