Multifractality and Hyperuniformity in Quasicrystalline Bose-Hubbard Models with and without Disorder

Abstract

Clarifying similarities and differences in physical properties between crystalline and quasicrystalline systems is one of central issues in studying quasicrystals. To contribute to this, we apply multifractal and hyperuniform analyses to 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 of local superfluid amplitude and boson density. In both Mott insulating and superfluid phases, the distributions are hyperuniform. Analyzing the order metric that quantifies the complexity of nonuniform spatial patterns, we find that both quasicrystals show a significant increase 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 concept to differentiate between crystalline and quasicrystalline bosonic systems. The order metric clarifies if the distribution of a physical quantity reflects the point distribution or not, and quantifies how complex the distribution is in comparison with the point distribution. Moreover, we introduce on-site random potentials into these 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. The same multifractality 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.

14 pages, 17 figures

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