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Length scale formation in the Landau levels of quasicrystals

Exotic tiling patterns of quasicrystals have motivated extensive studies of quantum phenomena such as critical states and phasons. Nevertheless, the systematic understanding of the Landau levels of quasicrystals in the presence of the magnetic field has not been established yet. One of the main obstacles is the complication of the quasiperiodic tilings without periodic length scales, thus it has been thought that the system cannot possess any universal features of the Landau levels. In this paper, contrary to these assertions, we develop a generic theory of the Landau levels for quasicrystals. Focusing on the two dimensional quasicrystals with rotational symmetries, we highlight that quasiperiodic tilings induce anomalous Landau levels where electrons are localized near the rotational symmetry centers. Interestingly, the localization length of these Landau levels has a universal dependence on n for quasicrystals with n-fold rotational symmetry. Furthermore, macroscopically degenerate zero energy Landau levels are present due to the chiral symmetry of the rhombic tilings. In this case, each Landau level forms an independent island where electrons are trapped at given fields, but with field control, the interference between the islands gives rise to an abrupt change in the local density of states. Our work provide a general scheme to understand the electron localization behavior of the Landau levels in quasicrystals.