The atomic arrangement of materials significantly influences their physical properties, requiring precise analysis of these structures. Atomic arrangements are classified into crystals, amorphous solids, and quasicrystals. The crystal exhibits periodic atomic patterns, while amorphous solids are characterized by disordered arrangements. However, quasicrystals lack periodicity but exhibit long-range order.

    Multifractality and hyperuniformity are two mathematical frameworks used to characterize the spatial patterns of atomic arrangements. The atomic structure of quasicrystals demonstrates multifractal properties, often referred to as self-similarity, while their global structure exhibits hyperuniformity. Fractal atomic arrangements are typically associated with disordered systems, such as amorphous solids, while uniformity characterizes periodic structures like crystals. The coexistence of these features in quasicrystals raises fundamental questions regarding their physical properties and quantum states.

    Recent experimental advancements have developed bosonic quasicrystalline systems using ultracold atoms, motivating further theoretical analysis. This study employed the Bose-Hubbard model, which describes a phase transition between Mott insulating and superfluid phases. Introducing local disorders, such as impurities, at the phase boundary leads to a novel quantum phase termed Bose glass. Hyperuniform and multifractal analyses were applied to investigate the characteristics of these phases, their transitions, and their differences from periodic systems.

   Both the Mott insulating and superfluid phases exhibited hyperuniformity without local disorder. Increasing the hopping amplitude (the probability of bosons transitioning between atomic sites) induced a phase transition from the Mott insulating to the superfluid phase. A substantial increase in the complexity of physical quantity distributions was observed near the phase transition in quasicrystals, a phenomenon not present in crystals.

   The Mott insulating and superfluid phases maintained their hyperuniformity in local disorder, while the Bose glass phase exhibited multifractal characteristics. These findings indicate that quantum states in quasicrystals can manifest as either hyperuniform or multifractal structures depending on the phase and disorder.

    The results provide a theoretical foundation for understanding quasicrystalline phenomena, including superconductivity, ferromagnetism, the Kondo effect, and quantum criticality. Hyperuniformity-based analyses have been applied across physics, life sciences, astronomy, and mathematics. Extending such analyses to quasicrystalline systems could enable the quantitative characterization of spatial patterns. This study aims to contribute to identifying novel quasicrystalline properties with potential applications that benefit humanity.

    (Written by Masahiro Hori on behalf of all the authors)

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