Effect of disorder on multifractality and hyperuniformity in quasicrystalline Bose-Hubbard models

Despite the aperiodic structures of quasicrystals, we sometimes find quantum phenomena similar to those of crystals. However, the difference of quantum phenomena in crystals and quasicrystals has yet to be clearly understood. To quantitatively distinguish them, we apply multifractality and hyperuniformity to the characterization of nonuniform spatial patterns in the Bose-Hubbard model on the Penrose and Ammann-Beenker tilings. Based on the mean-field approximation, we obtain real-space distributions in the expectation values of bosonic condensate and number operators. In both Mott insulating and superfluid phases, the distributions are hyperuniform. Moreover, analyzing the order metric that quantifies the complexity of nonuniform spatial patterns, we find that the Penrose and Ammann-Beenker tilings show a divergence of the order metric at a phase boundary between the Mott insulating and superfluid phases, in stark contrast to the case of a periodic square lattice. Our results suggest that hyperuniformity is a useful method to differentiate between crystalline and quasicrystalline Bosonic systems. Next, we introduce on-site random potentials into the quasicrystalline Bose-Hubbard models, leading to a Bose glass phase. Contrary to the Mott insulating and superfluid phases, we find that the Bose glass phase is multifractal. We find that the same multifractality also appears on a Bose glass phase in the periodic square lattice. Therefore, multifractality is common in a Bose glass phase irrespective of the periodicity of systems.