Length scale formation and the localization in the Landau levels of quasicrystals*

The quasicrystals have motivated extensive studies of quantum phenomena such as critical states. Nevertheless, the Landau levels of quasicrystals have been barely understood because of the complication of tilings without periodic length scales. Thus, it has been thought that the system cannot possess any universal features of the Landau levels. Contrary to these assertions, we discuss new quantum phenomena in quasicrystals where electrons localization behavior is distinct in short and large length scales under magnetic fields. In short length scales, electrons are completely localized forming islands with a certain radius, however, in large length scales, such localized islands are extensively distributed and interfere with each other under field control. In the two-dimensional rhombic quasicrystals with rotational symmetries, we derive exact flux conditions and the localization length for such electron localization. Furthermore, macroscopically degenerate zero-energy Landau levels are present due to the chiral symmetry of the rhombic tilings. This gives an important message that the critical state can be completely removable under a magnetic field and the emergence of such quantum states can be applied to control the fractality of electron wave functions in quasicrystals.